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arxiv |
Superfluids, vortices and spinning charged operators in 4d CFT
22 Jun 2019
Gabriel Cuomo *[email protected]
Institute of Physics
Theoretical Particle Physics Laboratory (LPTP)
EPFL
LausanneSwitzerland
Superfluids, vortices and spinning charged operators in 4d CFT
22 Jun 2019
We include vortices in the superfluid EFT for four dimensional CFTs at large global charge. Using the state-operator correspondence, vortices are mapped to charged operators with large spin and we compute their scaling dimensions. Different regimes are identified: phonons, vortex rings, Kelvin waves, and vortex crystals. We also compute correlators with a Noether current insertion in between vortex states. Results for the scaling dimensions of traceless symmetric operators are given in arbitrary spacetime dimensions.IntroductionConformal field theories (CFTs) play a key role in particle and condensed matter physics. As fixed points of the renormalization group flow, they act as landmarks in the space of quantum field theories (QFTs). Through the AdS/CFT correspondence[1,2], they promise to shed light on quantum gravity. They also describe critical points for second order phase transitions. Finally, CFTs are also among the few examples of interacting QFTs where exact results are available without supersymmetry. Recently, the bootstrap program[3,4]achieved much progress in the study of CFTs, both through numerical [5, 6] and analytical[7,8,9]techniques.Basic observables in CFTs are correlation functions of local operators in the vacuum. Despite this, sometimes one can make predictions for the CFT data defining the theory studying the dynamics of finite density states [10]. This is a consequence of the state/operator correspondence[11,12], which relates states in radial quantization to local operators with the same quantum numbers. So far, this idea has been mainly applied in the investigation of the superfluid phase in conformal field theories[10,13,14,15,16,17,18,19,20]. Indeed superfluids are the most natural candidates to describe states at large internal quantum numbers in CFTs. They admit a simple and universal effective field theory (EFT) description[21,22]which allows the computation of correlators in a perturbative expansion controlled by the charge density. The same strategy was recently applied also in the context of non-relativistic CFTs[23,24,25].As the angular momentum is increased, the superfluid starts rotating and vortices develop[26]. These can be included in the EFT as heavy topological defects[27,28,29]. In[30], this EFT was used to describe operators with large spin and large charge in three dimensional CFTs. In this work, we study the predictions of the vortex EFT for four dimensional CFTs.For traceless symmetric operators J =J = J 34 /2, the corresponding state passes through three distinct regimes, qualitatively similar to the CFT 3 case:• For 2 ≤ J 34 Q 1/3 the lightest operator corresponds to a phonon wave of angular momentum J in the superfluid. The scaling dimension is given by• For Q 1/3 J 34 ≤ Q, the minimal energy state is given by a single vortex ring, whose radius increases with J. Its energy isThe leading contribution in (1.5) comes from the first term, because of the logarithmic enhancement. The other terms can be interpreted as finite-size corrections due to the vortex extension and are functionally distinguished from the relative Q 1/3 /J corrections.• For Q J 34 Q 4/3 the superfluid forms a vortex crystal. The scaling dimension of the corresponding operator is given by(1.6)Mixed symmetric representations are conveniently parametrized in terms of J 34 , J 12 in (1.1). We write J ab to generically denote any of them. We find the following results: (2.18) corresponding to the charge density ρ = 3 2πR 2 J Q cos θ. The second term in (2.18) is the electrostatic energy of the crystal. The leading corrections arise from the vortex masses and the magnetic field fluctuations. As J → Q 3/2 the vortex velocities approach the relativistic regime and the EFT breaks down.
Summary of results
Let us first set our conventions for the four dimensional rotation group SO (4). Spinning operators in four dimensions are classified in representations labelled by two positive half-integer quantum numbers (J,J). These are related to the maximal values allowed for the Cartan generators J 34 and J 12 as
(J,J) = |J 34 − J 12 | 2 , |J 12 + J 34 | 2 . (1.1)
With no loss of generality, we assume J 34 ≥ J 12 ≥ 0. The main prediction of the superfluid EFT is the scaling dimension of the lightest scalar operator of charge Q in the spectrum. It is given by [13] ∆ 0 (Q) = αQ 4/3 + βQ 2/3 + . . . , (1.2) for Q 1; here α and β are independent Wilson coefficients. In this work, we compute the scaling dimension of the lightest operator as the spin is increased. As in [30], the EFT describes the regime where the spin is below the unitarity bound, J,J Q 4/3 , and cannot reach the regime analyzed by the analytic bootstrap [7,8,31,32,33,34,35,36,37,38,39]. To leading order in the charge and the spin, the results depend on the first coefficient in (1.2) and on an extra coefficientγ parametrizing the vortex tension.
• For 2 ≤ J 12 ≤ J 34 Q 1/3 the minimal energy state is given by two phonons propagating on the superfluid, with energy:
∆ = αQ 4/3 + J 34 (J 34 + 2) 3 + J 12 (J 12 + 2) 3 + O J 4 ab Q 2/3 . (1.7)
• For 1 ≤ Q − J 34 Q and 2 ≤ J 12 Q 1/3 , the lowest energy state corresponds to a Kelvin wave of spin J 12 propagating on a large vortex ring. The corresponding operator scaling dimension is given by: (1.11)
∆
These results apply to CFTs whose large charge sector can be described as a superfluid and which admit vortices. These are natural and simple conditions, hence we expect them to apply to a wide range of theories with a U (1) symmetry. Nonetheless, we cannot prove these assumptions. The rest of the paper is organized as follows. In section 2 we review the superfluid description of large charge operators as well as the vortex EFT in 2+1 dimensions. In 3 we formulate the effective field theory (EFT) for vortices in 3 + 1 dimensions. The results of this section are derived in sec. 4. In 5 we show how to make predictions for correlators involving a current insertion between two vortex states and in 6 we briefly comment on how the results (1.4) and (1.6) change in generic dimensions. Finally in 7 we draw our conclusions and comment on future research directions. Technical details are given in the appendices A, B, C.
Conventions and coordinates on S 3 : Lorentz indices µ, ν, . . . go from 0 to 3 and we use mostly minus metric signature sgn(g µν ) = {1, −1, −1, −1}. Spatial indices are written as i, j, . . . = 1, 2, 3 and are raised and lowered with a positive metric |g ij |. We use the notationḟ = ∂ 0 f for time derivatives. Indices a, b, . . . are used for the R 4 embedding of S 3 and go from 1 to 4. Embedding coordinates are denoted X a = X a . Calling X a (x) the R 4 coordinate corresponding to an S 3 point x, the chordal distance between two points x and x is given by:
∆X 2 (x, x ) = a X a (x) − X a (x ) 2 .
(1.12)
A convenient parametrization of S 3 is provided by Hopf coordinates, defined via the embedding: X 1 = R cos ξ sin η, X 2 = R sin ξ sin η, X 3 = R cos φ cos η, X 4 = R sin φ cos η. (1.13) This gives the following metric tensor ds 2 R 2 = dη 2 + sin 2 ηdξ 2 + cos 2 ηdφ 2 , η ∈ [0, π/2], ξ ∈ [0, 2π], φ ∈ [0, 2π]. (1.14) Notice that, for fixed η different from 0 and π/2, ξ and φ describe an S 1 × S 1 submanifold.
2 Review of previous results
Conformal superfluid
Let us first remind that, in a CFT, the state-operator correspondence relates eigenstates of the Hamiltonian H on S d with the set of local operators at any given point [11,12]. The quantum numbers of the state on S d and the corresponding operator are the same. In particular, the energy E is related to the scaling dimension of the latter as ∆ = E/R. The EFT description of CFTs at large quantum numbers is based on the assumption that the lightest scalar operator with U (1) charge Q in a d + 1 dimensional CFT corresponds to a state with homogeneous charge density on R × S d . For Q 1, the scale associated with the density of this state is parametrically bigger than the S d radius R and the CFT is expected to be in a "condensed matter phase". As argued in [13], the simplest possibility is that the CFT enters a superfluid phase. Technically, this is equivalent to assuming an effective description in terms of a U (1) Goldstone boson [21]. The effective Lagrangian is fixed by shift symmetry and Weyl invariance:
L/ √ g = c(∂χ) d+1 + c 1 (∂χ) d−1 R + d(d − 1) [∂ µ (∂χ)] 2 (∂χ) 2 + c 2 (∂χ) d−1 R µν ∂ µ χ∂ ν χ (∂χ) 2 + . . . . (2.1)
We use the notation (∂χ) = (∂ µ χ∂ µ χ) 1/2 and c, c 1 , c 2 are Wilson coefficients. Here R µ νρσ is the Riemann tensor on the cylinder R × S d . We assume c, c 1 , c 2 ∼ O(1), corresponding to the generic expectation for a strongly coupled system. On a homogeneous background at finite charge, the field takes the value χ = µt, where µ is the chemical potential of the system. To leading order in derivatives, it is related to the U (1) charge density j 0 as
j 0 = Q R d Ω d = c(d + 1)∂ 0 χ(∂χ) d−1 = (d + 1)cµ d ,(2.2)
where Ω d = 2π
d+1 2 /Γ d+1 2
is the S d volume. The chemical potential sets the cutoff of the EFT:
Λ ∼ µ ∼ Q 1/d R . (2.3)
By the state/operator correspondence, the ground state of (2.1) corresponds to the minimal energy state with charge Q. Its energy is determined by a semiclassical analysis and takes the form:
∆ 0 (Q) = αQ d+1 d 1 + β α Q − 2 d + . . . . (2.4)
Quantum corrections provide a Q 0 contribution. For even d, there is no local counterterm correcting this term, which is hence universal [10]. The Lagrangian (2.1) describes also excitations on the background. For instance, expanding χ = µt + π and working to leading order in derivatives we get
L/ √ g = cµ d−1 d(d + 1) 2 π 2 − 1 d ∂ i π|g ij |∂ j π + . . . . (2.5)
Quantizing the system, it follows that the spectrum can be organized as a Fock space in terms of single particle states with angular momentum J and energy given by
ω J = c s λ J , c 2 s = 1 d . (2.6) Here λ 2 J = J(J+d−1) R 2
are the eigenvalues of the Laplacian on S d and the sound speed c s is fixed by conformal invariance. Physically, these states correspond to phonons propagating in the superfluid and are associated with new operators in the CFT. The J = 1 mode has ω 1 = 1/R and corresponds to the creation of a descendant. A natural question is how the spectrum changes, as the spin J is increased. When the angular momentum is parametrically smaller than the cutoff (2.3), the spectrum is reliably described by phonons (2.6). The results (1.3) and (1.7) then follow. Increasing spin, one finds singular solutions with a non zero winding number, such as π = φ, where φ is the azimuthal angle. This corresponds to the fact that, for J Q 1/d , vortices develop in the superfluid and must be included in the effective description. This was done in [30] for a 2 + 1 dimensional CFT. The main goal of this work is to carry a similar analysis for a 3 + 1 dimensional conformal field theory. To build some intuition, we briefly review the results of the 2 + 1 dimensional EFT in the next section.
Vortices in 2+1 dimensions
In d = 2, the action (2.1) reads
L = c(∂χ) 3 . (2.7)
To study vortices, it is convenient to consider a dual description in terms of a gauge field. To this aim, we introduce an independent variable v µ ≡ ∂ µ χ and a Lagrange multiplier A µ to set the curl of v µ to zero:
L = cv 3 − 1 2π A µ µνρ √ g ∂ ν v ρ ,(2.8)
where µνρ / √ g is the antisymmetric Levi-Civita tensor. Integrating out v µ we get
L = −κF 3/2 (2.9) where F = F µν F µν and F µν = ∂ µ A ν − ∂ ν A µ . The coefficient κ is related to c as κ = 1 2 5/4 (3π) 3/2 √ c .
The U (1) current relates the two descriptions
j µ = 3c(∂χ)∂ µ χ = 1 4π µνλ √ g F νλ .
(2.10)
As a consequence, the charge density (2.2) translates into a homogeneous magnetic field F θφ = B sin θ = Q 2R 2 sin θ, which sets the cutoff of the EFT according to (2.3). The action (2.9) describes a propagating degree of freedom, given by the fluctuations of the magnetic field F θφ and which corresponds to the phonon in the original picture, together with a nonpropagating Coulomb field A 0 , which does not have any local analogue in the scalar formulation. As we will see, it is precisely this extra component which provides the leading coupling to the vortices.
In the EFT, vortices are heavy charged particles in the dual description (2.9). They are treated as 0 + 1 dimensional worldlines, whose spacetime trajectory is parametrized by a function X µ p (τ ) of a time parameter τ . The action of the superfluid plus vortices is fixed by the requirement of Weyl invariance and τ -reparametrization invariance; the lowest orders in derivatives take the form [30]
S = −κ d 3 xF 3/2 − p q p A µ dX µ p − p dτ √ F g µνẊ µ pẊ ν p F p j µẊ µ jẊ . (2.11)
The second term is the minimal coupling between the gauge field and a particle of charge q p ; this cannot be written in a local form in the scalar picture, showing the convenience of the gauge formulation. Notice that the charge q p corresponds to the Goldstone winding number and is hence quantized: q p ∈ Z. The third term is the action for a relativistic point particle in a superfluid 1 ; it is multiplied by an arbitrary function of jµẊ µ jẊ , since the superfluid velocity breaks Lorentz symmetry and provides an alternative condensed matter metric [26]. Working in the physical gauge X 0 p = τ , we notice that the leading term in time derivatives for the vortex lines arises from the second piece in (2.11). As we will self-consistently see, this implies that vortices move with non-relativistic velocities |˙ X| ∼ 1/ √ B. Hence we can neglect terms with two time derivatives in the last term, retaining only a constant contribution proportional to √ B which is interpreted as the vortex mass. This procedure is sometimes called lowest Landau level approximation in the literature [40,41,42,43,44].
The equations of motion (EOMs) deriving from (2.11) are
1 e 2 ∇ i f ij = p q pẊ j p δ 2 x i − X i p √ g , 1 e 2 ∇ i E i = p q p δ 2 x i − X i p √ g , (2.12) E i = (Ẋ p ) j F ji ,(2.13)
where E i = F i0 is the electric field and e 2 = 2 1/4 √ B 3κ . The particle EOMs (2.13) are first order in derivatives and imply that vortices move with drift velocity |˙ X p | ∼ | E/B| ∼ 1/ √ Q as anticipated. Consequently, particle velocities, as well as the magnetic field fluctuations sourced by them, are negligible and the only relevant interaction is the electrostatic one. We now look for static classical solutions of the EOMs (2.12) and (2.13). By the state/operator correspondence, these classical solutions will be associated to operators with the same quantum numbers. The spin and the scaling dimension of the corresponding operators are then determined classically from the energy momentum tensor. The scaling dimension of a state with n vortices reads 2
∆ = ∆ 0 (Q) + R 2e 2 d 2 x √ g E 2 +γ p √ 2R √ B = ∆ 0 (Q) − √ Q 12α p =r q p q r log Q∆X 2 (x p , x r ) +γn Q, (2.14)
where X a p = (sin θ p cos φ p , sin θ p sin φ p , cos θ p ) is the vortex coordinate in the R 3 embedding of S 2 and ∆X 2 (x p , x r ) = 3 a=1 X a p − X a r 2 is the chordal distance between two vortices. The first term in (2.14) is the energy of the homogeneous phase, given by (2.4) with d = 2:
∆ 0 (Q) = αQ 3/2 + βQ 1/2 + . . . . (2.15)
The second term is the energy stored in the electric field sourced by the vortices, which is further rewritten as a sum over pairwise contributions in the right-hand side; the log Q ∼ log Λ 2 contribution arises from the logarithmically divergent self-energy of the point charges. We also used that the Figure 1: A vortex-antivortex pair moving on the sphere at fixed distance; in the stereographic projection the motion corresponds to two circular orbits.
net charge on the sphere must be zero p q p = 0, as a consequence of Gauss law. Finally, the last term is the contribution of n vortex masses and is written in terms of an independent coefficientγ, assumed to be the same for all vortices. Similarly, the angular momentum is
J a = RB e 2 d 2 x √ g n i a ij √ gE j = − Q 2 p q p X a p . (2.16)
Here n i a is the Killing vector corresponding to the specified rotation, a = 1, 2, 3, and we used Gauss law to obtain the right-hand side. We can now discuss the consequences of the vortex EFT for the CFT spectrum. To this aim, notice that the self-energy contribution ∼ log Q in eq. (2.14) is proportional to p q 2 p and implies that vortices with |q| > 1 are energetically unfavored 3 . The two main results of [30] are:
• The lowest energy state for √ Q J ≤ Q consists of a vortex-antivortex pair rotating on the sphere, at a distance proportional to the spin ∆X/2 = J/Q (see fig. 1). The scaling dimension of the corresponding operator reads
∆ = ∆ 0 (Q) + √ Q 3α log J √ Q + 2γ Q + O Q × Q J 2 . (2.17)
The leading correction to the ground state energy arises from the second term as a consequence of the logarithmic divergence of the vortex self-energy. This depends on the same coefficient α appearing in (2.14). The vortex mass contribution, given by the last term in (2.17), depends on a new coefficient and scales as the first subleading term in the ground state energy (2.4).
Corrections to this formula arise from the particle velocities and the phonon field. As J → √ Q, the vortices become relativistic and the derivative expansion breaks down.
• For Q J Q 3/2 the lowest energy state corresponds to a vortex crystal phase. Its energy is found approximating the vortex distribution as a continuous distribution and then minimizing the energy at fixed angular momentum. The result is
∆ 0 (Q) = ∆ 0 (Q) + 1 2α
For J √ Q there are no vortices and spinning operators are described by phonons (2.6).
3 Formulation of the EFT
Dual gauge field
As in 2 + 1 dimensions, to write a local coupling between vortices and the superfluid we consider a dual description in terms of a gauge field. Following the steps in sec. 2.2, we rewrite the leading order Lagrangian (2.1) in d = 3 using a two form Lagrange multiplier A µν = −A νµ :
L = cv 4 − 1 4π A µν µνρσ √ g ∂ ρ v σ ,(3.1)
Integrating out v µ then gives
L = −κH 4/3 , H µνρ = ∂ µ A νρ + ∂ ν A ρµ + ∂ ρ A µν ,(3.j µ = 4c(∂χ) 2 ∂ µ χ = 1 12π µνρσ √ g H νρσ . (3.3)
Consequently, the homogeneous charge density j 0 = Q 2π 2 R 3 in the vacuum translates into a constant background field:
H ηξφ = −B sin η cos η, B ≡ Q πR 3 . (3.4)
The cutoff of the theory (2.3) is thus set by B 1/3 in the dual description. The action (3.2) is often called of Kalb-Ramond type and is invariant under the gauge transformations A µν → A µν + ∂ µ ξ ν − ∂ ν ξ µ , for an arbitrary vector ξ µ . The gauge redundancy allows imposing three gauge fixing conditions, since a gauge transformation generated by a total derivative ξ µ = ∂ µ α is trivial.
In the following, we shall be interested in fluctuations of the background (3.4). It is thus convenient to expand the gauge field in a background valueĀ µν plus fluctuations:
A µν =Ā µν + δA µν ,(3.5)
where a possible choice isĀ
ηξ =Ā ηφ = 0,Ā ξφ = − B 2 1 − cos 2 η . (3.6)
Fluctuations are conveniently parametrized in terms of two three vectors b i and a i defined as:
δA ij = √ g ijk b k , δA 0i = a i . (3.7)
We partially fix the gauge requiring ∇ i A ik = 0, which sets the curl of b i to zero. Then the Lagrangian to quadratic order in the fluctuation reads:
L 1 4e 2 f 2 + 1 2e 2 ḃ iḃ i − 1 3 ∇ i b i 2 , (3.8) where e 2 = ( √ 6B) 2/3 8κ and f 2 = f ij f ij with f ij = ∂ i a j − ∂ j a i . (3.9)
Following the gauge fixing, the field b i is purely longitudinal and corresponds to the phonon. Instead a i is a non-propagating degree of freedom, called the hydrophoton since the residual U (1) gauge invariance acts as a i → a i − ∂ i ξ 0 . Analogously to the Coulomb field in (2.9), the hydrophoton does not correspond to a local field in the original description and provides the leading coupling to the vortices.
String-vortex duality
Vortices in the dual description correspond to topological line defects, which are described as 1 + 1 dimensional strings embedded in the 3 + 1 dimensional spacetime [27]. The line element of a vortex p is parametrized by X µ p (τ, σ), where τ and σ are the world-sheet coordinates. We use the words "vortex" and "string" interchangeably. The Lagrangian is required to be Weyl invariant and reparametrization invariant for both τ and σ and is analogous to (2.11). The lowest order terms are given by
S = −κ d 4 x √ gH 4/3 − p λ p dτ dσA µν ∂ τ X µ p ∂ σ X ν p − p dτ dσH 2/3 |det(G αβ )|F p h αβ G αβ + . . . . (3.10)
The first term was discussed in the previous section. The second term is the leading coupling between a string of vorticity λ p ∈ Z and the gauge field. The last term is the generalized Nambu-Goto (NG) action for the vortex; in appendix C we derive its form via the coset construction. Here, the world-sheet metric is provided by:
G αβ = g µν ∂ α X µ p ∂ β X ν p , α, β = τ, σ. (3.11)
Since the superfluid velocity breaks Lorentz invariance, one can construct another independent symmetric world-sheet tensor, which can be chosen as
h αβ = ∂ α X µ ∂ β X ν j µ j ν j 2 . (3.12)
In general the NG action contains an arbitrary function of G αβ h αβ , where G αβ is the inverse of G αβ . Weyl invariance further fixes the power of H which multiplies it. Finally dots in (3.10) stands for higher derivative terms.
Consider now the physical gauge X 0 p = τ for vortices. Using (3.6), the second term in (3.10) is linear in time derivatives of the vortex line. As we will self-consistently see in the next section, this implies that vortices move with drift velocity |˙ X| ∼ f /B ∼ B −1/3 . Then, similarly to what we argued below (2.11), terms of the kind˙ X ·˙ X in the NG action can be treated as higher derivatives and we neglect them. The coupling of the phonon field to the strings is also negligible to leading order. In this regime, the action reduces to
S 1 e 2 d 4 x 1 4 f 2 + 1 2 ḃ iḃ i − 1 3 ∇ i b i 2 − p dτ dσ λ p Ā ij ∂ τ X i ∂ σ X j + a i ∂ σ X i p + γ p B 2/3 (∂ σ X) , (3.13)
where γ p = 6 1/3 F p (1) and we define
(∂ σ X) = |g ij |∂ σ X i p ∂ σ X j p . (3.14)
Notice that the phonon spectrum (2.6) to leading order is not affected by the presence of vortices.
4 Results of the EFT
Classical analysis
From the leading order action (3.13) the following equations of motion for the hydrophoton and the strings are derived
− 1 e 2 ∇ i f ij = p J j p ≡ p λ p dσ ∂ σ X j p δ 3 (x i − X i p ) √ g , (4.1) λ p f ik − B √ g ijkẊ j p ∂ σ X k p = γ p B 2/3 |g ij | D Dσ ∂ σ X j (∂ σ X) . (4.2)
Eq. (4.1) is analogous to Ampère's circuital law in magnetostatic, a vortex acting as an electric current J i p sourcing the field f ij . Eq. (4.2) is the string equation of motion. Notice that it is first order in time derivatives and implies that vortices move with drift velocity |˙ X| ∼ f /B ∼ B −1/3 . The right-hand side arises from the NG action and it is proportional to the covariant derivative of the line element D Dσ ∂σX j (∂σX) ; the left-hand side comes from the minimal coupling to the gauge field. As in sec. 2.2 the electrostatic problem required the net charge on the sphere to be zero, the 3 + 1 dimensional magnetostatic problem defined by (4.1) and (4.2) requires zero vorticity flux on every closed surface. To this aim, we only consider closed strings.
The energy and angular momentum associated to solutions of the EOMs are computed from the stress energy tensor T µν = 2 √ g δS δg µν :
T µν = κ H 2/3 4H µσρ H σρ ν + g µν H 2 + p γ p B 2/3 dτ dσ δ 4 (x µ − X µ p ) √ g |det(G αβ )| G αβ ∂ α X σ p ∂ β X ρ p g σµ g ρν . (4.3)
The classical energy of the state is found from
E = ∆ R = 3Q 4/3 8π 2/3 c 1/3 R + 1 4e 2 d 3 x √ gf 2 + p γ p B 2/3 dσ(∂ σ X). (4.4)
The first term is the energy of the homogenous ground state. The second term is the energy stored in the magnetostatic field f ij created by the vortices. Finally, the last term is the energy contribution from the tension and is proportional to the length of the string L p . Eq. (4.1) gives the field a i in terms of the string current:
a i (x) = e 2 p d 3 x G ij (x, x )J j p (x ), (4.5)
where G ij (x, x ) is the photon propagator on S 3 . In appendix A it is shown that the photon Green function on S d is
G ij (x, x ) = − ∂ i ∂ j u(x, x ) F (u(x, x )), F (u) = Γ(d − 2) (4π) d 2 Γ d 2 R d−2 2 F 1 1, d − 2; d 2 ; 1 − u 2R 2 , (4.6) up to an irrelevant gauge dependent term. Here u = 1 2 ∆X 2 (x, x ),
where ∆X 2 is the chordal distance between two points in embedding space. Then the scaling dimension of the corresponding operator can be written as
∆ = R E = αQ 4/3 + Re 2 2 p,p d 3 x √ g d 3 x g J j p (x)G jk (x, x )J k p (x ) + p γ p R B 2/3 L p , (4.7)
where α = 3 8π 2/3 c 1/3 . Notice the analogy with the structure of (2.14). The angular momentum (in units of 1/R) of the corresponding state is computed similarly:
J ab = RB 2e 2 d 3 x √ g n i ab ijk √ gf jk , (4.8)
where n ab is the Killing vector corresponding to a rotation in the (X a , X b ) plane. Using Ampère's law (4.1) and Stoke's theorem, it is conveniently rewritten as
1 2 J ab abcd = − RB 2 p λ p dσ p X p c (∂ σ X p d ) − X p d (∂ σ X p c ) = −RB p λ p dX p c ∧ dX p d ,(4.9)
where X p a are the vortex coordinates in the R 4 embedding of S 3 . The last equation on the righthand side is a formal notation for the area enclosed by the vortex projection in the (X c , X d ) plane. In the following we will study simple specific configurations.
Vortex rings
In nature, vortices often have a ring shape and move with a constant speed inversely proportional to the radius [26]. It is hence natural to look for vortex ring solutions of the EOMs (4.1) and (4.2). As we will see, a vortex ring generalizes the vortex-antivortex configuration in fig. 1. The simplest configuration one can study is a slowly moving vortex ring with unit negative charge λ = −1. We pick the gauge ξ = σ and consider a radius rR ≤ R ring in the (X 1 , X 2 ) plane in embedding space. The EOMs implies that the ring rotates with constant drift velocity v in the (X 3 , X 4 ) plane:
X 2 1 (t, σ) + X 2 2 (t, σ) = R 2 sin 2 η(t, σ) = R 2 r 2 = const. φ(t, σ) = vt + const. . (4.
10)
The precise value of v is fixed by eq. (4.1). From eq. (4.9) it follows that the only nonvanishing component of the angular momentum is given by:
J 34 = Qr 2 . (4.11)
In figure 2 the motion is depicted in stereographic coordinates, defined by the relation (x, y, z) = 1 1+X 1 (X 3 , X 4 , X 2 ). Eq. (4.10) corresponds to a ring orbiting around the z axis; as the angular momentum is increased, the ring size increases and its velocity decreases. For r → 1 the surface embedded by the ring in the stereographic projection extends to cover the whole plane and the vortex lies statically on the geodesic corresponding to the z axis. Fig. 2 qualitatively generalizes the 2 + 1 dimensional motion depicted in fig. 1. Using (4.7) we can calculate the energy of this configuration as:
E = αQ 4/3 /R + e 2 R 2 dξdξ J i (ξ)G ij x(ξ), x(ξ ) J j (ξ ) + γB 2/3 2πrR (4.12)
The only nontrivial contribution arises from the second term, corresponding to the magnetostatic self-energy of the string. It diverges due to the short distance behaviour of the hydrophoton propagator. We regulate the calculation working in d + 1 spacetime dimensions and get
E = αQ 4/3 /R + e 2 πR r 2π(3 − d) + r log 4πr 2 B 2/3 R 2 − γ E − 2ψ (3/2) + 9/2 + 1/3 log 6 4π − r 2π log(r + 1) − 1 π log(r + 1) + γB 2/3 2πrR. (4.13)
Details of the computation are given in appendix B.1. There is a divergent piece for d → 3 proportional to the vortex length, which renormalizes the string tension. The contribution logarithmically enhanced by the cutoff ∼ e 2 r log r 2 B 2/3 can be seen as a consequence of the renormalization group running of γ [27] induced by the hydrophoton. Collecting everything, the scaling dimension (4.7) for a vortex ring state reads
∆ = αQ 4/3 + ∆ V (Q, J 34 ), (4.14)
where we isolated the vortex contribution to the energy:
∆ V (Q, J) = 3 8α Q 1/6 J 1/2 log J/Q 1/3 − 3 4α Q 1/6 J 1/2 log 1 + J/Q − 3 2α Q 2/3 log 1 + J/Q +γQ 1/6 J 1/2 . (4.15)
Hereγ is a finite new coupling which absorbs all contributions proportional to r in (4.13). As in (2.17), the leading contribution arises because of the classical running of the tension induced by the magnetostatic self-energy and is given by the first term in (4.15). For J Q, the other contributions can be expanded in powers of the vortex length and to leading order effectively scale as Q 1/6 J 1/2 . Physically, this is understood noticing that the vortex energy density is set by e 2 ∼ Q 2/3 , hence for short vortices the energy can be estimated as the length times the energy density (neglecting the logarithmic running of the tension): 2πrR × e 2/3 ∼ Q 1/6 J 1/2 . However, as J → Q the functional dependence of the second and third term in eq. (4.15) deviates from this expectation, as a consequence of the vortex finite size. As the ring radius is decreased to inverse cutoff length r → 1/(ΛR), corresponding to J 34 → Q 1/3 , the magnetostatic field f ∼ e 2 /(Rr) becomes of the same order of the background field B and the vortex velocity approaches the relativistic regime. Hence subleading contributions to (3.13) become unsuppressed and the EFT breaks down. Eq. (4.14) can be identified as the minimal energy state at fixed angular momentum in its regime of validity.
We now study states with two vortices, one laying on the (X 1 , X 2 ) plane and the other on the (X 3 , X 4 ) plane in embedding space. Because of (4.9), these configurations are associated to operators in mixed symmetric representations of the SO(4) group. Consider first a radius R ring in the (X 1 , X 2 ) plane interacting with a ring of arbitrary size in the (X 3 , X 4 ) plane. In this geometry, the interaction does not affect the equations of motion and the solution takes a simple form vortex 1 :
X 2 1 (t, σ 1 ) + X 2 2 (t, σ 1 ) = r 2 1 = 1, σ 1 = ξ 1 ; vortex 2 : cos 2 η 2 (t, σ 2 ) = r 2 2 , ξ 2 (t, σ 2 ) = v 2 t, σ 2 = φ 2 . (4.16)
Focussing on negative unit charge vortices λ 1 = λ 2 = −1, this configuration corresponds to an operator in a mixed symmetric representation with spin given by
J 34 = Q, J 12 = Qr 2 2 .(4.17)
Since the electric currents J i sourced by the strings are orthogonal, the corresponding scaling dimension is found analogously to (4.14):
∆ = αQ 4/3 + ∆ V (Q, J 34 ) + ∆ V (Q, J 12 ).(4.18)
To leading order, a similar solution exists for 0 ≤ (1 − r 2 1 ) (RΛ) −2 , hence for 0 ≤ Q − J 34 Q 1/3 . As before, the consistency of the EFT requires J 12 Q 1/3 . In general, the mutual interaction affects non trivially the motion of the two vortex rings. One can, however, identify the logarithmically enhanced contributions analogous to the first term in (4.15) just from the free action. These indeed arise due to the tension running induced by the hydrophoton contribution to the vortex self-energy. For Q 1/3 J 12 , J 34 ≤ Q, the energy then reads:
∆ = αQ 4/3 + 3 8α Q 1/6 J 1/2 34 log J 34 /Q 1/3 + J 1/2 12 log J 12 /Q 1/3 + O Q 1/6 J 1/2 34 , Q 1/6 J 1/2 12 .
(4.19) This result holds as long as the minimal distance d between the two vortices is larger than the inverse of the cutoff:
d 2 R 2 ∼ (J 12 + J 34 − Q) 2 J 12 J 34 1 Q 2/3 . (4.20)
Vortex crystals
Since the magnetostatic self-energy of a single vortex is proportional to λ 2 , strings with |λ| ≥ 1 are energetically unfavored. Hence the minimal energy state for values of the angular momentum J 34 Q is made by n 1 vortices. We then approximate the vortex distribution with a continuous current density J i (x). The corresponding state is found minimizing the energy (4.4) at fixed angular momentum (4.8), giving the following density profile:
J ξ = 2 πR 2 J 34 Q , J φ = J η = 0. (4.21)
The leading contribution to the energy comes from the magnetostatic field and reads ∆ = αQ 4/3 + 3 4α
J 2 34 Q 4/3 . (4.22)
Physically, this state corresponds to a vortex crystal [45,46,47]. When J 34 → Q 4/3 , the magnetic field f approaches B, vortices become relativistic and the EFT breaks down. Similarly, the ground state for Q J 34 , J 12 Q 4/3 is provided by a vortex crystal, whose current density and energy are given by
J ξ = 2 πR 2 J 34 Q , J φ = 2 πR 2 J 12 Q , J η = 0,(4.
Quantization and Kelvin waves
Vortices in four dimensions are extended objects and can thus propagate Kelvin waves on them [27]. The corresponding states are then naturally associated to new operators of the CFT. To study them, we consider a single string of vorticity λ = −1. It is convenient to parametrize its coordinates via the following variables:
z(t, σ) = X 1 (t, σ) + iX 2 (t, σ) = R sin η(t, σ)e iξ(t,σ) ,(4.25)
w(t, σ) = X 3 (t, σ) + iX 4 (t, σ) = R cos η(t, σ)e iφ(t,σ) . (4.26)
These are related through the constraint |z| 2 + |w| 2 = 1. We pick the gauge ξ = σ and t = τ . Integrating out explicitly the hydrophoton from eq. (3.13), we find the single vortex action as
S 1−vortex = dtdσ i B 2 w * ẇ − γB 2/3 |∂ σ z| 2 + |∂ σ w| 2 + e 2 4 dtdσdσ ∂ σ ∂ σ ∆X 2 (σ, σ ) F ∆X 2 (σ, σ ) 2R 2 , (4.27)
where F is given in (4.6). Eq. (4.27) can be seen as the (nonlocal) action of a complex field w(t, σ) living on R × S 1 . It is manifestly invariant under the action of the unbroken rotation generators J 34 , corresponding to rotations around the vortex w → e iα w, and J 12 , corresponding to translations along the string σ → σ + α.
We expand for small fluctuations around the background w = 0, which describes a radius R ring in the (X 1 , X 2 ) plane with J 34 = Q. The action to quadratic order in w reads:
S 1−vortex dtdσ i B 2 w * ẇ − γB 2/3 − γB 2/3 2 |∂ σ w| 2 + γB 2/3 2 |w| 2 + S (2) non−local ,(4.28)
where S
non−local is found expanding the second line in (4.27). It follows that the vortex is quantized as a standard non-relativistic field:
w(t, σ) = 2 B ∞ n=−∞ a n 2π e −iωnt+inσ ,
[a n , a † m ] = 2πδ nm . (4.29)
As usual the a n annihilate the vacuum a n |0 = 0, and thus so does w(t, σ). The proper frequencies ω n are computed in appendix B.2 and read Rω n ≡ ∆ k (n) = π(n 2 − 1)
Q 1/3 3 8α log Q 2/3 − 2ψ n + 1 2 − 2γ E − 1 − log 64 +γ . (4.30)
Notice that the n = 0 mode decreases the energy, while the n = ±1 modes have ω ±1 = 0. To understand this, it is useful to compute the angular momentum in terms of ladder operators to order O(Q 0 ). The rotations generated by J 12 and J 34 are linearly realized and their generators are quadratic in terms of ladder operators:
J 34 = Q − n a † n a n 2π , J 12 = n n a † n a n 2π . (4.31)
The string realizes nonlinearly the full rotation group 4 . As a consequence, the broken components of the angular momentum are linear in the n = ±1 annihilation and creation operators: As shown in fig. 3, a Kelvin wave in stereographic coordinates takes the form of a solenoid, trapping the magnetic field inside. The string undergoes a helical motion analogous to the one of a wine opener. Notice that Kelvin waves carry less energy than phonons with the same angular momentum (2.6). It follows that a state obtained acting on the vacuum as
J 23 + J 14 = − Q 2π a −1 + a † −1 , J 23 − J 14 = Q 2π a 1 + a † 1 , J 31 + J 24 = i Q 2π a −1 − a † −1 , J 31 − J 24 = i Q 2π a 1 − a † 1 .(a † 0 ) m a † n |0 ≡ |J 34 = Q − m − 1, J 12 = n (4.34)
is the minimal energy state for the specified value of the angular momentum. This description applies in the linear regime m + 1 = Q − J 34 Q. When n = J 12 → Q 1/3 higher derivative terms become unsuppressed and the EFT breaks.
Higher order corrections
Corrections arise from higher derivative terms we neglected in (3.13) and are suppressed by powers of the cutoff scale (2.3). Following [30], here we comment on their form.
The first class of corrections was discussed in [10,13] and arises considering the effect of curvature terms in the superfluid and vortex action; these corrections are controlled by the sphere radius and hence scale as 1/(ΛR) 2 ∼ 1/Q 2/3 . These are present also in the absence of vortices.
Focus now on the single vortex state described in sec. 4.2. We find corrections controlled by the vortex length L ∼ J 34 /Q, which hence scale as 1/(ΛL) 2 ∼ Q 1/3 /J 34 . They arise from the terms we neglected in the NG action to write (3.10) and are proportional to (∇ i b i )/B, f 2 /B 2 and˙ X 2 . Higher derivatives of the string line element as well as the phonon contribution to the energy (3.2) belong to the same class. Similarly, there are corrections of the form Q 1/3 /J 34 , Q 1/3 /J 12 to eq. (4.18) for a two vortex state. Notice that the subleading Q 2/3 term in the ground state energy is bigger than the vortex contribution (4.15) for Q 1/3 J 34 Q. The latter gives instead the leading contribution for J 34 ∼ Q. The vortex contribution is anyway functionally distinguished from the ground state energy correction and is thus always calculable.
Let us now turn our attention to the Kelvin waves discussed in 4.4. The same corrections discussed for a vortex ring exists in this case. Furthermore, for n 1 higher derivative corrections to the single vortex action (4.28) become important. As typical for a non-relativistic field, these arise due to terms with two time derivatives, or, equivalently, with four space derivatives (suppressed by an extra H −2/3 factor by Weyl invariance) and scale as n 2 /Q 2/3 = J 2 12 /Q 2/3 . Notice that the relative corrections to the ground state energy of the vortex are bigger than the Kelvin wave energy (4.30) for J 2 12 /Q 1/3 Q 1/2 /J 1/2 34 ; however, these corrections are independent of J 12 , which enters only through (4.30).
Finally, the leading corrections to the energy of the vortex crystals states discussed in 4.3 arise both from the phonon contribution to the energy, which is proportional to (∇ i b i ) 2 /f 2 ∼ J ab /Q 4/3 2 , and from the free tension contribution, which gives Q/J corrections using (4.21) or (4.23). Here J ab stands for both J 34 and/or J 12 depending on the state.
Correlators
We now turn our attention to the study of correlators. As in [30], the most natural correlation function 6 which can be studied corresponds to a current insertion within two equal vortex states. In the EFT, this is determined through the following relations:
j 0 = Q 2π 2 R 3 , j φ = √ g 2π f ηξ , j ξ = − √ g 2π f ηφ . (5.1)
The hydrophoton field is obtained from (4.1), which, following the analogy with Ampère's law, can be conveniently rewritten in integral form as
1 2 C dx i ijk √ gf jk = −e 2 λ enc , (5.2)
where λ enc is the vorticity flux through the surface enclosed by the curve C. Using this, eq.s (5.1) can be used to make nontrivial predictions about the OPE coefficients of the theory. Consider first the traceless symmetric state corresponding to a radius R vortex in the (X 1 , X 2 ) plane, which has J 34 = Q and J 12 = 0. For this state, eq. (5.1) reads:
j 0 = Q 2π 2 R 3 , j φ = e 2 4π 2 R , j ξ = 0. (5.3)
The expectation value of a spin-1 parity even conserved operator in a traceless symmetric state |(J, J), J 34 = 2J, J 12 = 0 is [48]: Cutting off the sums at m √ J , we obtain the following predictions
(a m = Q 2π 2 , if m = 0, 0, if 1 ≤ m Q 1/3 ; b m = 3Q 2/3 8π 2 α , if m = 0, 0, if 1 ≤ m Q 1/3 .j 0 = Q 2π 2 R 3 , j φ = e 2 2π 2 R J Q cos 2 η, j ξ = 0.a m = Q 2π 2 , if m = 0, 0, if 1 ≤ m J/Q;b m = 3 8π 2 α J 34 Q 1/3 , if m = 0, 1, 0, if 2 ≤ m J/Q. (5.8)
Analogously, for the state (4.24) with Q J 12 , J 34 Q 4/3 , the EFT gives
j 0 = Q 2π 2 R 3 , j φ = e 2 2π 2 R J 34 Q cos 2 η, j ξ = e 2 2π 2 R J 12 Q sin 2 η.a m = Q 2π 2 , if m = 0, 0, if 1 ≤ m J 12 /Q; b m = 3 8π 2 α J 34 Q 1/3 , if m = 0, 1, 0, if 2 ≤ m J 12 /Q; c m = (−1) m 3 8π 2 α J 12 Q 1/3 , if m = 0, 1, 0, if 2 ≤ m J 12 /Q.
Vortices in arbitrary dimensions
Based on the considerations so far, as well as on the previous results of [30], it is not hard to understand the qualitative feature of the vortex EFT in generic dimensions. We give some brief comments here for completeness. We focus on the derivation of the scaling dimensions for traceless symmetric operators in generic higher dimensions. We first need to construct the dual of the d
S KR = − p λ p d d−1σ A µ 1 µ 2 ...µ d−1 ∂ τ X µ 1 p ∂ σ 1 X µ 2 p . . . ∂ σ d−2 X µ d−1 p . (6.2)
One can similarly write the Nambu-Goto like action for the membrane [51]; we do not report here the expression since its detailed form will not be needed in the following.
One can now proceed as in sec. 4. From the energy momentum tensor, one finds that the leading contribution to the vortex energy comes from the hydrophoton gauge field. Generalizing eq.s (2.16) and (4.9), the angular momentum is proportional to the volume enclosed by the vortex in embedding coordinates. For Q 1/d J ≤ Q, the minimal energy state corresponds to a single spherical vortex. The leading contribution to the vortex energy arises from the running of the tension, induced by the hydrophoton contribution to the self-energy as in (4.13). This can be computed using a flat space approximation for the gauge field Green function and a UV hard cutoff Λ ∼ Q 1/d /R to regulate the result:
∆ = ∆ 0 (Q) + d 2α(d + 1) J d−2 d−1 Q 1 d(d−1) log J/Q 1 d , Q 1/d J ≤ Q,(6.3)
where ∆ 0 (Q) is given by (2.4). We expect d dependent corrections of order J
d−2 d−1 Q 1 d(d−1) to (6.
3), similarly to (4.14); these contributions however will not be logarithmically enhanced by the cutoff. As in section 4.3, for Q J Q d+1 d we can identify the minimal energy state as a vortex crystal. Following the same steps which lead to (4.22), we find that the energy of this state is
∆ = ∆ 0 (Q) + d 4α J 2 Q d+1 d , Q J Q d+1 d . (6.4)
Eq.s (6.3) and (6.4) match the results obtained in [30] and in this paper for d = 2, 3.
Conclusions and future directions
Condensed matter phases often admit a simple effective description [22]. In CFTs, one can take advantage of this using the state/operator correspondence to study CFT data at large quantum numbers. In this work, these ideas were used to compute the scaling dimensions of operators of large internal charge and spin in a U (1) invariant CFT 4 . The results obtained for traceless symmetric operators can be seen as a generalization of those obtained in CFT 3 [30]; however the study of operators in mixed symmetric representations explored qualitatively distinct regimes, such as Kelvin wave propagation on a string (sec. 4.4). We also provided predictions for correlators of the U (1) current in between vortex states in sec. 5 and generalized the predictions for the scaling dimensions of traceless symmetric operators to arbitrary dimensions in eq.s (6.3) and (6.4). The most direct extension of this work would be a detailed analysis of higher order corrections, both in three and four dimensions. In particular, a refinement of the continuum approximation used in sec. 4.3 might allow for the study of collective excitations in the vortex crystal phase, corresponding to new CFT operators, possibly similar to the Tkachenko mode studied in [46,47].
Within the exploration of the superfluid phase of CFTs [10,13,14,15,16,17,18,19,20], the non-Abelian case still leaves some open questions. As argued in [13], the basic prediction for the scaling dimension of the lightest charged operator is insensitive to the non-Abelian nature of the symmetry group, which instead manifests itself via the existence of the so-called gapped Goldstones [52,53]. Massive Goldstones are crucially needed to close the current algebra of the non-Abelian group, like standard Goldstones, but at the same time have a fixed gap of order cutoff dictated by the symmetry. Their role in the large charge sector of CFTs and in more general finite density QFTs deserves further investigation [54].
Most of the existing results for large charge operators in CFTs are derived under the assumption that the CFT admits a superfluid phase. It is hence important to check whether this assumption applies in known theories. So far, most computations focussed on the prediction for the scaling dimension of the lightest charged operator. This has been verified in Monte-Carlo simulations of the O(2) [55] and O(4) [56] model and perturbatively for Monopole operators [59,60,61,62,63], to order O(N 0 ) in the CP N model [57] and at leading order in the number of flavors in QED 3 and the gauged Gross-Neveu model [58]. Relatedly, large charge states have been studied in AdS/CFT in the context of holographic superconductors [64,65,66]. We are currently addressing similar questions within the ε-expansion [67], which allows for extensive checks of the EFT predictions. Perhaps, the techniques explored in these works might also find application in a different context, such as the study of processes with many external legs within the Standard Model [68,69,70,71].
Despite their generality, superfluids are not the only possible description for large charge states in CFT. For instance, if one assumes the charge Q to be unbroken, a natural phase 7 which might describe the CFT is a Fermi liquid [74]. Different descriptions are also possible and are expected to apply in the presence of moduli spaces, which naturally allow for light degrees of freedom other than the Goldstone mode. This is the case for free massless theories and N = 2 superconformal field theories, where the large R-charge expansion is organized differently [75,76,77]. In [78,79] the possibility of a semiclassical but inhomogeneous phase in the O(4) model was also explored.
In [80], the question of how to find solutions to the crossing equations at large charge was addressed in connection with the existence of a macroscopic limit of correlators [81]. However, it remains an open question whether one can relate explicitly the large charge sector of CFTs with the spectrum of light operators. Perhaps relatedly, a bootstrap analysis recently connected large scaling dimensions tails of weighted spectral density of primary operators with light operators exchanged in the dual channel [82].
Finally, the predictions of analyticity and the existence of a perturbative expansion for large internal quantum numbers are reminiscent of the bootstrap results for large spin operators [7,8,31,32,33,34,35,36,37,38,39]. The physical picture behind those results is particularly clear in the dual AdS space 8 [8,31], where double-trace operators are associated to two widely separated objects. Because of the AdS geometry, these only interact weakly via the exchange of highly off-shell modes. A universal EFT description might exist in this case as well.
Consider the action of a massless vector field coupled to a conserved current J µ (in Euclidean signature):
S = d d x √ g 1 4 f µν f µν − a µ J µ , f µν = ∂ µ a ν − ∂ ν a µ . (A.1)
The gauge field on the equations of motion is given by
a µ (x) = d d x g G µν (x, x )J ν (x ), (A.2)
where G µν (x, x ) satisfies the equation
∇ µ ∂ µ G νν (x, x ) − ∂ ν G µν (x, x ) = −g νν (x) δ(x − x ) √ g + ∂ ν Λ ν (x, x ). (A.3)
Here Λ ν is a pure gauge term which drops from physical observables; primed and unmprimed indices refer, respectively, to the points x and x . Let us define the following biscalar
u = 1 2 (X − X ) 2 , (A.4)
where (X − X ) 2 is the chordal distance in embedding coordinates. Given the isometries of the sphere, it is possible to parametrize the propagator as
G νν (x, x ) = − (∂ ν ∂ ν u) F (u) + ∂ ν ∂ ν S(u). (A.5)
The last term is gauge dependent and drops from eq. (A.2). The following properties hold:
1. ∇ µ ∂ µ u = d(1 − u) , 2. g µν ∂ µ u∂ ν u = u(2 − u), 3. ∇ µ ∂ ν u = g µν (1 − u), 4. (∇ µ u)(∇ µ ∂ ν u) = (1 − u)∂ ν u, 5. (∇ µ u)(∇ µ ∂ ν ∂ ν u) = −∂ ν u∂ ν u,
These can be easily explicitly verified in stereographic coordinates, for instance. It follows
∇ µ ∂ µ G νν (x, x ) − ∂ ν G µν (x, x ) = − (∂ ν ∂ ν u) u(2 − u)F + (d − 1)(1 − u)F + (∂ ν u∂ ν u) (1 − u)F + (1 − d)F . (A.6)
By symmetry, we can write Λ ν (x, x ) = (∂ ν u)Λ(u). Then for x = x (A.3) gives two equations:
u(2 − u)F + (d − 1)(1 − u)F = −Λ, (A.7) (1 − u)F − (d − 1)F = Λ . (A.8)
We can integrate the second and plug the result in the first to obtain
(2 − u)uF (u) + d(1 − u)F (u) − (d − 2)F (u) = 0. (A.9)
This is just Klein Gordon equation for a scalar field of mass m 2 = d − 2 on S d . The solution is fixed requiring a power low singularity for u → 0 and regularity at the antipodal point u → 2 [87]:
F (u) = Γ(d − 2) (4π) d 2 Γ d 2 2 F 1 1, d − 2; d 2 ; 1 − u 2 , d > 2. (A.10)
The normalization is determined matching the short distance limit with a flat space propagator. Plugging in (A.5), we get eq. (4.6) in the main text.
B Vortex energy in dimensional regularization
To regulate the computation of the magnetostatic energy, it is convenient to work in d+1 spacetime dimensions. It is natural to modify the Lagrangian (3.2) in a way which preserves Weyl invariance:
L = −κH (d+1)/3 . (B.1)
The definition of H in terms of the two-form field A µν here is unchanged. Notice that working in arbitrary d with a 2-form field we loose the duality with a shift invariant scalar, hence this regularization breaks U (1) invariance at intermediate steps in the calculation 10 . A more responsible approach might be, perhaps, to promote A µν to a d − 2 form field, preserving Weyl invariance of the action. An investigation of this issue might be helpful in expanding the result of this paper to subleading orders. Expanding the action (B.1) to quadratic order gives
L f luct = 1 4e 2 (d) f ij f ij + 1 2e 2 (d) ḃ iḃ i − (d − 3) 3 (∇ i b i ) 2 , (B.2)
where we defined the electric coupling in d space dimensions as
e 2 (d) = √ 6B 2− d+1 3 2(d + 1)κ = e 2 1 − (d − 3) log B 1 3 + 1 4 + 1 6 log 6 + O (d − 3) 2 . (B.3)
The NG action discussed in section 3.2 is unchanged in d dimensions.
B.1 Vortex ring self-energy
Consider a single vortex moving on a trajectory given by (4.10). We want to compute the selfenergy contribution due to the hydrophoton, i.e. the second term in eq. (4.12). In Hopf coordinates (1.13) and in dimensional regularization, it reads:
E mag = e 2 (d) 2 R 2 dξdξ G ξξ (η, ξ, φ); (η, ξ , φ) = πRe 2 (d)R 3−d I(r, d), (B.4)
where we isolated the integral
I(r, d) = r 2 Γ(d − 2) (4π) d/2 Γ d 2 2π 0 dξ cos(ξ) 2 F 1 1, d − 2; d 2 ; 1 − 1 2 r 2 (1 − cos ξ) .
(B.5) 10 Conversely, a cutoff approach as in [30] breaks Weyl invariance.
In d = 3, the integral is logarithmically divergent for ξ → 0, corresponding to the interaction of an infinitesimal line element with itself. Setting 1 2 (1 − cos ξ) = y in (B.5), we get
I(r, d) = 2r 2 Γ(d − 2) (4π) d/2 Γ d 2 1 0 dy (1 − 2y) (1 − y)y 2 F 1 1, d − 2; d 2 ; 1 − r 2 y . (B.6)
The divergent part comes from the first term in the expansion of the hypergeometric function when the argument goes to one:
2 F 1 (a, b; c; 1 − z) z→0 − −− → 1 z a+b−c Γ(c)Γ(a + b − c) Γ(a)Γ(b) , a + b > c. (B.7)
We separate explicitly this contribution and recast the integral as:
I(r, d) = I div (r, d) + I reg (r, d), (B.8)
where the divergent piece is
I div (r, d) = 2r 2 Γ d 2 − 1 (4π) d/2 1 0 dy (1 − 2y) (1 − y)y 1 r 2 y d−2 2 = r 2π(3 − d) + r log 4πr 2 − γ E − 2ψ (3/2) 4π + O ((3 − d)) , (B.9)
and the regular part can be evaluated directly in d = 3, where it reads
I reg (r) ≡ I reg (r, 3) = r 2π 2 1 0 dy (1 − 2y) y √ 1 − y arcsin 1 − r 2 y 1 − r 2 y − π 2 . (B.10)
To compute the latter, it is convenient to use the following expansion
arcsin √ 1 − x 2 √ 1 − x 2 = ∞ m=0 −(−2) m+1 Γ m+3 2 2 (m + 1) 2 x m m! , 0 ≤ x ≤ 1. (B.11)
Interchanging sum and integral, the regular part gives
I reg (r) = r 2 4π 2 ∞ m=1 (−1) m+1 2π(m − 1)r m−1 m(m + 1) = r π − r 2π log(r + 1) − 1 π log(r + 1). (B.12)
Collecting everything and adding the tension contribution, we arrive at (4.13).
B.2 Kelvin waves frequency
The EOMs which derive from (4.28) give the oscillation frequency of Kelvin waves as:
1 2 Bω n = γ πB 2/3 R 2 n 2 − 1 + 2πe 2 (d)R 3−d R 2 δω I n , (B.13)
where the second term comes from the nonlocal piece of the action and is written in terms of the following integral:
δω I n = 1 2 dσ n 2 cos(nσ) − cos σ F (1 − cos σ) + cos 2 σ − cos(nσ) cos σ F (1 − cos σ) . (B.14)
Let us sketch the evaluation of (B.14). Changing variables as before, we write δω I n as the sum of the following two contributions:
I 1 (n) = Γ(d − 2) (4π) d/2 Γ(d/2) 1 0 dy y(1 − y) n 2 T n (1 − 2y) − (1 − 2y) 2 F 1 (1, d − 2; d/2; 1 − y), (B.15) I 2 (n) = − Γ(d − 1)/2 (4π) d/2 Γ d 2 + 1 1 0 dy(1 − 2y) y(1 − y) [(1 − 2y) − T n (1 − 2y)] 2 F 1 (2, d − 1; d/2 + 1; 1 − y).
(B.16) Here T n (x) = cos (n arccos(x)) is a Chebyshev polynomial. The divergent contributions are identified from the leading term of the Hypergeometric expansion (B.7) and can be evaluated using
1 0 dy T n (1 − 2y) y(1 − y) y m− 1 2 = √ πΓ(m) 1 2 − m n Γ m + n + 1 2 .
(B.17)
To evaluate the regular parts, we use the following results:
λ n ≡ 1 0 dy T n (1 − 2y) y(1 − y) arcsin √ 1 − y √ y √ 1 − y − π 2 √ y = π 2 ψ n 2 + 1 + 2ψ n + 1 2 − ψ n + 1 2 − 2ψ (n + 1) , (B.18) ρ n ≡ 1 0 dy 1 − 2y y(1 − y) T n (1 − 2y) 2 F 1 2, 2; 5 2 ; 1 − y − 3π 8y 3/2 + 3π 16y 1/2 = 3 2 π n 2 + 1 ψ n − 1 2 − ψ n − 1 2 + log 2 + 4n 4 + 6n 2 + 3n − 1 4n 3 − 4n 2 − n + 1 + 3 8 . (B.19)
Using Mathematica we computed these integrals explicitly for fixed integer values of n and identified their functional form; the result was then verified numerically and using the n → ∞ asymptotic expansion of the results (B.18) and (B.19). This indeed can be obtained explicitly truncating the series expansion of the Hypergeometric functions in the integrals and using (B.17). The regular contributions from I 1 (n) and I 2 (n) finally read
I reg 1 (n) = 1 4π 2 n 2 λ n − λ 1 , (B.20) I reg 2 (n) = 1 12π 2 (ρ n − ρ 1 ) − 1 64π 1 0 dy (1 − 2y) [T n (1 − 2y) − (1 − 2y)] y (1 − y) = 1 12π 2 (ρ n − ρ 1 ) + 3ψ n + 1 2 + 3γ E − 4 + log(64) − 6 4n 2 −1 96π . (B.21)
The second contribution in (B.21) arises since we subtracted the O 1/ √ y term in the expansion of the Hypergeometric function from the first piece, in order to use (B.19). Collecting everything and expanding for d → 3, we find the following remarkably simple result:
δω I n = n 2 − 1 8π(3 − d) + n 2 − 1 log π − 2ψ n+1 2 − γ E − 1 16π
.
(B.22)
Eq. (4.30) then follows.
C Nambu-Goto action from the coset construction
In [13], it was argued that two charged scalar operators insertions in d + 1 dimensions at x = 0 and x = ∞ induce a specific symmetry breaking pattern for the leading trajectory in the path integral. A similar logic can be applied when the operators have also large spin J. As in the scalar case, translations P µ , special conformal transformations K µ and dilatation D are broken, with the combination D + µQ left unbroken. Assuming the operator insertion to be polarized in the (x 1 , x 2 ) plane, the Lorentz generators J 1p , J 2p with p, q = 0, 3, . . . must necessarily be broken. A vortex corresponds to the regime where it is energetically favorable for the system to still be in an almost homogeneous state, rotations being broken by a localized region of size 1/j 0 ∼ R/Q 1/d in which the superfluid description breaks. This region naturally extends from 0 to ∞ along the directions orthogonal to the spin polarization, corresponding hence to a d − 1 dimensional membrane. In this regime, J 12 parametrizes rotation around the vortex and it is thus unbroken. We then identify the symmetry breaking pattern corresponding to a vortex as:
D = D + µ Q, J 12 , J pq unbroken, D, P µ , K µ , J mp broken. (C.1)
where we introduced the set of indices m, n = 1, 2 and p, q = 0, 3, . . . . In order to apply the coset construction in a curved manifold, it is convenient to think in terms of the generators acting in a local chart, denoted { D, P µ , K µ , J µν }. These are naturally associated with those acting on the plane considering the formal R → ∞ limit on R × S d [13]:
D = −R P 0 , J ij = J ij , J 0i = −R P i , P 0 = P 0 + D R + K 0 2R 2 , K 0 = 1 2 K 0 − R D + R 2 P 0 , P i = P 0 + J 0i R − K i 2R 2 , K i = 1 2 K i + R J 0i − R 2 P i . (C.2)
We then rewrite the symmetry breaking pattern (C.1) in terms of the hatted generators. Focussing on 2 + 1 and 3 + 1 dimensions, we get 2 + 1 :
P 0 = P 0 + µ Q, J 12 unbroken, P i , J 0i , K µ , Q broken;
3 + 1 :
P p = P p + µ Q, J 12 unbroken, P m , J 0i , J n3 , K µ , Q broken.
(C.3)
From (C.3) we can construct the Nambu-Goto action for the vortex via the coset construction [88,89], applied to the case of a membrane [51].
C.1 2+1 dimensions
Following [13], we gauge all spacetime symmetries and specify the manifold only at the end of computations. We henceforth do not consider special conformal transformations anymore and work with a covariant derivative in terms of three gauge connections:
D µ = ∂ µ + iẽ a µ P a + i 2 ω ab µ J ab + iA µ D. (C.4)
Inices a, b = 0, 1, . . . label the gauged Poincaré generators and should not be confused with spacetime indices µ, ν, . . . [90,91]. From (C.1), the coset of a vortex line in 2+1 dimensions is formally identical to the conformal superfluid one Ω = e iy aP a e iσD e iη i J 0i e iπQ = e iy a Pa e iσD e iη i J 0i e iχQ , χ = µt + π, (C.5)
The Maurer-Cartan (MC) one form reads
Ω −1 D µ Ω = iE a µ P a + ∇ a σD + ∇ a πQ + ∇ a η i J 0i + 1 2 Ω ij a J ij ; (C.6) where E a µ = e −σ e b µ Λ a b , ∇ a π = e σ e µ b Λ b a ∂ µ χ − µδ 0 a , ∇ a σ = e σ e µ b Λ b a (∂ µ σ + A µ ) . (C.7)
Here e a µ transforms as a vierbein. We introduced the Lorentz matrix (e −iη i J 0i ) a b = Λ a b . The expressions of ∇ a η i and Ω ij a are not needed here. One can also construct curvature invariants as
Ω −1 [D µ , D ν ]Ω = iE a µ E b µ T c ab P c + 1 2 R cd ab J cd + A ab D . (C.8)
Explicit expressions for these can be found in [13]. Finally, one needs to consider the projection of the MC one form onto the vortex world-line x µ (λ) [51]:
x µ Ω −1 D µ Ω = iE P 0 + ∇y i P i + ∇σD + ∇χQ + ∇η i J 0i + 1 2 Ω ij J ij , (C.9)
where E =ẋ µ e −σ e b µ Λ 0 b , ∇y i = E −1ẋµ e −σ e b µ Λ i b , ∇χ = E −1ẋµ ∂ µ χ, ∇σ = E −1ẋµ (∂ µ σ + A µ ) . (C.10) We can reduce the number of independent Goldstones setting to zero one or more of the invariants in (C.6), (C.8) or (C.9). When an algebraic solution exists, these conditions are called Inverse Higgs Constraints (IHCs) [92,93]. In this case, the same IHCs which lead to the superfluid action are imposed:
T a bc = 0, ∇ 0 π = 0, ∇ i π = 0, ∇ a σ = 0. (C.11)
The first is a torsion free condition and selects the the spin one-connection ω ab µ compatible with the metricĝ µν = e −2σ g µν . The others are used to eliminate A µ , σ and η i :
A µ = −∂ µ σ,
µe −σ = (e µa e ν a ∂ µ χ∂ ν χ) 1/2 ,
η i η tanh η = − e µ i ∂ µ χ e µ 0 ∂ µ χ , (C.12)
where η ≡ η i η i and (∂χ) = (e aµ e ν a ∂ µ χ∂ ν χ) 1/2 . We can now construct the leading order invariants in the world-line. Noticing that ∇χ = µ and ∇σ = 0, these are constructed out of the einbein E and the covariant derivative ∇y i as:
µE =ẋ µ ∂ µ χ, ∇y i ∇y i = 1 − (∂χ) 2ẋµẋ µ (ẋ µ ∂ µ χ) 2 . (C.13)
The most general NG action is then written in terms of an arbitrary function: S = µ dλ E f ∇y i ∇y i = dt ẋ µẋ µ (∂χ)F (ẋ µ ∂ µ χ) 2 (∂χ) 2ẋµẋ µ .
(C.14)
This is precisely the action used in [30].
C.2 3+1 dimensions
From (C.3), the coset is written as Ω = e iy a Pa e iσD e iη i J 0i e iξ n J n3 e iχQ . (C. 15) One can compute the MC one form as before Ω −1 D µ Ω = iE a µ P a + ∇ a σD + ∇ a χQ + ∇ a η i J 0i + ∇ a ξ n J n3 + Ω 12 a J 12 ; (C. 16) with
E a µ = e −σ e c µ Λ b c R a b , ∇ a χ = e σ e µ c Λ c b R b a ∂ µ χ, ∇ a σ = e σ e µ c Λ c b R b a (∂ µ σ + A µ ) . (C.17)
Here we introduced another Lorentz matrix (e −iξ n J n3 ) a b = R a b . Curvature invariants are written as before. The MC form projected on the vortex world-sheet X µ (τ, σ) reads:
∂ α X µ Ω −1 D µ Ω = iE p α P p + ∇ p y n P n + ∇ p σD + ∇ p χQ + ∇ p η i J 0i + ∇ p ξ n J n3 + Ω 12 p J 12 , (C. 18) where α = τ, σ and
E p α = ∂ α X µ e −σ e c µ Λ b c R p b , ∇ p y n = E α p ∂ α X µ e −σ e c µ Λ b c R n b , ∇ p χ = E α p ∂ α X µ ∂ µ χ, ∇ p σ = E α p ∂ α X µ (∂ µ σ + A µ ) .
(C. 19) Here E α p is the inverse of the world-sheet vierbein: E p α E α q = δ p q , E p α E β p = δ α β . As before, the IHCs (C.11) are imposed. Since [P 3 , J n3 ] ∼ P n , we can also eliminate ξ n imposing the following IHC
∇ 3 y n = 0 =⇒ ξ n ξ tan ξ = v n v 3 , (C.20)
where the vector v i is given by
v i = (∂ 3 X µ ∂ µ χ) ∂ 0 X µ e c µ Λ i c − (∂ 0 X µ ∂ µ χ) ∂ 3 X µ e c µ Λ i c (∂χ) −det(G αβ ) h αβ G αβ . (C.21)
Here G αβ and h αβ are:
G αβ = g µν ∂ α X µ ∂ β X ν , h αβ = ∂ µ χ∂ ν χ (∂χ) 2 ∂ α X µ ∂ β X ν . (C.22)
These expression agree with the previous definitions (3.11) and (3.12). Since ∇ p χ = µδ 0 p and ∇ p σ = 0, leading order invariants are built out of the following objects µ 2 det(E p α ) = (∂χ) 2 |det(G αβ )| G αβ h αβ , ∇ 0 y n ∇ 0 y n = 1 − 1 h αβ G αβ .
(C.23)
One finally writes the leading order action as S = µ 2 dτ dσ det E p α f (∇ 0 y n ∇ 0 y n ) = dτ dσ(∂χ) 2 |det(G αβ )|F [G αβ h αβ ]. (C.24)
Using (3.3), this agrees with the last line in (3.10).
J•
= αQ 4/3 + ∆ V (Q, J 34 ) + π(J 2 12 − 1)Q 1/3 3 8α log Q 2/3 − 2ψ J 12 + 1 2 − 2γ E − 1 − log 64 +γ + O 12 ≤ J 34 ≤ Q and (J 12 + J 34 − Q) 2 J 12 J 34 /Q 2/3, the minimal energy state is given by two vortex rings. When 1 ≤ Q − J 34 Q 1/3 the energy is given by the sum of the two free contributions ∆ = αQ 4/3 + ∆ V (Q, J 34 ) + ∆ V (Q, the result in the general case, which takes the same form only to logaFor Q J 12 ≤ J 34 Q 4/3 the superfluid arranges in a vortex lattice as in (1.6); the scaling dimension of the corresponding
.
2) where H = −H µνρ H µνρ and κ The U (1) current provides the relation between χ and A µν :
Figure 2 :
2The vortex ring orbit in stereographic coordinates.
4.31) we see that the n = 0 mode decreases J 34 (and the radius of the vortex) by one unit,
Figure 3 :
3A Kelvin wave in stereographic coordinates (x, y, z) = 1 1+X 1 (X 3 , X 4 , X 2 ).hence it corresponds to the quantization of the classical ring solution discussed in 4.2. Eq. (4.32) implies that the n = ±1 modes do not correspond to new states, but describe rotations of the string orientation and therefore have vanishing frequency. In this sense, their role is analogous to that of the J = 1 phonons in (2.6), describing descendants of the ground state. The modes with |n| ≥ 2 correspond to new solutions and are interpreted as Kelvin waves propagating on the vortex; in the CFT they correspond to operators with the following quantum numbers 5 J 34 = Q − 1, J 12 = n, ∆ = αQ 4/3 + ∆ V (Q, Q) + ∆ k (n).(4.33)
, J), 2J, 0|j φ (η, ξ, φ)|(J, J), 2J, 0 = R −3 2J m=0 b m sin 2m η, (J, J), 2J, 0|j ξ (η, ξ, φ)|(J, J), m and b m are arbitrary theory dependent real coefficients, subject to the constraint m b m = 0. Then the EFT gives
made only for m Q 1/3 since the EFT breaks for distances of order of the inverse cutoff (2.3) from the vortex. A similar analysis can be done for the vortex crystal states in (4.22) and (4.24). Consider first the traceless symmetric case Q J 34 Q 4/3 and J 12 = 0. Using (4.21), eq. (5.1) reads
holds on scales larger than the vortex separation ∼ 1/ √ J ∼ Q/J, on which the continuous approximation (4.21) can be used. It is then convenient to rewrite eq. (5.4) in Fourier basis (J, J), 2J, 0|j 0 (η, ξ, φ)|(J, J), 2J, 0 = R −3 2J m=0ã m cos (2m η) , (J, J), 2J, 0|j φ (η, ξ, φ)|(J, J), 2J, 0 = R −3 2J m=0b m cos (2m η) .
2 ,
2of generality, we assume J 12 ≤ J 34 . The three-point function of a spin-1 conserved operator in a mixed symmetric state |(J,J), J 34 , J 12 , with (J,J) = |J 34 −J 12 | |J 34 +J 12 | ,J), J 34 , J 12 |j φ (η, ξ, φ)|(J,J), J 34 , J 12 = R −3 2|J−J|+1 m=0 b m cos(2m η), (J,J), J 34 , J 12 |j ξ (η, ξ, φ)|(J,J), J 34 , J 12 = R −3 2|J−J|+1 m=0 c m cos(2m η).
m , b m and c m are real coefficients, which satisfy the constraints m (−1) m b m = m c m = 0 and b 2J+1 = −c 2J+1 . We then obtain the following results for the OPE coefficients:
+ 1 dimensional Lagrangian (2.1) in terms of a d − 1 form gauge field A. Proceeding as in sec. 3.1, this reads L = −κ |H · H| d+1 2d , H = dA. (6.1) As in (3.3), the gauge and the scalar description are related by * H ∝ j, where * stands for the Hodge dual. The action (6.1) can be expanded to quadratic order in terms of a non-propagating hydrophoton d − 2 gauge form and a longitudinal vector corresponding to the phonon. Vortices are d − 1 membranes which couple to the gauge field A through a Kalb Ramond like interaction. Calling X µ p (σ) their line elements, whereσ = (τ, σ 1 , . . .) parametrizes the membrane coordinates, this coupling reads
See appendix C for a derivation from the coset construction.2 Here we correct a typo in eq. (19) of[30].
Notice that vorticity is quantized q p ∈ Z.
This can be used to derive the NG string action, see appendix C.
In ∆ we neglect a ∼ Q −1/3 contribution from the vortex Casimir energy, which is independent of J 12 .
To leading order, scalar insertions read as in the homogeneous phase[13].
If this is consistent with conformal invariance[72,73].8 A similar picture was originally proposed in[83], with no reference to the gravity dual.9 A similar derivation in 4d de Sitter can be found in[84].
Acknowledgements I thank Anton de la Fuente, Alexander Monin, and Riccardo Rattazzi for useful discussions and valuable comments on the draft. My work is partially supported by the Swiss National Science Foundation under contract 200020-169696 and through the National Center of Competence in Research SwissMAP.A Photon propagator on the sphereHere we obtain the photon propagator on a d dimensional sphere following the simple method 9 of[85,86]. In this section we set R = 1.
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| {'fraction_non_alphanumeric': 0.0789722815437793, 'fraction_numerical': 0.07052905043922415, 'mean_word_length': 3.485852550491334, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 121, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We include vortices in the superfluid EFT for four dimensional CFTs at large global charge. Using the state-operator correspondence, vortices are mapped to charged operators with large spin and we compute their scaling dimensions. Different regimes are identified: phonons, vortex rings, Kelvin waves, and vortex crystals. We also compute correlators with a Noether current insertion in between vortex states. Results for the scaling dimensions of traceless symmetric operators are given in arbitrary spacetime dimensions.IntroductionConformal field theories (CFTs) play a key role in particle and condensed matter physics. As fixed points of the renormalization group flow, they act as landmarks in the space of quantum field theories (QFTs). Through the AdS/CFT correspondence[1,2], they promise to shed light on quantum gravity. They also describe critical points for second order phase transitions. Finally, CFTs are also among the few examples of interacting QFTs where exact results are available without supersymmetry. Recently, the bootstrap program[3,4]achieved much progress in the study of CFTs, both through numerical [5, 6] and analytical[7,8,9]techniques.Basic observables in CFTs are correlation functions of local operators in the vacuum. Despite this, sometimes one can make predictions for the CFT data defining the theory studying the dynamics of finite density states [10]. This is a consequence of the state/operator correspondence[11,12], which relates states in radial quantization to local operators with the same quantum numbers. So far, this idea has been mainly applied in the investigation of the superfluid phase in conformal field theories[10,13,14,15,16,17,18,19,20]. Indeed superfluids are the most natural candidates to describe states at large internal quantum numbers in CFTs. They admit a simple and universal effective field theory (EFT) description[21,22]which allows the computation of correlators in a perturbative expansion controlled by the charge density. The same strategy was recently applied also in the context of non-relativistic CFTs[23,24,25].As the angular momentum is increased, the superfluid starts rotating and vortices develop[26]. These can be included in the EFT as heavy topological defects[27,28,29]. In[30], this EFT was used to describe operators with large spin and large charge in three dimensional CFTs. In this work, we study the predictions of the vortex EFT for four dimensional CFTs.For traceless symmetric operators J =J = J 34 /2, the corresponding state passes through three distinct regimes, qualitatively similar to the CFT 3 case:• For 2 ≤ J 34 Q 1/3 the lightest operator corresponds to a phonon wave of angular momentum J in the superfluid. The scaling dimension is given by• For Q 1/3 J 34 ≤ Q, the minimal energy state is given by a single vortex ring, whose radius increases with J. Its energy isThe leading contribution in (1.5) comes from the first term, because of the logarithmic enhancement. The other terms can be interpreted as finite-size corrections due to the vortex extension and are functionally distinguished from the relative Q 1/3 /J corrections.• For Q J 34 Q 4/3 the superfluid forms a vortex crystal. The scaling dimension of the corresponding operator is given by(1.6)Mixed symmetric representations are conveniently parametrized in terms of J 34 , J 12 in (1.1). We write J ab to generically denote any of them. We find the following results: (2.18) corresponding to the charge density ρ = 3 2πR 2 J Q cos θ. The second term in (2.18) is the electrostatic energy of the crystal. The leading corrections arise from the vortex masses and the magnetic field fluctuations. As J → Q 3/2 the vortex velocities approach the relativistic regime and the EFT breaks down.', 'arxivid': '1906.07283', 'author': ['Gabriel Cuomo *[email protected] \nInstitute of Physics\nTheoretical Particle Physics Laboratory (LPTP)\nEPFL\nLausanneSwitzerland\n'], 'authoraffiliation': ['Institute of Physics\nTheoretical Particle Physics Laboratory (LPTP)\nEPFL\nLausanneSwitzerland'], 'corpusid': 189998761, 'doi': '10.1007/jhep02(2020)119', 'github_urls': [], 'n_tokens_mistral': 35846, 'n_tokens_neox': 29007, 'n_words': 16711, 'pdfsha': '5154d969c390c27e82e7b7da0532e0f7ce6b7860', 'pdfurls': ['https://arxiv.org/pdf/1906.07283v3.pdf'], 'title': ['Superfluids, vortices and spinning charged operators in 4d CFT', 'Superfluids, vortices and spinning charged operators in 4d CFT'], 'venue': []} |
arxiv |
Unidentified FRBs in archival data
1 Mar 2019
E F Keane
SKA Organization
Jodrell Bank Observatory
SK11 9DLMacclesfield, CheshireUK
D R Lorimer
Department of Physics and Astronomy
West Virginia University
P. O. Box 6315MorgantownWVUSA
Center for Gravitational Waves and Cosmology
Chestnut Ridge Research Building
West Virginia University
MorgantownWVUSA
F Crawford
Department of Physics and Astronomy
Franklin and Marshall College
PO Box 300317604LancasterPAUSA
Unidentified FRBs in archival data
1 Mar 2019Draft version March 4, 2019 Typeset using L A T E X RNAAS style in AASTeX62surveys -methods: data analysis FRB 010312
1 The relevant survey pointing is SMC018 034. 2 https://github.com/evanocathain/destroy gutted 3 http://sigproc.sourceforge.net/ 4 For example heimdall, a GPU-based single pulse search codehttps://sourceforge.net/projects/heimdall-astro/
Recently Zhang et al. (2019) reported the discovery of FRB 010312. This event occurred during a survey of the Magellanic Clouds and surrounding regions wherein the first fast radio burst, FRB 010724, was discovered (Lorimer et al. 2007). The reported signal-to-noise ratio (S/N) of FRB 010312 is 11, for a pulse width of 24(1) ms. Examining the data 1 with destroy 2 , and performing no RFI mitigation, we confirm the burst detection and obtain a S/N= 12 with a pulse width of 37 +17 −8 ms, at the reported dispersion measure. This implies a peak flux density of ∼ 150 mJy and specific fluence of
WHY MISSED?
Much more interesting than the above minutiae is why and how this event was missed in the previous analyses of these data. The survey was originally searched (Crawford et al. 2016) using seek from the sigproc 3 software suite. This algorithm uses down-sampling and smoothing processes in searching a time series, that can result in under-estimated S/N values; these steps were initially motivated as a time saving measure to overcome computational constraints that now no longer exist. The upshot is that the S/N values reported by seek can be degraded by as much as a factor of √ 2 (Keane & Petroff 2015). For FRB 010312, this exact scenario occurred and the detection S/N determined was 8 and based on this the candidate was dismissed as sub-threshold.
RE-PROCESSING ARCHIVAL DATASETS
Several archival datasets have only been searched using this and other pipelines that perform similarly or worse. Without re-processing these data with something optimised 4 , in the worst case the true detectable population of FRBs in a dataset could, in a Euclidean Universe, be 2 3/4 ≈ 1.65 times larger than one might initially estimate, i.e. a 10σ threshold would in fact be a 14σ threshold. Reprocessing these same data sets in search of radio pulsars as ever more sophisticated search techniques are developed, as RFI mitigation procedures improve and as computational constraints disappear has been successful over and over again (see Knispel et al. 2013 and references therein). This was, of course, true for the original FRB discovery and the same is likely true for FRB searches generally. As well illustrated by FRB 010312, there is potentially a large number of missed FRBs in datasets that have already been searched. The situation may even be more stark when we consider how difficult it is to mitigate radio frequency interference and that all FRB searches ever performed have looked for temporally symmetric broadband flat-spectrum pulses, and most FRBs do not look like that. We encourage further work using multiple search codes, where possible, and also searching a broad DM and pulse width range.
EFK would like to thank everybody who took part in FRB2019 in Amsterdam for a highly informative and enjoyable meeting. Endless reprocessing of FRB datasets is encouraged!
∼ 5.6 Jy ms if the signal occurred at boresight. As Zhang et al. point out, their detection brings this survey more in-line with other expectations for the FRB rate (Champion et al. 2016; Bhandari et al. 2018).
. Bhandari, MNRAS. 4753370MNRASBhandari et al., 2018, MNRAS, 475, 1427. Champion et al., 2015, MNRAS, 460, L30. Crawford et al., 2016, MNRAS, 460, 3370.
. & Keane, Petroff, Knispel, Science. 447777MNRASKeane & Petroff, 2015, MNRAS, 447, 2852. Knispel et al., 2013, ApJ, 774, 93. Lorimer et al., 2007, Science, 318, 777.
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arxiv |
THREE CIRCLES THEOREMS FOR HARMONIC FUNCTIONS
9 Jan 2016
Guoyi Xu
THREE CIRCLES THEOREMS FOR HARMONIC FUNCTIONS
9 Jan 2016
We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.2010 Mathematics Subject Classification. 35B40, 58J05, 53C23, 35A01. 1 2 GUOYI XU where ρ(x) = d(p, x) and p is some fixed point on M n . In [Yau87], the following conjecture was made: Conjecture 1.1 (S.-T. Yau). Let M n be a complete manifold with nonnegative Ricci curvature, then dim H d (M) < ∞ for any d > 0. P. Li and L.-F. Tam firstly proved the conjecture for linear growth harmonic function (d = 1) in [LT89], and they further verified the 2-dimensional case (n = 2) in [LT91]. In 1997, this conjecture was completely proved by Colding and Minicozzi [CM97a] (also see [Li97], [CM98a], [CM98b] and [LW99] for further developments). Although Yau's conjecture was confirmed, there are still several important questions about harmonic functions with polynomial growth remained open. It is well known that on any complete noncompact manifold, there always exist nonconstant harmonic functions (see [GW75]). A natural question is about the existence of nonconstant harmonic function with polynomial growth. Note any complete manifold with nonnegative Ricci curvature has at least linear volume growth (see [Yau75]). C. Sormani proved the following: Theorem 1.2 ([Sor00]). Let M n be a complete manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function of polynomial growth, then the manifold splits isometrically, M n = R × N n−1 .As observed in [Din04], there exists a suitable metric on R + × S n−1 , which has nonnegative Ricci curvature and linear volume growth, but can not split isometrically. From Theorem 1.2, it will not admit any nonconstant harmonic functions with polynomial growth. A concrete example is given in Section 4 (Example 4.8).Hence, to study the existence of nonconstant harmonic function with polynomial growth in nonnegative Ricci curvature context, we need to have some restriction on the volume growth of M n . To the author's knowledge, the following question is still open : Question 1.3. If (M n , g) is a complete manifold with Rc ≥ 0 and maximal volume growth, does there exist d ≥ 1 such that dim H d (M) ≥ 2? In other words, is there any nonconstant harmonic function with polynomial growth order at most d on M n ?In another direction, putting the existence problem in the positively curved context, L. Ni [Ni10] made the following conjecture: Remark 1.6. One novel thing in one of our Three Circles theorems (Theorem 3.2) is that it can be applied for the collapsed case, i.e. the case that the maximal volume growth assumption does not hold. Also Theorem 3.2 is dealing with the integral J u , whose domain is different from the original one considered in [Din04] (see Corollary 1.11 there). View from the domains of the integral appearing in those results, Theorem 3.2 is more like a three disks theorem, and Ding's technical tool is based on a three annuli theorem.As one application of our Three Circles Theorems, we proved the following theorem, which generalizes the existence result in [Din04].Theorem 1.7. Let (M n , g) be a complete manifold with nonnegative Ricci curvature, the tangent cone at infinity with renormalized limit measure is a unique metric
Introduction
In 1975, S.-T. Yau [Yau75] generalized the classical Liouville theorem to complete manifolds with nonnegative Ricci curvature. Specially, he proved that any positive harmonic function on such manifolds is constant. In [Che80], S.-Y. Cheng further proved that on such manifolds any harmonic function of sublinear growth must be constant. On complete manifolds, harmonic functions with polynomial growth are important analytic functions besides the Green's function and the heat kernel (note the latter two have singularities).
In the study of harmonic functions on complete manifolds, Yau considered the space of harmonic functions with polynomial growth:
H d (M) = {u| u(x)
is harmonic on M n ; |u(x)| ≤ K(ρ(x) + 1) d f or some K > 0} Conjecture 1.4 (L. Ni). Let (M n , g) be a complete Riemannian manifold of positive sectional curvature, the necessary and sufficient condition that M n admits nonconstant harmonic functions of polynomial growth is, that M n is of maximum volume growth.
For the corresponding conjecture on Kähler manifolds, L. Ni proved that if the manifold is complete Kähler with bounded nonnegative bisectional curvature and of maximum volume growth, it admits nonconstant holomorphic functions of polynomial growht (see Corollary 1 of [Ni05]). On the other hand, recently, G. Liu (see Theorem 2 of [Liub]) showed that if M is a complete noncompact Kähler manifold with positive bisectional curvature, and it admits a nonconstant holomorphic function with polynomial growth, then M is of maximal volume growth.
In Riemannian geometry context, Y. Ding [Din04] proved that on complete manifolds with Rc ≥ 0, maximal volume growth and the unique tangent cone at infinity, there exists nonconstant harmonic function with polynomial growth. Note when M n has nonnegative sectional curvature, from Theorem I.26 in [CCG + 10], the tangent cone at infinity of M n is the unique metric cone C(Y), where Y is a compact metric space. Hence Ding's existence result will imply the sufficient part of Conjecture 1.4. However we have some difficulties to verify the proof of Ding's existence result. For example, Lemma 1.1, Corollary 1.11 and Lemma 1.2 in [Din04] does not hold for zero function.
The main technical tool in [Din04] is a generalization of the monotonicity of frequency for harmonic functions on R n , which is a type of Three Circles Theorem in L 2 sense on complete manifolds (also see [Zha99] and [CDLM08] for related results).
The classical Hadamard's Three Circles Theorem was implied in the announcement [Had96] published in 1896, we state it as the following form, which is sort of consistent with our presentation in Theorem 3.2 and Theorem 3.4 of this paper. The classical Hadamard Three Circles Theorem for holomorphic functions had also been generalized to solutions of partial differential equations in different contexts by L. Simon [Sim83], J. Cheeger and G. Tian [CGT94], G. Liu [Liua]. In spite of our concerns about the argument in [Din04], partially motivated by the results there, we proved two modified Three Circles Theorems for harmonic functions in integral sense (see Theorem 3.2 and Theorem 3.4 in Section 3).
cone C(X) with the unique conic measure of power κ ≥ 2, and H 1 (X) > 0. Then inf{α| α ∈ D(M), α 0} < ∞ (1.2) and for any d > inf{α| α ∈ D(M), α 0},
dim H d (M) ≥ 2 (1.3)
The conic measure of power κ and D(M) will be defined in Section 2. Generally, we do not know the uniqueness of the renormalized limit measure with respect to one tangent cone at infinity of the manifold (compare Example 1.24 of [CC97]). However, from Theorem 5.9 of [CC97], if (M n , g) is a complete manifold with Rc ≥ 0 and maximal volume growth, then every tangent cone at infinity has its unique renormalized measure; and the measure is a multiple of Hausdorff measure H n , which is a conic measure of power n. Hence, Theorem 1.7 implies the existence result in [Din04] mentioned above. The uniqueness of tangent cone at infinity is an important and hard problem, which was addressed in [CGT94] and [Col14] for Ricci flat manifolds under various assumptions.
Remark 1.8. When (M n , g) is a complete manifold with Rc ≥ 0 and maximal volume growth, S. Honda [Honb] showed dim H d (M) = 1 for any d < inf{α| α ∈ D(M), α 0}. Hence the conclusion of Theorem 1.7 seems to be a sharp result except that the critical case d = inf{α| α ∈ D(M), α 0} is not clear yet.
To prove Theorem 1.7, we partially followed the strategy in [Din04]. Because our Three Circles Theorem (Theorem 3.2) works for the collapsed case too, we succeeded in proving the existence of nonconstant harmonic functions with polynomial growth in the collapsed case first time, although with additional assumptions.
More concretely, to construct one nonconstant harmonic function with polynomial growth, we firstly choose a suitable harmonic function of polynomial growth on the tangent cone at infinity of the manifold, where we used the assumptions that the manifold has the conic renormalized limit measure and the tangent cone at infinity is a metric cone.
Then we use the results of Cheeger [Che99], to get a sequence of approximate functions defined on a sequence of increasing geodesic balls exhausting the manifold, which are vanishing at the same fixed point. Solving the Dirichlet problem on those geodesic balls with the same boundary conditions as the corresponding approximate functions, this yields a family of harmonic functions defined on the exhausting domains of the manifold.
Because the sequence of harmonic functions constructed as above have the asymptotic growth behavior as the chosen harmonic function of polynomial growth on the tangent cone at infinity, we can get that the the ratio between the average integrals of those harmonic functions on bigger domain and smaller domain are uniformly bounded near infinity, where the bound depends on the growth rate of the harmonic function of polynomial growth on the tangent cone at infinity chosen above.
If we can get 'some induction estimate' of the ratios from outer domains to inner domains, the uniform polynomial bound of the family of harmonic functions will be obtained by the induction method. Then, after the suitable rescaling, using the well-known Cheng-Yau's gradient estimate for harmonic functions in [CY75], combining with the Arzela-Ascoli theorem, for some subsequence of these harmonic functions, we get the limit function defined on the whole manifold, which is harmonic function of polynomial growth. The nonconstancy of the limit function follows from its vanishing at the fixed point, and the non-vanishing of some local integral of the limit function, which resulted from the suitable chosen rescaling mentioned above.
Our Three Circles Theorem (Theorem 3.2) will play the role of the 'induction estimate' needed in the above argument. Starting from the eigenfunctions expansion of harmonic functions on the metric cone with conic measure, the key idea to prove the Three Circles Theorem, is to use the gap between the eigenvalues of the tangent cone's cross-section. When the tangent cone at infinity with renormalized limit measure is the unique metric cone with unique conic measure, this gap is implied by the discreteness of the spectrum of Laplace operator on the cross-section, and we get the last piece in the proof of Theorem 1.7.
When nonconstant harmonic functions with polynomial growth exist on complete manifolds, as proved in [CM97b], the bound of frequency is essential to describe the asymptotic structure of those functions (like the almost separation of variables). Hence a natural question is about the uniform bound of frequency of harmonic functions on manifolds. Based on the study in [CM97b], Colding and Minicozzi posed the following conjecture: CM97c]). Suppose that M n has nonnegative Ricci curvature and maximal volume growth. If u ∈ H d (M) for some d > 0, then the frequency of u is uniformly bounded. Remark 1.10. Besides [CM97b] and [CM97c], the frequency was also studied in [Alm00] and [GL86]. For more related reference about the frequency, the reader can consult Remark 2.16 in [CM97b]. We would like to point out that the monotonicity of the frequency in Euclidean space can be viewed as the quantitative version of the classical Three Circles Theorem.
Conjecture 1.9 ([
Roughly say, to get the uniform bound of the frequency, we only need to control the ratios of I(r) on concentric circles with increasing radii. Checking the results and techniques developed in [CM97b] carefully, the ratios have uniform bound on a sequence of concentric circles, whose radii are approaching the infinity. If a suitable Three Circles Theorem is available, the uniform bound of ratios can be obtained by the induction method similar as the former argument, which will imply the uniform bound of the frequency. Hence, using the Three Circles Theorem (Theorem 3.4) established in Section 3, we proved the following theorem: Theorem 1.11. Suppose that (M n , g) has nonnegative Ricci curvature and maximal volume growth, also assume the tangent cone at infinity of M n is unique. Then for u(x) ∈ H d (M), the frequency of u(x) is uniformly bounded by C(u, n, V M , d).
Remark 1.12. This theorem confirms Conjecture 1.9 with the additional assumption the uniqueness of the tangent cone at infinity of manifolds. In fact, we proved a stronger result which implies Theorem 1.11, see Theorem 5.5 in Section 5 for details.
As we mentioned before, from [CCG + 10], for any complete manifold with nonnegative sectional curvature, the tangent cone at infinity is a unique metric cone. From Theorem 1.7, we have the following corollary.
Corollary 1.13. Suppose that (M n , g) has nonnegative sectional curvature, the tangent cone at infinity with renormalized limit measure is a unique metric cone C(X) with the unique conic measure of power κ ≥ 2, and H 1 (X) > 0. Then (1.2) and (1.3) hold.
Remark 1.14. On non-negatively curved manifolds, maximal volume growth implies the uniqueness of the tangent cone at infinity and the conic renormalized limit measure of power κ = n and H n−1 (X) > 0, hence the sufficient part of Conjecture 1.4 is implied by Corollary 1.13.
On the other hand, there exists complete manifolds M n with positive sectional curvature, whose tangent cone at infinity C(X) has the unique renormalized limit measure, which is conic measure of power κ ≥ 2 and H 1 (X) > 0 (see Example 4.9). Hence by Corollary 1.13, Example 4.9 is a counterexample to the necessary part of Conjecture 1.4.
The organization of this paper is as the following. In Section 2, we stated some background facts about Gromov-Hausdorff convergence and Cheeger-Colding's theory, which are needed for later sections. We also recalled the definition of frequency function and the related formulas.
In Section 3, we proved two Three Circles Theorems, which are the key technical tools applicable for the existence and frequency problems respectively. For both theorems, the method is proof by contradiction and reduced the related analysis to the analysis on the tangent cone at infinity.
In Section 4, we constructed the nonconstant harmonic function of polynomial growth from the harmonic function on the tangent cones at infinity. And the Three Circles Theorem is used to guarantee the polynomial growth of the constructed harmonic function. We also constructed two example manifolds, which address the nonexistence and existence of harmonic functions with polynomial growth, under linear volume growth and at least quadratic volume growth assumptions respectively. Specially, one example is the first counterexample to the necessary part of Conjecture 1.4.
In Section 5, using the other Three Circles Theorem, combining the results and techniques developed in [CM97b], we proved the uniform bound of frequency. Some technical results in this section are well-known from [CM97b] in more general context, but we provide the details here to make our argument self-contained in this concrete case.
Background and notations
In this section, we always assume that (M n , g) is an n-dimensional complete manifold with Rc ≥ 0. We firstly review some background material about Gromov-Hausdorff convergence and analysis on limit spaces.
Let (M n i , p i , ρ i ) be a sequence of pointed Riemannian manifolds, where p i ∈ M n i and ρ i is the metric on M n i .
If (M n i , p i , ρ i ) converges to (M ∞ , p ∞ , ρ ∞ ) in the Gromov-Hausdorff sense, we write (M n i , p i , ρ i ) d GH −→ (M ∞ , p ∞ , ρ ∞ )
. See [Gro99] for the definition and basic facts concerning Gromov-Hausdorff convergence.
A metric space (M ∞ , p ∞ , ρ ∞ ) is a tangent cone at infinity of M n if it is a Gromov-Hausdorff limit of a sequence of rescaled manifolds (M n , p, r −2 j g), where r j → ∞. By Gromov's compactness theorem, [Gro99], any sequence r j → ∞, has a subsequence, also denoted as r j → ∞, such that the rescaled manifolds (M n , p, r −2 j g) converge to some tangent cone at infinity M ∞ in the Gromov-Hausdorff sense.
Let us recall that from Bishop-Gromov's volume comparison theorem, we can define the asymptotic volume ratio
V M = lim r→∞ V(r) r n (2.1)
where V(r) is the volume of the geodesic ball B(r) centered at p with radius r. And the above definition is independent of p, so we omit p there. If V M > 0, we say that (M n , g) has maximal volume growth. Note V M ≤ V n 0 (1) from Bishop-Gromov's volume comparison theorem, where V n k (r) is the volume of ball with radius r in the n-dimensional space form with sectional curvature equal to k.
Example of Perelman [Per97] shows that tangent cone at infinity is not unique in general even if the manifold with Rc ≥ 0 has maximal volume growth and quadratic curvature decay. Although the tangent cone at infinity may be not unique, under maximal volume growth assumption, Cheeger and Colding proved the following theorem characterizing it:
Theorem 2.1 ( [CC96]). Let M n be a complete manifold with Rc ≥ 0 and maximal volume growth, then every tangent cone at infinity M ∞ is a metric cone C(X), where X is a compact metric space and diam(X) ≤ π.
Note the metric on the metric cone C(X) is dr 2 + r 2 dX, where r ∈ [0, ∞). In the collapsed case (i.e. the maximal volume growth assumption does not hold), we can consider the renormalized measure on the limit space under the measured Gromov-Hausdorff convergence. As in Section 9 of [Che99], we have the following definition.
Definition 2.2. If ω i , ω ∞ are Borel regular measures on M n i , M ∞ , we say that (M n i , p i , ρ i , ω i ) converges to (M ∞ , p ∞ , ρ ∞ , ω ∞ ) in the measured Gromov-Hausdorff sense, if (M n i , p i , ρ i ) d GH −→ (M ∞ , p ∞ , ρ ∞ ), in addition, for any x i → x ∞ , (x i ∈ M n i , x ∞ ∈ M ∞ ), r > 0, we have ω i B i (x i , r) → ω ∞ B ∞ (x ∞ , r) where (M ∞ , ρ ∞ ) is a length space with length metric ρ ∞ , and B i (x i , r) = {z ∈ M n i | d ρ i (z, x i ) ≤ r} , B ∞ (x ∞ , r) = {z ∈ M ∞ | d ρ ∞ (z, x ∞ ) ≤ r}
For later use, we also set up the following Blow Down Setup: Note that (M n , g, µ) is a complete Riemannian manifold with Rc ≥ 0, where µ is the volume element determined by the metric g. We can define
(M i , p, ρ i , ν i ), where M i is the same differential manifold as M n , ρ i is the metric defined as ρ i = r −2 i g, {r i } ∞ i=1
is an increasing positive sequence whose limit is ∞, p is a fixed point on M i = M n , and ν i is the renormalized measure defined by
ν i (A) := B i (1) 1dµ i −1 A 1dµ i = r n i V(r i ) −1 µ i (A) (2.2) where A ⊂ M i , B i (1) := {z ∈ M i | d ρ i (z, p) ≤ 1}
, and µ i is the volume element determined by ρ i . Then by Gromov's compactness theorem (see [Gro99]) and Theorem 1.6 in [CC97], after passing to a suitable subsequence, we have
(M i , p, ρ i , ν i ) d GH −→ (M ∞ , p ∞ , ρ ∞ , ν ∞ ) in the measured Gromov-Hausdorff sense,
where ν ∞ is the renormalized limit measure defined as in Section 1 of [CC97].
Let Z be a metric space and let ν be a Borel measure on Z. As in Section 2 of [CC00a], we define the associated Hausdorff measure in codimension 1 (denoted as ν −1 ) as follows. Fix
δ > 0 and U ⊂ Z, let B = {B r i (q i )} be a covering of U with r i < δ, for all i. Put (ν −1 ) δ (U) = inf B i r −1 i ν B r i (q i ) (2.3) and ν −1 (U) = lim δ→0 (ν −1 ) δ (U) (2.4)
Definition 2.3. On a metric cone (C(X), dr 2 + r 2 dX), ν is called conic measure of power κ, and κ is a positive constant denoted as p(ν), if for any Ω ⊂⊂ C(X),
ν(Ω) = ∞ 0 r κ−1 dr X χ(Ω r )dν −1 (2.5) where Ω r = {z|z ∈ Ω, r(z) = r}, χ(·) is the characteristic function on C(X).
If (M n , g) is a complete manifold with Rc ≥ 0 and maximal volume growth, from Theorem 2.1 above and Theorem 5.9 in [CC97], every tangent cone at infinity of M n is a metric cone, with the unique corresponding renormalized limit measure, which is a conic measure of power n. In collapsing case, our definition of conic measure will play the role of co-area formula on metric cones in non-collapsing case, which was showed in Section 7 of [Honb].
Assume that (M n , g) is a complete manifold with Rc ≥ 0, all tangent cones at infinity are metric cones and every renormalized limit measure is conic measure, we define the set of all tangent cones at infinity of M n with renormalized limit measure as M (M) := {(C(X), ν)| C(X) is the metric tangent cone at infinity of M n , ν is the conic renormalized limit measure}.
From [CC00b] (also see [Che99]), there exists a self-adjoint Laplace operator ∆ (C(X),ν) on (C(X), ν) ∈ M (M). From (2.3) and (2.4), ν induces a natural measure ν −1 on X, which satisfies a volume doubling property. Similar argument as in [Din02] (see Section 4 there), weak Poincaré inequality also holds on (X, ν −1 ). Hence from [CC00b] (also see [Che99]), volume doubling property, weak Poincaré inequality and the rectifiability of the cross section X yields the existence of a selfadjoint positive Laplace operator ∆ (X,ν −1 ) on (X, ν −1 ).
When
H 1 (X) > 0 where H i is i-dimensional Hausdorff measure, L 2 (X)
is an infinite dimensional Hilbert space. Now from Rellich-type Compactness Theorem (Theorem 4.9 of [Hona], also see the Appendix of [Xu14]), similar as the standard elliptic theory on compact manifolds (see Chapter 6 in [War83] etc.), on compact metric measure space (X, ν −1 ), we have an orthonormal basis
{ϕ i (x)} ∞ i=1 for L 2 (X), and a sequence 0 = λ 1 < λ 2 ≤ λ 3 ≤ · · · , lim i→∞ λ i = ∞, such that ∆ (X,ν −1 ) ϕ i (x) = −λ i ϕ i (x) (2.6)
Now assume that (M n , g) is a complete manifold with Rc ≥ 0, all tangent cones at infinity are metric cones and every renormalized limit measure is conic measure, and H 1 (X) > 0. Then we have the following proposition:
Proposition 2.4. For any (C(X), ν) ∈ M (M), assume that conic measure ν is of power κ > 0, for any u(x) ∈ H 1 0 (C(X))
,
∆ (C(X),ν) u = ∂ 2 u ∂r 2 + κ − 1 r ∂u ∂r + 1 r 2 ∆ (X,ν −1 ) u (2.7)
Proof: For any w ∈ H 1 0 (C(X)), by definition of ∆ (C(X),ν) and ∆ (X,ν −1 ) (see Section 6 of [CC00b]), we can use integration by parts, combining with the definition of conic measure (2.5), then
C(X) ∆ (C(X),ν) u · wdν = − C(X) ∇u · ∇wdν = − ∞ 0 r κ−1 X ∇u · ∇wdν −1 dr = − ∞ 0 r κ−1 X (∇ r u + 1 r ∇ x u) · (∇ r w + 1 r ∇ x w)dν −1 (x) dr (2.8) = − ∞ 0 r κ−3 X ∇ x u · ∇ x wdν −1 (x) dr − ∞ 0 r κ−1 X ∇ r u · ∇ r wdν −1 (x) = ∞ 0 r κ−1 X 1 r 2 ∆ (X,ν −1 ) u · w + ∞ 0 r κ−1 X ∂ 2 u ∂r 2 + κ − 1 r ∂u ∂r · w = C(X) ∂ 2 u ∂r 2 + κ − 1 r ∂u ∂r + 1 r 2 ∆ (X,ν −1 ) u · wdν (2.9)
where (2.8) follows from the metric cone structure of C(X). From (2.9), we obtain (2.7).
The following corollary is similar as Theorem 1.11 of [CM97b] (also see [Che79]), for completeness we provide its proof here following the argument in [CM97b].
Corollary 2.5. If u is a harmonic function on
(C(X), ν) with respect to ∆ (C(X),ν) , then u(r, x) = ∞ i=1 c i r α i ϕ i (x) (2.10)
where c i , α j ≥ 0 are constants, and ϕ j , λ j = α j κ + α j − 2 are defined in (2.6).
Proof: We can assume that u(0) = 0. By the spectral theorem applied on (X, ν −1 ),
u(1, x) = ∞ j=0 a j ϕ j (x) (2.11)
where the convergence is in L 2 (X, ν −1 ) sense.
On the other hand, from Proposition 2.4, it is not hard to prove that
u(r, x) = ∞ j=0 a j r α j ϕ j (x) is a harmonic function on (C(X), ν), where α j (α j + κ − 2) = λ j and α j ≥ 0. Now consider the harmonic functioñ u(r, x) = u(r, x) −û(r, x)
From (2.11),ũ vanishes on ∂B 1 ⊂ C(X) and at the vertex 0. Then by the maximum principle,ũ ≡ 0. Hence (2.10) follows.
And we also define S (M) the spectrum at infinity of (M n , g) and D(M) the degree spectrum at infinity of (M n , g):
S (M) := {λ| λ = λ j (X, ν −1 ) f or some positive interger j and (C(X), ν) ∈ M (M)} D(M) := {α ≥ 0| α κ + α − 2 = λ f or some λ = λ j (X, ν −1 ) ∈ S (M) and κ = p(ν)}
We also define the convergence concept for functions on manifolds {M n i } as the following, it is called "uniform convergence in Gromov-Hausdorff topology", for simplification, sometimes it is written as "uniform convergence in G-H topology". Definition 2.6 (Uniform Convergence in G-H topology). Suppose
K i ⊂ M n i d GH −→ K ∞ ⊂ M ∞ Assume that { f i } ∞ i=1 are functions on M n i , f ∞ is a function on M ∞ . and Φ i : K ∞ → K i are ǫ i -Gromov-Hausdorff approximations, lim i→∞ ǫ i = 0. If f i • Φ i converge to f ∞ uniformly, we say that f i → f ∞ uniformly over K i d GH −→ K ∞ .
In the rest of this section, unless explicitly stated, (M n , g) is an n-dimensional complete manifold with Rc ≥ 0 and maximal volume growth. We restrict our discussion to the case of n ≥ 3, fix p ∈ M n , let G(x) denote the minimal positive Green's function on M n with singularity at p. And as in [CM97c], we will normalize G(x) by
∆G(x) = (2 − n)V n−1 1 (π)δ p (x) (2.12)
From [Var81] (also see [Li86]) and the maximal volume growth of the manifold, we know that G(x) exists. Set
b(x) = V M V n 0 (1) G(x) 1 2−n , ρ(x) = d(p, x) (2.13)
Note when M n is R n , the function b(x) is just the distance function ρ(x). We also use B(r) to denote the geodesic ball centered at p with radius r on M. And we have the following fact:
lim ρ(x)→0 b(x) ρ(x) = V M V n 0 (1) 1 2−n (2.14)
We collect some important facts about b(x) proved by Cheeger and Colding [CC96], Colding and Minicozzi [CM97c], Colding [Col12] in the following.
Theorem 2.7 ([CC96], [CM97c], [Col12]). lim r→∞ b(x)≤r |∇b| 2 − 1 2 dx Vol(b(x) ≤ r) = lim r→∞ b(x)≤r Hess(b 2 ) − 2g 2 dx Vol(b(x) ≤ r) = 0 ; (2.15) lim ρ(x)→∞ b(x) ρ(x) = 1 ; |∇b| ≤ 1 (2.16)
where g is the metric tensor on M n .
Let us recall the definition of frequency function in [CM97b], we firstly define:
I u (r) = r 1−n b(x)=r u 2 |∇b|dx (2.17) D u (r) = r 2−n b(x)≤r |∇u| 2 dx , F u (r) = r 3−n b(x)=r ∂u ∂n 2 |∇b|dx (2.18)
then the frequency function is defined by
F u (r) = D u (r) I u (r) (2.19) where u(x) is a harmonic function defined on {b(x) ≤ r}.
Using the fact that u is harmonic, differentiating (2.17), we get
I ′ u (r) = 2 D u (r) r ≥ 0 (2.20)
From (2.20), I 1 (r) is constant. Then by the fact (2.14), it is not hard to see that
I 1 (r) = nV M (2.21)
We further define two quantities which are technically easier to be dealt with, comparing with D u and F u .
E u (r) = r 2−n b(x)≤r |∇u| 2 |∇b| 2 dx , W u (r) = E u (r) I u (r) (2.22)
Sometimes for simplification, we omit the subscript u in I u (r), · · · , W u (r) when the context is clear, and use I(r), · · · , W (r) instead.
When M n is a complete manifold with Rc ≥ 0 and maximal volume growth, r j → ∞, assume that the rescaled manifolds (M n , p, r −2 j g) converge to some tangent cone at infinity M ∞ in the Gromov-Hausdorff sense, From Theorem 0.1 of [CM97c], and Theorem 3.21, Corollary 4.22 of [Din02], we have the following proposition:
Proposition 2.8. If K j and K ∞ are compact subsets of (M n , p, r −2 j g) and M ∞ respectively, suppose K j
d GH −→ K ∞ , then b j → b ∞ uniformly over K j d GH −→ K ∞ , where b j and b ∞ are defined by (2.13) on (M n , p, r −2 j g) and (M ∞ , p ∞ , ρ ∞ ) respectively, furthermore b ∞ = ρ ∞ .
Three Circles Theorems for harmonic functions
Different types of Three Circles Theorems were proved by Simon [Sim83], Cheeger and Tian [CGT94], Colding, DeLellis and Minicozzi [CDLM08] in different contexts. Also see Zhang [Zha99], Ding [Din04] for harmonic functions on manifolds and Liu [Liua] for holomorphic functions on Kähler manifolds.
However, to study Question 1.3, Conjecture 1.4 and Conjecture 1.9, we need to do some modification to get the Three Circles Theorems applicable on those problems.
Lemma 3.1. For {w i } ∞ i=1 , w i ≥ 0, if ∞ i=1 w i ≤ ∞ i=1 2 2(α−α i ) w i (3.1) then ∞ i=1 2 −2α i w i ≤ ∞ i=1 2 2(α−2α i ) w i (3.2)
where 0 = α 1 < α 2 ≤ α 3 ≤ · · · , and α > 0. Furthermore, the equality in (3.2) holds if and only if w i = 0 for all i satisfying α i α.
Proof: (3.1) is equivalent to
α i α 1 w i (1 − 2 2(α−α i ) ) ≤ w 1 (2 2α − 1) (3.3)
and (3.2) is equivalent to
α i α 1 2 −2α i w i (1 − 2 2(α−α i ) ) ≤ w 1 (2 2α − 1) (3.4) Note α i α 1 2 −2α i w i (1 − 2 2(α−α i ) ) ≤ α i α 1 2 −2α w i (1 − 2 2(α−α i ) ) (3.5) If α i α 1 w i (1 − 2 2(α−α i ) ) ≤ 0, from (3.5), α i α 1 2 −2α i w i (1 − 2 2(α−α i ) ) ≤ 0 ≤ w 1 (2 2α − 1) If α i α 1 w i (1 − 2 2(α−α i ) ) > 0, from (3.3) α i α 1 2 −2α i w i (1 − 2 2(α−α i ) ) ≤ 2 −2α w 1 (2 2α − 1) < w 1 (2 2α − 1)
Hence (3.4) is proved, and (3.2) is obtained. Check the above argument carefully, it is easy to find that the equality in (3.2) holds if and only if w i = 0 for all i satisfying α i α.
On (M n , g, µ) where µ is a Borel regular measure on M n , define the J-function of u as the following:
J u (r) = 1 µ(B(r)) B(r) u 2 dµ (3.6)
Unless otherwise mentioned, for J u (r) in (3.6), the measure µ will be assumed as the volume measure determined by the metric g.
Theorem 3.2. Let (M n , g) be a complete manifold with nonnegative Ricci curvature, assume that every tangent cone at infinity of M n is a metric cone C(X) with the conic renormalized limit measure of power κ ≥ 2, and H 1 (X) > 0. If α D(M), then there exists integer k 0 = k 0 (α) > 1, such that for r ≥ k 0 , and u(x) harmonic over B(r) ⊂ (M n , g),
(3.7) J u (r) ≤ 2 2α J u ( r 2 ) implies J u ( r 2 ) ≤ 2 2α J u ( r 4 ) (3.8)
Proof: By contradiction. If Theorem 3.2 is not true, then there exists a sequence {r l }, r l → ∞, and the corresponding harmonic functions u l such that the following inequalities hold:
J u l (r l ) ≤ 2 2α J u l ( r l 2 ) , J u l ( r l 2 ) > 2 2α J u l ( r l 4 ) (3.9)
Using the assumptions about tangent cones with renormalized limit measure, combining the knowledge about measured Gromov-Hausdorff convergence, without loss of generality (by choosing subsequence of {u l }), we assume (M n , p, ρ l , ν i ) converges to (C(Y), p ∞ , ρ ∞ , ν ∞ ) in the measured Gromov-Hausdorff sense as in Blow Down Setup of Section 2, and (C(Y), ν ∞ ) is a metric cone with conic measure.
Clearly (3.9) implies J u l ( r l 2 ) 0, definẽ
u l = u l J u l ( r l 2 ) 1 2
Look atũ l as the function on B l (1) ⊂ (M n , g l ), from (3.9)
J (l) u l (1) ≤ 2 2α J (l) u l ( 1 2 ) , J (l) u l ( 1 2 ) > 2 2α J (l) u l ( 1 4 ) (3.10)
where J (l) u l is the J-function ofũ l on manifold (M n , g l , ν l ). Also we have
J (l) u l ( 1 2 ) = Jũ l ( r l 2 ) = 1 (3.11)
From Theorem 1.2 in [LS84] and Cheng-Yau's gradient estimate in [CY75], we have the following estimates:
sup B (l) (1−θ) |ũ l | ≤ C(n, p, θ) J (l) u l (1) 1 2 ≤ C(n, p, θ, α) J (l) u l ( 1 2 ) 1 2 = C(n, p, θ, α) sup B (l) (1−θ) |∇ũ l | g l ≤ C(n, p, θ, α)
So for any θ ∈ (0, 1),ũ l and |∇ũ l | are uniformly bounded over B (l) (1 − θ). By Harnack's convergence theorem in the Gromov-Hausdorff sense (see [Din02], also [Xu14]), we get that {ũ l } converges uniformly on compact subsets of B ∞ (1) ⊂ C(Y) to u ∞ , and u ∞ is harmonic over B ∞ (1). Hence,
lim l→∞ J (l) u l ( 1 2 ) = J u ∞ ( 1 2 ) (3.12) where J u ∞ is the J-function of u ∞ on (C(Y), ρ ∞ , ν ∞ ) defined as in (3.6).
By u ∞ is harmonic over B ∞ (1) ⊂ C(Y), as in (2.10) we can write
u ∞ = ∞ i=1 c i r α i ϕ i (x) (3.13)
where {ϕ i (x)} are the eigenfunctions of ∆ Y on Y, also the orthonormal basis for
L 2 (Y), ∆ Y ϕ i (x) = −λ i ϕ i (x), λ i = α i (α i + n − 2) and α i ≥ 0.
From (3.10), we get
B ∞ (1) u 2 ∞ dν ∞ = lim k→∞ B ∞ 1− 1 k u 2 ∞ dν ∞ = lim k→∞ lim l→∞ B l 1− 1 k ũ 2 l dν l ≤ lim l→∞ B l (1)ũ 2 l dν l ≤ 2 2α lim l→∞ V(r l ) V( r l 2 ) B l ( 1 2 )ũ 2 l dν l = 2 2α+κ B ∞ ( 1 2 ) u 2 ∞ dν ∞ (3.14) B ∞ ( 1 2 ) u 2 ∞ dν ∞ ≥ 2 2α+κ B ∞ ( 1 4 ) u 2 ∞ dν ∞ (3.15)
in the last equality of (3.14) we used the assumption that the renormalized limit measure is conic measure of degree κ.
Plug (3.13) into (3.14) and (3.15), we get
∞ i=1 w i ≤ ∞ i=1 2 2(α−α i ) w i , ∞ i=1 2 −2α i w i ≥ ∞ i=1 2 2(α−2α i ) w i where w i = c 2 i 2α i +κ .
From the above two inequalities, by Lemma 3.1 and the assumption α D(M), we get w i = 0 and c i = 0, hence u ∞ ≡ 0. Taking limit in (3.11), combining (3.12) and u ∞ = 0, we obtain
1 = lim l→∞ J (l) u l ( 1 2 ) = J u ∞ ( 1 2 ) = 0
It is the contradiction, hence the conclusion is proved.
Recall we defined I u in (2.17) for harmonic functions u(x), we have the other Three Circles Theorems for I u , which will be useful for estimating the frequency of u(x). Before proving the theorem, we firstly need to control the C 0 and C 1 norm of u(x) by I u , which is achieved by the following lemma. After simplification, we get
D 1 − θ 2 )r ≤ C(n, θ)I(r) (3.19)
Assume that |∇u| 2 (x 0 ) = sup b≤(1−θ)r |∇u| 2 and b(x 0 ) = r 0 ≤ (1−θ)r. From (2.16), there exists r 1 = C(p, V M , θ)r > 0 such that B x 0 (r 1 ) ⊂ {b ≤ 1 − θ 2 r}, hence by Theorem 1.2 in [LS84],
|∇u| 2 (x 0 ) ≤ C(n) V B x 0 (r 1 ) B x 0 (r 1 ) |∇u| 2 ≤ C(n, p, θ, V M )r −2 D 1 −I u (r) ≤ 2 2α I u ( r 2 ) implies I u ( r 2 ) ≤ 2 2α I u ( r 4 ) (3.
22)
Proof: By contradiction. If Theorem 3.4 is not true, then there exists a sequence {r l }, r l → ∞, and the corresponding harmonic functions u l such that the following inequalities hold:
I u l (r l ) ≤ 2 2α I u l ( r l 2 ) , I u l ( r l 2 ) > 2 2α I u l ( r l 4 ) , u l (p) = 0 (3.23)
Without loss of generality (by choosing subsequence of {u l }), we can assume that
(M n , p, ρ l ) d GH −→ (C(X), p ∞ , ρ ∞ ) (3.24)
where ρ l = g l = r −2 l g is the rescaled metric, and C(X) is one tangent cone at infinity of (M n , g), which is a metric cone by Theorem 2.1.
Clearly (3.23) implies I u l ( r l 2 ) 0, definẽ u l = u l I u l ( r l 2 ) 1 2
Look atũ l as the function on B l (1) ⊂ (M n , g l ), we have
I (l) u l (1) ≤ 2 2α I (l) u l ( 1 2 ) , I (l) u l ( 1 2 ) > 2 2α I (l) u l ( 1 4 ) ,ũ l (p) = 0 (3.25)
where I (l) u l is the frequency function ofũ l on manifold (M n , g l ), and
I (l) u l ( 1 2 ) = Iũ l ( r l 2 ) = 1 (3.26)
From Lemma 3.3 and (3.25), we have the following estimates,
sup b l ≤(1−θ) |ũ l | ≤ C(n, p, θ, V M ) I (l) u l (1) 1 2 ≤ C(n, p, θ, V M , α) I (l) u l ( 1 2 ) 1 2 = C(n, p, θ, V M , α) sup b l ≤(1−θ) |∇ũ l | g l ≤ C(n, p, θ, V M , α)
where b l is the b(x) function defined as in (2.13) on (M n , p, g l ).
So for any θ ∈ (0, 1),ũ l and |∇ũ l | are uniformly bounded over {b l ≤ (1 − θ)}. By Harnack's convergence theorem in the Gromov-Hausdorff sense (see [Din02], also [Xu14]), we get that {ũ l } converges uniformly on compact subsets of {b ∞ < 1} ⊂ C(X) to u ∞ , and u ∞ is harmonic over {b ∞ < 1}. From Proposition 3.4 in [Honb], we have
lim l→∞ I (l) u l ( 1 2 ) = I u ∞ ( 1 2 ) (3.27)
By u ∞ is harmonic over {b ∞ < 1}, as in (2.10) we can write
u ∞ = ∞ i=1 c i r α i ϕ i (x) (3.28)
where {ϕ i (x)} are the eigenfunctions of ∆ X on X, also the orthonormal basis for
L 2 (X), ∆ X ϕ i (x) = −λ i ϕ i (x), λ i = α i (α i + n − 2) and α i ≥ 0. From (3.25), note b ∞ = ρ ∞ on C(X), again by Proposition 3.4 in [Honb] b ∞ =1 u 2 ∞ = lim k→∞ b ∞ = 1− 1 k u 2 ∞ = lim k→∞ 1 − 1 k n−1 lim l→∞ I (l) u l 1 − 1 k ≤ lim l→∞ I (l) u l (1) ≤ lim l→∞ 2 2α I (l) u l ( 1 2 ) = 2 2α+n−1 b ∞ = 1 2 u 2 ∞ (3.29) b ∞ = 1 2 u 2 ∞ ≥ 2 2α+n−1 b ∞ = 1 4 u 2 ∞ (3.30)
in the first inequality of (3.29) we used the fact I(r) is nondecreasing in r. Plug (3.28) into (3.29) and (3.30), we get
∞ i=1 w i ≤ ∞ i=1 2 2(α−α i ) w i , ∞ i=1 2 −2α i w i ≥ ∞ i=1 2 2(α−2α i ) w i
where w i = c 2 i . From the above two inequalities and Lemma 3.1, c i = 0, hence u ∞ ≡ 0.
Taking limit in (3.26), combining (3.27) and u ∞ = 0, we obtain
1 = lim l→∞ I (l) u l ( 1 2 ) = I u ∞ ( 1 2 ) = 0
It is the contradiction, hence the conclusion is proved.
The existence of harmonic functions with polynomial growth
In the following lemma, we assume that (M n , g) is a complete manifold with Rc ≥ 0, and every tangent cone at infinity of M n with renormalized limit measure is a metric cone C(X) with conic measure of power κ ≥ 2, and H 1 (X) > 0.
Lemma 4.1. Assume u ∞ is harmonic on C(X) ∈ M (M) and u ∞ (p ∞ ) = 0, then there exist R i → ∞, B(R i ) ⊂ M n , such that lim i→∞ d GH B i (1), B ∞ (1) = 0, where B i (1) ⊂ (M n , R −2 i g), B ∞ (1) ⊂ C(X)
, and u i harmonic on B p (R i ) = B i (1) satisfying the following property:
lim i→∞ |u i • Ψ ∞,i − u ∞ | L ∞ B ∞ (1) = 0 , u i (p) = 0 (4.1)
where Ψ ∞,i : B ∞ (1) → B i (1) is an ǫ i -Gromov-Hausdorff approximation, and lim i→∞ ǫ i = 0.
Remark 4.2. The above lemma was inspired by Theorem 2.1 of [Din04], however there is a small (but new) restriction on u i (u i (p) = 0) in our statement, which is crucial in our proof of Theorem 4.3.
Proof: It follows from Lemma 10.7 in [Che99], there exists Lipschitz functioñ
u i defined on (M n , R −2 i g) such that lim i→∞ |ũ i • Ψ ∞,i − u ∞ | L ∞ B ∞ (1) = 0 ; Lip(ũ i ) ≤ C (4.2)
where C is some positive constant independent of i, and
Lip( f ) := sup z∈B ∞ (1) lim inf r→0 sup d(z,y)=r | f (y) − f (z)| r
Letû i be the solution of the following Dirichlet problem:
(4.3) ∆û i = 0 , on B i (1) u i =ũ i , on ∂B i (1)
By (4.2) and Lemma 10.7 in [Che99],ũ i | ∂B i (1) is uniformly bounded. From maximum principle on B i (1),û i are uniformly bounded.
Let
x i ∈ ∂B i (1), x i → x ∞ ∈ ∂B ∞ (1)
. For any ǫ > 0, from (4.2) there exists δ ∈ (0, 1) such that
|ũ i (x) −ũ i (x i )| < ǫ 2 i f |x − x i | < δ
On the cone C(X) there is a unique ray starting from the pole p ∞ , passing through x ∞ . Pick a point q ∞ on this ray with
d ∞ (p ∞ , q ∞ ) > d ∞ (p ∞ , x ∞ ), then d ∞ (x, q ∞ ) > d(x ∞ , q ∞ ) , ∀x ∈ {z| d ∞ (z, x ∞ ) < δ} ∩B ∞ (1)
Hence we can choose q i → q ∞ such that
d ρ i (q i , x) ≥ d ρ i (q i , x i ) , ∀x ∈ {z| d ρ i (z, x i ) < δ} ∩B i (1) Consider w i (x) = d ρ i (q i , x i ) 2−n − d ρ i (q i , x) 2−n
By the Laplacian comparison theorem, ∆ i w i ≤ 0, and it is easy to see that
w i ≥ 0 on {z| d ρ i (z, x i ) < δ} ∩B i (1), w i (x i ) = 0.
Hence it is the barrier function defined as in Section 2.8 of [GT01].
Now let M = sup i sup x∈∂B i (1) |ũ i (x)| < ∞, using the fact lim i→∞ w i (y i ) = d ∞ (q ∞ , x ∞ ) 2−n − d ∞ (q ∞ , y ∞ ) 2−n i f y i → y ∞ (4.4)
hence there exists constant k > 0, which is independent of i, such that when i >> 1,
kw i (x) ≥ 2M i f |x − x i | ≥ δ
Then it is easy to check
ũ i (x i ) + ǫ 2 + kw i (x) −û i (x) ∂B i (1) ≥ 0 , ∆ ũ i (x i ) + ǫ 2 + kw i (x) −û i (x) ≤ 0 From maximum principle, in B i (1), u i (x i ) + ǫ 2 + kw i (x) ≥û i (x)
Similarly in B i (1), we havẽ
u i (x i ) − ǫ 2 − kw i (x) ≤û i (x)
Hence
|û i (x) −ũ i (x i )| ≤ ǫ 2 + kw i (x) x ∈ B i (1) (4.5)
Note k is independent of i, using the fact (4.4) again, we get δ 0 > 0, such that for From Harnack's convergence theorem in the Gromov-Hausdorff sense (see [Din02], also [Xu14]
x i ∈ ∂B i (1), d ρ i (x, x i ) ≤ δ 0 implies |û i (x) −ũ i (x i )| ≤ ǫ),û i converges to w ∞ on B ∞ (1), i.e. lim i→∞ |û i • Ψ ∞,i − w ∞ | L ∞ B ∞ (1) = 0 (4.6)
From (4.2), (4.3) and (4.6), we get that
w ∞ | ∂B ∞ (1) = u ∞ | ∂B ∞ (1) . From maximum principle on C(X), w ∞ = u ∞ on B ∞ (1). We get lim i→∞ |û i • Ψ ∞,i − u ∞ | L ∞ B ∞ (1) = 0 Choose u i (x) =û i (x) −û i (p), noteû i (p) → u ∞ (p ∞ ) = 0, then lim i→∞ |u i • Ψ ∞,i − u ∞ | L ∞ B ∞ (1) = 0
The conclusion is obtained. Proof: By assumption, there exists α 1 ∈ D(M), α 1 0 and α 1 < d. Hence there is C(X) ∈ M (M), and ϕ 1 (x) is the eigenfunction on X with respect to eigenvalue λ 1 = α 1 (α 1 + κ − 2), X |ϕ 1 | 2 = 1. Let u ∞ = r α 1 ϕ 1 (x) in Lemma 4.1, then choose {u i } from Lemma 4.1. We have the following lemma:
= i 0 (d − α 1 , r 0 ) > 0 such that if i ≥ i 0 , for any r ∈ [r 0 R i , R i ], (4.7) J u i (r) ≤ 2 2d · J u i ( r 2 )
Proof: By contradiction. If the lemma is not true, without loss of generality, we can assume that for some r 0 ∈ (0, 1), there exists a subsequence of {i} ∞ 1 , for simplicity also denoted as {i} ∞ 1 such that
J u i (r i ) > 2 2d · J u i ( r i 2 ) (4.8) where r i ∈ [r 0 R i , R i ].
Note R −1 i r i ∈ [r 0 , 1], without loss of generality, we can assume that there exists a subsequence of {i}, for simplicity also denoted as {i} such that
lim i→∞ R −1 i r i = c 0 ∈ [r 0 , 1] (4.9)
where c 0 is some constant.
Taking the limit in (4.8), from (4.9) and Lemma 4.1 we get
1 ν ∞ B ∞ (c 0 ) B ∞ (c 0 ) |u ∞ | 2 dν ∞ ≥ 2 2d ν ∞ B ∞ ( c 0 2 ) B ∞ c 0 2 |u ∞ | 2 dν ∞ (4.10)
From u ∞ = r α 1 ϕ 1 (x) and d > α 1 , (4.10) implies X |ϕ 1 (x)| 2 dx = 0, which is contradiction.
Note d D(M), from Theorem 3.2 and induction method, there exists k 0 = k 0 (d) such that for r ∈ k 0 2 , R i , (4.7) holds, where we choose i big enough such that
i ≥ i 0 (d − α 1 , r 0 ), R i > k 0 and u i 0. Now we defineǔ i (x) = u i (x) J u i ( k 0 2 ) (4.11)
then Jǔ i ( k 0 2 ) = 1 ,ǔ i (p) = 0 (4.12)
Note the scaling invariant property of (4.7), hence there exists i 0 > 0, if i ≥ i 0 , for r ∈ [ k 0 2 , R i ], Jǔ i (r) ≤ 2 2d · Jǔ i ( r 2 ), and we get
Jǔ i (r) ≤ 2 2d ln 2 r k 0 +1 Jǔ i (k 0 ) = r k 0 2d
· 2 2d Jǔ i (k 0 ) (4.13) By Theorem 1.2 of [LS84], d(x, p).
|ǔ i (x)| 2 ≤ C(n, p)Jǔ i 2ρ(x) (4.14) recall that ρ(x) =
From (4.13) and (4.14),
when i ≥ i 0 , for ρ(x) ∈ [ k 0 4 , R i 2 ], |ǔ i (x)| ≤ C(n, p, d, k 0 )ρ(x) d (4.15)
Combining with the Cheng-Yau's gradient estimate in [CY75] and the Arzela-Ascoli theorem, after taking suitable subsequence,ǔ i converges to a polynomial growth harmonic function u(x) on M n . From (4.12), we know that u(x) is not constant. The conclusion is proved.
Proof of Theorem 1.7: When the tangent cone at infinity of M n with renormalized limit measure is the unique metric cone with conic measure, denoted as (C(X), ν), then D(M) is a countable set by the fact that the spectrum of (X, ν −1 ) is a discrete set. Hence we can find d D(M) and d > inf{α| α ∈ D(M), α 0}, from Theorem 4.3, the conclusion is proved.
Lemma 4.5. Suppose (M n , g) has nonnegative sectional curvature, and for some fixed constants κ > 1, a 0 > 0, lim s→∞ V (B(x, s)) s κ = a 0 (4.16)
where the convergence in (4.16) is uniform for all x ∈ M n . Then the tangent cone at infinity of M n with renormalized limit measure is a unique metric cone C(X) with unique conic measure ν of power κ, H κ−1 (X) > 0 and κ ≥ 2 is an integer.
Remark 4.6. If (4.16) holds uniformly for all x ∈ M n , we will say (M n , g) has uniform asymptotic polynomial volume growth of degree κ.
Proof: Assume x i → x, r i → ∞, then B i (x i , r) → B ∞ (x, r), we have ν B ∞ (x, r) = lim i→∞ µ i B i (x i , r) µ i B i (p, 1) = lim i→∞ µ B(x i , r i r) µ B(p, r i ) = r κ lim i→∞ µ B(x i , r i r) a 0 (r i r) κ = r κ (4.17)
where the last equation follows from the uniform convergence of (4.16).
From the definition of Hausdorff dimension, using (4.17), we obtain that the Hausdorff dimension of C(X) is κ and H κ (C(X)) > 0. Because C(X) is a metric cone on metric space X, it is not hard to get that the Hausdorff dimension of X is (κ − 1) and H κ−1 (X) > 0.
By Theorem 5.5 of [CC00b] and the definitions of Ahlfors l-regular and νrectifiable (Definition s 5.1 and 5.3 in [CC00b]), where l is some non-negative number, using (4.17), we obtain that ν is Ahlfors κ-regular at all x ∈ C(X), and κ must be a non-negative integer. By assumption κ > 1, we proved that κ ≥ 2 is an integer.
From the Definition 2.3 and (4.17), it is straightforward to verify that ν is a conic measure of power κ.
From the above Lemma and Theorem 1.7, we have the following corollary.
Corollary 4.7. Suppose (M n , g) has nonnegative sectional curvature and uniform asymptotic polynomial volume growth of degree κ, and κ > 1, then (1.2) and (1.3) hold.
Example 4.8 (Ding's example). On R n , we define the warped product metric g = dr 2 + f 2 (r)dS n−1 , where S n−1 is the classical (n − 1)-dimensional unit sphere, f (r) is defined by modifying the famous symmetric mollifier e − 1 1−r 2 as the following :
(4.18) f (r) = a − b exp − 1 1− r+3 − 1 4 2 , 0 ≤ r < 1 − 3 − 1 4 a , r ≥ 1 − 3 − 1 where b = a · exp 1 1−3 − 1 2 , a = 1−3 − 1 2 2 2·3 − 1 4 . The number 3 − 1 4 is chosen in (4.18),
because it is the inflection point of the symmetric mollifier e − 1 1−r 2 . It is straightforward to check that Rc(g) ≥ 0 by the above definition of f (r) and the metric g is smooth.
And it is obvious that (R n , g) has linear volume growth and will not split isometrically. Hence by Theorem 1.2, there does not exist any nonconstant harmonic function of polynomial growth on (R n , g). Note that there are many choices of a, δ, {c i } 6 i=0 satisfying the above eight assumptions, and the following is one choice satisfying all the above assumptions: which implies that f (r) is a C 2 function on [0, ∞). And by (4.28), (4.29) and (4.30),
a = 1 2 √ 3 , δ = 2π √ 3 , c 0 = 13 4 , c 1 = 4 , c 2 = 1 , c 3 = 1 2 , c 4 = 1 4 , c 5 = 1 4 , c 6 = 1f(0) = h(0) , f ′ (0) = h ′ (0) , f ′′ (0) = h ′′ (0) (4.34)
And (4.34) combining with (4.33) yields that h is a also a C 2 function on [0, ∞).
From the definition of f (r), h(r) and the formula (4.22), it is easy to get Rm(∂r, X, ∂r, X) > 0 , Rm(∂r, Y, ∂r, Y) > 0 (4.35)
Now we consider (h(x) − f(x)) ′′ , using (4.31),
(h(x) − f(x)) ′′ = c 2 c 2 3 e −c 6 x e (c 6 −c 3 )x − 1 ≥ 0 , ∀x ≥ 0 (4.36)
On the other hand, from (4.34), (h − f) ′ (0) = 0. Then by (4.36)
(h − f) ′ (x) ≥ (h − f) ′ (0) = 0 , ∀x ≥ 0 (4.37)
Again, by (4.34) and (4.37), .37), (4.40) and (4.19), we obtain that when r > δ,
(h − f)(x) ≥ (h − f)(0) = 0 , ∀x ≥ 0 (4.38)Then h ′ (0) = f ′ (0) = c 2 c 3 ≤ 1 2 , by h ′′ (x) < 0, h ′ (x) < h ′ (0) ≤ 1 2 (4.40) From (4Rm(X 1 , X 2 , X 1 , X 2 ) > 0 (4.41)
From (4.38), (4.40) and (4.21), when r > δ,
Rm(Y 1 , Y 2 , Y 1 , Y 2 ) = h 2 4 − (h ′ ) 2 − 3f 2 h 4 > 3(h 2 − f 2 ) h 4 ≥ 0 (4.42) Now consider ϕ(x) := f 3 (x) − h 3 (x)f ′ (x)h ′ (x), note ϕ(0) = h 3 (0)(1 − f ′ (0) 2 ) > 0 (4.43)
On the other hand, using h ′′ < 0,
ϕ ′ (x) = 3f ′ f 2 − 3h 2 (h ′ ) 2 f ′ − h 3 f ′′ h ′ − h 3 f ′ h ′′ > h ′ h 2 − f ′′ h − 3f ′ h ′ (4.44) and −f ′′ h − 3f ′ h ′ = c 2 c 3 e −Rm(X, Y, X, Y) = ϕ(x) f(x) · h 4 (x)
> 0 (4.48) From (4.35), (4.41), (4.42) and (4.48), the metric g = dr 2 + f 2 (r)k 1 + h 2 (r)k 2 on M 8 has positive sectional curvature, where f , g are defined in (4.23) and (4.24). It is not hard to see that this metric also has the uniform asymptotic polynomial volume growth of degree 5 as in (4.16) and Remark 4.6. Then by Corollary 4.7, there exists nonconstant harmonic function of polynomial growth on (M 8 , g), but (M 8 , g) does not have maximal volume growth. This disproves the necessary part of Conjecture 1.4.
Uniform bound of frequency function
Much of argument in this section followed the detailed analysis about frequency function in [CM97b] (especially Proposition 3.3, Proposition 3.36, Proposition 4.11 and Lemma 7.1 there). We are providing details here again to make our argument self-contained and concrete enough for our purpose, some more general argument can be found in [CM97b].
In this section, I(r), D(r), E(r), F(r), F (r) and W (r) are defined as in Section 2 with respect to some nonconstant function u ∈ H d (M). Further assume that
u(p) = 0, where p ∈ M n .
The following Lemma is a weak version of a uniform Harnack inequality for harmonic function with polynomial growth. |u| ≤ C(n, d)r d−1 , ∀r ≥ 1 (5.2) From (2.16) and V(r) ≤ V n 0 (1)r n , there exists C 1 > 0 such that for any r > 0,
D(r) = r 2−n b(x)≤r
|∇u(x)| 2 dx ≤ C 1 (r 2d + 1) (5.3) By (5.1) amd (5.3),
C 1 2 (4n+1) j R 0 2d + 1 > 2 10nd j D(R 0 ) which implies 2 (2−2n) jd + 2 −10nd j > D(R 0 ) C 1 R 2d 0 let j → ∞ in the above, we get 0 ≥ D(R 0 ) C 1 R 2d 0 .
However, D(R 0 ) > 0 because u is nonconstant, which is the contradiction. Proof: From Theorem 2.7, for given δ > 0, there exists R = R(p, δ) > 0 such that for ρ(x) = r ≥ R,
ln b(x) ρ(x) ≤ δ , b(x)≤r |∇b| 2 − 1 2 dx ≤ δ 2 Vol(b(x) ≤ r) (5.6)
where δ > 0 is to be determined later. From Cauchy-Schwarz inequality,
b≤r |∇b| 2 − 1 ≤ δVol(b ≤ r) (5.7)
Then for s ∈ [r, 2 4n r],
|D(s) − E(s)| = s 2−n b≤s |∇u| 2 (1 − |∇b| 2 ) ≤ s 2−n sup b(x)≤s |∇u| 2 (x) · δVol(b ≤ s) ≤ C(δ, n)s 2 sup b(x)≤s |∇u| 2 (x) (5.8)
in the first inequality above we used (5.7), and we have lim δ→0 C(δ, n) = 0.
Without loss of generality, assume e δ ≤ 4 3 . From Theorem 1.2 in [LS84] and (5.6) above, There exists δ = δ(n, V M , ǫ, γ) such that (5.12) implies ln D(s) E(s) ≤ ǫ. Combining all the above, there exists R 1 = R 1 (p, δ) = R 1 (p, n, V M , ǫ, γ) satisfying our conclusion. From Theorem 2.7, given any δ > 0, there exists R = R(p, δ) > 0 such that for r = ρ(x) ≥ R, we have where C 1 (n) is the constant depending only on n.
sup b(x)≤2 4n r |∇u| 2 (x) ≤ sup B( 4 3 ·2 4n r) |∇u| 2 ≤ C(n) Vol(B( √ 3 · 2 4n r)) B( √ 3·2 4n r)) |∇u| 2 ≤ C(n) V M · √ 3 · 2ln b(x) ρ(x) ≤ δ , (5.16) b(y)≤r |∇b| 2 − 1 2 dy ≤ δ 2 Vol(b ≤ r) , (5.17) b(y)≤r Hess(b 2 ) − 2g 2 dy ≤ δ 2 Vol(b ≤ r) (
Proof: As in the proof of Lemma 5.2, we can choose R 3 = R 3 (p) > 0 such that for ρ(x) = r > R 3 2 , | ln b(x) ρ(x) | ≤ 1 2 ln 4 3 . Similar as (5.9), for r > R 3 , Proof: If u has no zero point, then by Yau's Liouville theorem [Yau75] u is constant, the conclusion is straightforward. Assume u(p) = 0 where p ∈ M is some fixed point. We will firstly prove the following claim: where r i > R 4 and s ∈ [r i , 2 2n r i ].
Using (5.28), for r ∈ [2r i , 2 2n r i ], we get I(r) ≤ 2 10d I( r 2 )
By 5d D(M), using Theorem 3.4 and induction method, there exists k 0 = k 0 (d) such that if r i > k 0 then
I(r) ≤ 2 10d I( r 2 ) , r ∈ [ k 0 2 , 2 2n r i ]
which implies I(2 4n+2 r) ≤ 2 10d 4n+3 I( r 2 ) for r ∈ [ k 0 2 , 2 −2−2n r i ]. Let R 5 = max{ k 0 2 , R 3 }, where R 3 is from Lemma 5.4. Note R 5 = R 5 (p, d). By Lemma 5.4, D(2 4n+1 r) ≤ C 1 (n)2 60nd D(r) , r ∈ [R 5 , 2 −2−2n r i ] (5.32) From (5.32), similar to the above argument to get (5.31), we get that there exists R = max R 5 , R 4 δ, p, n, V M , C 1 (n)2 60nd such that for r ∈ [R, 2 −2−2n r i ], F (s) ≤ 31d + C 1 (n) , s ∈ [R, 2 −2 r i ]
In the above inequality, let i → ∞, then for r ≥ R(δ, p, n, V M , d), F (r) ≤ C(n, d). If we fix δ = 1 4 ln 4 3 , then Claim 5.6 is proved. Because F (r) is continuous function of r, F (r) ≤ C on [0, R], where C is some constant depending on u and R. Combining the above results together, the conclusion of the theorem is proved. Now we prove Theorem 1.11 by using the above theorem. Proof of Theorem 1.11: If d < 1, from [Che80] u must be constant, then the conclusion follows trivially. Hence we assume d ≥ 1 in the rest of the proof. When the tangent cone at infinity of M n is unique, denoted as C(X), then D(M) is a countable set by the fact that the spectrum of C(X) is discrete. Because d ≥ 1, we can find d 0 ∈ [d, d + 1] such that 5d 0 D(M), note u ∈ H d (M) ⊂ H d 0 (M). From Theorem 5.5, F u (r) ≤ C(u, n, V M , d 0 ) ≤ C(u, n, V M , d), the conclusion is proved.
Theorem 1. 5 (
5Hadamard's Three Circles Theorem). If f (z) is a holomorphic function on {z| r 4 ≤ |z| ≤ r} ⊂ C, where r > 0 is some constant, then M(r/2) M(r/4) ≤ M(r) M(r/2) (1.1) where M(s) := max |z|=s {| f (z)|},
Lemma 3. 3 .
3Assume that (M n , g) is an n-dimensional complete manifold with Rc ≥ 0 and maximal volume growth, u(p) = 0 and u(x) is harmonic on {b ≤ r} ⊂ M n . Then for any θ ∈ (0, 1) and r > 0, sup b≤(1−θ)r |u| 2 ≤ C(n, p, θ, V M )I u (r) (3.16) sup b≤(1−θ)r |∇u| 2 ≤ C(n, p, θ, V M )r −2 I u (r) (3.17) Proof: From I ′ (r) = 2D(r) 2 D(s)ds ≤ r n−1 I(r) (3.18) by the definition of D(r), s n−2 D(s) is nondecreasing, and therefore (3.
b≤(1−θ)r |∇u| 2 ≤ C(n, p, θ, V M )r −2 I(r) (3.20) By integrating (3.20) along geodesics starting at p and using u(p) = 0,sup b≤(1−θ)r |u| 2 ≤ C(n, p, θ, V M )I(r)Theorem 3.4. Assume that (M n , g) is an n-dimensional complete manifold with Rc ≥ 0 and maximal volume growth, α D(M), then there exists integer k 0 = k 0 (α) > 1, such that for r ≥ k 0 , u(x) harmonic over B(r) ⊂ (M n , g) and u(p)
for any i >> 1. In fact, from (4.5), we can obtain thatû i are uniformly continuous near boundary of B i (1). Combining with the Cheng-Yau's gradient estimate,û i are uniformly continuous on B i (1).
Theorem 4. 3 .
3Let (M n , g) be a complete manifold with nonnegative Ricci curvature, assume that every tangent cone at infinity of M n with renormalized limit measure is a metric cone C(X) with conic measure of power κ ≥ 2, and H 1 (X) > 0. If there exists d D(M) and d > inf{α| α ∈ D(M), α 0}, then dim H d (M) ≥ 2.
Lemma 4. 4 .
4For any given positive constant r 0 ∈ (0, 1), there exists i 0
δ
If a, δ, {c i } 6 i=0 are positive constants satisfying the following eight assumptions:= a −1 sin −1 a(c 1 − c 2 ) , (4.27) c 0 − c 5 = c 1 − c 2 , 4 + c 5 c 6 = c 2 c 3 , (4.30) c 6 > c 3 , (4.31)c 0 ≥ 3c 2 + c 5 (4.32)
Define s(r) := 1 a sin(ar), by (4.25), (4.26) and (4.27), we have s(δ) = f(0) , s ′ (δ) = f ′ (0) , s ′′ (δ) = f ′′ (0) (4.33)
From ( 4 .
426), (4.28) and (4.32), we get 1 − c 2 2 c 2 3 c 2 c 2 3 = c 1 − c 2 = c 0 − c 5 ≥ 3c 2
c 3 x c 0 c 3 + c 3 c 4 x − 3c 4 − (c 3 c 5 + 3c 5 c 6 )e −c 6 x ≥ c 2 c 3 e −c 3 x c 0 c 3 − 3c 4 − (c 3 c 5 + 3c 5 c 6 ) = c 2 c 3 e −c 3 x c 0 c 3 − 3c 2 c 3 − c 5 c 3 ≥ 0 (4.45) in the last equation above we used (4.30), and in the last inequality we used (4.32). Combining (4.44) with (4.45), we obtain ϕ ′ (x) > 0 (4.46) From (4.43) and (4.46), ϕ(x) > 0 , ∀x ≥ 0 (4.47) By (4.47) and (4.20), when r > δ, x = r − δ > 0,
Lemma 5. 1 .
1For nonconstant u ∈ H d (M), there exists positive increasing sequence {r i } ∞ i=1 such that lim i→∞ r i = ∞ and for any i D(2 4n+1 r i ) ≤ 2 10nd D(r i ) Proof: By contradiction. If the conclusion does not hold, there exists R 0 ≥ 1, such that D(2 4n+1 r) > 2 10nd D(r) when r ≥ R 0 by induction we get that for any j = 1, 2, 3, · · · D(2 (4n+1) j R 0 ) > 2 10nd j D(R 0 ) (5.1) On the other side, by u ∈ H d (M) and Corollary 3.2 in Chapter 1 of [
≤D(r) ≤ r 2
2C(n, V M )r −2 D(2 4n+1 r) ≤ C(n, V M )γD(r)r −2 (5.9) in the last inequality we used (5.4).Note that s ∈ [r, 2 4n r], hence x)≤2 4n r|∇u| 2 (x) ≤ C(n, V M )γD(s)r −2 ≤ C(n, V M )γD(s)s −2 (5.11)in the last inequality we used s ≤ 2 4n r.By(5.8) and (5.11), we obtain |D(s) − E(s)| ≤ C(δ, n, V M )γD(s) where lim δ→0 C(δ, n, V M ) = 0. Hence E(s) D(s) ≥ 1 − γC(δ, n, V M ) (5.12)
Lemma 5. 3 .
3Given positive constants γ, ǫ, there exists R 2 = R 2 (ǫ, p, n, V M , γ) > 0 such that if r > R 2 andD(2 4n+1 r) ≤ γD(r) ln W ) ′ (t), 0}dt > −ǫ (5.14)Proof: Using the first variation formula of energy in the Appendix of [From Cauchy-Schwarz inequality, it is easy to get W ≥ 0. Hence we only need to bound the integrals of J and K.
5 J
52 e δ − 1 + C(n, V M , γ)δ ln(.16, it is easy to see D(s) ≥ E(s), hence K ≥ K 2 . Now,|K 2 | ≤ 2s 2−n sup b=s |∇u| 2 D(s) b(x)=s |∇b| − |∇b| −1 dx ≤ 2s 2−n · sup b≤2 4n r |∇u| 2 D(s) |∇b| − |∇b| −1 dx ≤ C(n, V M , γ)s n, V M , γ) Vol(b ≤ 2 4n r) r n δe nδ ≥ −C(n, V M , γ)δ (5.21)in the first inequality above we used the co-area formula, and (5.17) was used in the second inequality. From (5.20) and (+ K ≥ lim δ→∞ −C(n, V M , γ) e δ − 1 + δ = Hence there exists δ 0 = δ 0 (ǫ, n, V M , γ) satisfying (5.19), and if δ ≤ δ W ′ (t), 0}dt ≥ −ǫ Choose R 2 = max{R(p, δ 0 ), R 1 (ǫ, p, n, V M , γ)}, the conclusion is proved.Lemma 5.4. For p ∈ M, there exists R 3 = R 3 (p) > 0 such that if r > R 3 and I(2 4n+2 r) ≤ γI(
5.24) along geodesics starting at p and using u(p) ds = I(2 4n+2 r) − I(2 4n+1 r) ≤ I(2 4n+2 r) Note s n−2 D(s) is nondecreasing in s from the definition of D(r), we get 2 2 4n+1 r n−1 D(2 4n+1 r) ≤ 2 4n+2 r n−1 I(2 4n+2 r) Combining (5.25), simplifying the above inequality yields D(2 4n+1 r) ≤ 2 n−2 I(2 4n+2 r) ≤ 2 n−2 γI( r 2 ) = C 1 (n)γD(r) Theorem 5.5. Suppose that (M n , g) has nonnegative Ricci curvature and maximal volume growth. For u ∈ H d (M), if 5d D(M), then the frequency of u is bounded by C(u, n, V M , d).
Claim 5. 6 .
6There exists a constant R = R(p, n, V M , d) > 0 such that if r ≥ R, F (r) ≤ C(n, d).By Lemma 5.1, there exist r i → ∞, such that D(2 4n+1 r i ) ≤ 2 10nd D(r i )Choose δ > 0, such that δ < 1 2 ln 4 3 , by Lemma 5.2 and Lemma 5.3, there existsR 4 = R 4 (δ, p, n, V M , 2 10nd ) > 0 such that if r i > R 4 , then ln D(s) E(s) ≤ δ , ∀s ∈ [r i , 2 4n r i ] D(2 4n r i ) D(r i ) − ln F (2 4n r i ) F (r i ) ≤ ln D(2 4n r i ) D(2 4n+1 r i ) · 2 10nd + ln E(2 4n r i ) D(2 4n r i ) + ln D(r i ) E(r i ) − ln W (2 4n r i ) W (r i ) ≤ (10nd + n − 2) ln 2 + 3δ ≤ 12dn · ln 2 (5.29)in the second inequality from the end, we used (5.26) and (5.27).From(5.29), there exists s i ∈ [2 2n r i , 2 4n r i ] such that F (s i ) ≤ 3d (5.30) By (5.26), (5.27) and (5.30), for r i > R 4 , s ∈ [r i , 2 2n r i ], W (s) ≤ 3de 2δ ≤ 4d Combining with (5.26), we get F (s) ≤ W (s)e δ < 5d (5.31)
Lemma 5.2 (Equivalence of E and D). For ǫ > 0, there exists R 1 = R 1 (ǫ, p, n, V M , γ) > 0 such that for r ≥ R 1 , if then for s ∈ [r, 2 4n r],D(2 4n+1 r) ≤ γD(r)
(5.4)
ln
D(s)
E(s)
≤ ǫ
(5.5)
5.18)We choose δ > 0 such that then (5.16) implies that for s ≥ R,From Lemma 5.2, there exists R 1 = R 1 (ǫ, p, n, V M , γ) such that if r ≥ max R, R 1 , for s ∈ [r, 2 4n r], ln D(s) E(s) ≤ δ, hence J 1 ≤ 2 s e δ − 1. From (5.11), (5.18), the Cauchy-Schwarz inequality and the Bishop-Gromov volume comparison theorem, we get taking integral on [r, 2 4n r], where r ≥ max R, R 1 ,e 2δ <
4
3
(5.19)
{b ≤ s} ⊂ B
4
3
s ,
B(
√
3s) ⊂ {b ≤ 2s}
Now we estimate J,
|J| ≤
2
s
D(s) − E(s)
E(s)
+
1
s
·
s 2 sup b≤s |∇u| 2
E(s)
· s −n
b≤s
Hess(b 2 ) − 2g
= J 1 + J 2
J 2 ≤
1
s
C(n, V M )γ
D(s)
E(s)
· δV n
0 (1)e nδ ≤
1
s
C(n, V M , γ)δe [(n+1)δ] ≤
C(n, V M , γ)
s
δ
in the last inequality we used (5.19).
Hence |J| ≤ 2
s e δ − 1 + C(n,V M ,γ)
s
δ,
[Din04]to us in 2012. We are grateful to Shouhei Honda for his detailed comments and enthusiastic suggestions on the paper. Last but not least, we particularly thank Gang Liu for carefully reading the earlier version of the paper and pointing out some gaps, and we benefit from several long conversations with him.Example 4.9 (Counterexample of Ni's Conjecture). Let us start from the generalized Hopf fibration of S 7 as the following:where S 3 , S 7 , S 4 carry the metrics g S 3 , g S 7 , 1 4 g S 4 ; π is a Riemannian submersion with totally geodesic fibers and k 1 = g S 3 , k 2 = π * 1 4 g S 4 ; g S n denotes the canonical metric of curvature ≡ 1 on S n .Then for metric g = dr 2 + f 2 (r)k 1 + h 2 (r)k 2 on M 8 , which is diffeomorphic to R 8 , from (8.13) in[CC97]and Section 2 in[BKN12], we haveIn the following a, δ, {c i } 6 i=0 are positive constants to be determined later, setr > δ where f and h are defined as the following:It is easy to see that the metric g = dr 2 + f 2 (r)k 1 + h 2 (r)k 2 has positive sectional curvature when 0 ≤ r ≤ δ, in the following we will try to find suitable constants a, δ, {c i } 6 i=0 such that the C 2 -metric g has positive sectional curvature when r > δ.
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| {'fraction_non_alphanumeric': 0.1060893470790378, 'fraction_numerical': 0.04912714776632302, 'mean_word_length': 3.2356194690265485, 'pattern_counts': {'":': 0, '<': 25, '<?xml version=': 0, '>': 97, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 128, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.2010 Mathematics Subject Classification. 35B40, 58J05, 53C23, 35A01. 1 2 GUOYI XU where ρ(x) = d(p, x) and p is some fixed point on M n . In [Yau87], the following conjecture was made: Conjecture 1.1 (S.-T. Yau). Let M n be a complete manifold with nonnegative Ricci curvature, then dim H d (M) < ∞ for any d > 0. P. Li and L.-F. Tam firstly proved the conjecture for linear growth harmonic function (d = 1) in [LT89], and they further verified the 2-dimensional case (n = 2) in [LT91]. In 1997, this conjecture was completely proved by Colding and Minicozzi [CM97a] (also see [Li97], [CM98a], [CM98b] and [LW99] for further developments). Although Yau's conjecture was confirmed, there are still several important questions about harmonic functions with polynomial growth remained open. It is well known that on any complete noncompact manifold, there always exist nonconstant harmonic functions (see [GW75]). A natural question is about the existence of nonconstant harmonic function with polynomial growth. Note any complete manifold with nonnegative Ricci curvature has at least linear volume growth (see [Yau75]). C. Sormani proved the following: Theorem 1.2 ([Sor00]). Let M n be a complete manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function of polynomial growth, then the manifold splits isometrically, M n = R × N n−1 .As observed in [Din04], there exists a suitable metric on R + × S n−1 , which has nonnegative Ricci curvature and linear volume growth, but can not split isometrically. From Theorem 1.2, it will not admit any nonconstant harmonic functions with polynomial growth. A concrete example is given in Section 4 (Example 4.8).Hence, to study the existence of nonconstant harmonic function with polynomial growth in nonnegative Ricci curvature context, we need to have some restriction on the volume growth of M n . To the author's knowledge, the following question is still open : Question 1.3. If (M n , g) is a complete manifold with Rc ≥ 0 and maximal volume growth, does there exist d ≥ 1 such that dim H d (M) ≥ 2? In other words, is there any nonconstant harmonic function with polynomial growth order at most d on M n ?In another direction, putting the existence problem in the positively curved context, L. Ni [Ni10] made the following conjecture: Remark 1.6. One novel thing in one of our Three Circles theorems (Theorem 3.2) is that it can be applied for the collapsed case, i.e. the case that the maximal volume growth assumption does not hold. Also Theorem 3.2 is dealing with the integral J u , whose domain is different from the original one considered in [Din04] (see Corollary 1.11 there). View from the domains of the integral appearing in those results, Theorem 3.2 is more like a three disks theorem, and Ding's technical tool is based on a three annuli theorem.As one application of our Three Circles Theorems, we proved the following theorem, which generalizes the existence result in [Din04].Theorem 1.7. Let (M n , g) be a complete manifold with nonnegative Ricci curvature, the tangent cone at infinity with renormalized limit measure is a unique metric", 'arxivid': '1601.02066', 'author': ['Guoyi Xu '], 'authoraffiliation': [], 'corpusid': 119134763, 'doi': '10.1007/s00208-016-1366-5', 'github_urls': [], 'n_tokens_mistral': 29238, 'n_tokens_neox': 25019, 'n_words': 14542, 'pdfsha': '747cf2c772b2686b81b27b335aabc3b4633803f7', 'pdfurls': ['https://arxiv.org/pdf/1601.02066v1.pdf'], 'title': ['THREE CIRCLES THEOREMS FOR HARMONIC FUNCTIONS', 'THREE CIRCLES THEOREMS FOR HARMONIC FUNCTIONS'], 'venue': []} |
arxiv |
A local-global principle for unipotent characters
12 Jan 2023
Damiano Rossi
A local-global principle for unipotent characters
12 Jan 2023
We obtain an adaptation of Dade's Conjecture and Späth's Character Triple Conjecture to unipotent characters of simple, simply connected finite reductive groups of type A, B and C. In particular, this gives a precise formula for counting the number of unipotent characters of each defect d in any Brauer ℓ-block B in terms of local invariants associated to e-local structures. This provides a geometric version of the local-global principle in representation theory of finite groups. A key ingredient in our proof is the construction of certain parametrisations of unipotent generalised Harish-Chandra series that are compatible with isomorphisms of character triples.
Introduction
The local-global conjectures are currently some of the most interesting and challenging problems in representation theory of finite groups. Among others, these include the McKay Conjecture [McK72], the Alperin-McKay Conjecture [Alp76] and Alperin's Weight Conjecture [Alp87] all of which can be deduced by a deeper statement known as Dade's Conjecture [Dad92], [Dad94], [Dad97]. The latter also implies the celebrated Brauer's Height Zero Conjecture introduced in [Bra56] and whose proof has recently been completed in [MNSFT22] and [Ruh22a] while relying on a combined effort of many other authors.
In this paper, we are particularly interested in Dade's Conjecture which, for every prime number ℓ, suggests a precise formula for counting the number of irreducible characters of a finite group, with a given ℓ-defect and belonging to a given Brauer ℓ-block, in terms of the ℓ-local structure of the group itself. This conjecture has been further extended in [Spä17] where the Character Triple Conjecture was formulated by introducing a compatibility with N -block isomorphisms of character triples, hereinafter denoted by ∼ N , as defined in [Spä17,Definition 3.6]. This notion plays a fundamental role in many aspects of group representation theory and, as we will see later, gives us a way to control the representation theory of local subgroups. Furthermore, it was exploited to reduce Dade's Conjecture to finite quasi-simple groups as explained in [Spä17,Theorem 1.3].
Our aim is to adapt and prove the two conjectures described in the previous paragraph to the case of unipotent characters of finite reductive groups. The approach considered here is inspired by ideas introduced by the author in [Ros22c] and provides further evidence for the conjectures formulated in that paper [Ros22c,Conjecture C and Conjecture D]. In particular, the ℓ-local structures considered above are replaced by more suitable e-local structures arising from the geometry of the underlying algebraic group that are compatible with the framework of Deligne-Lusztig theory. Therefore, our results also suggest the existence of an e-local-global principle for the representation theory of finite reductive groups.
More precisely, let G be a simple, simply connected group of type A, B or C which is defined over an algebraically closed field of positive characteristic p and let F ∶ G → G be a Frobenius endomorphism endowing G, as a variety, with an F q -structure for some power q of p. We denote by G F the finite reductive group consisting of the F q -rational points on G. Furthermore, we fix an odd prime ℓ different from p and denote by e the multiplicative order of q modulo ℓ. We let L e (G, F ) denote the set of e-chains of (G, F ) of the form σ = {G = L 0 > L 1 > ⋅ ⋅ ⋅ > L n } where each L i is an e-split Levi subgroup of (G, F ). The final term of the e-chain σ is denoted by L(σ) = L n , while σ ∶= n is the length of σ. Observe that the latter induces a partition of the set L e (G, F ) into the sets L e (G, F ) ± consisting of those e-chains σ that satisfy (−1) σ = ±1. Furthermore, notice that G F acts by conjugation on the set L e (G, F ) and indicate by G F σ the stabiliser of the e-chain σ. It follows directly from the definition that this action preserves the length of e-chains and, in particular, it restricts to an action of G F on the set L e (G, F ) >0 of e-chains of positive length. Now, to each non-negative integer d and Brauer ℓ-block B of the finite group G F , we associate a set L d u (B) ± consisting of quadruples (σ, M, µ, ϑ) where σ is an e-chain belonging to L e (G, F ) ± , (M, µ) is a unipotent e-cuspidal pair of (L(σ), F ) such that M does not coincide with G, and ϑ is an irreducible character of the e-chain stabiliser G F σ belonging to the character set Uch d (B σ , (M, µ)) defined by the choice of d, B, σ and (M, µ) as described in Definition 5.5. Once again, the group G F acts by conjugation on L d u (B) ± and we indicate the corresponding set of G F -orbits by L d u (B) ± G F . Moreover, for every such orbit ω, we denote by ω • the corresponding G F -orbit of pairs (σ, ϑ) such that (σ, M, µ, ϑ) ∈ ω for some unipotent e-cuspidal pair (M, µ).
With the above notation, we are now able to state our first main result. For simplicity, in the next theorem we assume that the prime ℓ does not divide the greatest common divisor (q ± 1, n + 1) whenever (G, F ) is of type A n (±q) and where A n (−q) denotes 2 A n (q) as usual. Observe however that this assumption can be removed as explained in Remark 5.7 (see Theorem 5.9 for the more general statement).
Theorem A. For every Brauer ℓ-block B of G F and every non-negative integer d, there exists an
Aut F (G F ) B -equivariant bijection Λ ∶ L d u (B) + G F → L d u (B) − G F such that X σ,ϑ , G F σ , ϑ ∼ G F X ρ,χ , G F ρ , χ
for every ω ∈ L d u (B) + G F , any (σ, ϑ) ∈ ω • , any (ρ, χ) ∈ Λ(ω) • and where X ∶= G F ⋊ Aut F (G F ) and Aut F (G F ) is the group of automorphisms described in Section 3.1.
The above theorem provides an adaptation of Späth's Character Triple Conjecture to the framework of Deligne-Lusztig theory for the unipotent characters of finite reductive groups. Theorem A also offers further evidence for the validity of [Ros22c,Conjecture D], in fact the set L d u (B) ± introduced above is a subset of the set of quadruples L d (B) ± considered in [Ros22c,Conjecture D] which is identified by only selecting unipotent e-cuspidal pairs (M, µ) among those appearing in such quadruples.
Next, we obtain a formula for counting the number of unipotent characters of ℓ-defect d in the Brauer ℓ-block B in terms of local invariants associated to e-local structures. For each e-chain σ of (G, F ) with positive length, we define k d u (B σ ) to be the number of characters belonging to one of the character sets Uch d (B σ , (M, µ)) for some unipotent e-cuspidal pair (M, µ) of (L(σ), F ) up to G F σ -conjugation (see also (5.10)). Furthermore, let k d u (B) and k d c,u (B) be the number of irreducible characters with ℓ-defect d and belonging to the Brauer ℓ-block B that are unipotent and unipotent e-cuspidal respectively. Then, by using the bijection given by Theorem A we can determine the difference k d u (B) − k d c,u (B) in terms of an alternating sum involving the terms k d u (B σ ) arising from the e-local structure G F σ .
Theorem B. For every Brauer ℓ-block B of G F and every non-negative integer d, we have
k d u (B) − k d c,u (B) = σ (−1) σ +1 k d u (B σ )
where σ runs over a set of representatives for the action of G F on L e (G, F ) >0 .
We point out that the restriction on the prime ℓ made for simplification before Theorem A only concerns the condition on isomorphisms of character triples and hence does not affect Theorem B. As before, this result provides an adaptation of Dade's Conjecture to the framework of Deligne-Lusztig theory for the unipotent characters of finite reductive groups and gives new evidence in favour of [Ros22c,Conjecture C]. The necessity for the introduction of the corrective term k d c,u (B) in the equality of Theorem B can be understood as an analogue to the exclusion of the case of blocks with central defect in the statement of Dade's Conjecture or, depending on the formulation under consideration, of the case where d = 0. We refer the reader to the more detailed discussion given in the paragraph following Definition 5.2. Finally, we mention that Theorem B also provides evidence for a positive answer to a question recently posed by Broué [Bro22a].
It is particularly interesting to notice that, to the author's knowledge, Theorem B cannot be obtained directly using techniques available at the present time, but only as a consequence of the existence of G F -block isomorphisms of character triples as those considered in Theorem A. In fact, while Deligne-Lusztig theory allows us to control the representation theory of finite reductive groups, it is not sufficient to control the representation theory of e-chain stabilisers G F σ . However, observe that the stabiliser G F σ contains the finite reductive group L(σ) F as a normal subgroup. Therefore, we can first use Deligne-Lusztig theory to study the characters of L(σ) F and then apply Clifford theory via G F -block isomorphisms of character triples to control the characters of G F σ (see Proposition 4.5 and Proposition 5.6 for further details).
In order to achieve the latter step, we need to make Deligne-Lusztig theory and, more precisely, e-Harish-Chandra theory for unipotent characters compatible with G F -block isomorphisms of character triples. This ideas was first suggested by the author in [Ros22c, Parametrisation B] and further studied in [Ros22d]. Our next result, which is a key ingredient in the proofs of Theorem A and Theorem B, establishes this conjectured parametrisation in the unipotent case under the assumption specified above. This can also be seen as an extension of the parametrisation introduced by Broué, Malle and Michel in [BMM93, Theorem 3.2 (2)] to the language of G F -block isomorphisms of character triples.
Theorem C. For every unipotent e-cuspidal pair (L, λ) of (G, F ) there exists an
Aut F (G F ) (L,λ) - equivariant bijection Ω G (L,λ) ∶ E G F , (L, λ) → Irr N G (L) F λ that preserves the ℓ-defect of characters and such that X χ , G F , χ ∼ G F N Xχ (L), N G (L) F , Ω G (L,λ) (χ)
for every χ ∈ E(G F , (L, λ)) and where X ∶= G F ⋊ Aut F (G F ).
The proof of Theorem C, and therefore of Theorem A and Theorem B, partially relies on certain conditions on the extendibility of characters of e-split Levi subgroups that were first introduced to settle the inductive conditions for the McKay Conjecture and the Alperin-McKay Conjecture, and then further studied in the context of Parametrisation B of [Ros22c] (see the exact statement given in [Ros22d, Definition 5.2]). These conditions were obtain, under certain assumptions, for groups of type A, B and C in the papers [BS20], [Bro22b] and [Bro-Ruh] respectively. Nonetheless, a version of these results is expected to hold in general and hence we believe that the above theorems, obtained here for types A, B and C with respect to an odd prime ℓ, will extend to the general case as well.
Structure of the paper
The paper is organised as follows. In Section 2 we introduce the necessary notation and recall the main definitions and results used throughout the paper. Furthermore, in Section 2.4 we introduce the notion of pseudo-unipotent character (see Definition 2.2) and prove a result on the regularity of blocks covering those containing such characters. Next, in Section 3 we start working towards a proof of Theorem C. First, in Section 3.1 we consider certain equivariance properties that can be established in the presence of extendibility conditions for characters of e-split Levi subgroups.
Here, we also present a candidate for the bijection Ω G (L,λ) required by Theorem C. Next, in Section 3.2 we construct the required G F -block isomorphisms of character triples. Using these results, we can then prove Theorem C in Section 3.3. The following step is to extend the parametrisation of unipotent e-Harish-Chandra series in the group G, as given by Theorem C, to a parametrisation of pseudo-unipotent e-Harish-Chandra series in F -stable Levi subgroups K of (G, F ). This is done in Theorem 4.4. Once this is established, in Section 4.2 we exploit the theory of G F -block isomorphisms to obtain bijections above e-Harish-Chandra series that are required to control the representation theory of the e-chain stabilisers G F σ . A more detailed analysis of the characters of G F σ is carried out in Section 5.1. In particular, we obtain a parametrisation of the character sets Uch d (B σ , (M, µ)) in Proposition 5.6. Finally, in Section 5.2 and Section 5.3 we apply these results to prove Theorem A and Theorem B respectively.
2 Notation and background material 2.1 Characters and blocks of finite groups We recall some standard notation from representation theory of finite groups as can be found in [Isa76] and [Nav98], for instance. Let Irr(G) the set of ordinary irreducible characters. If N ⊴ G and ϑ ∈ Irr(N ), then we denote by Irr(G ϑ) the set of irreducible characters of G that lie above ϑ. More generally, if S is a subset of irreducible characters of N , then we denote by Irr(G S) the union of the sets Irr(G ϑ) for ϑ ∈ S, that is, the set of irreducible characters of G that lie above some character in the set S.
Next, we denote by G ϑ the stabiliser of the irreducible character ϑ ∈ Irr(N ) under the conjugacy action of G and say that ϑ is G-invariant if G = G ϑ . In this case, we say that (G, N, ϑ) is a character triple. These objects provide important information in the study of Clifford theory and play a crucial role in many aspects of the local-global conjectures. Of paramount importance is the introduction of certain binary relations on the set of character triples. We refer the reader to [Nav18, Chapter 5 and 10] and [Spä18] for a more detailed introduction to these ideas and for the necessary background on projective representations. The binary relation considered here was introduced in [Spä17, Definition 3.6] and is known as N -block isomorphism of character triples, denoted by ∼ N . This equivalence relation has further been studied in [Ros22a].
In order to construct N -block isomorphisms of character triples, it is often useful to prove certain results on the extendibility of characters. Here, we introduce the notion of maximal extendibility (see [MS16,Definition 3.5]) that will be considered in the following sections. Let N ⊴ G be finite groups and consider S a subset of irreducible characters of N . Then, we say that maximal extendibility holds for the set S with respect to the inclusion N ⊴ G if every character ϑ ∈ S extends to its stabiliser G ϑ . More precisely, we can specify an extension map
Λ ∶ S → ∐ N ≤H≤G Irr(H) (2.1)
that sends each character ϑ ∈ S to an extension Λ(ϑ) of ϑ to the stabiliser G ϑ .
Next, we consider modular representation theory with respect to a fixed prime number ℓ. For χ ∈ Irr(G), there exist unique non-negative integers d(χ), called the ℓ-defect of χ, such that ℓ d(χ) = G ℓ χ(1) ℓ and where for an integer n we denote by n ℓ the largest power of ℓ that divides n. For any d ≥ 0, let Irr d (G) be the set of irreducible characters χ of G that satisfy d(χ) = d and denote by k d (G) its cardinality. Associated to the prime ℓ, we also have the set of Brauer ℓ-blocks of G. Each block is uniquely determined by the central functions λ B (see [Nav98,p. 49]). For every χ ∈ Irr(G), we denote by bl(χ) the unique block that satisfies χ ∈ Irr(bl(χ)). We conclude this introductory section with an analogue of [Isa76, Problem 5.3] for blocks that will be used in the sequel.
Lemma 2.1. Let H ≤ G be finite groups and consider blocks b of H and B of G. If ζ is a linear character of G, then:
(i) there are blocks b ⋅ ζ H of H and B ⋅ ζ of G satisfying
Irr(b ⋅ ζ H ) = {ψζ H ψ ∈ Irr(b)} and Irr(B ⋅ ζ) = {χζ χ ∈ Irr(B)}; (ii) If b G = B, then (b ⋅ ζ H ) G = B ⋅ ζ.
Proof. The first point is [Riz18, Lemma 2.1]. Next, let g ∈ G and denote by Cl G (g) the G-conjugacy class of g and by Cl G (g) + the corresponding conjugacy class sum in the group algebra. Since the intersection Cl G (g) ∩ H is a union of H-conjugacy classes, we can find h 1 , . . . , h n ∈ Cl G (g) ∩ H such that
Cl G (g) ∩ H = n ∐ i=1 Cl H (h i )
and where n is zero if Cl G (g) ∩ H is empty. In particular, observe that ζ(h i ) = ζ(g) since λ is a class function of G. Now, using the notation of [Nav98, p.87] we obtain
λ B⋅ζ (Cl G (g) + ) = λ B (Cl G (g) + ) ζ(g) = λ G b (Cl G (g) + ) ζ(g) = n i=1 λ b (Cl H (h i ) + ) ζ(g) = n i=1 λ b (Cl H (h i ) + ) ζ H (h i ) = n i=1 λ b⋅ζ H (Cl H (h i ) + ) = λ G b⋅ζ H (Cl G (g))
where for every algebraic integer α of C we denote by α its reduction modulo a maximal ideal containing the prime ℓ (see [Nav98,Chapter 2]). This shows that B ⋅ ζ = (b ⋅ ζ H ) G and we are done.
Finite reductive groups and unipotent characters
Let G be a connected reductive group defined over an algebraic closure of a field of positive characteristic p different from ℓ and consider a Frobenius endomorphism F ∶ G → G associated with an F q -structure for a power q of p. The set of F q -rational points on the variety G is denoted by G F and is called a finite reductive group. By abuse of notation we also refer to the pair (G, F ) as a finite reductive group.
Let L be a Levi subgroup of a parabolic subgroup P of G and assume that L (but not necessarily P) is F -stable. Using ℓ-adic cohomology, Deligne-Lusztig [DL76] and Lusztig [Lus76] defined a Z-linear map
R G L≤P ∶ ZIrr L F → ZIrr G F with adjoint * R G L≤P ∶ ZIrr G F → ZIrr L F .
The exact definition can be found in [CE04,Section 8.3]. These maps are known to be independent of the choice of the parabolic subgroup P in almost all cases (see [BM11] and [Tay18]) and, in particular, in those considered in this paper. Therefore, we will always omit P and denote R G L≤P simply by R G L . Next, using Deligne-Lusztig induction we define the unipotent characters of G F . These are the irreducible characters χ of G F that appear as an irreducible constituent of the virtual character R G T (1 T ) for some F -stable maximal torus T of G. The set of unipotent characters of G F is denoted by Uch(G F ) and its cardinality by k u (G F ). Similarly, if B is a block of G F and d a non-negative integer, then k d u (B) denotes the cardinality of the intersection Uch(G F ) ∩ Irr d (B).
e-Harish-Chandra theory for unipotent characters
Denote by e the multiplicative order of q modulo ℓ, if ℓ is odd, or modulo 4, if ℓ = 2. In this section, we collect the main results of e-Harish-Chandra theory for unipotent characters. This was first introduced by Fong and Srinivasan [FS86] for classical groups and then further developed by Broué, Malle and Michel [BMM93] for unipotent characters. The compatibility of this theory with Brauer ℓ-blocks was described by Cabanes and Enguehard in [CE94] for good primes and completed by Enguehard [Eng00] for bad primes. These results also provide a description of the characters belonging to unipotent blocks (see [CE94,Theorem (iii)]). Another description of these characters was provided by the author in [Ros22c] under certain resctrictions on the prime ℓ (see also [Ros22c,Remark 4.14] for a comparison between the two descriptions). We refer the reader to the monographs [CE04] and [GM20] for a more complete account of this beautiful theory.
The theory of Φ e -subgroups that constitutes the foundation of e-Harish-Chandra theory was introduced in [BM92]. Following their terminology, we say that an F -stable torus S of G is a Φ e -torus if its order polynomial is a power of the e-th cyclotomic polynomial, that is, if P (S,F ) = Φ n e for some integer n and where Φ e denotes the e-th cyclotomic polynomial (see [CE04,Definition 13.3]). Then, we say that a Levi subgroup L of G is an e-split Levi subgroup if there exists a Φ e -torus S such that L = C G (S). More precisely, we say that L is an e-split Levi subgroup of (G, F ) to emphasise the role of the Frobenius endomorphism F . Observe that, for any torus T, there exists a unique maximal Φ e -torus of T denoted by T Φe (see [CE04, Proposition 13.5 3.4]). Then, it can be shown
that an F -stable Levi subgroup L of G is e-split if and only if L = C G (Z ○ (L) Φe ) (see, for instance, [GM20, Proposition 3.5.5]).
Next, recall that (L, λ) is a unipotent e-cuspidal pair of (G, F ) if L is an e-split Levi subgroup of (G, F ) and λ ∈ Irr(L F ) satisfies * R L M (λ) = 0 for every e-split Levi subgroup M < L. A character λ with the property above is said to be a unipotent e-cuspidal character of L F . We denote by CP u (G, F ) the set of unipotent e-cuspidal pairs of (G, F ) and by k c,u (G F ) the number of unipotent e-cuspidal characters of G F . Moreover, we define the e-Harish-Chandra series associate to the e-cuspidal pair (L, λ) to be the set of irreducible constituents of the virtual character R G L (λ), denoted by E(G F , (L, λ)).
Uch G F = ∐ (L,λ) E G F , (L, λ)
where (L, λ) runs over a set of representatives for the action of G F on the set of unipotent ecuspidal pairs of (G, F ) as explained in [BMM93, Theorem 3.2 (1)]. This is a well known fact and will be used throughout the paper without further reference. As a consequence of the partition above, it now remains to parametrise the unipotent e-Harish-Chandra series. If (L, λ) is a unipotent e-cuspidal pair, we denote by W G (L, λ) F ∶= N G (L) F λ L F the corresponding relative Weyl group. Then, [BMM93, Theorem 3.2 (2)] parametrises the characters in an e-Harish-Chandra series in terms of the characters in the relative Weyl group by showing the existence of a bijection
Irr W G (L, λ) F → E(G F , (L, λ)).
(2.2)
In Section 3 we reformulate (2.2) in order to obtain Theorem C.
Unipotent e-Harish-Chandra series are also used to parametrise the so-called unipotent blocks, that is, those blocks that contain unipotent characters. This is the main result of [CE94]. More precisely, if ℓ is odd and good for G, with ℓ ≠ 3 if 3 D 4 is an irreducible rational component of (G, F ), then for every ℓ-block B of G F there exists a unipotent e-cuspidal pair (L, λ), with (L, λ) unique up to G F -conjugation, such that all the irreducible constituents of R G L (λ) belongs to the block B. In this case, we write B = b G F (L, λ) and we also have
Uch(G F ) ∩ Irr (b G F (L, λ)) = E(G F , (L, λ)).
Moreover, [CE94, Proposition 3.3 (ii) and Proposition 4.2] imply that bl(λ) G F = B.
Pseudo-unipotent characters
We denote by (G * , F * ) a group in duality with (G, F ) with respect to a choice of an F -stable maximal torus T of G and an F * -stable maximal torus
T * of G * . If τ ∶ G sc → [G, G]
is a simply connected covering (see [GM20, Remark 1.5.13]), then there exists an isomorphisms between the abelian groups
Z (G * ) F * → Irr G F τ (G sc ) z ↦ẑ G according to [CE04, (8.19)]. Notice that, if L is an F -stable Levi subgroup of G, then its dual L * is an F * -stable Levi subgroup of G * and we have Z(G * ) F * ≤ Z(L * ) F * .
In particular, every element z ∈ Z(G * ) F * defines a linear characters ofẑ L and restriction of characters yields the equality
(ẑ G ) L F =ẑ L .
In the next definition, we consider characters that are obtained by multiplying these linear characters with unipotent characters.
Definition 2.2. Let (K, F ) be a finite reductive group and consider a Levi subgroup of L ≤ K and an irreducible character θ ∈ Irr(L F ). We say that θ is (K, F )-pseudo-unipotent if there exists an element z ∈ Z(K * ) F * such that θẑ L is unipotent. Moreover, for every unipotent character λ ∈ Uch(L F ), we denote by ps K (λ) the set of (K, F )-pseudo-unipotent characters of L F of the form λẑ L for some z ∈ Z(K * ) F * . Moreover, we denote by ps K (L F ) the set of all (K, F )-pseudo unipotent characters of L F . When the group K coincides with L, we denote the set of characters ps L (L F ) simply by ps(L F ).
In accordance with the terminology introduced above, we say that an e-Harish-Chandra series of (K, F ) is pseudo-unipotent if it is of the form E(K F , (L, ν)) for some ν ∈ ps K (λ) and where (L, λ) is a unipotent e-cuspidal pair of (K, F ). In this case, we also say that (L, ν) is a pseudounipotent e-cuspidal pair. We define the union of all the series associated to characters in [CE04,(8.20)], we deduce that the elements of the pseudo-unipotent e-Harish-Chandra series E(K F , (L, λẑ)) are exactly the irreducible characters of the form ϕẑ K for some unipotent character ϕ ∈ E(K F , (L, λ)). Moreover, we point out that λ is the unique unipotent character in the set ps K (λ) according to [CE04,Proposition 8.26]. Similarly, the unipotent characters in the set E(K F , (L, ps K (λ))) are those in the series E(K F , (L, λ)).
ps K (λ) by E(K F , (L, ps K (λ))). Since R K L (λẑ L ) = R K L (λ)ẑ K for every z ∈ Z(K * ) F * by
Our next lemma, shows that blocks covering pseudo-unipotent characters are regular as defined in [Nav98,p.210].
Lemma 2.3. Let L be an F -stable Levi subgroup of G and suppose that ℓ is odd and good for G. For every L F ≤ H ≤ N G (L) F and every character ϑ ∈ Irr(H) lying above some pseudo-unipotent character in ps(L F ), the block bl(ϑ) is L F -regular. In particular, the Brauer induced block bl(ϑ) H is defined and is the unique block of H covering bl(ϑ).
Proof. Let ϕ ∈ Uch(L F ) and z ∈ Z(L * ) F * such that ϕẑ L lies below the character ϑ and chose a unipotent e-cuspidal pair (M, µ) of L such that ϕ ∈ E(L F , (M, µ)). In particular, bl(ϕ) = b L F (M, µ) according to [CE94].
If Q ∶= Z(M) F ℓ , then M F = C G F (Q) according to [CE94, Propo- sition 3.3 (ii)]. Moreover, observe that [CE94, Proposition 4.2] implies that bl(ϕ) = b L F (M, µ) = bl(µ) L F while [Riz18, Lemma 2.1]
implies that bl(ϕ) and bl(ϕẑ L ) have the same defect groups. Now, applying [Nav98, Lemma 4.13 and Theorem 9.26], we can find defect groups D ϑ , D ϕ and D µ of bl(ϑ), bl(ϕ) and bl(µ) respectively with the property that
D µ ≤ D ϕ ≤ D ϑ . Since Q ≤ O ℓ (M F ) ≤ D µ by [Nav98, Theorem 4.8], we deduce that Q ≤ D ϑ and hence C H (D ϑ ) ≤ C H (Q) = M F ≤ L F .
By [Nav98,Lemma 9.20] we conclude that the block bl(ϑ) is L F -regular. The second part of the lemma now follows from [Nav98, Theorem 9.19].
Compatibility with isomorphisms of character triples
The aim of this section is to show how the bijection (2.2) can be made compatible with isomorphisms of character triples and with the action of automorphisms. This property was first suggested by the author in [Ros22c, Parametrisation B] and further studied in [Ros22d]. Our Theorem C gives a solution of this conjectured result for unipotent e-Harish-Chandra series and groups of type A, B and C. Before proceeding further, we show how the parametrisation (2.2) can be reformulated in a more convenient form. For this, let (L, λ) be a unipotent e-cuspidal pair of (G, F ) and assume thatλ is an extension of λ to the stabiliser N G (L) F λ . Then, by applying Gallagher's theorem [Isa76, Corollary 6.17] and the Clifford correspondence [Isa76, Theorem 6.11] we obtain a bijection
Irr W G (L, λ) F → Irr N G (L) F λ η ↦ λ η N G (L) F
and therefore (2.2) holds if an only if there exists a bijection
E(G F , (L, λ)) → Irr N G (L) F λ .
(3.1)
This new reformulation will allow us to introduce the aforementioned compatibility with isomorphisms of character triple isomorphisms.
Equivariance and maximal extendibility
In this section, we consider some equivariance properties for the parametrisation (3.1) which are related to maximal extendibility (see (2.1)) of unipotent characters.
As in the previous sections, consider a connected reductive group G with a Frobenius endomorphism F ∶ G → G defining an F q -structure on G. We denote by Aut F (G F ) the set of those automorphisms of G F obtained by restricting some bijective morphism of algebraic groups σ ∶ G → G that commutes with F to the set of F q -rational points G F . Notice that the restriction of such a morphism σ to G F , which by abuse of notation we denote again by σ, is an automorphism of the finite group G F . We refer the reader to [CS13, Section 2.4] for further details. In particular, observe that any morphism σ with the properties above is determined by its restriction to G F up to a power of F and hence it follows that Aut F (G F ) acts on the set of F -stable closed connected subgroups of G. Then, given an F -stable closed connected subgroup H of G, we can define the set Aut F (G F ) H consisting of those automorphisms σ as above that stabilise the algebraic group H.
L, λ) of (G, F ) there exists an Aut F (G F ) (L,λ) - equivariant bijection I G (L,λ) ∶ Irr W G (L, λ) F → E G F , (L, λ) such that I G (L,λ) (η)(1) ℓ = G F ∶ N G (L, λ) F ℓ ⋅ λ(1) ℓ ⋅ η(1) ℓ for every η ∈ Irr(W G (L, λ) F ).
Proof. This follows from the proof of [CS13,Theorem 3.4]. See also [Ros22d,Theorem 3.4].
As explained at the beginning of this section, if the character λ extends to the stabiliser N G (L) F λ , then we can use the bijection (2.2) to obtain (3.1). A similar argument can be used to include the equivariance property described above and obtain an equivariant version of (3.1). Observe that, by the discussion on automorphisms above, it follows that the group Aut F (G F ) acts on the set of ecuspidal pairs (L, λ) and therefore we can define the stabiliser Aut F (G F ) (L,λ) . Furthermore, recall that we denote by d(χ) the ℓ-defect of an irreducible character χ.
Corollary 3.2. Let (L, λ) be a unipotent e-cuspidal pair of (G, F ) and suppose that λ has an ex-
tension λ ◇ ∈ Irr(N G (L) F λ ) which is additionally Aut F (G F ) (L,λ) -invariant. Then there exists an Aut F (G F ) (L,λ) -equivariant bijection Ω G (L,λ) ∶ E G F , (L, λ) → Irr N G (L) F λ such that d(χ) = d Ω G (L,λ) (χ)
for every χ ∈ E(G F , (L, λ)).
Proof. Consider the bijection I G (L,λ) given by Proposition 3.1 and define the map
Ω G (L,λ) ∶ E G F , (L, λ) → Irr N G (L) F λ I G (L,λ) (η) ↦ (λ ◇ η) N G (L) F
for every η ∈ Irr(W G (L, λ) F ) and where λ ◇ is the extension of λ to N G (L) F λ given in the statement. This is a well defined bijection by the Clifford correspondence [Isa76, Theorem 6.11] and Gallagher's theorem [Isa76, Corollary 6.17]. Moreover, for every α ∈ Aut F (G F ) such that (L, λ) α = (L, λ) and
every η ∈ Irr(W G (L, λ) F ) we have (λ ◇ η) N G (L) F α = (λ ◇ η) α N G (L) F = (λ ◇ η α ) N G (L) F because α stabilises λ ◇ . On the other hand I G (L,λ) (η) α = I G (L,λ) (η α )
by the properties of I G (L,λ) and hence we conclude that
Ω G (L,λ) is Aut F (G F ) (L,λ) -equivariant. Fur- thermore, if we consider η ∈ Irr(W G (L, λ) F ) and define the characters χ ∶= I G (L,λ) (η) and ψ ∶= (λ ◇ η) N G (L) F , then the degree formula from Proposition 3.1 implies that ℓ d(χ) = G F ℓ χ(1) ℓ = N G (L, λ) F ℓ λ(1) ℓ ⋅ η(1) ℓ = N G (L) F ℓ ψ(1) ℓ = ℓ d(ψ)
and hence we deduce that d(χ) = d(ψ) as required.
Next, we consider a regular embedding G ≤G as defined in [CE04, (15.1)]. Then,G is a connected reductive group with connected centre and whose derived subgroup coincides with that of G, that is,
[G,G] = [G, G].
In particular, observe thatG = Z(G)G, that G is normal inG and that the quotientG G is an abelian group. Moreover, for every Levi subgroup L of G, we deduce thatL ∶= Z(G)L is a Levi subgroup ofG and that L ≤L is again a regular embedding. These observations will be used throughout this paper without further reference.
We also recall that, according to [DM91,Proposition 13.20], restriction of characters yields a bijection between the unipotent characters ofG F and those of G F . In particular, every unipotent character of G F isG F -invariant. Using this observation, we can compare the relative Weyl groups inG F with those in G F . Lemma 3.3. Let (L, λ)be a unipotent e-cuspidal pair of (G, F ), setL = LZ(G) and consider a unipotent extensionλ of λ toL F . Then,
NG(L) F λ = NG(L) F λ and we have WG(L,λ) F ≃ W G (L, λ) F . Proof. Sinceλ extends λ, it is clear that the stabiliser NG(L) F λ is contained in NG(L) F λ .
On the other hand, let x ∈ NG(L) F λ and observe thatλ x is a unipotent character ofL F that restricts to λ x = λ. Then, [DM91,Proposition 13.20] implies thatλ x =λ and therefore x ∈ NG(L) F λ . From this, we also conclude that NG(L) F λ =L F N G (L) F λ and therefore that WG(L,λ) F ≃ W G (L, λ) F .
As a consequence of the lemma above, we show that when λ extends to its stabiliser N G (L) F λ , then every irreducible character of N G (L) that lies above λ is NG(L) F -invariant and extends to NG(L) F . Corollary 3.4. Let (L, λ) be a unipotent e-cuspidal pair of (G, F ) and suppose that λ has an extension λ ◇ ∈ Irr(N G (L) F λ ). Then every character of N G (L) F lying above λ extends to NG(L) F .
Proof. To start, we fix a unipotent extensionλ of λ toL F and recall that NG(L) F λ = NG(L) F λ according to Lemma 3.3. Then, applying [Spä10, Lemma 4.1 (a)] we deduce that there exists an extensionλ ◇ of λ ◇ to NG(L) F λ that also extendsλ. Consider now an irreducible character ψ of N G (L) F lying above λ. By Gallagher's theorem [Isa76, Corollary 6.17] and the Clifford correspondence [Isa76, Theorem 6.11], it follows that there exists an irreducible character η of the relative Weyl group W G (L, λ) F such that ψ is induced from the irreducible character ψ 0 ∶= ηλ ◇ . Moreover, by using Lemma 3.3, we have WG(L,λ) F ≃ W G (L, λ) F . Then, η, viewed as a character of N G (L) F λ , admits an extension, sayη, to NG(L) F λ . Now, defineψ 0 ∶=ηλ ◇ and observe thatψ 0 lies aboveλ. By the Clifford correspondence, it follows that the characterψ of NG(L) F induced from ψ 0 is irreducible and therefore, applying [Isa76, Problem 5.2], we conclude thatψ extends ψ. The proof is now complete.
We can now construct a parametrisation of unipotent e-Harish-Chandra series in the groupG F which agrees with the bijection Ω G (L,λ) from Corollary 3.2 via restriction of characters.
Proposition 3.5. Let (L, λ) be a unipotent e-cuspidal pair of (G, F ) and suppose that λ has an extension λ ◇ ∈ Irr(N G (L) F λ ) which is additionally Aut F (G F ) (L,λ) -invariant. Ifλ is a unipotent extension of λ toL F , then there exists a bijectionΩG (L,λ) making the following diagram commute
E G F , (L,λ) Irr NG(L) F λ E G F , (L, λ) Irr N G (L) F λ ΩG (L,λ) ResG F G F Res NG(L) F N G (L) F Ω G (L,λ)
and where Ω G (L,λ) is the bijection given by Corollary 3.2.
Proof. First, observe that λ has an extensionλ toL F according to [DM91,Proposition 13.20]. Moreover, restrictions fromG F to G F induces a bijection from the set E(G F , (L,λ)) to E(G F , (L, λ)) according to [CE94, Proposition 3.1]. Next, consider a character ψ ∈ Irr(N G (L) F ) lying above λ and observe that ψ admits an extensionψ 0 ∈ Irr(NG(L) F ) by Corollary 3.4. Letλ 0 be an irreducible constituent of the restrictionψ 0,L F and notice thatλ 0 is an extension of λ sinceL F L F is abelian. Now, Gallagher's theorem [Isa76, Corollary 6.17] implies that there exists a linear character ν ∈ Irr(L F L F ) such thatλ 0 ν =λ. Since NG(L) F N G (L) F ≃L F L F we can identify ν with its extension to NG(L) F . Then, it follows that the characterψ ∶=ψ 0 ν is an extension of ψ to NG(L) F lying aboveλ. Then the assignment ψ ↦ψ defines a bijection between Irr(N G (L) F λ)
and Irr(NG(L) F λ ) whose inverse is given by restriction of characters. We can now definẽ
ΩG (L,λ) (χ) ∶=ψ for everyχ ∈ E(G F , (L,λ)) andψ ∈ Irr(NG(L) F λ ) whenever Ω G (L,λ) (χ G F ) =ψ N G (L) F .
Construction of G F -block isomorphisms of character triples
From now on, we assume that G is simple, simply connected and of type A, B or C. Furthermore, we suppose that ℓ is odd and denote by e the order of q modulo ℓ.
We now give a more explicit construction of the group of automorphism Aut F (G F ). Fix a maximally split torus T 0 contained in an F -stable Borel subgroup B 0 of G. This choice corresponds to a set of graph automorphisms γ ∶ G → G and a field endomorphism F 0 ∶ G → G. More precisely, if we consider the set of simple roots ∆ ⊆ Φ(G, T 0 ) corresponding to the choice T 0 ⊆ B 0 , then we have an automorphism γ ∶ G → G given by γ(x α (t)) ∶= x γ(α) (t) for every t ∈ G a and α ∈ ±∆ and where γ is a symmetry of the Dynkin diagram of ∆, while F 0 (x α (t)) ∶= x α (t p ) for every t ∈ G a and α ∈ Φ(G, T 0 ). Here, we denote by x α ∶ G a → G a one-parameter subgroup corresponding to α ∈ Φ(G, T 0 ). We define the subgroup A of Aut F (G F ) generated by the graph and field automorphisms described above.
In addition, we choose our regular embedding G ≤G to be defined in such a way that the graph and field automorphisms extends toG (see, for instance, [MS16, Section 2B]). In particular, the group A acts via automorphisms onG F and we can form the external semidirect productG F ⋊ A which acts on G F . It turns out thatG F ⋊ A and Aut F (G F ) induce the same set of automorphisms on the finite group G F (see, for instance, [GLS98, Section 2.5]).
Throughout this section, we consider a fixed unipotent e-cuspidal pair (L, λ) of (G, F ) and a unipotent extensionλ of λ toL F (whose existence is ensured by [DM91,Proposition 13.20]) where, as always, we defineL ∶= LZ(G). In the next lemma, we show that the hypothesis of Corollary 3.2 is satisfied under our assumptions.
Lemma 3.6. There exists an extension λ ◇ of λ to N G (L) F λ that is (G F A) (L,λ) -invariant.
λ ◇ of λ to N G (L) F λ which is (G F A) (L,λ) -invariant. Since (G F A) (L,λ) = L(G F A) (L,λ) it suffices to show that λ ◇ isL F -invariant.
However, the latter assertion follows immediately from the fact that λ ◇ extends to NG(L) F λ according to Lemma 3.3 and [Spä10, Lemma 4.1 (a)].
As an immediate consequence of the lemma above, we deduce that every character of N G (L) F lying above λ extends to NG(L) F . This can be considered as a local analogue of [DM91, Proposition 13.20].
Lemma 3.7. Every irreducible character of N G (L) F lying above λ extends to NG(L) F .
Proof. This follows from Corollary 3.4 whose hypothesis is satisfied by Lemma 3.6.
We point out that, under our assumptions, every irreducible character of N G (L) F lying above λ extends to its stabiliser in NG(L) F because the quotient NG(L) F N G (L) F is cyclic according to [GM20, Proposition 1.7.5]. However, in the lemma above we are also showing, using independent methods, that each such character is NG(L) F -invariant.
Using Lemma 3.6, we can now define bijections Ω ∶= Ω G (L,λ) andΩG (L,λ) as described in Corollary 3.2 and Proposition 3.5 respectively. In what follows, we consider the sets of characters G ∶= E(G F , (L, λ)), L ∶= Irr(N G (L) F λ),G ∶= E(G F , (L,λ)) andL ∶= Irr(NG(L) F λ ). Our next aim is to show that the parametrisation Ω is compatible with G F -block isomorphisms of character triples. We start by checking the group theoretic properties required for the existence of such isomorphisms (see [Spä17, Remark 3.7 (i)]).
Lemma 3.8. For every χ ∈ G and ψ ∶= Ω(χ) ∈ L we have (
G F A) L,χ = (G F A) L,ψ andG F A χ = G F (G F A) L,ψ .
Proof. We argue as in the proof of [Ros22c, Lemma 4.2]. To start, we observe that
(G F A) (L,λ),χ = (G F A) (L,λ),ψ since the map Ω is (G F A) (L,λ) -equivariant. Set U (χ) ∶= (G F A) L,χ and U (ψ) ∶= (G F A) L,ψ .
First, consider x ∈ U (χ) and observe that according to [BMM93, Theorem 3.2 (1)] there exists y ∈ N G (L) F such that (L, λ) xy = (L, λ). In particular, xy ∈ (G F A) (L,λ),χ = (G F A) (L,λ),ψ and hence x ∈ U (ψ) since ψ y = ψ. This shows that U (χ) ≤ U (ψ). On the other hand, suppose that x ∈ U (ψ). By Clifford's theorem there exists y ∈ N G (L) F such that λ xy = λ and so xy ∈
(G F A) L,ψ = (G F A) L,χ . Since χ y = χ we deduce that x ∈ U (χ) and hence U (χ) = U (ψ). To conclude, it is enough to show thatG F A χ = G F U (χ). First, notice that G F U (χ) ≤G F A χ since χ isG F -invariant.
On the other hand, for x ∈G F A χ we know that (L, λ) x is G F -conjugate to (L, λ) thanks to [BMM93, Theorem 3.2 (1)]. Therefore, we obtain x ∈ G F U (χ) and as explained above this concludes the proof.
We now apply Lemma 3.8 to show that the mapΩ satisfies some useful equivariance properties. Before doing so, we need to introduce some notation. For this purpose, consider a pair (G * , F * ) dual to (G, F ) and a pair (G * , F * ) dual to (G, F ). Let i * ∶G * → G * be the surjection induced by duality from the inclusion G ≤G and observe that Ker(i * ) = Z(G * ) since G is simply connected (see [CE04, Section 15.1]). As shown in [CE04,(15.2)], there exists an isomorphism
Ker(i * ) F → Irr G F G F (3.2) z ↦ẑG
Furthermore, if L is an F -stable Levi subgroup of G and z ∈ Ker(i * ), then we defineẑL to be the restriction ofẑG toL F andẑ NG(L) to be the restriction ofẑG to NG(L) F . We set K ∶= Ker(i * ) and obtain an action of the group K on the characters ofG F ,L F and NG(L) F as defined in [Ros22d, Definition 2.1]. Moreover, we consider the external semidirect product (G F A)⋉K given by defining z x as the unique element of K corresponding to the character (ẑG) x of the quotientG F G F via the isomorphism specified in (3.2), whenever x ∈G F A and z ∈ K. Then, for every F -stable Levi subgroup L of G, we obtain an action of (G F A) L ⋉ K on the irreducible characters ofL F and NG(L) F . We denote by ((G F A) L ⋉ K)λ the stabiliser ofλ ∈ Irr(L F ). In particular, it follows that ((G F A) L ⋉ K)λ acts on the sets of charactersG andL. Next, we show that the bijectionΩ is compatible with this action.
Lemma 3.9. The bijectionΩ is (NG(L) F (G F A) (L,λ) ⋊ K)λ-equivariant.
Proof. Letχ ∈G andψ ∈L. By the definition ofΩ, we haveΩ(χ) =ψ if and only if Ω(χ) = ψ where χ ∶=χ G F and ψ ∶=ψ N G (L) F . Now, if we consider g ∈ NG(L) F , x ∈ (G F A) (L,λ) and z ∈ K such that (gx, z) stabilisesλ, then we obtainΩ
χ (gx,z) =ψ (gx,z) if and only if Ω χ (gx,z) G F = ψ (gx,z) N G (L) F . (3.3)
However, since the restriction ofχ (gx,z) to G F coincides with χ x and the restriction ofψ (gx,z) to N G (L) F coincides with ψ x , we deduce that the equality in (3.3) holds by the equivariant properties of Ω as described in Corollary 3.2.
One of the main ingredients for the construction of the projective representations needed to obtain G F -block isomorphisms of character triples is given by the following two lemmas on maximal extendibility.
Lemma 3.10. Maximal extendibility holds for G with respect to the inclusion G F ⊴ G F A, that is, every character χ ∈ G extends to G F A χ . The local version of the lemma above is a consequence of the results obtained in [BS20].
Lemma 3.11. Maximal extendibility holds for L with respect to the inclusion N G (L) F ⊴ (G F A) L , that is, every character ψ ∈ L extends to (G F A) L,ψ .
Proof. As in the proof of Lemma 3.10, it is enough to prove the result in the case where G is of type A. In fact, if G is of type B or C, then the quotient (G F A) (L,ψ) N G (L) F is cyclic because it it a subquotient of A. Now, if G is of type A the result follows from [BS20, Theorem 1.2].
Finally, we can start constructing isomorphisms of character triples for the bijection Ω. As a first step, we obtain a weaker isomorphism, know as G F -central isomorphism of character triples and denoted by ∼ c G F , whose requirements are given by [Spä17, Remark 3.7 (i)-(iii)] and replacing the condition on defect groups by imposing that C G (N ) ≤ H 1 ∩ H 2 with the notations used there. We refer the reader to [Ros22b, Definition 3.3.4] for a precise definition.
Proposition 3.12. For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have
G F A χ , G F , χ ∼ c G F (G F A) L,ψ , N G (L) F , ψ .
Proof. We start by constructing projective representations associated with χ and ψ. According to Proposition 3.5 we can find a unipotent extensionχ ∈G of χ toG F . Furthermore, by Lemma 3.10 there exists an extension χ ′ of χ to G F A χ . LetD glo be a representation ofG F affordingχ and D ′ glo a representation of G F A χ affording χ ′ . Now, [Spä12, Lemma 2.11] implies that
P glo ∶ G F A χ → GL χ(1) (C) defined by P glo (x 1 x 2 ) ∶=D glo (x 1 )D ′ glo (x 2 )
for every x 1 ∈G F and x 2 ∈ G F A χ is a projective representation associated with χ. Next, observe thatψ ∶=Ω(χ) ∈L is an extension of ψ to NG(L) F and consider an extension ψ ′ of ψ to (G F A) L,ψ given by Lemma 3.11. LetD loc be a representation of NG(L) F affordingψ and D ′ loc a representation of (G F A) L,ψ affording ψ ′ . Once again, [Spä12, Lemma 2.11] shows that the map
P loc ∶ G F A L,ψ → GL ψ(1) (C)
given by P loc (x 1 x 2 ) ∶=D loc (x 1 )D ′ loc (x 2 ) for every x 1 ∈ NG(L) F and x 2 ∈ (G F A) L,ψ is a projective representation associated with ψ. We denote by α glo and α loc the factor set of P glo and P loc respectively. As explained in the proof of [Ros22d, Theorem 4.3], in order to prove that α glo coincides with α loc via the isomorphismG
F A χ G F ≃ (G F A) L,ψ N G (L) F , it suffices to show that (µ glo x ) NG(L) F = µ loc x (3.4)
for every x ∈ (G F A) L,χ and where µ glo x ∈ Irr(G F G F ) and µ loc x ∈ Irr(NG(L) F N G (L) F ) are determined by Gallagher's theorem (see [Isa76,Corollary 6.17]) via the equalitiesχ = µ glo xχ x andψ = µ loc xψ x respectively. Because (G F A) L,χ = N G (L) F (G F A) (L,λ),χ , we may assume that x stabilises λ. Let z ∈ K such that µ glo
x =ẑG and observe that (x, z) is an element of (G F A) (L,λ),χ ⋊ K that stabilisesχ. Then, applying [BMM93, Theorem 3.2 (1)], we deduce thatλ andλ (x,z) are NG(L) Fconjugate and we may choose g ∈ NG(L) F such thatλ = (λ (x,z) ) g =λ (xg,z) . In other words (xg, z) ∈ NG(L) F (G F A) (L,λ) ⋊ K λ and thus Lemma 3.9 implies that the equalityχ =χ (xg,z) holds if and only ifψ =ψ (xg,z) . From this, we immediately deduce the equality required in (3.4).
Next, denote by ζ glo and ζ loc the scalar functions associated to P glo and P loc respectively. To conclude the proof, it remains to show that ζ glo and ζ loc coincide on C (G F A)χ (G F ) = Z(G F ). As in the proof of [Ros22d,Theorem 4.3], it is enough to show that the restrictions ofχ andψ to Z(G F ) are multiples of a common irreducible constituent. This follows from the fact that unipotent characters contain the center in their kernel. In fact, on one hand, 1 Z(G F ) is the unique irreducible constituent ofχ Z(G F ) becauseχ is unipotent. On the other hand,ψ lies aboveλ and, since Z(G F ) ≤ Z(L F ) andλ is unipotent, we deduce that 1 Z(G F ) is the unique irreducible constituent ofψ Z(G F ) . This completes the proof.
We conclude this section by verifying the remaining condition [Spä17, Remark 3.7 (iv)] and obtain the required G F -block isomorphisms of character triples for the map Ω.
Proposition 3.13. For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have
G F A χ , G F , χ ∼ G F (G F A) L,ψ , N G (L) F , ψ .
Proof. By Proposition 3.12 it is enough to check the block theoretic requirement given by [Spä17, Remark 3.7 (ii) and (iv)]. First, observe that under our assumption [CE94, Proposition 3.3 (ii)] shows that
L F = C G F (E) where E ∶= Z(L) F ℓ . In particular, N J (L) = N J (E) for every G F ≤ J ≤ G F . Furthermore,
Proof of Theorem C
Proof of Theorem C. The hypothesis of Corollary 3.2 is satisfied under our restrictions on G according to Lemma 3.6 and therefore we obtain an Aut F (G F ) (L,λ) -equivariant bijection
Ω G (L,λ) ∶ E G F , (L, λ) → Irr N G (L) F λ
that preserves the ℓ-defect of characters. Next, observe that the groupsG F A and X ∶= G F ⋊ Aut F (G F ) induce the same automorphisms on G F according to the description given in [GLS98, Section 2.5]. Then, by applying [Spä17, Theorem 5.3] and Proposition 3.13, we conclude that
X χ , G F , χ ∼ G F N X (L) ψ , N G (L) F , ψ
for every χ ∈ E(G F , (L, λ)) and where ψ ∶= Ω G (L,λ) (χ) and the proof is now complete.
Consequences of Theorem C
In this section, we collect some consequences of Theorem C. First, we extend the parametrisation obtained in Theorem C from unipotent e-Harish-Chandra series of the simple group G to pseudounipotent (see Definition 2.2) e-Harish-Chandra series of the Levi subgroups of G. More precisely, for every F -stable Levi subgroup K of G, we construct a parametrisation of the e-Harish-Chandra series associated to e-cuspidal pairs of the form (L, λ) for some (K, F )-pseudo-unipotent character λ ∈ ps K (L F ). In a second step, we construct character bijections above this parametrisation by exploiting results on isomorphisms of character triples (see Corollary 4.6). This will allow us to control the characters of e-chain stabilisers lying above pseudo-unipotent characters (see Proposition 5.6).
Parametrisation of pseudo-unipotent characters of Levi subgroups
Let K be an F -stable Levi subgroup of G and set K 0 ∶= [K, K]. Observe that since the group G is simply connected, the subgroup K 0 is also simply connected according to [MT11,Proposition 12.14]. In addition, under our assumption on the type of G, we deduce that the simple components of K 0 can only be of some of the types A, B or C.
Proposition 4.1. For every unipotent e-cuspidal pair (L 0 , λ 0 ) of (K 0 , F ) there exists a defect preserving Aut F (K F 0 ) (L 0 ,λ 0 ) -equivariant bijection
Ω K 0 (L 0 ,λ 0 ) ∶ E K F 0 , (L 0 , λ 0 ) → Irr N K 0 (L 0 ) F λ 0 such that Y ϑ , K F 0 , ϑ ∼ K F 0 N Y ϑ (L 0 ), N K 0 (L 0 ) F , Ω K 0 (L 0 ,λ 0 ) (ϑ) for every ϑ ∈ E(K F 0 , (L 0 , λ 0 )) and where Y ∶= K F 0 ⋊ Aut F (K F 0 ).
Proof. Notice that K 0 is the direct product of simple algebraic groups K 1 , . . . , K n and that the action of F permutes the simple components K i . Denote the direct product of the simple components in each F -orbit by H j for j = 1, . . . , t. The (H j , F ) are the irreducible rational components of (K, F ) and we have
K F 0 = H F 1 × ⋯ × H F t . Similarly, if we define the intersections M j ∶= L 0 ∩ H j , then we have a decomposition L F 0 = M F 1 × ⋯ × M F t .
In particular, we can write λ 0 = µ 1 × ⋯ × µ t with µ j ∈ Irr(M F j ). In this case, notice that (M j , µ j ) is a unipotent e-cuspidal pair of (H j , F ). Next, suppose that H j = H j,1 × ⋅ ⋅ ⋅ × H j,m j and observe that H F j ≃ H F m j j,1 . By the discussion at the beginning of this section we know that H j,1 is a simple, simply connected group of type A, B or C and hence it satisfies the assumptions of Theorem C. Then, via the isomorphism H F j ≃ H F m j j,1 , we obtain an
Aut F (H F j ) (M j ,µ j ) -equivarint bijection Ω H j (M j ,µ j ) ∶ E H F j , (M j , µ j ) → Irr N H j (M j ) F µ j
that preserves the defect of characters and such that
Y j,ϑ , H F j , ϑ ∼ H F j N Y j,ϑ (M j ), N H j (M j ) F , Ω H j (M j ,µ j ) (ϑ) (4.1) for every ϑ ∈ E(H F j , (M j , µ j )) and where Y j ∶= H F j ⋊ Aut F (H F j ).
Since the characters in the sets E(K F 0 , (L 0 , λ 0 )) and Irr(N K 0 (L 0 ) F λ 0 ) are direct products of characters belonging to the sets E(H F j , (M j , µ j )) and Irr(N H j (M j ) F µ j ) respectively, we obtain a bijection
Ω K 0 (L 0 ,λ 0 ) ∶ E K F 0 , (L 0 , λ 0 ) → Irr N K 0 (L 0 ) F λ 0 by setting Ω K 0 (L 0 ,λ 0 ) (ϑ 1 × ⋅ ⋅ ⋅ × ϑ t ) ∶= Ω H 1 (M 1 ,µ 1 ) (ϑ 1 ) × ⋅ ⋅ ⋅ × Ω Ht (Mt,µt) (ϑ t )
for every ϑ j ∈ E(H F j , (M j , µ j )). Finally, arguing as in the proof of [Ros22c, Proposition 6.5], we deduce that the bijection Ω K 0 (L 0 ,λ 0 ) preserves the defect of characters, is Aut F (K F 0 ) (L 0 ,λ 0 ) -equivariant, and, using (4.1), it induces the K F 0 -block isomorphisms of character triples required in the statement.
In our next result, we replace the automorphism group Y ∶= K F 0 ⋊ Aut F (K F 0 ) with the group of automorphisms of G F stabilising K, that is, X ∶= (G F ⋊ Aut F (G F )) K . To do so, we apply the so-called Butterfly theorem [Spä17, Theorem 5.3] which basically states that, for any finite group G, the notion of G-block isomorphism of character triples only depends on the automorphisms induced on G.
Corollary 4.2. Let (L 0 , λ 0 ) be a unipotent e-cuspidal pair of (K 0 , F ). The map Ω K 0 (L 0 ,λ 0 ) given by Proposition 4.1 is Aut F (G F ) K,(L 0 ,λ 0 ) -equivariant and satisfies
X ϑ , K F 0 , ϑ ∼ K F 0 N X ϑ (L 0 ), N K 0 (L 0 ) F , Ω K 0 (L 0 ,λ 0 ) (ϑ) (4.2) for every ϑ ∈ E(K F 0 , (L 0 , λ 0 )) and where X ∶= (G F ⋊ Aut F (G F )) K .
Proof. First, observe that Aut F (G F ) K is contained in Aut F (K F 0 ) because K 0 is an F -stable characteristic subgroup of K. In particular, we deduce that the map In our next result, we exploit this idea in order to lift the bijections given by Proposition 4.1 to the Levi subgroup K. Consequently, we extend the parametrisation of unipotent e-Harish-Chandra series given by Theorem C for the simple group G to a parametrisation of e-Harish-Chandra series associated to (K, F )-pseudo-unipotent characters for every F -stable Levi subgroup K of G. First, we need a preliminary lemma.
Ω K 0 (L 0 ,λ 0 ) is Aut F (G F ) K,(L 0 ,λ 0 ) - equivariant. Next,
Lemma 4.3. Let (L, λ) be a unipotent e-cuspidal pair of (K, F ) and define X ∶= Theorem 4.4. For every unipotent e-cuspidal pair (L, λ) of (K, F ) there exists a defect preserving Aut F (G F ) K,(L,λ) -equivariant bijection
(G F ⋊Aut F (G F )) K . If K F ≤ H ≤ N G (L) F and Q is an ℓ-radical subgroup of N H (L), then C X (Q) ≤ N X (L).Ω K (L,λ) ∶ E K F , (L, ps K (λ)) → Irr N K (L) F ps K (λ) such that X χ , K F , χ ∼ K F N Xχ (L), N K (L) F , Ω K (L,λ) (χ)
for every χ ∈ E(K F , (L, ps K (λ))) and where X ∶= (G F ⋊ Aut F (G F )) K .
Proof. Recall that K 0 = [K, K] and define L 0 ∶= L ∩ K 0 and λ 0 the restriction of λ to L F 0 . Observe that (L 0 , λ 0 ) is a unipotent e-cuspidal pair of (K 0 , F ). Let z ∈ Z(K * ) F * and consider a character χ belonging to E(K F , (L, λẑ L )). Since the restriction of λẑ L to L F 0 coincides with λ 0 , [GM20, Corollary 3.3.25] implies that χ lies above some character in E(K F 0 (L 0 , λ 0 )). On the other hand, suppose that χ ∈ Irr(K F ) lies above some χ 0 ∈ E(K F 0 , (L 0 , λ 0 )). By [CE94, Proposition 3.1] the character χ 0 has an extension χ ′ ∈ E(K F , (L, λ)) and hence, using Gallagher's theorem [Isa76,Corollary 6.17] and [CE04,(8.19)], we can find z ∈ Z(K * ) F * such that χ = χ ′ẑ K . Since χ ′ẑ K is a character of E(K F , (L, λẑ L )) according to [CE04,(8.20)], we conclude that
E K F , (L, ps K (λ)) = Irr K F E K F 0 , (L 0 , λ 0 ) . (4.3)
Next, suppose that ψ ∈ Irr(N K (L) F λẑ L ). In this case, ψ lies above the restriction of λẑ L to L F 0 which coincides with λ 0 . In particular, there exists some ϕ ∈ Irr(N K 0 (L 0 ) F λ 0 ) such that ψ lies above ϕ. On the other, if χ lies above such a character ϕ ∈ Irr(N K 0 (L 0 ) F λ 0 ), then it lies above λ 0 and therefore we can find z ∈ Z(K * ) F * such that ψ ∈ Irr(N K (L) F λẑ L ). This shows that
Irr N K (L) F ps K (λ) = Irr N K (L) F Irr N K 0 (L 0 ) F λ 0 . (4.4)
Finally, consider the map Ω K 0 (L 0 ,λ 0 ) given by Proposition 4.1. Then, the result follows from (4.3) and (4.4) by applying [Ros22c, Proposition 6.1 and Remark 6.2] as explained in the proof of [Ros22c, Corollary 6.10] and using the K F -block isomorphisms of character triples obtained in Corollary 4.2.
Here, we consider A ∶= G F ⋊ Aut F (G F ), A 0 ∶= N A (L), K ∶= K F 0 , K 0 = N K 0 (L) F = N K 0 (L 0 ) F , G ∶= G F , X ∶= (G F ⋊ Aut F (G F )) K , S ∶= E(K F 0 , (L 0 , λ 0 )), S 0 ∶= Irr(N K 0 (L 0 ) F λ 0 ), V ∶= (G F ⋊ Aut F (G F )) K,S and U ∶= (G F ⋊ Aut F (G F )) K,L,Y 0 .
Observe that the condition on defect groups required by [Ros22c, Proposition 6.1] is satisfied by Lemma 4.3.
Above e-Harish-Chandra series
We now further extend Theorem C by lifting the character bijections from Theorem 4.4 with respect to normal inclusions. for every χ ∈ Irr(H E(K F , (L, ps K (λ)))) and where X ∶= (G F ⋊ Aut F (G F )) K .
Proof. We apply [Ros22c,Proposition 6.1] to the bijection given by Theorem 4.4. We consider A ∶= Before proceeding further, we point out an interesting analogy with another important character correspondence. The Glauberman correspondence plays a fundamental role in the study of the localglobal counting conjectures and lies at the heart of most reduction theorems. In its most basic form, it states that for every finite ℓ-group L acting on a finite ℓ ′ -group K, there exists a bijection
G F ⋊ Aut F (G F ), G ∶= G F , K ∶= K F , A 0 ∶= N A (L), X ∶= N A (K), S ∶= E(K F , (L, ps K (λ))), S 0 ∶= Irr(N K (L) F ps K (λ)), U ∶= X 0,λ , Vf L ∶ Irr L (K) → Irr(N K (L))
between the set of L-invariant characters of K and the characters of the normaliser N K (L) (see, for instance, [Nav18, Section 2.3]). A very deep result due to Dade [Dad80] and recently reproved by Turull [Tur08], shows that, if K and L are subgroups of a finite group G and KP ≤ H ≤ KN G (L), then the Glauberman correspondence f L can be lifted to a character correspondence for H, that is, there exists a bijection
f H L ∶ Irr (H χ) → Irr (N H (L) f L (χ)) (4.5)
for every χ ∈ Irr L (K). On the other hand, the parametrisation of unipotent e-Harish-Chandra series obtained by Broué, Malle and Michel [BMM93, Theorem 3.2] lies at the centre of the proofs of the local-global counting conjectures for finite reductive groups. It is interesting to note that our methods yield a character bijection above e-Harish-Chandra series which is analogous to (4.5) in the context of the Glauberman correspondence. This is an immediate consequence of Proposition 4.5. for every χ ∈ E(K F , (L, ps K (λ))).
Proof. This follows immediately from the proof of Proposition 4.5 by following the construction made in [Ros22c, Proposition 6.1].
Towards Theorem A and Theorem B
Finally, we apply the results obtained in the previous sections to prove Theorem A which is our main result. Then, we obtain Theorem B as a corollary by applying the e-Harish-Chandra theory for unipotent characters developed by Broué, Malle and Michel [BMM93] and by Cabanes and Enguehard [CE94]. Before doing so, we introduce the relevant notation and prove some preliminary results.
Preliminaries on e-chains
Our first aim is to define e-local structures for finite reductive groups that play a role analogue to that of ℓ-chains in the context of Dade's Conjecture and the Character Triple Conjecture. The connection between the set of e-chains and that of ℓ-chains has already been studied in [Ros22c, Section 7.2]. These results provide a way to obtain Dade's Conjecture and the Character Triple Conjecture as a consequence of [Ros22c, Conjecture C and Conjecture D]. The possibility to use different types of chains is crucial in the study of Dade's Conjecture and has been introduced by Knörr and Robinson [KR89]. Their results were insipred by previous studies conducted by many authors including Brown [Bro75] and Quillen [Qui78] who analised the homotopy theory of associated simplicial complexes.
Definition 5.1. We denote by L e (G, F ) the set of e-chains of the finite reductive group (G, F ), that is, chains of the form
σ = {G = L 0 > L 1 > ⋅ ⋅ ⋅ > L n }
where n is a non-negative integer and each L i is an e-split Levi subgroup of (G, F ). We denote by σ ∶= n the length of the e-chain σ and by L(σ) its last term. Furthermore, we define L e (G, F ) >0 to be the set of e-chains having length strictly larger than 0.
Observe that the notion of length defined above, induces a partition of the set L e (G, F ) into echains of even and odd length. More precisely, we denote by L e (G, F ) ± the subset of those e-chains σ ∈ L e (G, F ) that satisfy (−1) σ = ±1.
In what follows, given an e-chain σ and an e-split Levi subgroup M of (L(σ), F ), we denote by σ+M the e-chain obtained by adding M at the end of σ. We also allow the possibility that M = L(σ), in which case we have σ + L(σ) = σ. Vice versa, we denote by σ − L(σ) the e-chain obtained by removing the last term L(σ) from σ. In this way we obtain (σ + M) − L(σ + M) = σ where as usual L(σ + M) denotes the final term of the e-chain σ + M. Here, we use the convention that σ 0 − L(σ 0 ) = σ 0 = σ 0 + G where σ 0 = {G} is the trivial e-chain.
Next, consider the action of G F on the set of e-chains L e (G, F ) induced by conjugation: for every g ∈ G F and σ = {L i } i , we define
σ g ∶= G = L 0 > L g 1 > ⋅ ⋅ ⋅ > L g n .
It follows from this definition that the stabiliser G F σ coincides with the intersection of the normalisers N G (L i ) F for i = 1, . . . , n. Similarly, we can define an action of Aut F (G F ) on L e (G, F ) and give an analogous description of the chains stabilisers Aut F (G F ) σ . In particular, notice that the last term of the chain satisfies L(σ) F ⊴ G F σ . Using this observation, we can use the results of Section 4.2 to control the characters of G F σ that lie above pseudo-unipotent series of L(σ).
Definition 5.2. For every e-chain σ ∈ L e (G, F ) we denote by CP u (σ) the set of unipotent ecuspidal pairs (M, µ) ∈ CP u (L(σ), F ) that satisfy M < G. Furthermore, for any such pair (M, µ) ∈ CP u (σ), we define the character set
Uch(G F σ , (M, µ)) ∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Irr G F σ E L(σ) F , M, ps L(σ) (µ) L(σ) > M (5.1) Irr G F σ E L(σ) F , M, ps L(σ−L(σ)) (µ) L(σ) = M (5.2)
The need to distinguish the cases (5.1) and (5.2) will become apparent in the proofs of Proposition 5.6 and Theorem 5.9 below. Observe that in the definition above, we are excluding the degenerate case where G = L(σ) = M and therefore the chain σ − L(σ) in the case (5.2) is always defined. To understand the reason why we are excluding this case, we can consider an analogy with Dade's Conjecture. For every finite group G, recall that k(G) denotes the number of its irreducible characters and that, for any non-negative integer d, the symbol k d (G) denotes the number of those irreducible characters of ℓ-defect d. The local-global counting conjectures provide a way to determine the global invariants k d (G) in terms of ℓ-local structures. This idea was made precise by Isaacs and Navarro [IN20]. According to their definitions, the block-free version of Dade's Conjecture can be stated by saying that the functions k d are chain local for every d > 0. Consequently, and because a sum of chain local functions is chain local, we deduce that the difference k − k 0 = ∑ d>0 k d is a chain local function. On the other hand, using the fact that groups admitting a character of ℓ-defect zero have trivial ℓ-core, it is easy to see that k 0 is not chain local. The exclusion of the case G = L(σ) = M can be explained by interpreting these observations in the context of unipotent characters. Recall that k u (G F ) and k c,u (G F ) denote the number of unipotent characters of G F and unipotent e-cuspidal characters of G F respectively. If ℓ does not divide the order of Z(G F ), then [CE94] implies that the unipotent e-cuspidal characters of G F have defect zero. Therefore, as in the case of Dade's Conjecture, the global invariant we want to determine e-locally is the difference
k u (G F ) − k c,u (G F ).
Finally, notice that k c,u (G F ) is exactly the number of unipotent e-cuspidal pairs (M, µ) of L(σ) satisfying G = L(σ) = M.
In the following lemma, we show that if the set Uch(G F σ , (M, µ)) is non-empty then (M, µ) is uniquely defined up to G F σ -conjugation.
Lemma 5.3. Let σ ∈ L e (G, F ) and consider two unipotent e-cuspidal pairs (M, µ) and (K, κ) in CP u (σ). If the sets Uch(G F σ , (M, µ)) and Uch(G F σ , (K, κ)) have non-trivial intersection, then (M, µ) and (K, κ) are G F σ -conjugate.
Proof. Suppose that ϑ is a character belonging to Uch(G F σ , (M, µ)) and Uch(G F σ , (K, κ)). If we set L ∶= L(σ), then we can find elements s, t ∈ Z(L * ) F * and characters ϕ ∈ E(L F , (M, µ)) and ψ ∈ E(L F , (K, κ)) such that ϑ lies above ϕŝ L and ψt L . By Clifford's theorem, we deduce that ϕŝ L = (ψt L ) g for some g ∈ G F σ . Furthermore, sinceŝ is a linear character, we obtain that ϕ = ψ g (t L ) g (ŝ L ) −1 . Since both ϕ and ψ g are unipotent characters of L F , using [CE04, Proposition 8.26] we deduce that (t L ) g (ŝ L ) −1 = 1 L and therefore ϕ = ψ g . But then, [BMM93, Theorem 3.2(1)] shows that (M, µ) and (K, κ) g are L F -conjugate and the result follows.
Next, we describe the block theory associated to characters in the sets introduced in Definition 5.2.
Lemma 5.4. Let σ ∈ L e (G, F ) and consider a unipotent e-cuspidal pair (M, µ) ∈ CP u (σ) and a character ϑ ∈ Uch(G F σ , (M, µ)). Then: (i) the block bl(ϑ) is L(σ) F -regular;
(ii) if the character ϑ lies above a given ϕẑ L(σ) ∈ E(L(σ) F , (M, µẑ M )) for some z ∈ Z(L(σ) * ) F * , then we have bl(ϕẑ L(σ) ) = bl(µẑ M ) L(σ) F and bl(ϑ) = bl(ϕẑ L(σ) ) G F σ = bl(µẑ M ) G F σ (iii) the induced block bl(ϑ) G F is defined.
Proof. The first point follows from Lemma 2.3 by choosing L = L(σ) and H = G F σ . Furthermore, in the case of (5.2) observe that L(σ) ≤ L(σ − L(σ)) and hence Z(L(σ − L(σ)) * ) ≤ Z(L(σ) * ). Therefore, we can always find ϕ and z as in the statement of (ii). Since ϕ is an irreducible constituent of the virtual character R .14] implies that bl(µẑ M ) G F is well defined and so is bl(ϑ) G F by (ii) and transitivity of block induction. This concludes the proof.
Using the lemma above, we can now define the following character set. This yields the e-local object through which we can determine the number of unipotent characters in a given block of B of G F and with a given defect d ≥ 0 (see Section 5.3).
Definition 5.5. Let B be a block of G F and d a non-negative integer. For every e-chain σ ∈ L e (G, F ) and unipotent e-cuspidal pair (L, λ) ∈ CP u (σ) we define the character set
Uch d (B σ , (M, µ)) ∶= ϑ ∈ Uch G F σ , (M, µ) d(ϑ) = d, bl(ϑ) G F = B .
where bl(ϑ) G F is defined according to Lemma 5.4 (iii). Furthermore, we denote the cardinality of this set by k d u (B σ , (M, µ)) ∶= Uch d (B σ , (M, µ)) .
To conclude this section, we show that Proposition 4.5 can be used to parametrise the character sets from Definition 5.5. such that
X σ,ϑ , G F σ , ϑ ∼ G F σ X σ+M,ϑ , G F σ+M , Ω B,d σ,(M,µ) (ϑ)
for every ϑ ∈ Uch d (B σ , (M, µ)) and where X ∶= G F ⋊ Aut F (G F ).
Proof. First, observe that if M coincides with the last term L(σ) of the chain σ, then we have σ + M = σ which implies Uch d (B σ , (M, µ)) = Uch d (B σ+M , (M, µ)). In this case the result holds by defining Ω B,d σ,(M,µ) as the identity. Therefore, we can assume that M < L(σ) and define ρ ∶= σ+M. Now, according to (5.1) we have
Uch G F σ , (M, µ) = Irr G F σ E L(σ) F , (M, ps L(σ) (µ)) . (5.3)
On the other hand, noticing that M coincides with the last term L(ρ) of the chain ρ and that ρ−L(ρ) = σ, we obtain the equality E(L(ρ) F , (M, ps L(ρ−L(ρ)) (µ))) = ps L(σ) (µ). Then, observing that G F ρ = N G F σ (M), we can apply (5.2) to obtain the equality
Uch G F ρ , (M, µ) = Irr N G F σ (M) ps L(σ) (µ)) . (5.4)
Next, we apply Proposition 4.5 by choosing the groups in that statement to be H = G F σ , K = L(σ) and (L, λ) = (M, µ). By (5.3) and (5.4), we deduce that there exists an Aut F (G F ) σ,(M,µ)equivariant bijection
Ω L(σ),G F σ (M,µ) ∶ Uch(G F σ , (M, µ)) → Uch(G F ρ , (M, µ)). (5.5)
Moreover, using the H-block isomorphisms given by Proposition 4.5 together with [Spä17, Lemma 3.8 (b)], we deduce that
X σ,ϑ , G F σ , ϑ ∼ G F σ X ρ,ϑ , G F ρ , Ω L(σ),G F σ (M,µ) (ϑ) (5.6)
for every ϑ ∈ Uch d (G F σ , (M, µ)). To conclude, observe first that Ω
L(σ),G F σ (M,µ)
sends characters of defect d to characters of defect d. Moreover, by the transitivity of block induction and using (5.6), we deduce that
bl(ϑ) G F = bl Ω L(σ),G F σ (M,µ) (ϑ) G F .
This shows that the bijection from (5.5) sends characters in the set Uch d (B σ , (M, µ)) to characters in the set Uch d (B σ+M , (M, µ)) and therefore it restricts to a bijection, denoted by Ω B,d σ,(M,µ) , satisfying the properties required in the statement. This completes the proof.
We conclude this section with a remark on the isomorphisms of character triples obtained in Proposition 5.6.
Remark 5.7. Suppose that ℓ does not divide q ± 1 if G is of type A(±q). In this case, every e-split Levi subgroup L of G satisfies L = C ○ G (Z(L) F ℓ ) according to [CE04,Proposition 13.19]. This fact can be used to show that the G F σ -block isomorphisms of character triples given by Proposition 5.6 can be extended to G F -block isomorphisms of character triples. First, we claim that C G F X σ,ϑ (D) ≤ X σ,ϑ (5.7)
for every irreducible character ϑ of G F σ and every ℓ-radical subgroup D of G F σ+M . Define Q i ∶= Z ○ (L i ) F ℓ for every e-split Levi subgroup L i appearing in the chain σ. Then, using the fact that D is ℓ-radical, we obtain the inclusions Q i ≤ O ℓ (G F σ ) ≤ D. Therefore, every element x ∈ G F X σ,ϑ that centralises D centralises also each Q i and hence normalises each L i . It follows that C G F X σ,ϑ (D) ≤ (G F X σ,ϑ ) σ = X σ,ϑ as required by (5.7). We can now apply [Ros22a, Lemma 2.11] to the G F σ -block isomorphisms given by Proposition 5.6 to show that X σ,ϑ , G F σ , ϑ ∼ G F X σ+M,ϑ , G F σ+M , Ω B,d σ,(M,µ) (ϑ)
for every ϑ ∈ Uch d (B σ , (M, µ)).
Proof of Theorem A
We are finally ready to prove our main theorem which provides a bijection for unipotent characters in the spirit of the Character Triple Conjecture [Spä17, Conjecture 6.3]. In this section, we prove a slightly stronger result that provides further information on the type of e-chains and isomorphisms of character triples. In the following definition we introduce the analogue of the set C d (B) ± considered in the Character Triple Conjecture as defined in [Spä17, p. 1097].
Definition 5.8. Let B be a block of G F and consider a non-negative integer d. We define the set L d u (B) ± = (σ, M, µ, ϑ) σ ∈ L e (G, F ) ± , (M, µ) ∈ CP u (σ), ϑ ∈ Uch d (B σ , (M, µ)) .
The conjugacy action of G F induces an action of G F on L d u (B) ± defined by (σ, M, µ, ϑ) g ∶= (σ g , M g , µ g , ϑ g ) for every element g ∈ G F and (σ, M, µ, ϑ) ∈ L d u (B) ± . We denote by L d u (B) ± G F the corresponding set of G F -orbits of tuples. Moreover, for every such orbit ω, we denote by ω • the corresponding G F -orbit of pairs (σ, ϑ) such that (σ, M, µ, ϑ) ∈ ω for some (M, µ) ∈ CP u (σ). In other words, if we indicate by (σ, M, µ, ϑ) the G F -orbit of (σ, M, µ, ϑ), then (σ, M, µ, ϑ)
• is the G F -orbit of the pairs (σ g , ϑ g ).
In a similar way, if Aut F (G F ) B denotes the set of those automorphisms α ∈ Aut F (G F ) that stabilise B, then we can define (σ, M, µ, ϑ) α ∶= (σ α , M α , µ α , ϑ α ) for every α ∈ Aut F (G F ) B and (σ, M, µ, ϑ) ∈ L d u (B). In this way, we obtain an action of the group Aut F (G F ) B on the set L d u (B) ± and on the corresponding set of orbits L d u (B) ± G F .
Theorem 5.9. For every block B of G F and every non-negative integer d, there exists an Aut F (G F ) Bequivariant bijection Λ ∶ L d u (B) + G F → L d u (B) − G F . Moreover, for every ω ∈ L d u (B) + G F , any (σ, ϑ) ∈ ω • and any (ρ, χ) ∈ Λ(ω) • we have σ = ρ ± 1 and X σ,ϑ , G F σ , ϑ ∼ J X ρ,χ , G F ρ , χ with J = G F σ , if σ = ρ − 1, or J = G F ρ , if σ = ρ + 1, and where X ∶= G F ⋊ Aut F (G F ).
Proof. Define A ∶= Aut F (G F ) and observe that X = G F ⋊ A. In a first step, we construct an equivariant bijection between triples of the form (σ, M, µ). More precisely, let S denote the set of such triples (σ, M, µ) with σ ∈ L e (G, F ) and (M, µ) ∈ CP u (σ). We define a map ∆ ∶ S → S Observe that the chain σ −M is always defined since M < G by the definition of CP u (σ). Moreover, it is clear from the definition above that the map ∆ is A-equivariant and satisfies ∆ 2 = Id. Therefore, observing that σ ± M = σ ± 1, we conclude that ∆ restricts to an A-equivariant bijection
∆ ∶ S + → S −
where S ± denotes the set of those triples (σ, M, µ) of S that satisfy σ ∈ L e (G, F ) ± . Furthermore, notice once again that if ∆((σ, M, µ)) = (ρ, K, κ), then σ = ρ ± 1. such that X σ,ϑ , G F σ , ϑ ∼ G F σ X ρ,χ , G F ρ , χ (5.9) for every ϑ ∈ Uch d (B σ , (M, µ)) and where χ is the image of ϑ. Consequently, if U where the last equality holds by [BMM93, Theorem 3.2 (1)] and recalling that every pair (M, µ) ∈ CP u (σ 0 ) satisfies M < G = L(σ 0 ). Next, Theorem 5.9 implies that the sets L d u (B) + G F and L d u (B) − G F have the same cardinality and therefore we conclude from (5.11) and (5.12) that
k d u (B) − k d c,u (B) + σ∈L+ σ≠σ 0 k d u (B σ ) = σ∈L+ k d u (B σ ) = σ∈L− k d u (B σ ). (5.13)
Finally, noticing that (−1) σ +1 = ∓1 for every σ ∈ L ± , we can rewrite (5.13) as
k d u (B) − k d c,u (B) = σ∈L−∪L+ (−1) σ +1 k d u (B σ )
which is exactly the equality in the statement of Theorem B.
on e-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.3 Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Furthermore, if H ≤ G and b is a block of H, then b G denotes the block of G obtained via Brauer's induction (when it is defined). If B is a block of G and d ≥ 0, then let Irr d (B) be the set of irreducible characters belonging to the block B and having defect d. The cardinality of Irr d (B) is denoted by k d (B).
Unipotent characters where parametrised by Broué, Malle and Michel [BMM93, Theorem 3.2] by using e-Harish-Chandra theory. Their description can be divided into two parts. First, each unipotent character lies in a unique e-Harish-Chandra series, that is,
Now, let ℓ be a prime number not dividing q and denote by e the order of q modulo ℓ or q modulo 4 if ℓ = 2. In order to control the action of automorphism on unipotent e-Harish-Chandra series, we exploit a result ofCabanes and Späth. More precisely, in [CS13, Theorem 3.4] it was shown that the parametrisation given by Broué, Malle and Michel in [BMM93, Theorem 3.2 (2)] commutes with the action of those automorphisms in the set Aut F (G F ). Notice that the statement of [CS13, Theorem 3.4] only considers unipotent e-cuspidal pairs (L, λ) where L is a minimal e-split Levi subgroups (which is enough for the purpose of dealing with the McKay Conjecture). However, their proof works for the general case as well.Proposition 3.1. For every unipotent e-cuspidal pair (
for every block C 0 of N J (L) and every defect group D of C 0 we have E ≤ O ℓ (N J (L)) ≤ D and hence CG F (D) ≤ NG(L) F . Now, [KS15, Theorem B] implies that for every block C of NG(L) F covering C 0 , the induced blocks B ∶= CG F and B 0 ∶= C J 0 are well-defined and B covers B 0 . Letχ ∈G be an extension of χ and setψ ∶=Ω(χ). By Lemma 2.3 the block ofC ofψ coincides with the induced block bl(λ) NG(L) F . Furthermore, by [CE94, Proposition 4.2] we know that the blockB ofχ coincides with bG F (L,λ) = bl(λ)G F . Then, by the transitivity of block induction we getB =CG F . Consider now G F ≤ J ≤G F as in the previous paragraph and notice that bl(χ J )is the unique block of J covered byB. Now, since bl(ψ N J (L) ) is covered byC, we deduce that bl(ψ N J (L) ) J is covered byB and thereforebl (χ J ) = bl ψ N J (L) J . (3.5)As explained in the proof of [Ros22d, Theorem 4.8] we can now use (3.5) together with Proposition 3.12 to conclude the proof via an application of [Spä17, Theorem 4.1 (i)].
to obtain (4.2), we apply [Spä17, Lemma 3.8 and Theorem 5.3] to the isomorphism of character triples given by Proposition 4.1 as explained in the proof of [Ros22c, Corollary 6.8]. Isomorphisms of character triples play a fundamental role in representation theory of finite groups and in the study of the local-global conjectures. One of the most important consequences of the existence of isomorphisms of character triples is the possibility to lift character bijections. For instance, the main result of [NS14], shows how to apply this technique to construct bijections above characters of height zero in the context of the Alperin-McKay Conjecture [NS14, Theorem B]. The main consequence of this result, which follows from an argument introduced by Murai [Mur12], is a reduction theorem for the celebrated Brauer's Height Zero Conjecture [NS14, Theorem A]. This strategy ultimately lead to the solution of Brauer's conjecture[Ruh22a] and[MNSFT22]. For other applications of isomorphisms of character triples see[Tur17],[NSV20],[Ros22a],[Ruh22b] [Ros23] and [MR22, Proposition 1.1].
Proof. Let E ∶= Z(L) F ℓ and observe that L = C ○ G (E) according to [CE94, Proposition 3.3 (ii)]. Now, since O ℓ (N H (L)) is the smallest ℓ-radical subgroup of N H (L) [Dad92, Proposition 1.4], we deduce that E ≤ O ℓ (N H (L)) ≤ Q and it follows that C X (Q) ≤ C X (E) ≤ N X (L) as wanted.
Proposition 4 . 5 .
45Consider the setup of Theorem 4.4 and let K F ≤ H ≤ N G (K) F . Then, there exists a defect preserving Aut F (G F ) H,K,(L,λ) -equivariant bijection Ω K,H (L,λ) ∶ Irr H E K F , (L, ps K (λ)) → Irr (N H (L) ps K (λ)) such that (N X (H) χ , H, χ) ∼ H (N X (H, L) χ , N H (L), ψ)
Corollary 4. 6 .
6Consider the setup of Theorem 4.4 and let K F ≤ H ≤ N G (K) F . Then, there exists a bijection Ψ H χ ∶ Irr (H χ) → Irr N H (L) Ω K (L,λ) (χ)
M
(µ), it follows from [CE94, Proposition 4.2] (whose assumptions are satisfied by [CE94, Proposition 3.3 (ii)]) that bl(ϕ) = b L(σ) F (M, µ) = bl(µ) L(σ) F . Then, sinceẑ M is the restriction of the linear characterẑ L(σ) to M F , we deduce from Lemma 2.1 that bl(ϕẑ L(σ) ) = bl(µẑ M ) L(σ) F . Now, [Nav98, Theorem 9.19] implies that bl(ϑ) = bl ϕẑ L(second point follows by the transitivity of block induction. Finally, set Q ∶= Z(M) F ℓ and observe that QC G F (Q) = M F ≤ N G F (Q) by [CE94, Proposition 3.3(ii)]. Then, [Nav98, Theorem 4
Proposition 5. 6 .
6Let B be a block of G and d a non-negative integer. If σ ∈ L e (G, F ) and (M, µ) is a unipotent e-cuspidal pair in CP u (σ) then there exists an Aut F (G F ) B,σ,(M,µ) -equivariant bijection Ω B,d σ,(M,µ) ∶ Uch d (B σ , (M, µ)) → Uch d (B σ+M , (M, µ))
+ M, M, µ) , L(σ) > M (σ − M, M, µ) , L(σ) = M.
fix an A B -transversal T + in S + and observe that the image of T + under the map ∆, denoted by T − , is an A B -transversal in S because of the equivariance property of ∆. Consider (σ, M, µ) ∈ T + and write ∆((σ, M, µ)) = (ρ, M, µ). In what follows, we may assume without loss of generality that L(σ) > M and that ρ = σ + M, otherwise we repeat the arguments verbatim by replacing (σ, M, µ) with (ρ, M, µ). By Proposition 5.6 we obtain an A B,σ,(M,µ) -equivariant bijection Ω B,d σ,(M,µ) ∶ Uch d (B σ , (M, µ)) → Uch d (B ρ , (M, µ))
A B,(σ,M,µ) -transversal in the character set Uch d (B σ , (M, µ)), then its image, denoted by U (ρ,M,µ) − , under the bijection above is an A B,(ρ,M,µ) -transversal in the character set Uch d (B ρ , (M, µ)) because A B,(σ,M,µ) = A B,(ρ,M,µ) . Now, by the discussion in the previous paragraph and using Lemma 5.3, we conclude that the sets of G F -orbits L + ∶= (σ, M, µ, ϑ) (σ, M, µ) ∈ T + , ϑ ∈ U (σ,M,µ) + and L − ∶= (ρ, M, µ, χ) (ρ, M, µ) ∈ T − , χ ∈ U (ρ,M,µ) − are A B -transversals in the sets L d u (B) + G F and L d u (B) − G F respectively. Finally, we can define the bijection Λ by setting Λ (σ, M, µ, ϑ) x ∶= (ρ, M, µ, χ) x is trivial under our assumptions. Consequently, Irr d (B) ∩ E(G F , (M, µ)) = k d u (B) − k d c,u (B)
Proof. If G is of type B or C then the result follows from [Isa76, Corollary 11.22] since A is cyclic. Then, we can assume that G is of type A in which case the result follows from [CS17, Theorem 4.1]
(see also [Mal08, Theorem 2.4]).
∶= X S and J ∶= H.Notice that the conditions (i)-(iii) of
[Ros22c, Proposition 6.1] are satisfied by [BMM93, Theorem 3.2 (1)]. Furthermore, the requirements
about defect groups are satisfied by Lemma 4.3. Therefore, as explained in [Ros22c, Proposition
6.11], we obtain the claimed result by applying [Ros22c, Proposition 6.1 and Remark 6.2].
Using (5.8) and (5.9) together with the definition of Λ, we conclude that the properties required in the statement are satisfied and the proof is now complete. Now, as a consequence of Theorem 5.9 and Remark 5.7, we can finally prove Theorem A.Proof of Theorem A. Assume that ℓ does not divide q ± 1 whenever (G, F ) is of type A(±q). Consider the bijection Λ from Theorem 5.9 and chose ω ∈ L dIn either cases, applying Remark 5.7, we deduce thatProof of Theorem BOur final goal is to obtain a counting argument for unipotent characters as a consequence of Theorem 5.9. Recall that Dade's Conjecture provides a way to determine the number of characters in a given ℓ-block B and with a given defect d in terms of ℓ-local structures. Theorem B provides an adaptation of this idea to the unipotent characters of finite reductive groups by means of e-local structures compatible with e-Harish-Chandra theory (see Definition 5.5). For every σ ∈ L e (G, F ) we definewhere (M, µ) runs over a set of representatives for the action of G F σ on CP u (σ). Moreover, recall that k d u (B) and k d c,u (B) denote the number of irreducible characters belonging to the block B and with defect d that are unipotent and unipotent e-cuspidal respectively.Proof of Theorem B. To start, we determine the cardinality of the sets of G F -orbits L d u (B) ± G F . By applying Lemma 5.3, we obtainwhere σ runs over a set of representatives, say L ± , for the action of G F on L e (G, F ) ± and (M, µ) runs over a set of representatives for the action of G F σ on CP u (σ). Next, we isolate the contribution given by the trivial chain σ 0 ∶= {G} ∈ L e (G, F ) + to the sum in (5.11). In this case, we have L(σ 0 ) = G and hence ps L(σ) (µ) = {µ} for every (M, µ) ∈ CP u (σ 0 ) because the center Z(G * ) F *
. M , ) , ) ∈ L − Satisfying, every x ∈ A B and every (σ, M, µ, ϑ) ∈ L + and (ρ, M, µ, χ) ∈ L − satisfying ∆(σ, M, µ) =
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. Dipartimento Di Matematica E Informatica U, A Dini, Firenze , [email protected] DI MATEMATICA E INFORMATICA U. DINI, VIALE MORGAGNI 67 A, FIRENZE, ITALY Email address: [email protected]
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arxiv |
THE HEISENBERG PRODUCT: FROM HOPF ALGEBRAS AND SPECIES TO SYMMETRIC FUNCTIONS
28 Apr 2015
Marcelo Aguiar
Walter Ferrer Santos
Walter Moreira
THE HEISENBERG PRODUCT: FROM HOPF ALGEBRAS AND SPECIES TO SYMMETRIC FUNCTIONS
28 Apr 2015arXiv:1504.06315v2 [math.RA]
Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded-product the Heisenberg product, that for the highest and lowest degrees produces the classical external and internal products (and their namesakes in different contexts). In order to define it, we start from the two opposite more general extremes: species in the "commutative context", and endomorphisms of Hopf algebras in the "non-commutative" environment. Both specialize to the space of commutative symmetric functions where the definitions coincide. We also deal with the different coproducts that these objects carry -to which we add the Heisenberg coproduct for quasi-symmetric functions-, and study their Hopf algebra compatibility particularly for symmetric and non commutative symmetric functions. We obtain combinatorial formulas for the structure constants of the new product that extend, generalize and unify results due to Garsia, Remmel, Reutenauer and Solomon. In the space of quasisymmetric functions, we describe explicitly the new operations in terms of alphabets. 7 2.2. The generating function of the Heisenberg product of two species 9 3. The Heisenberg product of symmetric functions 10 3.
1. Introduction 1.1. General description of the paper. The diagram in Figure 1 displays the array of spaces we consider in this work and the table in Figure 2 displays the variety of products that had been therein defined and given due consideration by many authors.
In all these levels we define a new product, that we call the Heisenberg product that is not graded and that at the smallest and larger degrees produce respectively the internal and external products (or their namesakes as depicted in Figure 2). This is what we call the i nterpolation property of the Heisenberg product.
The name we choose for the new product, comes from the one that it usually carries at the level of Hopf algebras. Observe that we divided the spaces into three groups, marked with different kinds of boxes in the diagram. In the first group, marked with square boxes, we start by constructing the Heisenberg product in the category of species -or equivalently of representations of the symmetric groups, and we translate it to the space of symmetric functions. We use the Grothendieck group functor and the Frobenius characteristic map to move from categories to symmetric functions. This work is covered in Sections 2 and 3. In the space of symmetric functions we give an explicit combinatorial formula for the Heisenberg product in the linear basis of the complete homogeneous symmetric functions. Later this formula will be the main tool used to relate the commutative with the non commutative environment in Section 8. We also find an explicit formula for the structure constants of the Heisenberg product in the basis of power sums. The existence of these simple formulae poses the problem of finding an explicit description for the structure constants on the basis of Schur functions. The answer would contain as extreme cases the Littlewood-Richardson rule and (a still unknown) rule for the Kronecker coefficients.
In the second group, marked with oval boxes, we start by considering the Heisenberg product in the space of endomorphisms of a graded Hopf algebra in Section 4. This product is well known in the theory of Hopf algebras. In Section 6 we restrict it, via Schur-Weyl duality, to the vector space linearly generated by permutations, on one side, and on the other side to what we define as the space of descents (or Garsia-Reutenauer) endomorfisms of a Hopf algebra in Section 5. These two constructions are confluent when we restric both to the space of non-commutative symmetric functions in Section 7. Moreover, in this Section we present two different proofs of the restriction of the Heisenberg product to the space of descents (non commutative symmetric functions) -and suggest two others, relating to work of Brown, Mahajan, Schocker and others-. The first proof follows along the line of results of Garsia and Reutenauer characterizing the descents via Schur-Weyl duality. In this perspective, the invariance of the space of descents within the space of permutations with respect to the Heisenberg product, becomes a result in Hopf algebra theory. The second proof that has a combinatorial content, is based upon an explicit calculation of the Heisenberg product of elements of the cannonical basis X α , where α is a composition. By the interpolation property of the Heisenberg product, the combinatorial formula thus obtained, contains as special cases rules for the product in Solomon's descent algebra and for the external product of two basis elements of Σ. One readily verifies that the former is precisely the well-known rule of Garsia, Remmel, Reutenauer, and Solomon, and the latter is just the usual formula for the external product of the basic elements of the space of descents. In that manner, the apparently disconnected combinatorial descriptions of two different products -the Solomon and the external product-, become connected by the introduction of the new Heisenberg product.
The square boxes and the oval boxes have as common ground the bold square box that contains the space of symmetric functions. We show in Section 8 that via the usual projection of non-commutative symmetric functions onto symmetric functions the two definitions of Heisenberg product coincide in the latter space . This -together with the formal similarity of the constructions-justifies the use of the same name in both contexts.
Moreover, the spaces of non-commutative and commutative symmetric functions carry a well-known coalgebra structure. We show that this structure is compatible with the Heisenberg product, producing in both cases related Hopf algebras. This appears in Section 9 and 10.
Finally, the remaining vector space appearing in Table 1, is the space of quasisymmetric functions, which is dual to the space of non-commutative symmetric functions. We construct by means of this duality, the Heisenberg coproduct in quasi-symmetric functions in Section 11. In this situation, this coproduct together with the usual product, endow the quasi-symmetric functions with a Hopf algebra structure.
In the Appendix, we present the proofs of three technical lemmas, that in order to spare the reader of some nonessential distractions, were omitted in the places that the statements appeared.
Terminology and general notations.
In all the spaces we consider there are at least two well-known products. Although they are closely related by the inclusions, projections, and isomorphisms in Figure 1, mathematical developments have given them non consistent names in many cases. Figure 2 shows the more standardt nomenclature. The new product we introduce in this article, will be called in all the different contexts, the Heisenberg product.
We work over a field of characteristic zero, that will be denoted as k. When dealing with results in Hopf algebra theory, we adopt the usual notations in the area as presented for example in [17]. We adopt Sweedler's convention for the comultiplication and write ∆(h) = h 1 ⊗ h 2 , moreover S in general will be the antipode. An element of the Hopf algebra is said to be primitive if ∆(h) = h ⊗ 1 + 1 ⊗ h. The space of primitive elements is denoted as Prim(H).
The group of the permutations of n elements will be denoted as S n , [n] := {1, · · · , n} and a permutation σ : [n] → [n] frequently will be written in word format as: σ(1), · · · , σ(n).
If (a 1 , · · · , a r ) is a composition of n we write (a 1 , · · · , a r ) |= n. If we allow some of the elements a i to be zero, we call it a weak composition or a pseudo composition. Two compositions with the same elements but in a different order are considered as different compositions but the same partition.
For the basic notations and results in the theory of representations of symmetric groups we refer the reader to Zelevinski's [22]. Part 1. The commutative context
The Heisenberg product of species
We introduce the notion of Heisenberg product of two species, generalizing the ordinary Cauchy and Hadamard products. We follow the notation and terminology of [1] and [4].
Let Set × be the category of finite sets with bijections among them as morphisms. Let Vect or Vect k be the category of vector spaces over a fixed field.
Definition 2.1. The category of species, denoted as Sp, is the abelian category of functors from Set × into Vect: Sp = Vect Set × -with abelian structure induced by that of Vect-. An object p : Set × → Vect of Sp is called a species. The elements of Sp(p, q) -i.e. the morphisms of Sp-, are the natural transformations between p and q. The evaluation of the species p over a set I is a vector space denoted as p[I], and for a bijection f the effect of the species p is denoted as p
p[f | S ] ⊗ q[f | T ] : p[S] ⊗ q[T ] → p[S ′ ] ⊗ q[T ′ ]
defines p # q for arrows. Hence, p # q ∈ Sp.
This definition involves in particular two special situations: the case of the Cauchy product and of the Hadamard product of species. Indeed, consider the two "limit" situations in terms of the size of the intersection S ∩T in the above sum:
(p · q)[I] = I=S∪T ∅=S∩T p[S] ⊗ q[T ] = I=S⊔T p[S] ⊗ q[T ], (p × q)[I] = I=S∪T S∩T =I p[S] ⊗ q[T ] = p[I] ⊗ q[I],
where the symbol S ⊔ T stands for the disjoint union of S and T .
Since we can write:
(p # q)[I] = (p × q)[I] + I=S∪T ∅ =S∩T =I p[S] ⊗ q[T ] + (p · q)[I],
we say informally that the Heisenberg product "interpolates" between the Cauchy and the Hadamard products -p · q and p × q respectively-.
The triples (Sp, ·, i) and (Sp, ×, e) endowed with the obvious associative constraint, are additive monoidal categories with i and e as unit objects.
In the next theorem we present a generalization of this fact proving that (Sp, #, i) is also monoidal and additive. Theorem 2.3. The functor # : Sp × Sp → Sp together with the natural associativity constraint and with the unit object i, endows the category Sp, with an additive monoidal structure.
Proof. Let p, q, and r be three species. We prove that (p # q) # r [I] = p #(q # r) [I] for all finite sets I.
We have
(p # q) # r [I] = I=S∪T (p # q)[S] ⊗ r[T ] = I=S∪T S=U ∪V p[U] ⊗ q[V ] ⊗ r[T ],(2)p #(q # r) [I] = I=S ′ ∪T ′ p[S ′ ] ⊗ (q # r)[T ′ ] = I=S ′ ∪T ′ T ′ =U ′ ∪V ′ p[S ′ ] ⊗ q[U ′ ] ⊗ q[V ′ ],(3)
which implies that (2) and (3) coincide. From the definition (1) it is clear that
p # i = i # p = p.
In addition to the relation via interpolation of the Cauchy, Hadamard and Heisenberg products, they are also related by a natural isomorphism:
(p · e) × (q · e) ∼ = (p # q) · e,(4)
The required isomorphism (4) appears naturally when we evaluate each side on a finite set I. We have:
(p · e) × (q · e) [I] = I=S⊔T p[S] ⊗ k ⊗ I=S ′ ⊔T ′ q[S ′ ] ⊗ k (p # q) · e [I] = I=J⊔K (p # q)[J] ⊗ e[K] = J⊆I J=S∪S ′ p[S] ⊗ q[S ′ ] ⊗ k,
and clearly both spaces are naturally isomorphic.
2.1. The Heisenberg product of representations of the symmetric group. The language of species and the more classical language of representations of the symmetric group are essentially the same -see for example [4]-. Using the equivalence of the categories, we translate the Heisenberg product to the category of representations.
Therefore, we obtain an associative product at the level of the representations of the symmetric groups, which interpolates between the Kronecker and the induction product.
Let Rep(S n ) be the category whose objects are finite dimensional representations of S n and whose morphisms are S n -module homomorphisms. We consider the category
R = n≥0 Rep(S n ).(5)σ · x = p[σ](x).
Using the functor F we can express the Heisenberg product in terms of representations. The explicit construction uses the induction and restriction of representations to certain subgroups defined as follows.
Let p, q be non-negative integers. Given permutations σ ∈ S p and τ ∈ S q , let σ × τ ∈ S p+q be the permutation
(σ × τ )(i) = σ(i) if 1 ≤ i ≤ p, τ (i − p) + p if p + 1 ≤ i ≤ p + q.(6)
This operation gives an embedding of S p × S q into S p+q called the parabolic embedding. Let n be an integer satisfying max(p, q) ≤ n ≤ p + q. Define the set S p × n S q = S n−q × S p+q−n × S n−p , and consider the embeddings
S p × n S q ֒→ S n , (σ, ρ, τ ) → σ × ρ × τ,(7)S p × n S q ֒→ S p × S q , (σ, ρ, τ ) → (σ × ρ, ρ × τ ).(8)
Definition 2.5. The Heisenberg product of representations is the functor # : R×R → R defined for V ∈ Rep(S p ) and W ∈ Rep(S q ) as
V # W = p+q n=max(p,q) Ind Sn Sp×nSq Res Sp×Sq Sp×nSq (V ⊗ W ).(9)
Let (V # W ) n denote the component of degree n in (9) and the top component, that is, when n = p + q. In this case, the embedding (8) is the identity and (7) is the standard parabolic embedding S p × S q ֒→ S p+q . Then, we have that:
(V # W ) p+q = Ind S p+q Sp×Sq (V ⊗ W ),
which is the usual induction product of representations [8,22].
On the other hand, when n = p = q, the embedding (7) is the identity and (8) is the diagonal embedding S n ֒→ S n × S n . Therefore,
(V # W ) n = Res Sn×Sn Sn (V ⊗ W ),
which is the Kronecker product of representations [8,22].
The Heisenberg product contains terms of intermediate degrees between max(p, q) and p + q; in this sense it "interpolates" between the Kronecker and induction products. It is a remarkable fact that, as the Kronecker and induction products, the Heisenberg product is associative.
Moreover, it can be lifted to other settings ( permutations, non-commutative symmetric functions, and dually, quasi-symmetric functions) and for all the instances of our construction of the new product, these properties hold.
The next theorem proves that the operation defined in (9) is the translation of the Heisenberg product in species defined in (1). Note that although the functor F is an equivalence of categories, the language of species is considerably cleaner than the language of representations. The lengthy verifications in Theorem 2.6 ilustrate this claim.
Theorem 2.6. The Heisenberg product of representations endows R with an additive monoidal structure with unit objet the representation -denoted as i-that is k in degree zero and zero elsewhere. Moreover, the functor F given in Theorem 2.4 is monoidal, i.e.
F (p # q) ∼ = F (p) # F (p) , F (i) ∼ = i(10)
for species p and q.
Proof. It is enough to verify that the Heisenberg product of representations defined by (9) satisfies (10), since F is already an equivalence of categories and Sp is a tensor category with the Heisenberg product and with unit i. Fix i, j, and n, three non-negative integers such that max(i, j) ≤ n ≤ i + j. We claim that we have an isomorphism in R:
[n]=S∪T # S=i # T =j p[S] ⊗ q[T ] ∼ = Ind Sn S i ×nS j Res S i ×S j S i ×nS j p[i] ⊗ q[j] .(11)
Once this isomorphism is established, taking the direct sum over i and j, we obtain the n-th coordinate of F (p # q) in the left hand side, and the n-th coordinate of the product F (p) # F (q) on R in the right hand side.
The following fact can be proved easily. Let A and B be finite totally ordered sets.
Given decompositions A = A 1 ⊔ · · · ⊔ A n and B = B 1 ⊔ · · · ⊔ B n , with # A i = # B i for i = 1, . . . , n, there is only one bijection f : A → B such that f (A i ) = B i and f i = f | A i : A i → B i is increasing, for all i = 1, .
. . , n. We call f the canonical bijection between A and B induced by the partitions.
To establish the isomorphism we consider the following definitions. Given S and T such that [n] = S ∪ T , let S ′ = S \ T and T ′ = T \ S. If # S = i and # T = j, then let f S,T : [n] → [n] be the canonical bijection induced by the following partitions of [n]:
S ′ ⊔ (S ∩ T ) ⊔ T ′ and [n − j] ⊔ [n − j + 1, i] ⊔ [i + 1, n]
and let f S ′ , f S∩T , and f T ′ , be the restriction to the corresponding subsets. From the monotonicity conditions for f S,T , we get that f −1 S,T belongs to S i × n S j . We consider the standard identification of the induction module Ind G H (V ) with the tensor product kG ⊗ kH V . Let u ∈ p[S] and v ∈ q[T ], and define the map
p[S] ⊗ q[T ] ψ −→ Ind Sn S i ×nS j Res S i ×S j S i ×nS j p[i] ⊗ q[j] u ⊗ v −→ f −1 S,T ⊗ p[f S ′ ⊔ f S∩T ](u) ⊗ q[f S∩T ⊔ f T ′ ](v)(12)
and extend it to the direct sum in (11). For a permutation σ ∈ S n , the action of σ in u ⊗ v is, according to Theorem 2.4,
σ · (u ⊗ v) = p[σ | S ](u) ⊗ q[σ | T ](v).(13)
Observe that σ · (u × v) ∈ p σ(S) ⊗ q σ(T ) . The application of the map ψ yields
ψ σ · (u ⊗ v) = f σ(S),σ(T ) ⊗ α(u) ⊗ β(v)(14)
where
α = p[f σ(S ′ ) ⊔ f σ(S)∩σ(T ) ]p[σ | S ] and β = q[f σ(S)∩σ(T ) ⊔ f σ(T ′ ) ]q[σ | T ]. Since we can decompose σ | S into σ | ′ S ⊔ σ | S∩T , then by the functoriality of p we get that α = p (f σ(S ′ ) σ | S ′ ) ⊔ (f σ(S∩T ) σ | S∩T ) .
Letσ S ′ andσ S∩T be the only bijections such that the following diagrams commute
S ′ f S ′ / / σ | S ′ [i]σ S ′ σ(S ′ ) f σ(S ′ ) / / [i] S ∩ T f S∩T / / σ | S∩T [n − j + 1, i]] σ S∩T σ(S ∩ T ) f σ(S∩T ) / / [n − j + 1, i](15)
We conclude that α can be rewritten as
α = p (σ S ′ ⊔σ S∩T )(f S ′ ⊔ f S∩T ) = p[σ S ′ ⊔σ S∩T ] p[f S ′ ⊔ f S∩T )],
and we proceed similarly with β. In accordance with (13) we deduce that:
α(u) ⊗ β(v) = (σ S ′ ⊔σ S∩T ⊔σ T ′ ) · p[f S ′ ⊔ f S∩T ](u) ⊗ q[f S∩T ⊔ f T ′ ](v) .
Note that the permutationσ S ′ ⊔σ S∩T ⊔σ T ′ clearly belongs to S i × n S j . In equation (14), since the tensor product of f −1 σ(S),σ(T ) with α(u) ⊗ β(v) is performed with respect to this subgroup, we can move the permutation to the left factor where we get f −1 σS,σT (σ S ′ ⊔σ S∩T ⊔ σ T ′ ) = σf −1 S,T . This equality results again from the diagrams (15). This is precisely the definition of the action of σ on the image of the map ψ.
The map ψ is invertible, since for any element σ ⊗ (x ⊗ y),
we decompose σ = ξ(α × β × γ), where α × β × γ ∈ S n−j × S i+j−n × S n−i = S i × n S j and ξ is increasing in the intervals [n − j], [n − j + 1, i], and [i + 1, n]. Define the disjoint sets A = ξ [n − j] , B = ξ [n − j + 1, i] , C = ξ [i + 1, n] . Then, let S = A ⊔ B and T = B ⊔ C. It is straightforward to find u ⊗ v in p[S] ⊗ q[T ] such that ψ(u ⊗ v) = σ ⊗ (x ⊗ y).
Similarly, this process applied to the image of ψ in (12) yields back u ⊗ v.
2.2.
The generating function of the Heisenberg product of two species. The generating function associated to a species p is the formal series
F p (x) = n≥0 dim k p[n]
x n n! .
The generating series associated to the Cauchy product p · q of two species is the usual (Cauchy) product of the power series F p and F q . Similarly, the generating series of p × q is the Hadamard product of the generating series of p and q. Explicitly, if F p (x) = n≥0 a n x n /n! and F q (x) = n≥0 b n x n /n!, then
F p · q (x) = n≥0 i+j=n n i a i b j x n n! and F p × q (x) = n≥0 a n b n x n n! .
The classical names for these operations among formal series justify the names for the Cauchy and Hadamard products of species.
Theorem 2.7. The generating function of the Heisenberg product of two species p and q is
F p # q (x) = n≥0 i,j≤n n≤i+j n n − i, n − j, i + j − n a i b j x n n! , where n n − i, n − j, i + j − n = n! (n − i)!(n − j)!(i + j − n)! .
Proof. In the definition of the Heisenberg product of two species -Definition (1) In this subsection we recall some basic facts about the relations between species -viewed as such or as representations of the family of all symmetric groups-and the space of symmetric functions. Let K(S n ) be the Grothendieck group or representation group of the category of finite dimensional S n -modules, and call K and K the groups:
K = n≥0 K(S n ) ⊆ K = n≥0 K(S n ).
Consider the ring of polynomials k[x 1 , · · · , x n ] in n variables in which the symmetric group S n acts by permuting the variables. Call Λ k n the subring consisting of the homogeneous polynomials of degree k which are invariant under the action of S n . When m ≥ n, Λ k m proyects naturally onto Λ k n via the homomorphism ρ k m,n : Λ k m → Λ k n which maps the first n variables to themselves, and the other variables to 0. The space Λ k is defined as the inverse limit of the system considered above.
For Λ k the space of symmetric functions of degree k, define
Λ = k≥0 Λ k ⊆ Λ = k≥0 Λ k ,
called the space of symmetric functions and its completion, respectively (see [14]).
Observe that Λ and Λ are subspaces of k[x 1 , x 2 , · · · ] and k x 1 , x 2 , · · · respectively. We recall the following definition of special elements in Λ.
(1) The elementary symmetric functions are defined by the generating series:
r≥0 e r (x 1 , x 2 , · · · )t r = i≥1 (1 + x i t),
(2) The complete homogeneous symmetric functions are defined by:
r≥0 h r (x 1 , x 2 , · · · )t r = i≥1 (1 − x i t) −1 ,(3)
The power sums are defined by:
r≥0 p r (x 1 , x 2 , · · · )t r = i≥1 x i (1 − x i t) −1 .
The above defined functions are elements of k x 1 , x 2 , · · · , if we want to consider the corresponding elements in k[x 1 , x 2 , · · · x n ] we simply set 0 = x n+1 = x n+2 = · · · .
These, can in turn be defined in terms of the monoidal symmetric functions. A partition -finite or almost finite-α = (a 1 , a 2 , · · · ) with a 1 ≥ a 2 ≥ · · · ≥ 0 determines a monomial x α = x a 1 1 x a 2 2 · · · . The monomial symmetric function associated to α and denoted as m α is
m α = { α : α α} x α ,
where α stands for a composition and the symbol α α means that the mentioned composition produces the given partition α by permutation of the entries. For example m (21) = i =j x 2 i x j . We have the following equalites: Elementary symmetric functions: e r = m (1 r ) where (1 r ) is the partition of r formed only by 1's. Complete homogenous symmetric functions: h r = {α:|α|=r} m α where the sum is taken over all the partitions of r. Power sums: p r = m (r) , where (r) is the partition (r, 0, 0 · · · ).
For an arbitrary partition α = (a 1 , a 2 , · · · ) we define: e α = e a 1 e a 2 · · · , h α = h a 1 h a 2 · · · , p α = p a 1 p a 2 · · · .
When α runs over all partitions, the set of all functions m α form a Z-basis of Λ, and the same happens with the sets of the e α 's or the set of the h α 's. The set of the p α 's form a Q-basis of Λ Q .
The Frobenius characteristic map is the linear isomorphism
ch : K ⊗ Z k → Λ, ch(V ) = 1 n! σ∈Sn χ V (σ)p cycle(σ) ,
where V is a representation of S n , χ V its character and p cycle(σ) is the power sum associated the partition of n defined by to the cycle-type of σ. The map ch restricts to an isomorphisms of K and Λ. See [14,Proposition I.7.3] for proofs of the isomorphisms. The above result, yields another perspective regarding the complete homogeneous symmetric functions.
Indeed, if α = (a 1 , . . . , a r ) is a composition of n and
S α = S a 1 × · · · × S ar ,
it can be viewed as a subgroup of S n by iterating (6). These are the so called parabolic subgroups of S n . Let h α denote the permutation representation of S n corresponding to the action by multiplication on the quotient S n /S α . The isomorphism class of h α does not depend on the order of the parts of α, hence we will consider the representations h α for α running over the partitions of n.
If we denote the trivial S α -module by 1 (we omit the dependence on α for clarity), then the representation h α can also be expressed as
h α = Ind Sn Sα (1).(16)
The following equality holds ch(h α ) = h α (see [14,Proposition I.7.3]).
The Heisenberg product of complete homogeneous symmetric functions.
In Section 2 and in Subsection 3.1 we have established a path between the objects described in the diagram below:
Sp F / / R + 3 K ⊃ K ⊗ Z k / / K k ch / / Λ,
where the double arrow means the application of the Grothendieck functor.
Using the universal property of the Grothendick group functor, it is clear that in order to translate the Heisenberg product from R to K it is enough to verify that it is compatible with direct sums.
It is easy to make this verification in the category of species, where the colimits are defined pointwisely as the colimits of vector spaces. Then, the distributive property of the tensor product with respect to direct sums shows that (p ⊕ q) # r = (p # r) ⊕ (q # r) for p , q , r ∈ R.
Hence the operation # can be defined in K, and the subgroup K is clearly #-closed since the definition of the Heisenberg product involves only a finite number of summands. Now, by composition with the Frobenius characteristic isomorphism ch : K k → Λ we obtain an associative product on symmetric functions, which we call Heisenberg product of symmetric functions.
The next theorem gives an explicit formula for the Heisenberg product in the linear basis of Λ formed by the complete homogenous symmetric functions. This theorem, besides providing a combinatorial rule useful for computations, will later be used to make the connection with the Heisenberg product of non-commutative symmetric functions in Section 8.
In order to express the coefficients of the Heisenberg product of two complete homogenous symmetric functions, we need to define a particular set of plane partitions as follows. Let α = (a 1 , . . . , a r ) p and β = (b 1 , . . . , b s ) q be two compositions and n an integer with max(p, q) ≤ n ≤ p + q. Let a 0 = n − p, b 0 = n − q, and let M n α,β be the set of all (s + 1) × (r + 1)-matrices M = (m ij ) 0≤i≤s 0≤j≤r with non-negative integer entries and such that
• the sequence of column sums is (a 0 , a 1 , . . . , a r ),
• the sequence of row sums is (b 0 , b 1 , . . . , b s ),
• the first entry is m 00 = 0.
We illustrate these conditions as follows:
0 m 01 · · · m 0r n − q m 10 m 11 · · · m 1r b 1 . . . . . . . . . . . . . . . m s0 m s1 · · · m sr b s n − p a 1 · · · a r
Let p(M) be the partition of n whose parts are the non-zero m ij .
Theorem 3.1.
There is an associative product # in Λ, interpolating between the internal and external products, which can be expressed in the basis (h α ) of complete homogeneous functions as
h α # h β = p+q n=max(p,q) M ∈M n α,β h p(M ) .(17)
For example, using such theorem we get
h (2,1) # h 3 = h (2,1) + h (1,1,1,1) + h (2,1,1) + h (2,2,1) + h (2,1,1,1) + h (3,2,1) ,
where the external product is recognized in the last term and the internal product in the first one, together with additional terms of degrees four and five. The existence of this operation poses the problem of finding an explicit description for its structure constants on the basis of Schur functions. The answer would contain as extreme cases the Littlewood-Richardson rule and (a still unknown) rule for the Kronecker coefficients.
Proof of Theorem 3.1. We prove that the following formula holds in the category R:
h α # h β = p+q n=max(p,q) M ∈M n α,β h p(M ) ,
where the representations h α are the induced representations defined in (16). The application of the Grothendieck group functor and the Frobenius characteristic immediately yields (17).
We fix n in the range max(p, q) ≤ n ≤ p + q. The n-summand of h α # h β is, according to (9),
(h α # h β ) n = Ind Sn Sp×nSq Res Sp×Sq Sp×nSq (h α ⊗ h β ) = Ind Sn Sp×nSq Res Sp×Sq Sp×nSq Ind Sp×Sq Sα×S β (1). (18)
Consider the composition of the first two functors Res Sp×Sq Sp×nSq Ind Sp×Sq Sα×S β in the right hand side of (9). We use Mackey's formula to interchange them (see [21]), as follows.
Let Υ ⊂ S p × S q be a complete set of representatives of the family of double cosets
(S p × n S q ) \ (S p × S q ) / (S α × S β ). For each v ∈ Υ, define υ (S α × S β ) = υ −1 (S α × S β )υ and S α × υ n S β = (S p × n S q ) ∩ υ (S α × S β ).(19)
The following diagram ilustrates the relative position of these groups and subgroups
S n S p × S q S p × n S q e e e e ❑ ❑ ❑ ❑ ❑ 6 6 6 6 ♥ ♥ ♥ ♥ ♥ S α × S β g g g g S α × υ n S β h h h h 7 7 7 7 ♥ ♥ ♥ ♥ ♥
In this situation Mackey's formula reads as the equality
Res Sp×Sq Sp×nSq Ind Sp×Sq Sα×S β (1) = υ∈Υ Ind Sp×nSq Sα× υ n S β Res Sα×S β Sα× υ n S β (1).
Using the transitivity of the induction functor and the property that it commutes with coproducts we deduce that (18) can be written as
(h α # h β ) n = Ind Sn Sp×nSq Res Sp×Sq Sp×nSq (h α ⊗ h β ) = υ∈Υ Ind Sn Sα× υ n S β (1).(20)
In Lemma 3.2 we construct a bijection υ → M υ between Υ and M n α,β with the property that S p(Mυ) = S α × υ n S β . Then (20) becomes
(h α # h β ) n = υ∈Υ Ind Sn Sα× υ n S β (1) = υ∈Υ Ind Sn S p(Mυ ) (1) = M ∈M n α,β h p(M ) ,
proving the theorem.
Lemma 3.2. In the notations of Theorem 3.1, there is a bijection Υ ∼ = M n α,β given by υ → M υ , such that S p(Mυ) = S α × υ n S β . Proof. The proof of this rather technical lemma is postponed until the appendix 13.1.
With the help of the coproduct, one can produce a simple relation between the Heisenberg product and the external and internal products in symmetric functions. The coproduct in this situation is dual to the external product and can be defined on the generators of the basis of complete homogeneous symmetric functions as:
∆(h a ) = i+j=a h i ⊗ h j .(21)
The identity that follows was suggested to the authors by A. Zelevinski and does not hold for the space of non-commutative symmetric funcions (see comment after Theorem (7.4)).
Lemma 3.3 (A. Zelevinski)
. Assume that f, g ∈ Λ, then:
f # g = f 1 ·(f 2 * g 1 ) · g 2 ,(22)
where ∆(f ) = f 1 ⊗ f 2 and ∆(g) = g 1 ⊗ g 2 .
Proof. The identity (22) follows from formula (17), by collecting the first row and first column of the matrix M as (h α ) 1 and (h β ) 2 , respectively, and the remaining submatrix of M is precisely the internal product of the second tensorand (h α ) 2 of the coproduct of h α with the first tensorand (h β ) 1 of the coproduct of h β .
3.3. The Heisenberg product of power sums. The power sums (p λ ) λ⊢n,n≥0 form a linear basis of Λ over Q. In this subsection we give an explicit formula for the Heisenberg product in this basis. Given two partitions λ and µ, denote by λµ the concatenation and reordering of λ and µ. For example, if λ = (3, 2, 1, 1) and µ = (2, 2, 1), then λµ = (3, 2, 2, 2, 1, 1, 1).
Theorem 3.4. The Heisenberg product in the basis of power sums can be expressed as
p λ # p µ = αγ=λ γβ=µ z(γ) p αγβ ,(23)
where z(γ) is the order of the stabilizer of the conjugacy class of a permutation of cycletype γ:
z(γ) = r r mr m r !,(24)
being m r the number of times r occurs in γ.
Proof. The coproduct in the basis of power sums is determined by requiring the functions p n , with n a non-negative integer, to be primitive elements:
∆(p n ) = 1 ⊗ p n + p n ⊗ 1. More explicity, ∆(p λ ) = αβ=λ p α ⊗ p β .
Then, formula (22) reads
p λ # p µ = α 1 α 2 =λ β 1 β 2 =µ p α 1 ·(p α 2 * p β 1 ) · p β 2 . But p α 2 * p β 1 = z(α 2 )δ α 2 ,β 1 p α 2 -see [14,
Chapter I (7.12)]-. Since the external product of power sums is done by concatenating the partitions, we obtain the result of the theorem.
As a particular case, assume that λ and µ are partitions of n. Note that there is a term in degree n only when λ = µ, otherwise γ would never be the empty partition and the degree of p αγβ would be strictly greater than n. Therefore, the only term in degree n is
z(λ) p λ , if λ = µ; 0, otherwise;
which is the expression of the internal product in the basis of power sums. On the other hand for any partitions λ and µ, when γ is the empty partition, we obtain the term of largest degree, namely p αβ , since z(γ) = 1 in this case. This gives the external product p λ · p µ = p λµ .
Note that the coefficients of Formula (23) in the basis of power sums are not necessarily the numbers z(γ). Indeed, the partition λ may be decomposed, in general, in more than one way as λ = αγ, since the operation of concatenation of partitions involves a reordering of the final result. For example, let (1 n ) be the partitions with n parts equal to 1. Then,
p (1 u ) # p (1 v ) = u+v n=max(u,v) u n − v v n − u (u + v − n)! p (1 n ) .(25)
In this case, the partitions of Formula (23) are α = (1 n−v ), β = (1 n−u ), and γ = (1 u+v−n ). The number of possible decompositions of (1 u ) into two partitions of length n − v and u + v − n is u n−v , and the same argument for (1 v ) yields the second binomial coefficient. The remaining factor of the coefficient is z
(γ) = z (1 u+v−n ) = (u + v − n)!, according to Formula (24). From the explicit expression h (n) (x) = x a 1 i 1 x a 2 i 2 · · · x a k i k -where (a 1 , · · · ,
a k ) ranges over all possible permutations of the parts of α = (ℓ 1 , · · · , ℓ k ) for all partitions α of n-, it is clear that h (1 u ) = p (1 u ) . Hence, Formula (25) can also be deduced from Theorem (3.1). We use this method in (44) for non-commutative symmetric functions.
H g ✽ ✽ ✽ ✽ ✽ ✽ H f B B ✝ ✝ ✝ ✝ ✝ ✝ g • f / / H H ⊗ H f ⊗g / / H ⊗ H m H ∆ O O f ⋆ g / / H (26)
Definition 4.1. The Heisenberg product of endomorphisms -denoted as f # g, for f, g ∈ End(H)-is defined by the diagram:
H ⊗3 cyclic / / H ⊗3 1⊗m ❆ ❆ ❆ ❆ ❆ ❆ H ⊗2 ∆⊗1 > > ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ H ⊗2 1⊗g H ⊗2 f ⊗1 O O H ⊗2 m~⑤⑤ ⑤ ⑤ ⑤ ⑤ ⑤ H ∆`❇ ❇ ❇ ❇ ❇ ❇ ❇ f # g / / H (27)
where the map cyclic :
H ⊗3 → H ⊗3 is x ⊗ y ⊗ z → y ⊗ z ⊗ x.
The associativity of the Heisenberg product follows from the Hopf algebra axioms and its unit is the map ιε.
In explicit terms one has:
(f # g)(h) = f (h 1 ) 2 g h 2 f (h 1 ) 1 .(28)
We call End f (H) the subspace of End(H) of finite rank linear homomorphisms, i.e. the image of the canonical inclusion H * ⊗ H ֒→ End(H). It is clear that the three above operations restrict to End f (H) -observe that for the linear generators of End f (H) the Heisenberg product takes the following form:
α|h#β|ℓ = α(h 1 ⇀ β)|h 2 ℓ, where (α|h)(h ′ ) = α(h ′ )h for α ∈ H * , h, h ′ ∈ H-. It is also clear that if H is finite dimensional End f (H) = End(H)
, is endowed with a coproduct given by the tensor products of the coproducts in H * and H. This coproduct, is compatible with the convolution product but not with the others.
Remark 4.2.
The Heisenberg product appears in the literature in different settings (see for example [17]). Given a Hopf algebra H and a H-module algebra A, the Heisenberg product is defined as the operation on the space A ⊗ H given as:
(a ⊗ h) #(b ⊗ k) = a(h 1 · b) ⊗ h 2 k.(29)K -if k ∈ K on h ∈ H the action is denoted as (k, h) → k · h : K × H → H-. We have that for all k ∈ K and h, ℓ ∈ H, ∆(k · h) = k 1 · h 1 ⊗ k 2 · h 2 and k · (hℓ) = (k 1 · h)(k 2 · ℓ). The action of K on H, induces a right action of K on H * via the formula: α ∈ H * , h ∈ H, k ∈ K, (α ↼ k)(h) = α(k · h)
. With respect to this action and if H is finite dimensional, H * becomes a right K-module bialgebra. In general H * is only a K-module algebra.
Definition 4.3.
Assume that H is a K-module bialgebra for a certain Hopf algebra K. We define an action of K on End(H) as follows: if k ∈ K and f ∈ End(H), (k · f )(h) = k 1 · f (S(k 2 )h), -S : K → K denotes the antipode-. Explictly, the action on the generators of End f (H) is the following: for α|h ∈ End f (H) and k ∈ K : k · (α|h) = (α ↼ Sk 2 )|k 1 · h. (1) The K-invariant elements for this action, i.e. the elements f ∈ End(H) such that for all k ∈ K, k · f = ε(k)f , are the K-equivariant homomorhisms. They form a vector subspace of End(H) denoted as End K (H). Clearly, the K-action is compatible with composition. (2) In the case that K is cocommutative, the K-action is also compatible with the convolution product. Then End(H) and End f (H) are K-module algebras with convolution and End K (H), End f,K (H) are subalgebras. (3) In the case that H is finite dimensional and K is cocommutative, the coproduct of End(H) is compatible with the action of K.
Indeed, if α ∈ H * , h ∈ H, k ∈ K, we have that ∆(k · (α|h)) = ∆((α ↼ Sk 2 )|k 1 · h) = (α ↼ Sk 2 ) 1 |(k 1 · h) 1 ⊗ (α ↼ Sk 2 ) 2 |(k 1 · h) 2 = (α 1 ↼ Sk 4 )|(k 1 · h 1 ) ⊗ (α 2 ↼ Sk 3 )|(k 2 · h 2 ) = (α 1 ↼ Sk 2 )|(k 1 · h 1 ) ⊗ (α 2 ↼ Sk 4 )|(k 3 · h 2 ) = k 1 · (α 1 |h 1 ) ⊗ k 2 · (α 2 |h 2 ).
Lemma 4.5.
(1) In the situation above, if K is cocommutative and H is a K-module bialgebra, then End(H) endowed with the Heisenberg product is a K-module algebra and
End f (H) is a K-subalgebra. (2) Moreover, End K (H) and End f,K (H) are #-subalgebras of End(H).
Proof. We prove only the assertion concerning the Heisenberg product in End f (H) as is the one we use in the applications. The rest of the proof is left to the reader. Consider α, β ∈ H * , h, ℓ ∈ H and k ∈ K.
We first compute:
k · (α|h#β|ℓ) = k · (α(h 1 ⇀ β)|h 2 ℓ) = (α(h 1 ⇀ β)) ↼ Sk 2 |k 1 · (h 2 ℓ) = (α(h 1 ⇀ β)) ↼ Sk 3 |(k 1 · h 2 )(k 2 · ℓ) = (α ↼ Sk 4 )((h 1 ⇀ β) ↼ Sk 3 )|(k 1 · h 2 )(k 2 · ℓ),(30)
next:
k 1 · (α|h)#k 2 · (β|ℓ) = (α ↼ Sk 2 )|k 1 · h)#(β ↼ Sk 4 )|k 3 · ℓ = (α ↼ Sk 2 )((k 1 · h) 1 ⇀ (β ↼ Sk 4 ))|(k 1 · h) 2 k 3 · ℓ = (α ↼ Sk 3 )(k 1 · h 1 ⇀ (β ↼ Sk 5 ))|(k 2 · h 2 )(k 4 · ℓ) = (α ↼ Sk 3 )(k 4 · h 1 ⇀ (β ↼ Sk 5 ))|(k 1 · h 2 )(k 2 · ℓ) .(31)
Now, the equality of (30) and (31) can be deduced from the following calculation: take
β ∈ H * , h, r ∈ H and k ∈ K, (k 1 · h ⇀ (β ↼ Sk 2 ))(r) = (β ↼ Sk 2 )(rk 1 · h) = β((Sk 2 ) · (r(k 1 · h))) = β((Sk 3 · r)(Sk 2 k 1 · h)) = β((Sk · r)h) = ((h ⇀ β) ↼ Sk)(r).
4.2.
The case of endomorphisms of graded Hopf algebras. Assume that H = n≥0 H n is a graded connected bialgebra, i.e. for all n, m ∈ N, H n H m ⊂ H n+m , ∆(H n ) ⊆ p+q=n H p ⊗ H q and H 0 = k -conectivity condition-. It is well known that in this situation H is a Hopf algebra, and that the antipode preserves the degree. In the case that each H n is finite dimensional, end(H) = end f (H) = n (H * n ⊗ H n ), that can be endowed with with a coproduct defined as below.
Definition 4.7. In the situation above, take α|h ∈ H * n ⊗ H n , if ∆(α) = p+q=n α p ⊗ α q and ∆(h) = r+s=n h r ⊗ h s , with α p ∈ H * p , α q ∈ H * q , h r ∈ H r , h s ∈ H s ; then ∆(α|h) = a+b=n α a |h a ⊗ α b |h b .
Remark 4.8. It is clear that the composition and convolution product defined in End(H) restricts to the chain of subspaces considered above. Moreover, in the case that the H n are finite dimensional, end(H) endowed with the convolution product and the above defined coproduct is a graded bialgebra.
The behaviour of the Heisenberg product in the graded case is described in the proposition that follows, that plays a central role in our constructions. to end(H). Moreover, if f ∈ End(H p ) and g ∈ End(H q ) then
f # g ∈ p+q n=max(p,q) End(H n )(32)
and the top and bottom components of f # g are
(f # g) p+q = f ⋆ g and, if p = q, (f # g) p = g • f.(33)
(2) In the case that f = α|k ∈ End(H p ) and g = β|ℓ ∈ End(H q ), we have:
α|k#β|ℓ = 0≤n≤min(p,q) α(k n ⇀ β)|k p−n ℓ
, if ∆(k) = n k n ⊗k p−n where k n ∈ H n and k p−n ∈ H p−n . Hence the Heisenberg product of End(H) also restricts to end f (H).
Proof. (1) Let h ∈ H n . The coproduct of h is ∆(h) = a+b=n h a ⊗ h b ,
with h a ∈ H a and h b ∈ H b . Using the formula (28) we obtain:
(f # g)(h) = a+b=n f (h a ) 2 g h b f (h a ) 1 .(34)
Suppose that f and g belong to end(H). The computation of the degree of every term in the sum yields
deg f (h a ) 2 g h b f (h a ) 1 = deg f (h a ) 2 + deg g h b f (h a ) 1 = deg f (h a ) 2 + deg h b f (h a ) 1 = deg f (h a ) 2 + deg(h b ) + deg f (h a ) 1 = deg f (h a ) + deg(h b ) = a + b = n,
proving that f # g is in end(H). We can refine the previous analysis as follows. Assume that f ∈ End(H p ) and g ∈ End(H q ). Then, Expression (34) is zero unless
a = p and b + deg f (h a ) 1 = q.(35)
Adding these two equations we get that n = a + b ≤ p + q. On the other hand, p = a ≤ a + b = n and q = b + deg f (h a ) 1 ≤ b + a = n, hence max(p, q) ≤ n. This proves (32). If we set n = p + q in (35) (2) The equality α|k#β|ℓ = 0≤n≤min(p,q) α(k n ⇀ β)|k p−n ℓ where ∆(k) = k n ⊗ k p−n follows immediately from the explicit formulae (34) and from the considerations of (1) with the corresponding bounds for the degrees.
Thus and as expected, the Heisenberg product interpolates between the composition and convolution products. The analogous interpolation property at all other noncommutative levels (permutations and non-commutative symmetric functions) is a consequence of this general result. (1) Assume that that K is a commutative Hopf algebra and that H is graded connected K-module Hopf algebra as above. Assume also that the action of K preserves the grading. In this situation one can consider the chain of subspaces of End(H) that follows:
End(H) ⊇ End gr (H) ⊇ end(H) ⊇ end f (H) | | | | End K (H) ⊇ End gr,K (H) ⊇ end K (H) ⊇ end f,K (H).
(2) In this context is clear that all the subspacs considered above are closed under the composition, convolution and Heisenberg products. In particular the following holds: if f ∈ End K (H p ) and g ∈ End K (H q ) then
f # g ∈ p+q n=max(p,q) End K (H n ).(36)
The Heisenberg product of Garsia-Reutenauer endomorphisms
In this section we define certain distinguished subspace of endomorphisms of the Hopf algebra H, that we call the Garsia-Reutenauer endomorphisms. Then we show that the Heisenberg product in End(H) (Section 4) can be restricted to this special subspace.
These endomorphisms are characterized in terms of their action on products of primitive elements of H.
The motivation for the definition is that in the case that H is the tensor algebra of a vector space, an important result of Garsia and Reutenauer -see [7]-relates this subspace with the space of non-commutative symmetric functions via Schur-Weyl duality (Lemma 6.1 and Theorem 7.2).
G(h 1 , . . . , h n ) = Span(h σ(1) · · · h σ(n) | σ ∈ S n ),
or in other words, G(h 1 , . . . , h n ) is the subspace generated by the products of the form h σ(1) · · · h σ(n) for σ ∈ S n .
For later use, we record the explicit expressions of the comultiplication in elements that are products of primitives h 1 · · · h n .
We consider the set of (p, q)-shuffles -that is denoted as Sh(p, q)-. A (p, q)-shuffle is a permutation ξ ∈ S p+q such that ξ(1) < · · · < ξ(p) and ξ(p + 1) < · · · < ξ(p + q) .
The comultiplication of h 1 · · · h n is given as:
∆(h 1 · · · h n ) = p+q=n ξ∈Sh(p,q) h ξ(1) · · · h ξ(p) ⊗ h ξ(p+1) · · · h ξ(p+q) .(37)
The following lemma lists some of the basic properties of the subspaces G(h 1 , . . . , h n ). (1) , . . . , h ξ(k) ) and a (2) ξ ∈ G(h ξ(k+1) , . . . , h ξ(n) ). Definition 5.3. Let H be an arbitrary Hopf algebra. The space of Garsia-Reutenauer endomorphisms of H -denoted as Σ(H)-is:
∆(a) = k+ℓ=n ξ∈Sh(k,ℓ) a (1) ξ ⊗ a (2) ξ , where a (1) ξ ∈ G(h ξΣ(H) = f ∈ End(H) f G(h 1 , . . . , h n ) ⊆ G(h 1 , . . . , h n ) for all h 1 , . . . , h n ∈ Prim(H) .
The subspace considered above, plays for End(H) the same role that the subspace of descents plays for the algebra of permutations. Proof. Given a primitive element h, we have ιε(h) = 0, hence the unit of the Heisenberg product is in Σ(H).
Take two endomorphisms f and g in Σ(H), and let h 1 , . . . , h n ∈ Prim(H). Then, we have by definition -see (37), (27) and (28)-:
(f # g)(h 1 · · · h n ) = k+ℓ=n ξ∈Sh(k,ℓ) f (h ξ(1) · · · h ξ(k) ) 2 g h ξ(k+1) · · · h ξ(n) f (h ξ(1) · · · h ξ(k) ) 1 . (38) As f (h ξ(1) · · · h ξ(k) ) ∈ G(h ξ(1) · · · h ξ(k) ), it follows from Lemma 5.2 that ∆ f (h ξ(1) · · · h ξ(k) ) = r+s=k η∈Sh(r,s) a (1) η ⊗ a (2) η , with a(1)
η ∈ G(h ξη(1) , . . . , h ξη(r) ) and a (2) η ∈ G(h ξη(r+1) , . . . , h ξη(k) ). Hence, we rewrite (38) as
(f # g)(h 1 · · · h n ) = k+ℓ=n ξ∈Sh(k,ℓ) a (2) η g(h ξ(k+1) · · · h ξ(n) a (1) η ).
But the argument of g belongs to G(h ξ(k+1) , . . . , h ξ(n) , h ξη(1) , . . . , h ξη(r) ). Using that g ∈ Σ(H) and using part (1) of Lemma 5.2 we obtain that (f # g)(h 1 · · · h n ) ∈ G(h ξη(r+1) , . . . , h ξη(k) , h ξ(k+1) , . . . , h ξ(n) , h ξη(1) , . . . , h ξη(r) )
⊆ G(h 1 , . . . , h n ),
proving that f # g ∈ Σ(H).
Remark 5.5. It is easy to show that Σ(H) is also a subalgebra of End(H) with respect to the composition and convolution products.
In the situation that H is a graded connected bialgebra, we can produce an homogeneous and equivariant version of the above results.
We , and in the case that K is a commutative bialgebra that acts on H, by homogenous bialgebra endomorphisms -see Section 4-we define:
Σ(H) ∩ End K (H) = Σ K (H) ⊇ Σ gr,K (H) ⊇ σ K (H).
As all the objects described above are defined as intersections, it is clear that they are closed under composition, convolution and Heisenberg product.
The Heisenberg product of permutations
In order to translate the Heisenberg product from endomorphisms of Hopf algebras to permutations we specialize the constructions of Section 4 and apply the methods related to the Schur-Weyl duality theorem -see [13]-.
Let
T (V ) = n≥0 V ⊗n
be the tensor algebra of a finite dimensional vector space V . It is a graded connected Hopf algebra with product defined by concatenation and with coproduct uniquely determined by the condition that the algebra generators -the elements v ∈ V -are primitive:
∆ : v → 1 ⊗ v + v ⊗ 1 for v ∈ V .(39)
As v 1 ⊗ · · · ⊗ v n = v 1 · · · v n , we omit the tensors when writing elements of T (V ). The general linear group GL(V ) acts on V and hence on each V ⊗n diagonally. Schur-Weyl duality -as presented for example in [6] or [13]-guarantees that the only endomorphisms of T (V ) which commute with the action of GL(V ) are (linear combinations of) permutations.
Let S = n≥0 kS n be the direct sum of all symmetric group algebras. The product in S is defined on permutations as the usual composition -denoted by σ • τ or στ -when σ and τ belong to the same homogeneous component of S, and is 0 in any other case. The identity in S n is denoted by Id n .
Lemma 6.1 (Schur-Weyl duality). In the notations above, let Ψ be the map
Ψ : S → end GL(V ) T (V ) ,
defined by sending σ ∈ S n to the endomorphism Ψ(σ) of T (V ), which in degree n is given by the right action of σ on V ⊗n :
v 1 · · · v n Ψ(σ) −→ v σ(1) · · · v σ(n)
and is 0 in the other homogeneous components. Then, Ψ is an homogeneous isomorphism of vector spaces.
Definition 6.2. The Heisenberg product of permutations # : S × S → S is defined by the commutativity of the diagram below:
S × S Ψ×Ψ / / # end GL(V ) T (V ) × end GL(V ) T (V ) # S Ψ / / end GL(V ) T (V )
Compare the above definition with the results by Malvenuto and Reutenauer in [15] where the authors deal with the convolution product in S. The considerations of Section 4, guarantee that the same methods can be applied in the situation treated in Definition 6.2 for the Heisenberg product.
This method, presented in [15] and used above, is important because it could be applied to other dualities than Schur-Weyl, i.e. to centralizer algebras of other groups (or Hopf algebras) acting on a tensor algebra.
It also can be applied to other products of endomorphisms, a remarkable case being that of the Drinfel'd product, which is studied in [18].
Next we exhibit an explicit formula for the Heisenberg product of two permutations, that at the lowest and highest degree, yield the usual formulae for the composition and the Malvenuto-Reutenauer products respectively -see the table appearing in Figure 2-. We establish the following notation for the (p, q) shuffle of maximal length:
β p,q = 1 2
· · · p p + 1 p + 2 · · · p + q q + 1 q + 2 · · · q + p 1 2 · · · q .
Notice in particular that β p,q = β −1 q,p . Theorem 6.3.
(1) Let σ ∈ S p and τ ∈ S q . Then, the Heisenberg product in S can be expressed as
σ # τ = p+q n=max(p,q)
ξ∈Sh(p,n−p) η∈Sh(p+q−n,n−q) ξ (ση) × Id n−p β 2n−p−q,p+q−n (Id n−q × τ ).
(40)
(2) When n = p + q:
(σ # τ ) n = σ ⋆ τ = ξ∈Sh(p,q) ξ(σ × τ ),
where σ × τ ∈ S p+q via the standard inclusion:
(σ × τ )(i) = σ(i) if 1 ≤ i ≤ p , p + τ (i − p) if p + 1 ≤ i ≤ p + q .
When n = p = q:
(σ # τ ) n = στ.
Proof.
(1) Using (37) and (38) for the endomorphisms Ψ(σ) and Ψ(τ ) induced by the permutations σ ∈ S p and τ ∈ S q , respectively, we obtain:
Ψ(σ) # Ψ(τ ) (v 1 · · · v n ) = r+s=n ξ∈Sh(r,s) Ψ(σ)(v ξ(1) · · · v ξ(r) ) 2 Ψ(τ ) v ξ r+1 · · · v ξn Ψ(σ)(v ξ(1) · · · v ξ(r) ) 1 .
The only non-zero terms occur when r = p, hence
Ψ(σ) # Ψ(τ ) (v 1 · · · v n ) = ξ∈Sh(p,n−p) Ψ(σ)(v ξ(1) · · · v ξ(p) ) 2 Ψ(τ ) v ξ(p+1) · · · v ξ(n) Ψ(σ)(v ξ(1) · · · v ξ(p) ) 1 = ξ∈Sh(p,n−p) u+v=p η∈Sh(u,v) v ξση(u+1) · · · v ξση(p) Ψ(τ )(v ξ(p+1) · · · v ξ(n) v ξση(1) · · · v ξση(u) ) = ξ∈Sh(p,n−p) η∈Sh(p+q−n,n−q) v ξση(p+q−n+1) · · · v ξση(p) v ξτ (p+1) · · · v ξτ (n) v ξσητ (1) · · · v ξσητ (p+q−n)
= ξ∈Sh(p,n−p) η∈Sh(p+q−n,n−q) Ψ ξ (ση) × Id n−p β 2n−p−q,p+q−n (Id n−q × τ ) (v 1 · · · v n ), which proves the first part of the theorem.
(2) This part follows directly. Observe that for the case n = p + q we obtain the product of permutations as defined by Malvenuto-Reutenauer in [16,15]. In the case n = p = q and since the action of S n on V ⊗n is from the right, the composition of permutations corresponds to composition of endomorphisms in the opposite order -compare with the results of Proposition 4.9-.
For example, writing the permutations in word format we obtain that: Notice that the red terms correspond to the Malvenuto-Reutenauer product.
The Heisenberg product of non-commutative symmetric functions
The descent set of a permutation σ ∈ S n is the subset of [n − 1] defined by
Des(σ) = i ∈ [n − 1] | σ(i) > σ(i + 1) .
Given J ⊆ [n − 1], define B J as the set of permutations σ ∈ S n with Des(σ) ⊆ J, and consider the following elements of kS n :
X J = σ∈B J σ.(41)
The family of subsets of [n−1] is in bijective correspondence with the set of compositions of n. Recally that α = (a 1 , · · · , a r ) is a composition of n if all the a i are positive integres and r i=1 a i = n, in this situation we write α = (a 1 , · · · , a r ) |= n. The bijection between compositions and subsets is:
(a 1 , a 2 , . . . , a r ) ←→ {a 1 , a 1 + a 2 , . . . , a 1 + · · · + a r−1 }.
For instance, if n = 9, then X (1,2,4,2) = X {1,3,7} and X (2,4,2,1) = X {2,6,8} . Definition 7.1. Let Σ n = k{X α : α |= n} ⊆ kS n , be the subspace linearly spanned by X α with α composition of n and Σ = n≥0 Σ n .
An important result of Garsia and Reutenauer characterizes the elements of S whose images by Ψ -in the notations of Lemma 6.1-belong to σ GL(V ) T (V ) . in terms of their action on the tensor algebra. Recall the Definition 5.3 of σ GL(V ) T (V ) .
Theorem 7.2 (Garsia-Reutenauer, [7]). Let Ψ be isomorphism defined in Lemma 6.1, then the following diagram commutes:
Σ ⊆ Ψ| Σ S Ψ σ GL(V ) T (V ) ⊆ end GL(V ) T (V ) ,
and the map Ψ| Σ is surjective.
A fundamental result of Solomon [20] states that Σ n is a subalgebra of the symmetric group algebra kS n with the composition product. This is Solomon's descent algebra. It is also well-known that Σ is closed under the external product [9,10,15]; in fact, X (a 1 ,...,ar) · X (b 1 ,...,bs) = X (a 1 ,...,ar,b 1 ,...,bs) .
The space Σ with the external product is the algebra of non-commutative symmetric functions.
The following theorem generalizes these two situations in view of the interpolation property of the Heisenberg product of permutations. Proof. This a direct result from Theorem 7.2, Theorem 5.4, and Schur-Weyl duality (Lemma 6.1).
The next theorem gives another version of the same result with an explicit description of the value of X α # X β for α |= p and β |= q. The structure coefficients of X α # X β are expressed in terms of the matrices M n α,β defined in Subsection 3.2. In [18] appears another combinatorial proof which extends a proof of Schocker in [19] for the composition product.
Moreover, another proof of Theorem 7.3 can be obtained by extending the Heisenberg product to the Coxeter complex of the symmetric group (that is, the faces of the permutahedron). This makes a connection with recent work of Brown, Mahajan, Schocker, and others on this aspect of the theory of descent algebras [5,2,19].
Recall the following definition: take α = (a 1 , . . . , a r ) p, β = (b 1 , . . . , b s ) q two compositions and let n be an integer, max(p, q) ≤ n ≤ p + q. Let a 0 := n − p, b 0 := n − q, and M n α,β be the set of all integral (s + 1) × (r + 1)-matrices with non negative entries M = (m ij ) 0≤i≤s 0≤j≤r such that: the sequence of column sums is (a 0 , a 1 , . . . , a r ), the sequence of row sums is (b 0 , b 1 , . . . , b s ) and the first entry is m 00 = 0. To visualize these conditions we write the diagram:
0 m 01 · · · m 0r n − q m 10 m 11 · · · m 1r b 1 . . . . . . . . . . . . . . . m s0 m s1 · · · m sr b s n − p a 1 · · · a r
Theorem 7.4. Let α p and β q be two compositions. Then
X α # X β = p+q n=max(p,q) M ∈M n α,β X c(M )(43)
where c(M) is the composition whose parts are the non-zero entries of M, read from left to right and from top to bottom.
Observe that even though this formula is similar to the one we had for symmetric functions (17), the occurrence of the compositions as indices of the basis makes the connection between the Heisenberg product and the external and Solomon products considerably subtler than in the commutatative context. In particular, a formula like (22) does not longer hold. Moreover, the above expression will allow us to make the connection with the Heisenberg product of the representations of the symmetric group. This point is taken up in Section 8.
By the interpolation property of the Heisenberg product, Theorem 7.4 contains as special cases rules for the product in Solomon's descent algebra and for the external product of two basis elements of Σ. One readily verifies that the former is precisely the well-known rule of Garsia, Remmel, Reutenauer, and Solomon as given in [7, Proposition 1.1], while the latter is the one appearing in (42).
As an example we have the following formula
X (1 u ) # X (1 v ) = u+v n=max(u,v) u n − v v n − u (u + v − n)! X (1 n ) ,(44)
where (1 n ) is the composition of n with n parts equal to 1. The coefficients arise by inspection of all the possible ways to fill out the entries of the matrix M in a diagram as below with row and column sums as prescribed.
0 ? · · · ? n − v ? ? · · · ? 1 . . . . . . . . . . . . . . . ? ? · · · ? 1 n − u 1 · · · 1
Similarly, one verifies that:
X #(n) (1) = n k=1 S(n, k)X (1 k ) ,
where the S(n, k) are the Stirling numbers of the second kind.
Proof of Theorem 7.4. Let us take a fixed integer n between max(p, q) and p + q. To the compositions α = (a 1 , . . . , a r ) and β = (b 1 , . . . , b s ) we associate the following sets:
E n 0 = [p + 1, n], F n 0 = [1, n − q], E n 1 = [1, a 1 ], F n 1 = n − q + [1, b 1 ], E n 2 = [a 1 + 1, a 1 + a 2 ], F n 2 = n − q + [b 1 + 1, b 1 + b 2 ], . . .
. . . E n r = [a 1 + · · · + a r−1 + 1, p], F n s = n − q + [b 1 + · · · + b s−1 + 1, q]. Observe that the family of intervals {E n j } j∈{0,...,r} and {F n i } i∈{0,...,s} are partitions of [1, n]. It is also clear that σ ∈ B α if and only if σ × Id n−p is increasing in E n j for all j ∈ {0, . . . , r}. Similarly, τ ∈ B β if and only if Id n−q × τ is increasing in F n i for all i ∈ {0, . . . , s}. Observe, also, that # E n j is the j-th coordinate of the pseudo-composition (n − p, a 1 , . . . , a r ), and # F n i is the i-th coordinate of (n − q, b 1 , . . . , b s ). Given η ∈ Sh(p + q − n, n − q) and τ ∈ B β , call ϕ η,τ = (η × Id n−p )β 0 (Id n−q × τ ) and define the matrix
M η,τ = #(F n i ∩ ϕ −1 η,τ E n j ) 0≤i≤s, 0≤j≤r
where we have abbreviated β 2n−p−q,p+q−n = β 0 . In this situation M η,τ ∈ M n α,β . Indeed, if we call m ij = #(F n i ∩ ϕ −1 η,τ E n j ), for i = j = 0 we have that
ϕ η,τ [1, n − q] = η[p + q − n + 1, p] ⊆ [1, p],
which shows that the intersection F n 0 ∩ ϕ −1 η,τ E n 0 is empty, and then m 00 = 0. The sum m 0j + · · · + m sj equals the number of elements of E n j , which is, as noted before, the j-th entry of the composition (n − p, a 1 , . . . , a r ). The same argument applies to the sum of the rows. In this manner, sending τ → M η,τ we define a map B β → M n α,β . Take M = {m ij } ∈ M n α,β and let B η,n β (M) the corresponding fiber of this map:
B η,n β (M) = τ ∈ B β #(F n i ∩ ϕ −1 η,τ E n j ) = m ij for all j ∈ {0, . . . , r}, i ∈ {0, . . . , s} . Therefore, we have a partition of B β = M ∈M n α,β B η,n β (M).
For ξ ∈ Sh(p, n − p) and η ∈ Sh(p + q − n, n − q), let us denote g n ξ,η (σ, τ ) = ξ (ση) × Id n−p β 0 (Id n−q × τ ), the n-term in the sum (40). The function g n ξ,η is bilinear, and we can write X α # X β = n ξ,η g n ξ,η (X α , X β ).
From now on as n is fixed we will omit it in the notations of the sets and the functions. Next we show that
ξ,η g ξ,η (X α , X β ) = M ∈M α,β X c(M ) .
For this, we write
ξ,η g ξ,η (X α , X β ) = ξ,η g ξ,η σ∈Bα σ, M ∈M α,β τ ∈B η β (M ) τ = M ∈M α,β ξ,η σ∈Bα τ ∈B η β (M ) g ξ,η (σ, τ ).(45)
If we denote by S α,β (M) the set of elements (ξ, η, σ, τ ) such that ξ ∈ Sh(p, n − p), η ∈ Sh(p + q − n, n − q), σ ∈ B α and τ ∈ B η α (M); then the map ψ : S α,β (M) → B c(M ) given by ψ(ξ, η, σ, τ ) = g ξ,η (σ, τ ) is a bijection. We prove this fact in Lemma 7.6. In this situation, if we group together the last three sums of (45) we obtain
ξ,η g ξ,η (X α , X β ) = M ∈M α,β X c(M ) ,
which concludes the proof of the theorem.
In the following two lemmas we assume the notations of the previous theorem. Their proofs, being rather technical are presented in the Appendix. See: 13.2 and 13.3. Lemma 7.5. For η ∈ Sh(p+q−n, n−q), τ ∈ B β and for all i = 0, . . . , s and j = 0, . . . , r, the sets F i ∩ ϕ −1 η,τ E j are disjoint intervals. Moreover, in each of these intervals the function ϕ η,τ is increasing and has image either contained in [1, p] or contained in [p + 1, n]. Lemma 7.6. For M ∈ M α,β , the map ψ : S α,β (M) → B c(M ) , which sends (ξ, η, σ, τ ) into g ξ,η (σ, τ ), is a bijection.
From non-commutative to commutative symmetric functions
In the previous four sections of Part 2: Non-commutative context; we constructed the following commutative diagram of algebras -endowed with their respective Heisenberg products-(omitting the part that includes the space Λ):
Λ Σ π o o ⊆ Ψ| Σ S Ψ σ GL(V ) T (V ) ⊆ end GL(V ) T (V ) ⊆ end T (V ) .
In the part of the above diagram that excludes Λ, all the Heisenberg products are induced by the one defined in the larger space end T (V ) . Recall also that the vertical isomorphisms are due to Schur-Weyl duality and to the results of [7] as mentioned in Section 7, Theorem 7.2.
We want to incorporate into this diagram the space of symmetric functions Λ equipped also with the Heisenberg product, this is expressed in the dotted arrow that is to be defined.
For each n ≥ 0 we define the linear map π n : Σ n → Λ n by its values on the basis {X α : α composition of n} as: π n (X α ) = h α , where α is the partition of n obtained by reordering the entries of the composition α. Let us denote by π : Σ ։ Λ the map induced in the direct sums.
It is well known that π is a morphism if we endow Σ with the Solomon product and Λ with the internal product, it is also a morphism when we endow both spaces with the external products.
The theorem that follows generalizes these compatibilities by proving that the map π : Σ ։ Λ is a morphism with respect to the Heisenberg products in Λ and Σ, as constructed in Sections 3 and 7, respectively.
π(X α # X β ) = h α # h β .
Proof. Comparing the explicit formulae-see equations (17) and (43)-it is clear that it is enough to construct for each n a bijection ψ : M n α,β → M n α, β , such that for all M ∈ M n α,β :
p ψ(M) = c(M) (46) Let σ and τ be two permutations which reorder into partitions the compositions α = (a 1 , . . . , a r ) and β = (b 1 , . . . , b s ), respectively, i.e.: α = (a σ(1) , . . . , a σ(r) ), β = (b τ (1) , . . . , b τ (r) ).
Define ψ(M) as the matrix obtained from M by permuting its columns with the permutation Id 1 × σ and its rows with the permutation Id 1 × τ . Clearly ψ(M) is in M n α, β and the map ψ is a bijection.
Moreover, since the (unordered) entries of M and ψ(M) are the same, we get Equation (46).
Compatibility of the coproduct with the Heisenberg product
In various of the spaces depicted in the table in Figure 1, coproducts can be introduced, and frequently they are compatible with the products we are considering. For brevity we concentrate in the consideration of the compatibility of the coproducts with the Heisenberg product at the level of permutations, of non commutative and of commutative symmetric functions. We start by considering a coproduct that is called the convolution (or external) coproduct.
We start by defining this coproduct at the level of S. Let n be a positive integer and decompose it as n = p + q. Then, as Sh −1 (p, q) is a set of representatives for the left cosets of S p × S q ⊆ S n , given σ ∈ S n there is a unique triple (ξ, σ p , σ ′ q ) such that: ξ ∈ Sh(p, q), σ p ∈ S p , σ ′ q ∈ S q , and
σ = (σ p × σ ′ q )ξ −1 .(47)
The coproduct ∆ : S → S ⊗ S is defined on σ ∈ S n as
∆(σ) = n p=0 σ p ⊗ σ ′ q ,(48)
where σ p and σ ′ q are as in (47). If follows directly from the formula (48) above (see [16]) that:
∆(X (a 1 ,...,ar) ) = b i +c i =a i 0≤b i ,c i ≤a i X (b 1 ,...,br) ⊗ X (c 1 ,...,cr) ,(49)
where for a pseudopartition α, α indicates that parts equal to zero have been omitted. It is then clear that the comultiplication ∆ can be restricted to the space of descents: i.e. that ∆(Σ) ⊆ Σ ⊗ Σ.
Remark 9.1. In [16, Thèoréme 5.3, Remarque 5.15] it is proved that equipped with the convolution product and the above coproduct, S becomes a graded connected Hopf algebra and it is also shown that ∆ is not compatible with the composition of permutations. Also in [3] a more recent and detalied study of (S, ⋆, ∆) is presented. Taking into account that the Heisenberg product, interpolates between the convolution (or Malvenuto-Reutenauer) and the composition product at the level of the permutations, we cannot expect it to be compatible with the coproduct considered above.
These operations are better behaved if we restrict our attention to the non commutative symmetric functions.
Next we prove that ∆ is compatible with the Heisenberg product in Σ, and in particular this implies that for descents the composition product is compatible with the comultiplication.
Theorem 9.2. The space (Σ, #, ∆) is a cocommutative Hopf algebra.
Proof. It is enough to prove the Heisenberg-multiplicativity of ∆ on elements of the form X α with α a composition of p for different p's.
Let α and β be compositions of p and q, respectively. We use Formula (43) to compute
∆(X α # X β ) = n M ∈M n α,β ∆(X c(M ) ) = p+q n=max(p,q) M ∈M n α,β γ+γ ′ =c(M ) X γ ⊗ X γ ′ .(50)
On the other hand,
∆(X α ) # ∆(X β ) = γ+γ ′ =α X γ ⊗ X γ ′ # δ+δ ′ =β X δ ⊗ X δ ′ = γ+γ ′ =α δ+δ ′ =β (X γ # X δ ) ⊗ (X γ ′ # X δ ′ ) = γ+γ ′ =α δ+δ ′ =β n,n ′ M ∈M n δ,γ M ′ ∈M n ′ δ ′ ,γ ′ X c(M ) ⊗ X c(M ′ ) .(51)
We show that the sums (50) it is clear that ( n, M , γ, γ ′ ) is a quadruple of indices appearing in the sum (50) and that the corresponding summands of (50) and (51) are the same.
Moreover, it is clear that the above correspondence between the indices of the sums is bijective.
Consider now the space of symmetric functions Λ and the proyection π : Σ → Λ. In Subsections 3.2 and 3.3, equations (21) and (23) we defined a coproduct on Λ similar to the one defined above and dual to the external product. We then gave its expression on the natural basis of complete homogeneous functions and power sums.
In particular we have that -see the notations of Subsection 3.2 and 3.3-: ∆(h a ) = i+j=a h i ⊗ h j and ∆(p n ) = 1 ⊗ p n + p n ⊗ 1.
Being π(X α ) = h α , and using the first of the above equalities and Theorem 9.2, we conclude the following result. Proof. We have that:
∆(π(X α )) = ∆(h α ) = h α 1 ⊗ h α 2 = π(X α 1 ) ⊗ π(X α 2 ) = (π ⊗ π)(∆(X α )).
Since π is Heisenberg multiplicative, we conclude that the coproduct and the Heisenberg product are compatible in the space Σ. The compatibility of the coproduct and the Heisenberg product in Σ induces the compatibility in Λ.
This theorem generalizes known results on (Λ, ·, ∆) and (Λ, * , ∆) -see for example [8]-.
Isomorphisms between the Heisenberg, convolution, and composition products
We have mentioned until now, two kinds of relations between the three products we have been dealing with: one is the interpolation connection and the other the formula appearing in Lemma 3.3.
In the spaces of symmetric functions there are further relations between the Heisenberg, internal and external products.
First we show that the external and Heisenberg products are isomorphic (in the commutative and non-commutative situations), but the isomorphism is not degreepreserving.
Similarly we prove that the Heisenberg and internal products are isomorphic in the commutative context -degrees not preserved-, but the isomorphism is only valid in the completion of the space. (X (a 1 ,...,ar)
) = X (a 1 ) # · · · # X (ar )(52)
is an isomorphism of Hopf algebras (which does not preserve the gradings).
Proof. Since the Heisenberg product has the external product as the only term in the upper degree, the matrix of the linear map ψ in the basis of the X α 's is triangular with 1 in the diagonal. Hence ψ is invertible and it is multiplicative because the external product in the basis of the X α 's is the concatenation of the compositions. We finish by proving that ψ is comultiplicative:
∆ ψ(X α ) # ψ(X β ) = (ψ ⊗ ψ)∆(X α · X β ).(53)
Clearly, it is enough to prove (53) on the algebra generators X (a) for all non-negative integers a. For the right hand side of (53) we have:
∆(X (a 1 ) · X (a 2 ) ) = ∆(X (a 1 ,a 2 ) ) = a+b=a 1 a ′ +b ′ =a 2 X (a,a ′ ) ⊗ X (b,b ′ ) .
Applying the map ψ ⊗ ψ and using formula (43) to compute ψ(X (a,a ′ ) ) = X (a) # X (a ′ ) and ψ(X (b,b ′ ) ) = X (b) # X (b ′ ) (note that we assume ψ(X (0) ) to be the identity) we get
(ψ ⊗ ψ)∆(X (a 1 ,a 2 ) ) = a+b=a 1 a ′ +b ′ =a 2 n,m X (n−a ′ ,n−a,a+a ′ −n) ⊗ X (m−b ′ ,m−b,b+b ′ −m) .(54)
On the other hand, taking into account that ψ(X α ) = X α for partitions with only one part, the left hand side of (53) is:
∆ X (a 1 ) # X (a 2 ) = k c 1 +c ′ 1 =k−a 2 c 2 +c ′ 2 =k−a 1 c 3 +c ′ 3 =a 1 +a 2 −k X (c 1 ,c 2 ,c 3 ) ⊗ X (c ′ 1 ,c ′ 2 ,c ′ 3 ) ,(55)
By collecting together the terms in (54) with n + m = k and interchanging the sums, it is easy to see that (54) and (55) are the same expression.
Corollary 10.2. The map (Λ, ·, ∆) → (Λ, #, ∆) given by
h (a 1 ,...,ar) → h (a 1 ) # · · · # h (ar )(56)
is an isomorphism of Hopf algebras (which does not preserve the gradings).
The Heisenberg and internal products are also isomorphic at the level of Λ.
Theorem 10.3. The map ( Λ, #) → ( Λ, * ) given by
f → f · n≥0 h (n)(57)
is an isomorphism of algebras.
Proof. This isomorphism follows from the isomorphism (4) in the category of species. Note that the species e corresponds to the object (1 0 , 1 1 , . . . ) in the category R, where 1 n is the trivial S n -module. Applying the Grothendieck group construction and then the Frobenius map ch, we deduce that e maps into the element n≥0 h (n) in Λ.
Remark 10.4. We give negative answers to three questions on possible extensions of the above results.
(1) The isomorphism (Σ, ·) ∼ = (Σ, #) of Theorem 10.1 does not extend to an isomorphism between ( Σ, ·) and ( Σ, #). Indeed, in case it did extend:
ψ(X (1) + X (1,1) + X (1,1,1) + · · · ) = X (1) + X (1) # X (1) + X (1) # X (1) # X (1) + · · · ,(58)
each of the terms in the infinite sum appearing in the right hand side of (58), contributes with a factor of degree 1 (namely, X (1) ); this infinite sum is not a well-defined element of Σ. The maps ϕ : f → f · n≥0 X (n) and ψ : f → n≥0 X (n) · f are not isomorphisms between ( Σ, #) and ( Σ, * ) because they are not not multiplicative. Indeed, using the rule (43) we obtain that:
X (3) # X (3) = X (3) + X (1,1,2) + X (2,2,1) + X (3,3) .
Then:
ϕ(X (3) # X (3) ) = n≥0 X (3,n) + n≥0 X (1,1,2,n) + n≥0 X (2,2,1,n) + n≥0 X (3,3,n) .(59)
On the other hand, computing ϕ(X (3) ) * ϕ(X (3) ) using Solomon's rule, gives:
ϕ(X (3) ) * ϕ(X (3) ) = n,m X (3,n) * X (3,m) = n X (3,n) * X (3,n) = n≥0
X (3,n) + X (2,1,1,n) + X (1,2,2,n) + X (3,3,n) .
We can see that (59) and (60) are different since, for example, the term X (2,1,1,n) appears in (60) but there is no term in (59) whose index is a composition starting with 2, 1, 1. A similar argument can be applied to show that ψ is also not multiplicative.
Part 3. Quasi-symmetric functions
The internal and external coproduct of quasi-symmetric functions
In this section we consider space of quasi-symmetric functions Q, dual to the noncommutative symmetric functions. First, in preparation to the introduction by dualization a new coproduct: the Heisenberg coproduct we recall the basic definitions of the internal and external coproducts.
In Sections 7, 8 and 9 we dealt with the following diagram:
Σ ⊆ π S Λ ,
and endowed the different spaces with #, the Heisenberg product and ∆, the external coproduct. In this part we introduce other coproducts, and to avoid confusions we rename the external coproduct as ∆ · (it was called simply ∆ when considered in (48) and (21)). We proved the compatibility of (#, ∆ · ) at the level of Σ and Λ. This compatibility is consistent with the known results about of the external (or Malvenuto-Reutenauer) product and the coproduct at the three levels of the above diagram, and about the compatibility with the internal (and Solomon) product at the levels of Λ and Σ (compare with Remark 9.1).
Of the three Hopf algebras: (Σ, ·, ∆ · ), (S, ⋆, ∆ · ) and (Λ, ·, ∆ · ) the second and third are self dual (see for example [9,10,11,12,15]).
Hence, we can complete the picture adding the space Q that is the graded dual of Σ, and that will fit into the following commutative diagram of Hopf algebras:
Σ ⊆ π S F Λ π * ⊆ Q
The diagram is self dual with respect to the antidiagonal. The new maps F and π * are the duals of the inclusion of Σ in S and of the projection of Σ onto Λ, respectively. The map F is described in detail in [3] and will not be used here. Note that the space S is a Hopf algebra with respect to the Malvenuto-Reutenauer product (as noted in Remark 9.1) and not with respect to the composition product.
Next we recall the definition of Q. Let X = {x 1 , x 2 , . . .} be an alphabet, i.e. a countable set, totally ordered by x 1 < x 2 < · · · . Let k[[X]] be the algebra of formal power series on X and Q = Q(X) the subspace linearly spanned by the elements
M α = i 1 <···<ir x a 1 i 1 · · · x ar ir(61)
as α = (a 1 , . . . , a r ) runs over all compositions of n, for n ≥ 0. The space Q is a graded subalgebra of k[[X]] known as the algebra of quasi-symmetric functions (see [10]). It is clear that any symmetric function is quasi-symmetric, hence we have the inclusion of algebras Λ ⊆ Q. In [15] it is proved that this map is π * defined above as the dual of the projection π : Σ ։ Λ (see Section 8).
The algebra Q carries two coproducts ∆ * and ∆ · which are defined via evaluation of quasi-symmetric functions on alphabets. Let Y be another alphabet. We can view the disjoint union X+Y and the Cartesian product X×Y as alphabets as follows: on X+Y we keep the ordering among the variables of X and among the variables of Y, and we require that every variable of X precede every variable of Y. On X × Y we impose the reverse lexicographic order:
(x h , y i ) ≤ (x j , y k )
means y i < y k or (y i = y k and x h < x j ).
The coproducts are defined by the formulas
∆ * f (X) = f (X × Y) and ∆ · f (X) = f (X + Y),
together with the identification Q(X, Y) ∼ = Q(X) ⊗ Q(X) (separation of variables). Consider the following pairing between the homogeneous components of degree n of Q and Σ:
M α , X β = δ α,β .(62)
It is known [9,10,15] that this pairing identifies the product of quasi-symmetric functions with the coproduct (49) of Σ, and the coproducts ∆ * and ∆ · with the internal and external products of Σ. In other words,
f g, u = f ⊗ g, ∆(u) , ∆ * f, u ⊗ v = f, uv , ∆ · f, u ⊗ v = f, u · v ,
for any f, g ∈ Q and u, v ∈ Σ. Here we set f ⊗ g, u ⊗ v = f, u g, v .
12. The Heisenberg coproduct of quasi-symmetric functions Let ∆ # be the coproduct of Q dual to the Heisenberg product of Σ:
∆ # f, u ⊗ v = f, u # v .
Since the Heisenberg product is a sum of terms of various degrees (32), the Heisenberg coproduct is a finite sum of the form
∆ # (f ) = i f i ⊗ f ′ i with 0 ≤ deg(f i ) and deg(f ′ i ) ≤ deg(f ) ≤ deg(f i ) + deg(f ′ i ). The terms corresponding to deg(f ) = deg(f i ) = deg(f ′ i ) and to deg(f ) = deg(f i ) + deg(f ′ i )
are the coproducts ∆ * (f ) and ∆ · (f ), respectively.
Let 1 + X denote the alphabet X together with a new variable x 0 smaller than all the others and with the property x k 0 = x 0 for any natural k. Let (1 + X) × (1 + Y) − 1 be the Cartesian product of the alphabets 1 + X and 1 + Y with reverse lexicographic ordering and with the variable (x 0 , y 0 ) removed. We can suggestively denote (1 + X) × (1+Y)−1 by X+Y+XY, although the order is given properly by the former expression.
The following result was obtained in conversation with Arun Ram.
Theorem 12.1. For any f ∈ Q,
∆ # f (X) = f (X + Y + XY).
Proof. We have to show that, with respect to the pairing (62),
M γ (X + Y + XY), X α ⊗ X β = M γ , X α # X β(63)
for all γ, α and β compositions of n, p and q, respectively. Let us fix a composition γ of n and let k the length of γ. Denote the set of indices of M γ (X + Y + XY) by
Y = (i 1 , j 1 ), . . . , (i k , j k ) (i 1 , j 1 ) < · · · < (i k , j k ) .
Consider the set A α,β = M ∈ M n α,β | w(M) = γ and define the map ψ : Y → α,β A α,β as follows: given (i 1 , j 1 ) < · · · < (i k , j k ), let M = ( m ij ) be a matrix of zeros big enough to set m j ℓ i ℓ = γ ℓ (as usual in these proofs, we start the indices in 0). Then, remove all zero rows and columns, except those with index 0; let us call M to the result. Since (0, 0) is not a possible index, we have m 00 = 0. Thus, M ∈ M n α,β where α is the composition obtained by adding all the rows of M but the first, and analogously with β and the rows of M.
The map ψ is surjective, since, given some M ∈ A α,β , we can build a sequence of indices in Y by reading the nonzero entries of M, say m uv , and considering the pairs (v, u) lexicographically ordered. Therefore, we can write j 1 ), . . . , (i k , j k ) . Collecting together the x's and y's establishes a bijection between the terms of the last sum indexed over ψ −1 (M) and the terms of M α (X)M β (Y). Indeed, take a term from this product, given by indices i r 1 < · · · < i r k and j s 1 < · · · < j s ℓ , and build the pairs (j su , i rv ) such that m v,u = 0. We also have to consider the pairs (0, i rv ) and (j su , 0) according to nonzero entries in the first row and column of M. Ordering these indices it is clear that they belong to ψ −1 (M) and this is the inverse process of grouping x's and y's.
M γ (X + Y + XY) = q∈Y (xy) γ q = α,β M ∈A α,β q∈ψ −1 (M ) (xy) γ q where (xy) γ q denotes the monomial (x i 1 y i 1 ) γ 1 · · · (x i k y i k ) γ k for q = (i 1 ,
Then, we can write
M γ (X + Y + XY) = α,β M ∈A α,β M α (X)M β (Y) = α,β # A α,β M α (X)M β (Y).
which obviously implies the equation (63).
We can express the dual of the isomorphism in Theorem 10.1 in term of alphabets in the full dual of Σ, which is Q = n≥0 Q n . The pairing , : Σ × Q → k is defined by f, g = n f n , g n n where f n and g n are the restrictions of f and g to the homogeneous components of degree n, and , n is the pairing defined in (62).
For this, given an alphabet X we define its exponential, e(X), by
e(X) = X + X (2) + X (3) + · · ·
where the divided power X (n) is the set
X (n) = (x i 1 , x i 2 , . . . , x in ) ∈ X n x i 1 < x i 2 < · · · < x in .(64)
We endow e(X) with the reverse lexicographic order. With this notations the following equation holds:
e(X + Y) = 1 + e(X) 1 + e(Y) − 1
where the equality is considered as ordered sets. Indeed, denote by (x) k the monomial x i 1 · · · x i k with i 1 < · · · < i k . Then, given (x) k (y) ℓ < (x ′ ) k ′ (y ′ ) ℓ ′ in e(X + Y), it is immediate to see that either (y) ℓ < (y ′ ) ℓ ′ or (y) ℓ = (y ′ ) ℓ ′ and (x) k < (x ′ ) k ′ , which is the definition of the order in the left hand side. Clearly, the same argument applies in the other direction.
Theorem 12.2. The dual of the isomorphism ψ from (Σ, ·, ∆ · ) to (Σ, #, ∆ · ) of Theorem 10.1 with respect to the pairing , , is the isomorphism ψ * from ( Q, ·, ∆ # ) to ( Q, ·, ∆ · ) given by ψ * (f ) = f e(X) .
Proof. We have to show that ψ(X γ ), f = X γ , ψ * (f ) . Observe that, from the definition of the pairing, it is enough to prove this equation for each degree. Moreover, it is enough to prove it for the generators of the algebra (Σ, ·) since, for g and g ′ generators
ψ(g · g ′ ), f = ψ(g) # ψ(g ′ ), f = ψ(g) ⊗ ψ(g ′ ), f (X + Y + XY) = i ψ(g), f i (X) ψ(g ′ ), f ′ i (Y) = i g, f i e(X) g ′ , f ′ i e(Y) = g ⊗ g ′ , f e(X) + e(Y) + e(X) e(Y) = g ⊗ g ′ , f e(X + Y) = g ⊗ g ′ , ∆ f (e(X)) = g · g ′ , f e(X) .
Next, we prove the duality for the set of generators X (n) for n ≥ 0, and for f = M α where α is a composition of n. In this case we have ψ(X (n) ) = X (n) and the equation X (n) , M α = X (n) , M α e(X) = δ (n),α is immediately verified.
Endowed with the coproduct ∆ # , the algebra Q is a graded connected Hopf algebra, in duality with the graded connected Hopf algebra (Σ, #, ∆).
We finish this Section by expressing the antipode of this Hopf algebra in terms of the alphabets.
First, define the evaluation of quasi-symmetric functions on the the opposite of an alphabet X by the equation
M α (−X) = (−1) r i 1 ≥···≥ir x a 1 i 1 · · · x ar ir ,(65)
for any composition α = (a 1 , . . . , a r ) (compare with the definition of M α in (61)). Next, we define the alphabet
X * = X + X 2 + X 3 + · · ·(66)
as the disjoint union of the Cartesian powers X n under reverse lexicographic order. For instance (
x 3 , x 1 , x 2 ) < (x 2 , x 2 ) < (x 1 , x 3 , x 2 ).
Theorem 12.3. The antipode of the Hopf algebra of quasi-symmetric functions (Q, ·, ∆ # ) is:
S # (f ) = f (−X) * .
Proof. By Theorem 12.1, it is enough to prove that M α X + (−X) * + X(−X) * = 0 for any alphabet X and for any composition α. We ilustrate the argument for a composition with only one part: α = (a), the argument for a composition with several parts is essentially the same.
By selecting variables from each of the three alphabets X, (−X) * , and X(−X) * , we can write
M (a) X + (−X) * + X(−X) * = x a i + r (−1) r (x i 1 · · · x ir ) a + r (−1) r x a j (x i 1 · · · x ir ) a .
It is easy to see that the first sum cancel with the terms with r = 1 of the second sum, while the remaining terms of the second sum cancel with the last sum.
Part 4. Appendix
The proofs
In this Appendix we provide the postponed proofs of the technical lemmas used in the paper.
Proof of Lemma 3.2.
Proof. To define the bijection Υ → M n α,β , we start by splitting the interals [1, p] and [1, q] as below:
E 1 = [1, a 1 ], F 1 = [1, b 1 ], E 2 = [a 1 + 1, a 1 + a 2 ], F 2 = [b 1 + 1, b 1 + b 2 ], . . . . . . E k = [a 1 + · · · + a k−1 + 1, p], F s = [b 1 + · · · + b s−1 + 1, q],
where α = (a 1 , . . . , a k ) and β = (b 1 , . . . , b s ). Given an element v = σ × τ ∈ S p × S q we consider the shuffles ζ α (σ) ∈ Sh(α) and ζ β (τ ) ∈ Sh(β) characterized by the equations
σ = ζ α (σ)u, τ = ζ β (τ )v,(67)
with u ∈ S α and v ∈ S β . To simplify the notation, we write ζ α = ζ α (σ) and ζ β = ζ β (τ ). We further split each interval E i and F j as below:
E i = E ′ i ⊔ E ′′ i , F j = F ′ j ⊔ F ′′ j , such that ζ α (E ′ i ) ⊆ [1, n − q], ζ β (F ′ j ) ⊆ [1, p + q − n], ζ α (E ′′ i ) ⊆ [n − q + 1, p], ζ β (F ′′ j ) ⊆ [p + q − n + 1, q]
, for i = 1, . . . , k and j = 1, . . . , s. Observe that with these definitions we have the decomposition of the interval [1, n] into
[1, n − q] = k i=1 ζ α (E ′ i ),(68)[n − q + 1, p] = k i=1 ζ α (E ′′ i ) = s j=1 n − q + ζ β (F ′ j ) ,(69)[p + 1, n] = s j=1 n − q + ζ β (F ′′ j ) .(70)
Define the matrix M σ×τ of dimension (k + 1) × (s + 1) whose entries are
m 00 = 0, m i0 = # E ′ i , for i = 1, . . . , k, m 0j = # F ′′ j , for j = 1, . . . , s, m ij = # ζ α (E ′′ i ) ∩ n − q + ζ β (F ′ j ) otherwise.
The matrix M σ×τ belongs to M n α,β . Assume that i = 0. Since
ζ α (E ′′ i ) ⊆ [n − q + 1, p] ⊆ s j=1 n − q + ζ β (F ′′ j ) , we get s j=0 m ij = # E ′ i + s j=1 # ζ α (E ′′ i ) ∩ n − q + ζ β (F ′ j ) = # E ′ i + # ζ α (E ′′ i ) ∩ s j=1 n − q + ζ β (F ′ j ) = # E ′ i + # ζ α (E ′′ i ) = # E i = a i .
On the other hand, if i = 0, then, by (70), the sum of m 0j for j = 0, . . . , s, coincides with #[p + 1, n] = n − p.
Next we show that the matrix M σ×τ does not depend on the choice of representative of the coset (S p × n S q )v. Let x ∈ S n−q , y ∈ S p+q−n , and z ∈ S n−p , so that x×y×z ∈ S p × n S q . Consider the representative v ′ = σ ′ × τ ′ where σ ′ = (x × y)σ and τ ′ = (y × z)τ.
Let ζ ′ α and ζ ′ β the shuffles associate to v ′ . As
ζ α (E ′ i ) ⊆ [1, n − q], then (x × y) ζ α (E ′ i ) = x ζ α (E ′ i ) . But we also have x ζ α (E ′ i ) = ζ ′ α (E i ) = ζ ′ α (Ẽ ′ i ) ⊔ ζ ′ α (Ẽ ′′ i ), where E i =Ẽ ′ i ⊔Ẽ ′′ i is the decomposition of E i corresponding to the shuffle ζ ′ α , that satisfies ζ ′ α (Ẽ ′ i ) ⊆ [1, n−q] and ζ ′ α (Ẽ ′′ i ) ⊆ [n − q + 1, p]. In summary, x ζ α (E ′ i ) ⊆ ζ ′ α (Ẽ ′ i ).
Interchanging the roles of ζ α and ζ ′ α we obtain an equality, which implies that m i0 = # E ′ i = #Ẽ ′ i = m ′ i0 , where m ′ ij are the entries of the matrix M v ′ . This proves the equality of the first row of the matrices. The argument for the other rows is similar.
The matrix M v do not depend on the choice of representative of v(S α × S β ), since the shuffles satisfying (67) are the same for all the elements on this coset. In conclusion, the matrix M v depends only on the double cosets (S p × n S q )v(S α × S β ).
Next we show that the parabolic subgroup S p(Mv) is S α × υ n S β . An element of S α × υ n S β can be written as x × y × z where
x × y = ζ α (σ a 1 × · · · × σ a k )ζ −1 α , y × z = ζ β (τ b 1 × · · · × τ bs )ζ −1 β . Evaluating at ζ α (E ′ i ) we deduce that ζ α σ a i (E ′ i ) = x(E ′ i ) and conclude that σ a i (E ′ i ) = E ′ i . Proceeding in a similar manner with the other decompositions we obtain
σ a i (E ′ i ) = E ′ i , τ b j (F ′ j ) = F ′ j ,(71)σ a i (E ′′ i ) = E ′′ i , τ b j (F ′′ j ) = F ′′ j ,(72)
for all i = 1, . . . , k and j = 1, . . . , s.
This decomposition can be further refined. Evaluating as above at the subsets X ij = ζ α (E ′′ i ) ∩ ζ β (F ′ j ), we obtain the equality ζ α σ a i ζ −1 α (X ij ) = y(X ij ) = ζ β τ b j ζ −1 β (X ij ) . Now, ζ α σ a i ζ −1 α (X ij ) ⊆ ζ α (E ′′ i ) and also ζ β τ b j ζ −1 β (X ij ) ⊆ ζ β (F ′ j ). From the above equality we conclude that ζ α σ a i ζ −1 α (X ij ) ⊆ ζ α (E ′′ i ) ∩ ζ β (F ′ j ), and then σ a i ζ −1 α (X ij ) ⊆ ζ −1 α (X ij ). This inclusion is actually an equality, since both sets have the same cardinality. Therefore, we get the following refinment of (71)
σ a i (E ′ i ) = E ′ i , σ a i ζ −1 α (X ij ) = ζ −1 α (X ij ), τ b j (F ′′ j ) = F ′′ j , τ b j ζ −1 β (X ij ) = ζ −1 β (X ij )
. Note that # X ij = m ij , and thus the previous decomposition shows that x×y ×z belongs to S p(M ) .
The map υ → M υ is invertible, since from the entries of the matrix M υ we can recover the shuffles ζ α and ζ β , which are in the same double coset as υ.
13.2. Proof of Lemma 7.5.
Proof. As η and τ are fixed throughout this lemma, we write ϕ = ϕ η,τ . Let x, y ∈ F i ∩ ϕ −1 E j with x < y. Consider z such that x < z < y. Therefore, x, y ∈ F i and, since F i is an interval, we conclude that z ∈ F i .
On the other hand ϕ(x), ϕ(y) ∈ E j . Since τ ∈ B β , then Id × τ is increasing in F i :
(Id × τ )(x) < (Id × τ )(z) < (Id × τ )(y).
In order to prove that ϕ(z) also belongs to E j , we consider the following cases:
(1) Assume that j = 0. Then, ϕ(x), ϕ(y) ∈ E 0 = [p + 1, n]. Since (η × Id) is the identity on that interval, this implies that β 0 (Id × τ )(x) and β 0 (Id × τ )(y) are in [p + 1, n]. But β −1 0 [p + 1, n] = [n−q + 1, 2n−p −q] and β 0 is increasing in that set. Therefore, the three terms in (73) belong to [n − q + 1, 2n − p − q] and, applying (η × Id)β 0 , which is increasing on this set, we obtain that ϕ(x) < ϕ(z) < ϕ(y).
(2) Assume that j > 0. Consider the cases:
(a) Assume i = 0. In this case we have x, z, y ∈ F 0 = [1, n − q]. Then, applying Id × τ | F 0 = Id we continue in the same set. The permutation β 0 sends increasingly [1, n − q] into [p + q − n + 1, p]. In this last interval, η is also increasing. Thus, the inequality (73) implies that ϕ(x) < ϕ(z) < ϕ(y). (b) Assume i > 0. We have that x, y, z ∈ F j ⊂ [n − q + 1, n]. Applying Id × τ we have that the terms of (73) are also in [n − q + 1, n]. If (Id × τ )(x) ∈ [n−q+1, 2n−p−q], then β 0 (Id×τ )(x) ∈ [p+1, n] and ϕ(x) ∈ [p+1, n] = E 0 , which contradicts the assumption j > 0. Therefore, the terms in (73) belong to [2n − p − q + 1, n]. The permutation β 0 maps increasingly this interval into [1, p + q − n], and η is also increasing in that image. Thus, we conclude that ϕ(x) < ϕ(z) < ϕ(y). In all the cases we obtain that ϕ(x) < ϕ(z) < ϕ(y), and since ϕ(x) and ϕ(y) belong to the interval E j , we deduce that ϕ(z) ∈ E j . This proves that F i ∩ ϕ −1 E j is an interval.
Notice that along the way we also proved that ϕ is increasing in the intervals F i ∩ϕ −1 E j as well as the assertions concerning the images.
The fact that the intervals F i ∩ ϕ −1 E j are disjoint follows immediately from the fact that the sets E j , for j = 0, . . . , r, and the sets F i , for i = 0, . . . , s, are disjoint. This finishes the proof.
13.3. Proof of Lemma 7.6.
Proof. For the matrix M = {m ij }, denote by s ij the sum of the entries m kℓ of M for (k, ℓ) ≤ (i, j) with respect to the lexicografical order of pairs. We define R 00 = [1, s 00 ] and R ij = [s kℓ , s ij ] where s ij covers s kℓ . Observe that some of the intervals R ij may be empty. Also note that # R ij = m ij .
The sequence (R 00 , R 01 , . . . , R sr ) is a pseudo-partition of the interval [n] and γ ∈ B c(M ) if and only if γ is increasing in R ij for all i ∈ {0, . . . , s} and j ∈ {0, . . . , r}.
Since M ∈ M α,β and therefore, j #(R ij ) = j m ij = # F i , it follows that
F i = j R ij .(74)
Moreover, if η ∈ Sh(p + q − n, n − q) and τ ∈ B η β , then F i ∩ ϕ −1 η,τ E j = R ij . This can be seen from the fact both sets are intervals with the same cardinal and from the following relation:
j (F i ∩ ϕ −1 η,τ E j ) = F i = j R ij .
In particular, we deduce that ϕ η,τ is increasing in R ij . Given (ξ, η, σ, τ ) ∈ S α,β (M), we will show that g ξ,η (σ, τ ) ∈ B c(M ) . To prove this, since ϕ η,τ | R ij is increasing and ϕ η,τ R ij ⊆ E j , we observe that (σ × Id)(η × Id)β 0 (Id × τ ) | R ij is also increasing. According to Lemma 7.5, the images of R ij under the previous permutation are in [1, p] or [p + 1, n], where ξ is increasing. Therefore, left multiplying by ξ we deduce that g ξ,η (σ, τ ) is increasing in R ij , which proves that it belongs to B c(M ) .
We prove now that ψ is bijective. Given γ ∈ B w(M ) , we show that there exists a unique quadruple (ξ, η, σ, τ ) ∈ S α,β (M) such that ψ(ξ, η, σ, τ ) = γ.
Assume there exists such a quadruple. Using the fact that E j = i ϕ η,τ R ij , we deduce that
ξ(σ × Id)E j = γ i R ij .(75)
This proves the uniqueness of the permutation ξ(σ × Id), in other words, it is the only permutation which maps E j increasingly into the set on the right side; and this implies the uniqueness of ξ and σ. Therefore, we have that (η × Id)β 0 (Id × τ ) = (σ × Id) −1 ξ −1 γ. Thus, η is characterized by the image of [1, n − q] under the permutation on the right, which is η[p + q − n + 1, p]. The uniqueness of τ follows immediately. Given γ ∈ B c(M ) , to construct (ξ, η, σ, τ ) we note that
#(E j ) = i m ij = # i R ij = # γ i R ij ,(76)
and, thus, we can construct a permutation µ such that (75) is verified, increasingly mapping E j into γ i R ij . This permutation can be written as µ = ξ(σ × µ ′ ) with ξ ∈ Sh(p, n − p), σ ∈ S p and µ ′ ∈ S n−p . Since µ is increasing on E 0 = [p + 1, n] we conclude that µ ′ = Id n−p , and from the monotony on E j with j > 0 we deduce that σ ∈ B α . In the same way as before, we construct η by mapping the interval [1, n − q] and for this, we will show that (σ × Id) −1 ξ −1 γ is increasing in F i for all i.
In particular, for i = 0, we obtain the desired property to define η. We then consider β −1 0 (η × Id) −1 (σ × Id) −1 γ, which equals Id × τ for some τ ∈ S p . Using (77) for i > 0 we conclude that τ ∈ B β ; and it follows from (76) that the constructed τ belongs to B η β (M). It remains to prove (77). Take x 1 , x 2 ∈ F i with x 1 < x 2 . Then, x 1 ∈ R ij 1 and x 2 ∈ R ij 2 for some j 1 ≤ j 2 . Assume that j 1 = j 2 , then γ(x 1 ) < γ(x 2 ). In this case, we have γ(x 1 ) = ξ(σ × Id)(e 1 ) and γ(x 2 ) = ξ(σ × Id)(e 2 ) with e 1 , e 2 ∈ E j . Since σ is increasing in E j we obtain that e 1 < e 2 as desired.
On the other hand, if j 1 < j 2 , then e 1 ∈ E j 1 and e 2 ∈ E j 2 and the conclusion follows easily as all the elements of E j 1 are smaller than those of E j 2 .
Figure 1 .
1Relation among the spaces where the Heisenberg structure is introduced.
Figure 2 .
2Standard terminology and symbols for the products
[f ]. The image of the set [n] = {1, 2, . . . , n} is written, for simplicity, p [n] = p[n], and by convention p[0] = p[∅].
For future reference we consider the following species:(1) The species one i: i[∅] = k and i[I] = 0 for I = ∅, (2) The exponential species e: e[I] = k for I = ∅. Definition 2.2. The Heisenberg product of species is the functor # : Sp × Sp → Sp given by: (p # q)[I] = I=S∪T p[S] ⊗ q[T ]. (1) Given two finite sets I and J, and a bijection f : I → J, we obtain a bijection (S, T ) → f (S), f (T ) between pairs (S, T ) with I = S ∪ T , and pairs (S ′ , T ′ ) with J = S ′ ∪ T ′ . The map (p # q)[f ] : (p # q)[I] → (p # q)(J), induced by the maps
-, the pairs of sets (S, T ) such that [n] = S ∪ T , # S = i, and # T = j, are in bijection with the decompositions [n] = U ⊔ W ⊔ V with # U + # W = i and # W + # V = j. Indeed, just take U = S \ T , V = T \ S, and W = S ∩T . Clearly, we obtain # U = n−i, # V = n−j, and # W = i+j −n. The multinomial coefficient in the formula of F p # q stands precisely for the number of possible ways to choose the decomposition U, V, W . 3. The Heisenberg product of symmetric functions 3.1. Species, representations of symmetric groups and symmetric functions.
Part 2 .
2The non-commutative context 4. The Heisenberg product of endomorphisms Let (H, m, ∆, ι, ε, S) be an arbitrary Hopf algebra, where m : H ⊗ H → H is the product, ∆ : H → H ⊗ H is the coproduct, ι : k → H is the unit, ε : H → k is the counit, and S : H → H is the antipode. The space End(H) of linear endomorphisms of H carries several associative products. Let f, g ∈ End(H). Composition and convolution are respectively defined by the diagrams:
Definition 4. 6 .
6Consider the following chain of linear subspaces of End(H), where End gr (H) is the subspace of the linear endomorphisms of H that preserve the degree: End(H) ⊇ End gr (H) = n End(H n ) ⊇ n End(H n ) := end(H) ⊇ n End f (H n ) := end f (H).
then we get deg f (h a ) 1 = 0, and (34) reduces to the convolution diagram in (26). If we set n = p = q, then deg(h b ) = deg f (h a ) 2 = 0, and (34) reduces to g f (h) = (g • f )(h), which is the composition product.
Definition 5. 1 .
1Let H be an arbitrary Hopf algebra. If h 1 , . . . , h n ∈ H, define
Lemma 5. 2 .
2For any h 1 , . . . , h n ∈ H we have:(i) If a ∈ G(h 1 , . . . , h k ) and b ∈ G(h k+1 , . . . , h n ), then ab ∈ G(h 1 , . . . h n ). (ii) If a ∈ G(h 1 , .. . , h n ) and h 1 , . . . , h n ∈ Prim(H), then
Theorem 5. 4 .
4If H is a Hopf algebra, the space Σ(H) of Garsia-Reutenauer endomorphisms is a subalgebra of End(H) with respect to the Heisenberg product.
define the following chain of subspaces of Σ(H) : Σ gr (H) := Σ(H) ∩ End gr (H) ⊇ σ(H) := Σ(H) ∩ end(H)
Theorem 7 . 3 .
73The subspace Σ ⊆ S is closed under the Heisenberg product.
Theorem 8 . 1 .
81For any pair of compositions α and β:
For instance : ∆(52413) = ( ) ⊗ (52413) + (1) ⊗ (4132) + (21) ⊗ (321) + (231) ⊗ (21) + (2413) ⊗ (1) + (52413) ⊗ ( ).
and (51) are the same as follows: take an octuple of indices corresponding to the sum (51): (γ, γ ′ , δ, δ ′ , n, n ′ , M, M ′ ) and construct the quadruple n + n ′ , M + M ′ , c(M), c(M ′ ) . Denote by col(M) (row(M)) the vector whose entries are the sum of the columns (rows) of the matrix M. Since col(M + M ′ ) = col(M) + col(M ′ ) = n − |γ| γ + n ′ − |γ ′ | γ ′ = (n + n ′ − p)α, where |ζ| is the sum of the parts of a composition ζ, and similarly with row(M + M ′ ) = (n + n ′ − q)β, we see that M + M ′ ∈ M n α,β . As c(M) + c(M ′ ) = c(M + M ′ ), if we set ( n, M, γ, γ ′ ) = n + n ′ , M + M ′ , c(M), c(M ′ )
Corollary 9 . 3 .
93The space of symmetric functions Λ equipped with the operations (#, ∆) is a cocommutative Hopf algebra and the map π : Σ → Λ is a morphism of Hopf algebras.
Theorem 10 . 1 .
101The map ψ : (Σ, ·, ∆) → (Σ, #, ∆) given by ψ
( 2 )
2The isomorphism ( Λ, #) ∼ = ( Λ, * ) of Theorem 10.3 does not restrict to an isomorphism between (Λ, #) and (Λ, * ). Indeed, the element 1 ∈ Λ maps to n≥0 h (n) which is in Λ but not in Λ.(3) A similar isomorphism to (57) cannot be established at the level of Σ = n≥0 Σ n .
is an equivalence of abelian categories. The action of the permutation group S n in the space p[n] is given byTheorem 2.4 ([4]). The functor F : Sp → R given by
F (p) = p[0], p[1], . . . , p[n], . . .
If A = H * and H acts on A by translations then (29) corresponds to (27) via the canonical inclusion H * ⊗ H ֒→ End(H). Note that in the definition of the Heisenberg product in End(H) there are no restrictions about the dimensions. 4.1. The case of K-equivariant endomorphisms. We need an equivariant version of the above construction.Assume that K is another bialgebra and that H is a K-module bialgebra : H is endowed with a left action of the algebra
MARCELO AGUIAR, WALTER FERRER SANTOS, AND WALTER MOREIRA
Infinitesimal bialgebras, pre-Lie and dendriform algebras, Hopf algebras. Marcelo Aguiar, Lecture Notes in Pure and Appl. Math. 237DekkerMarcelo Aguiar, Infinitesimal bialgebras, pre-Lie and dendriform algebras, Hopf algebras, Lecture Notes in Pure and Appl. Math., vol. 237, Dekker, New York, 2004, pp. 1-33.
Hopf monoids in species and associated hopf algebras. Marcelo Aguiar, Swapneel Mahajan, 191to appearMarcelo Aguiar and Swapneel Mahajan, Hopf monoids in species and associated hopf algebras, to appear (2004), 191.
Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. Marcelo Aguiar, Frank Sottile, Adv. Math. 1912Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permu- tations, Adv. Math. 191 (2005), no. 2, 225-275.
F Bergeron, G Labelle, P Leroux, Combinatorial species and tree-like structures. Cambridge University PressF. Bergeron, G. Labelle, and P. Leroux, Combinatorial species and tree-like structures, Cambridge University Press, 1998.
Semigroup and ring theoretical methods in probability, Representations of finite dimensional algebras and related topics in Lie theory and geometry. Kenneth S Brown, Fields Inst. Commun. 40Amer. Math. SocKenneth S. Brown, Semigroup and ring theoretical methods in probability, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 3-26.
P Etingof, O Golberg, S Hensel, T Liu, A Schwendner, D Vaintrob, E Yudovina, Introduction to representation theory. P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina, Introduc- tion to representation theory, http://www-math.mit.edu/ etingof/replect.pdf, 2011.
A decomposition of Solomon's descent algebra. A M Garsia, C Reutenauer, Adv. Math. 772A. M. Garsia and C. Reutenauer, A decomposition of Solomon's descent algebra, Adv. Math. 77 (1989), no. 2, 189-262.
Hopf algebras of symmetric functions and class functions. Ladnor Geissinger, Combinatoire et représentation du groupe symétrique. Actes Table Ronde C.N.R.S., Univ. Louis-Pasteur StrasbourgStrasbourg; BerlinSpringer579Ladnor Geissinger, Hopf algebras of symmetric functions and class functions, Combinatoire et représentation du groupe symétrique (Actes Table Ronde C.N.R.S., Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Springer, Berlin, 1977, pp. 168-181. Lecture Notes in Math., Vol. 579.
Noncommutative symmetric functions. I M Gelfand, D Krob, A Lascoux, B Leclerc, V S Retakh, J-Y Thibon, Adv. Math. 1122I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, and J-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995), no. 2, 218-348.
Multipartite P -partitions and inner products of skew Schur functions. Ira M Gessel, Combinatorics and algebra. 34Amer. Math. SocContemp. Math.Ira M. Gessel, Multipartite P -partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983), Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289-317.
12. , Symmetric functions, noncommutative symmetric functions and quasisymmetric functions ii. M Hazewinkel, Acta Aplicandae Mathematica. Acta Aplicandae MathematicaM. Hazewinkel, Symmetric functions, noncommutative symmetric functions and quasisymmetric functions, Acta Aplicandae Mathematica (2003). 12. , Symmetric functions, noncommutative symmetric functions and quasisymmetric functions ii, Acta Aplicandae Mathematica (2005).
The Schur lectures. Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity free actions and beyond. Dekker8Bar-Ilan Univ. , Ramat GanIsrael Math. Conf. Proc.Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity free actions and beyond., The Schur lectures (1992), Israel Math. Conf. Proc., Bar-Ilan Univ., vol. 8, Dekker, Bar-Ilan Univ. , Ramat Gan., 1995, pp. 1-182.
I G Macdonald, Symmetric functions and Hall polynomials. New YorkOxford Science PublicationsOxford Mathematical Monographs. With contributions by A. ZelevinskyI. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications.
Duality between quasi-symmetric functions and the Solomon descent algebra. C Malvenuto, C Reutenauer, J. Algebra. 1773C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967-982.
Claudia Malvenuto, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Univ. du Québecà Montréal. Produits et coproduits des fonctions quasi-symétriques et de l'algèbre des descentsClaudia Malvenuto, Produits et coproduits des fonctions quasi-symétriques et de l'algèbre des descents, Ph.D. thesis, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Univ. du Québecà Montréal, 1993.
Susan Montgomery, Published for the Conference Board of the Mathematical Sciences. DC82WashingtonSusan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washing- ton, DC, 1993.
Walter Moreira, Products of representations of the symmetric group and non-commutative versions. Texas A&M UniversityPh.D. thesisWalter Moreira, Products of representations of the symmetric group and non-commutative versions, Ph.D. thesis, Texas A&M University, 2008.
The descent algebra of the symmetric group, Representations of finite dimensional algebras and related topics in Lie theory and geometry. Manfred Schocker, Fields Inst. Commun. 40Amer. Math. SocManfred Schocker, The descent algebra of the symmetric group, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 145-161.
A Mackey formula in the group ring of a Coxeter group. Louis Solomon, J. Algebra. 412Louis Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), no. 2, 255-264.
Representation theory of finite groups: algebra and arithmetic. S Weintraub, Graduate Studies in Mathematics. 59American Mathematical SocietyS. Weintraub, Representation theory of finite groups: algebra and arithmetic, Graduate Studies in Mathematics, vol. 59, American Mathematical Society, Providence, RI, 2003.
Representations of finite classical groups. Andrey V Zelevinsky, Lecture Notes in Mathematics. 869Springer-VerlagE-mail address: maguiar@math. cornell.edu E-mail address: [email protected] E-mail address: [email protected] V. Zelevinsky, Representations of finite classical groups, Lecture Notes in Mathematics, vol. 869, Springer-Verlag, Berlin, 1981. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]
| {'fraction_non_alphanumeric': 0.10301312149937493, 'fraction_numerical': 0.02018919371191763, 'mean_word_length': 3.0771360918248587, 'pattern_counts': {'":': 0, '<': 45, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 5, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 241, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded-product the Heisenberg product, that for the highest and lowest degrees produces the classical external and internal products (and their namesakes in different contexts). In order to define it, we start from the two opposite more general extremes: species in the "commutative context", and endomorphisms of Hopf algebras in the "non-commutative" environment. Both specialize to the space of commutative symmetric functions where the definitions coincide. We also deal with the different coproducts that these objects carry -to which we add the Heisenberg coproduct for quasi-symmetric functions-, and study their Hopf algebra compatibility particularly for symmetric and non commutative symmetric functions. We obtain combinatorial formulas for the structure constants of the new product that extend, generalize and unify results due to Garsia, Remmel, Reutenauer and Solomon. In the space of quasisymmetric functions, we describe explicitly the new operations in terms of alphabets. 7 2.2. The generating function of the Heisenberg product of two species 9 3. The Heisenberg product of symmetric functions 10 3.', 'arxivid': '1504.06315', 'author': ['Marcelo Aguiar ', 'Walter Ferrer Santos ', 'Walter Moreira '], 'authoraffiliation': [], 'corpusid': 119174178, 'doi': '10.1007/s40863-017-0072-x', 'github_urls': [], 'n_tokens_mistral': 38555, 'n_tokens_neox': 34480, 'n_words': 20467, 'pdfsha': 'ab590a3e3edfeb0d52118b7fcb62ad9bc2d8ac4a', 'pdfurls': ['https://arxiv.org/pdf/1504.06315v2.pdf'], 'title': ['THE HEISENBERG PRODUCT: FROM HOPF ALGEBRAS AND SPECIES TO SYMMETRIC FUNCTIONS', 'THE HEISENBERG PRODUCT: FROM HOPF ALGEBRAS AND SPECIES TO SYMMETRIC FUNCTIONS'], 'venue': []} |
arxiv |
Enhancement of the Kondo effect in a quantum dot formed in a full-shell nanowire
Aleksandr E Svetogorov
Department of Physics
University of Basel
Klingelbergstrasse 82CH-4056BaselSwitzerland
Daniel Loss
Department of Physics
University of Basel
Klingelbergstrasse 82CH-4056BaselSwitzerland
Jelena Klinovaja
Department of Physics
University of Basel
Klingelbergstrasse 82CH-4056BaselSwitzerland
Enhancement of the Kondo effect in a quantum dot formed in a full-shell nanowire
(Dated: January 31, 2023)
We analyze results of a recent experiment [D. Razmadze et al., Phys. Rev. Lett., 125, 116803 (2020)] on transport through a quantum dot between two full-shell nanowires and show that the observed effects are caused by the Kondo effect enhancement due to a nontrivial geometry (magnetic flux in a full-shell nanowire) rather than the presence of Majorana bound states. Moreover, we propose that such a setup presents a unique and convenient system to study the competition between superconductivity and the Kondo effect and has significant advantages in comparison to other known approaches, as the important parameter is controlled by the magnetic flux through the full-shell nanowire, which can be significantly varied with small changes of magnetic field, and does not require additional gates. This competition is of fundamental interest as it results in a quantum phase transition between an unscreened doublet and a many-body Kondo singlet ground states of the system. arXiv:2301.12442v1 [cond-mat.supr-con]
I. INTRODUCTION
Semiconducting nanowires with full superconducting shell were recently introduced as possible realizations of topological superconductors, which may host Majorana bound states (MBSs) [1]. MBSs in turn have non-Abelian statistics, which could be exploited to develop a topologically protected qubit [2,3]. In nanowires with a thin shell Little-Parks effect [4] results in modulation of the superconducting order parameter with the applied magnetic flux. In case of a thin nanowire (with a diameter smaller than the superconducting coherence length) the Little parks effect is destructive resulting in a lobe structure of the order parameter as a function of the flux [5,6]. The idea of combining effectively onedimensional superconductors with a vortex is already fascinating by itself: the vortex can induce a phase winding of the superconducting order parameter and result in nontrivial properties, such as the well-known Caroli-de Gennes-Matricon bound states in 2D case [7]. Further experiments with full-shell nanowires were performed [8][9][10], including those where the observed zero-bias anomalies were shown to have non-topological nature [11]. However, a recent experiment [12] with a gate-controlled quantum dot (QD) in a full-shell hybrid interferometer showed non-trivial features in addition to the zero-bias peak. Without a magnetic flux (zeroth lobe) through the shell a change of occupation of the dot (from even to odd) leads to a change of a sign of the supercurrent through the dot, which is known as 0 − π phase transition [13][14][15][16]. However, with a flux around one magnetic flux quantum threading the superconducting shell (first lobe) the effective Josephson junction seems to stay in the 0 phase even for the odd occupation of the QD, in agreement with theoretical predictions [17][18][19][20][21] for a Josephson junction based on QD between two topological superconductors hosting MBSs. Such peculiar behaviour could be seen as evidence for the presence of MBSs in the system. Nevertheless, the authors were not completely satisfied with the interpretation, as they could not explain some of the observed features, such as an enhancement of the super-current in the odd state in the first lobe and no signs of fractional Josephson effect expected in the presence of MBSs were observed.
In this paper we argue that the results of this experiment can be interpreted as an enhancement of the Kondo effect [22][23][24][25][26] in the first lobe, including an enhancement of the supercurrent. It was predicted theoretically [24,[27][28][29] and shown experimentally [30][31][32] that if the Kondo effect can develop on an odd-occupied QD between superconducting leads, the ground state is a many-body Kondo singlet instead of a doublet (unscreened electron), which restores the 0 phase behaviour and enhances the critical supercurrent. The phase transition is determined by the ratio of two energy scales: the Kondo temperature T K and superconducting gap ∆. In this paper we show that if the superconducting order parameter acquires a phase winding around the shell, the relevant energy scale is not the absolute value of the gap |∆| but an effective gap which is significantly suppressed due to destructive interference, which in turn leads to the enhancement of the Kondo effect. Our interpretation is further supported by the fact that a bright zero-bias feature and no π phase is observed at the closing of the zeroth lobe (around half the flux quantum), where no topological superconductivity is expected. As a result, we can claim that due to the detailed experimental data provided by Razmadze et al. [12] it is possible to establish a non-trivial effect of the phase winding on the coherent transport and the ground state properties of a QD between two full-shell nanowires, which has not been predicted before. Further experimental studies of the effect can enrich the understanding of the underlying fundamental physics as only elaborate numerical approaches have been developed to quantitatively capture the QDbased Josephson junction behaviour in the competition regime [24,29,[33][34][35] (for review see [36]). Schematic representation of Cooper-pair tunneling through a QD electron state (green sphere) between two hollow-cylinder superconductors (blue) representing the shells and accumulation layers of underlying nanowires. The tunneling amplitude t(θ) (between the dot and each shell) is angledependent, as the electron wave function on the dot is shifted from the shell axis due to gating; the superconducting order parameter on the shell has angle dependence -nonzero phase winding (n = 0 for large enough magnetic flux Φ > Φ0/2).
II. MODEL
In this paper we focus on a QD formed in an uncovered (etched superconductor) and gated region of a full-shell nanowire. Assuming that electrons in the nanowire form an accumulation layer at the boundary with the superconducting shell [1], we use a hollow-shell approximation -transport through the system is mostly determined by Cooper pairs propagation along the shell and tunneling between the shells through the QD. We assume the shell to be much thinner than the magnetic penetration length and the diameter to be smaller than the superconducting coherence length, which results in no quantization of the magnetic flux through the shell but reentrant lobe structure due to the destructive Little-Parks effect: superconductivity is fully suppressed when the flux around odd half-integer multiples of a flux quantum is applied [6,12]. As a further simplification we model the shell as a hollow cylinder (in experiments it has rather a hexagonal crosssection) [1,37] with radius R, then the total magnetic flux through the shell is Φ = πR 2 B. We focus on the single-level QD limit (large level spacing), the number of electrons can be considered fixed by the applied gate due to strong Coulomb interactions on the QD (strong spatial confinement), which is a common experimental situation. The simplest model to describe such a system is the superconducting impurity Anderson (SIA) model [38] modified by the magnetic flux though the shell [37]:
H = H D + s=l/r (H s + V s SD ) ,(1)H D = σ ( d + σV Z ) d † σ d σ + U n ↑ n ↓ ,(2)H s = dxdθ σ ψ † σθx p 2 2m s − µ s ψ σθx + ∆ s (Φ)e −inθ+iφs ψ † σθx ψ † −σθx + h.c. ,(3)V s SD = dθ σ t s (θ)ψ † σθx d σ /(2π) + h.c.(4)
Here d is the bare dot energy level, which is controlled by gate voltage, V Z = µ B gB/2 is the Zeeman field (g is Landé g-factor and µ B is Bohr magneton), σ corresponds to electron spin up/down state (±1 if it is a coefficient, ↑ / ↓ if it is an index); d † σ is the QD electron creation operator (with spin σ), n σ = d † σ d σ is the occupation number operator, U is the charging energy (which is the largest energy scale in the system so that the singlelevel approximation is valid); ψ σθx are the shell operators (we omit s = l/r index for the left/right shell), x is the coordinate along the shells, θ is the angle around the shell. The superconducting order parameter in the shells ∆ l/r (Φ, θ) = ∆(Φ) l/r e −inθ+iφ l/r depends on magnetic flux Φ as well as angle θ, φ l/r is the phase at the same angle θ = 0 to the left/right of the QD, n denotes the number of phase windings defined by an integer of the ratio of magnetic flux to flux quantum (Φ 0 = π/e, we set = 1 throughout the paper): n = Φ/Φ 0 ; p 2 /(2m s ) and µ s are the kinetic term and the chemical potential in the respective shells. The last term describes tunneling between the left/right shell and the QD, the tunneling amplitude is given by t s (θ)/(2π) and dependends on the azimuthal angle θ, as the electron state on the QD is shifted away from the shell axis due to the dot gating. We work in the zero-temperature limit (the temperature in relevant experiments is well below characteristic energies). The main difference from [37] is the presence of two superconducting shells instead of one, which adds up and introduces an additional important parameter, namely the phase difference φ = φ l − φ r between the shells (at the same angle θ). The effective Hamiltonian can be obtained by integrating out the shells and introducing the dot Green function as
G(ω) = [ω − H ef f (ω)] −1
:
H ef f (ω) = H ef f,↑ (ω) ⊕ H ef f,↓ (ω) H ef f,σ (ω) = d + σV z 0 0 − d + σV z + Σ U σ + Σ S ,(5)
where the first term corresponds to the bare QD. The last two terms are self-energies from the Coulomb interaction U on the dot and from the proximity effect induced by the superconducting shells, respectively. The proximity effect contributes as
Σ S = − 2 Γ S √ ∆ 2 − ω 2 ω ∆ ef f cos φ 2 ∆ ef f cos φ 2 ω . (6) Here Γ S = πρ s π −π dθ 2π t (θ) 2
stands for the tunneling energy scale averaged over the angle θ around the shell (the contribution comes from hopping from QD to shell and back with amplitude t(θ)/(2π)), ρ s is the density of states at the Fermi energy, φ is the phase difference between left and right shell (at fixed angle θ). We assume symmetric tunneling t r (θ) = t l (θ) = t(θ) to the left/right shell, as corresponding results can be easily generalized for an asymmetric case [39]. Even more nontrivial contribution comes in the numerator of the off-diagonal terms (see Appendix A):
∆ ef f ≈ (1 − δ 0n ) 2∆ π −π t (θ) e iθn dθ 2π π −π t (θ) dθ 2π + δ 0n |∆| . (7)
One can see that for n = 0 and axially symmetric tunneling (t(θ) = const), ∆ ef f = 0! One should note that Eq. (7) is approximate, it works well for weak dependence of tunneling amplitude on θ; for general case see analyses in Appendix A.
The effect can be interpreted as destructive interference, which can be seen from a schematic of a Cooperpair tunneling trajectory between two superconducting hollow cylinders through a QD electronic state, Fig. 1; for axially symmetric case a Cooper pair has equal probability to tunnel between all possible angle positions on superconducting shells. If there is a phase winding around the shells, each trajectory corresponds to some phase difference ∆φ ∈ [0, 2πn], then summing over all trajectories gives exactly zero (which can be written as zero effective gap ∆ ef f ). However, the effective QD state (the bound state wave-function) is rather not centred on the shell axis, as the QD is gated from one side, which results in θ-dependent tunneling amplitude t(θ) and, therefore, nonzero ∆ ef f . As was discussed in [37], the angledependence of tunneling is rather weak as the tunneling is determined by the tails of the wave-function under the superconducting shell, which is screening the electrical field form the gate. The most straightforward consequence of such a destructive interference is the reduction of the Josephson effect by a factor ∆ ef f /∆. However, that is not the whole story. First of all, as it was already briefly discussed, if a flux around half flux quantum is applied, superconductivity is completely suppressed due to Little-Parks effect. Second, even an S-QD-S junction without magnetic flux shows nontrivial behaviour such as 0 − π phase transition. This phase transition was extensively studied theoretically, starting with the first predictions for transition induced by changing the occupation of the QD [13][14][15][16]40] and followed by more sophisticated regimes, when the Kondo effect may play a significant role [24,29,[33][34][35]41]. These effects are due to strong Coulomb interactions on the QD represented by Σ U σ term in the effective Hamiltonian. In [37] it was calculated in Hartree-Fock-Bogoliubov approximation [27,42] (the lowest U -order expansion)
Σ U σ ≈ U n −σ d σ d −σ d † σ d † −σ − n σ ,(8)
which cannot capture the Kondo effect. The latter was studied with powerful numerical approaches [24,[33][34][35] as of now no reliable analytical approach capable of tackling the problem in the most interesting regime of competition between the Kondo effect and superconductivity has been developed. Fully analytical methods are available only for special limiting cases such as Hartree-Fock-Bogoliubov approximation [27,42] and perturbation in cotunneling [43,44] through Yu-Shiba-Rusinov (YSR) state [45][46][47] analogs for ∆ T K or slave-boson mean field approaches in the opposite limit [48,49]. Nevertheless, it is well established that if the Kondo temperature T K (characteristic energy scale) is large enough in comparison to the superconducting gap ∆, the electron on a dot can form a Kondo singlet with quasiparticles in the superconductor (Kondo cloud) and, therefore, the junction stays in the 0 phase even in the odd sector, the cotunneling process is enhanced which in turn increases the supercurrent [25,29,35]. The Kondo temperature depends on tunneling amplitude, Coulomb interaction, and bare QD level energy [50]:
T K ∼ √ U Γ exp π d 2Γ 1 + d U , Γ = 2 Γ s .(9)
The exact proportionality factor 0.28 is well defined only in the middle of the odd occupation region ( d = −U/2) [36,51]. An important question is how the superconducting phase winding affects this competition.
III. ANALYSES OF THE EXPERIMENT
In this section we analyze the results of an experiment performed on a S-QD-S junction with a flux through the full-shell nanowire with the goal to establish topological superconductivity [12]. Several non-trivial features were reported that could potentially indicate the presence of Majorana fermions. First, the differential conductance in a voltage-bias configuration was measured, then the current-phase relation (CPR) was probed in a SQUID geometry. The even-occupied regime does not show anything unexpected: the differential conductance clearly shows a gap-closing around half flux quantum due to the destructive Little-Parks effect and a gap-reopening in the first lobe (Fig. 2c in [12]). Current-bias measurements in the SQUID geometry show a trivial Josephson effect in both lobes. In the odd-occupied regime a bright zero-bias peak develops at the closing of the zeroth lobe (around half flux quantum), which the authors identify as a Kondo peak. In the destructive regime no peak is visible (superconductivity is fully suppressed in the shells).
The zero-bias feature reappears in the first lobe, but less bright and a bit broadened. In the current-bias measurement a π phase is absent in the odd-occupied regime in the first lobe ( Fig. 4 in [12]), while it is present in the zeroth lobe (supercurrent reversal). As it was theoretically predicted, a full-shell nanowire could potentially acquire topological properties in the first lobe [1], which could explain the absence of the π phase and zero-bias peak by hybridization of a dot electron with MBSs [17][18][19][20][21]. However, current-bias measurements in SQUID geometry show the absence of the π phase in the center of the odd sector already at the closing of the zeroth lobe (data in supplemental material of [12], Fig. S8). An enhancement of the supercurrent in the odd state is clearly visible in comparison to the even state, which is a typical feature of the Kondo effect in S-QD-S junctions [24,29,52]. And all these features are qualitatively similar to the ones observed in the first lobe (nicely captured in Figs. 4e-f in [12]): the CPR of the SQUID is given by a sinusoid, but in the odd state the critical current is larger (higher average value), no phase shift is observed. That suggests that the observed effects are of the same origin. As we have shown in the previous section the superconducting phase winding introduces a new important energy scale -∆ ef f , which plays the role of the effective gap for the QD and which is reduced in comparison to |∆| due to destructive interference. In case of nonzero phase winding n > 0 off-diagonal elements of Σ S [see Eq. (8)] are smaller by a factor of ∆ ef f /∆, which suggests that in such a system the relevant parameter for the quantum phase transition between the unscreened doublet to the manybody Kondo singlet ground states is ∆ ef f /T K . A more formal way to see that is to perform a renormalization group (RG) analyses: the RG flow starts at large energy cutoff, off-diagonal terms of Σ U σ get renormalized due to off-diagonal elements of Σ S , see Appendix C. As a result, we were able to deduce that in the first lobe for odd occupation of the QD T K > ∆ ef f and the ground state is a Kondo singlet, which explains the 0 phase behaviour and the supercurrent enhancement as well as zero-bias peak in the differential conductance. Another question arising is whether the Zeeman effect can play a significant role, because the Zeeman field can split the Kondo peak if the g-factor is large enough [53][54][55]. However, it was shown that due to spin-orbit interaction the effective g-factor on a long QD can be renormalized (towards small values) [56][57][58]. That significantly complicates theoretical comparison of Zeeman energy V Z = µ B gB and Kondo temperature T K . In Appendix B we provide some simple analyses of the experimental data.
Another distinctive feature observed in experiment [12] is a change of dissipation between zeroth and first lobes: one can see a strong hysteresis in supercurrent though the SQUID in the zeroth lobe, which indicates underdamped junction regime corresponding to low dissipation. On contrary, in the first lobe no hysteresis is seen; this effect is independent of the QD occupation, therefore, it is not caused by the Kondo effect itself. We suggest that the higher damping can be attributed to lower effective gap (and described in terms of subgap states induced by the vortex [59]). The different junction dissipation regime in two lobes also implies a different ratio of critical and switching current. In the underdamped regime the switching current (which is measured in the experiment) can be significantly lower than the actual critical current, as the macroscopic quantum phase tunneling cannot be neglected, while in the first lobe strong dissipation (overdamped regime) should result in the switching current being in good correspondence with the critical current.
The experimental setup [12] appears to be a very convenient and unique device to study competition between superconductivity and the Kondo effect at relatively low magnetic fields without additional gates to control the tunneling amplitude due to destructive Little-Parks effect. The regime of competition is specifically interesting to study experimentally as no analytical approach exists to provide quantitative description of the system in this regime. The setup allowed the scientists to measure the CPR and differential conductance in the middle of the odd occupation sector all the way from the doublet ground state to the Kondo singlet smoothly varying the superconducting gap by changing the magnetic flux from zero to a half flux quantum, which does not even require going into the first lobe. The well resolved CPR close to the phase transition (at 40 mT and 45 mT for the first device; Fig. S8 in [12],) has a drastic difference, which is an outstanding feature of the change in the character of the ground state and is in perfect qualitative agreement with theoretical predictions (numerics). We suggest that a measurement of the CPR at different values of flux with smaller steps around the transition could be sufficient to establish the transitions between 0, 0 , π and π phases of such a QD-based junction [27,42] in the middle of the odd parity sector (before such transitions have been observed only as a function of gate voltage [31,60]). For this we recommend to fabricate an asymmetric SQUID with an ancilla junction being independent of the flux through the shell (i.e. a separate SIS junction) and having slightly larger critical current so that the CPR of S-QD-S junction is directly observed (large difference in critical currents between the junctions forming the SQUID decreases the contrast of the picture). Further study of the first lobe can provide better understanding of the Kondo cloud formation due to destructive interference. Moreover, it could be interesting to study the effect in shells of larger radius (or thinner shell, so that the superconducting coherence length is shorter than the shell's radius [6]), when Little-Parks effect does not suppress superconducting gap to zero. In that regime suppression of the gap at half flux quantum may not be enough to enhance the Kondo effect, however, the phase winding at higher magnetic fields can still reduce the effective gap to the values below the Kondo temperature, which would result in a zero-bias peak only in the first (or higher) lobe.
IV. CONCLUSIONS AND OUTLOOK
We provided a coherent interpretation for the results observed in an experiment [12] on a transport through a QD between two full-shell nanowires. Due to accurate and sufficient data presented, we were able to establish the effect of superconducting order parameter phase winding on a ground state of the QD and attribute it to the Kondo effect. We showed that the qunatum phase transition between a doublet and a many-body Kondo singlet ground state is controlled by a parameter ∆ ef f /T K |∆|/T K in case of a superconducting phase winding. We discussed the consequences of this transitions and suggested experiments to test the exist-ing theoretical results on the regime of competition of the Kondo effect and superconductivity. Finally, theoretical analyses of the results suggest that the conductance enhancement due to the Kondo effect in a vortex may as well explain zero-bias anomalies observed in different systems, such as the vortexes localized at magnetic impurities of some superconductors [61,62], which requires further studying.
Acknowledgements.
We thank Wolfgang Belzig, Mikhail Pletyukhov, Charles Marcus and Saulius Vaitiekėnas for fruitful discussions. This project received funding from the European Union's Horizon 2020 research and innovation program (ERC Starting Grant, grant agreement No 757725).
where the index s stands for left/right superconductor (we assume |∆| and t to be the same for the left/right shells). As discussed in [37], the approach can be extended to the hollow-cylinder model, which allows us to include a magnetic flux in the model. Here we briefly sketch the main steps of that approach, the difference with [37] being that here we consider superconducting shells both to the right and to the left of the QD, therefore, we need to take a relative phase into account as well. One needs to sum over all the possible positions on the left/right shell, which is done by introducing angledependent tunneling amplitudes t(θ)/(2π) as well as order parameters ∆ s = |∆|e iφs−inθ , where n is the number of the phase windings n = Φ/Φ 0 due to the flux through the shell, see Fig. 1. As was reported in [37] for the axially symmetric case t(θ) = const, the off-diagonal elements for any n = 0 are zero, which can be seen as destructive interference. Realistically, the wave function of an electron on the QD cannot be treated as axially symmetric (with respect to the shell axis) due to gating from one side and due to inhomogeneities. However, we can still assume the tunneling to have a relatively weak angle dependence (as discussed in [37] the tunneling is determined by the wave-function's tails underneath the shell, which are not very sensitive to the gating): t(θ) = t 0 + δt(θ). It is convenient to decompose this tunneling amplitude into harmonics t(θ) = m t m e −imθ , where t m = π −π t(θ)e imθ dθ 2π . Then, one can get the proximity term in the form of Eq. (A1) by integrating the shell contribution over the angle θ. The simplest estimation comes from expanding t 2 (θ) = t 2 0 + 2t 0 δt(θ) + .... The off-diagonal term takes the form
Σ S 01 ≈ − s 2πρ S t 0 |∆| e iφs √ ∆ 2 − ω 2 2π 0 t(θ)e −inθ dθ 2π .(A2)
Let us derive the term in a more accurate fashion. The thin superconducting shell Hamiltonian can be written in cylindrical coordinates as [1,63]
H s = k 2 s + k 2 r + (k θ + eAτ z ) 2 2m s − µ s τ z + + |∆| (cos [−inθ + iφ s ] τ x + sin [−inθ + iφ s ] τ y ) ,(A3)
where k s , k r and k θ are the longitudinal, radial, and tangential components of the momentum operator, m s is the effective electron mass, µ s is the chemical potential in the shell; A = 1 2eR Φ Φ0 is the vector potential, τ i are Pauli matrices acting in Nambu-Gorkov space. One can introduce a generalized angular momentum J z = −i∂ θ + 1 2 nτ z + 1 2 σ z , which is conserved (commutes with Hamiltonian) [1,59], then the eigenvalues m J are good quantum numbers and can take half-integer values:
m J − 1 2 − 1 2 n ∈ Z.(A4)
The resulting angular momentum number m is integer and fixed by m J and n in each spin and Nambu sec-tor (note that it is exact only in the absence of radial spin-orbit interaction [59]). Due to the term 1 2 nτ z in the definition of the generalized angular momentum, for a fixed spin the Hamiltonian is not block-diagonal in the (m, −m) Nambu sectors, but in the (m, −m − n) sectors [37]. The retarded Green function for the shell (decoupled from the QD) between two positions θ and θ is then given by [37]
g s (ω, θ, θ ) = m e −im(θ−θ ) D m,n ω + ks + [m+n−Φ/(2Φ0)] 2 2msR 2 ∆e −iφs+inθ ∆e iφs−inθ ω − ks − [m+Φ/(2Φ0)] 2 2msR 2 e −in(θ−θ ) ,(A5)
where
D m,n = ω 2 − ∆ 2 − ( ks − L m ) 2 − (ω − ks + L m ) Φ n − 2 Φ n L m ,(A6)
with
L m = m + Φ 2Φ0 2 2m s R 2 and Φ n = n − Φ Φ0 2 2m s R 2 .(A7)
Then the proximity effect of the shells on the QD can be described by the self-energy [37] Σ S (ω) = s dk s dθ 2π dθ 2π t(θ)g s (ω, θ, θ )t(θ ). (A8)
In the middle of the first lobe Φ/Φ 0 = n = 1 the result can be written in the form of Eq. (A1) but with t replaced by t 0 = π −π t(θ) dθ 2π , which is just the tunneling amplitude averaged over θ, and |∆| in the nnumerator of the offdiagonal terms replaced by
∆ ef f = m t −m t m+n t 2 0 ∆ .(A9)
In case of weak modulation compared to the angleindependent tunneling (t 0 |δt(θ)|) one gets for n = 0 However, these terms do not change the qualitative picture, therefore, Eq. (A2) gives a reasonable estimation, which makes ∆ ef f given by (A10) a relevant energy scale instead of |∆|.
∆ ef f ≈ 2 ∆ π −π t (θ) e iθn dθ 2π π −π t (θ) dθ 2π |∆| ,(A10)
The spin-orbit interaction was not taken into account in the simplified hollow-cylinder model, as it can play a role only in the semiconducting nanowire itself, therefore, it should not be crucial for the shell modelling and can only affect tunneling process to the QD states (introducing small spin-dependent corrections to the tunneling amplitudes [64]) and QD parameters. For the latter we can use empirical effective parameters, i.e. in the main text we stated that the effective Zeeman splitting is reduced due to spin-orbit interactions so that the effective g factor is small [56][57][58]. Therefore, we claim that including spin-orbit interactions as well as more realistic (i.e. hexagonal) geometry into consideration should not affect the result qualitatively. ric (left/right) tunneling, Σ U 01 (∞) = 0, U (∞) = U . In case of no phase winding, which was studied in [29], ∆ ef f = |∆| and Γ s = Γ/2, while in our case (the first lobe), the effective gap is reduced, which implies that the flow is slow and the off-diagonal term Σ U 01 (Λ) does not reach the critical value for the quantum phase transition into the π phase, which can be seen from the expression for the supercurrent [29]:
J = 1 2π Γ S 2 ∆ 2 ef f sin φ D Λ=0 (iω)(ω 2 + |∆| 2 ) − 2 Γ S ∆ ef f Σ U 01 (0) sin(φ/2) D Λ=0 (iω) ω 2 + |∆| 2 dω 2π .(C4)
If the first term dominates after the renormalization of Σ U 01 , which is the case of low ∆ ef f , the junction is in the 0 phase, otherwise in the π phase. One should note that Eq. (C4) is exact, the 0 or π phase behaviour of the QD-based junction is determined by the ratio of two competing terms, however, it is Σ U 01 (0) in the second term which cannot be calculated exactly in the limit of strong interactions.
FIG. 1. Schematic representation of Cooper-pair tunneling through a QD electron state (green sphere) between two hollow-cylinder superconductors (blue) representing the shells and accumulation layers of underlying nanowires. The tunneling amplitude t(θ) (between the dot and each shell) is angledependent, as the electron wave function on the dot is shifted from the shell axis due to gating; the superconducting order parameter on the shell has angle dependence -nonzero phase winding (n = 0 for large enough magnetic flux Φ > Φ0/2).
which is the same as Eq. (A2) [if one substitutes ∆ ef f for ∆ in the numerator of the off-diagonal terms of Eq. (A1)].Away from the first lobe center (Φ/Φ 0 = 1) the offdiagonal elements have additional terms in the denomi-
Appendix A: Effective gapThe effective Hamiltonian(5)was derived in[37]within the SIA model and consists of three parts: bare QD energy, Coulomb interactions, and proximity effect from the shell. For a usual S-QD-S junction (no phase winding) the latter is given by[27,29]Appendix B: Analysis of Kondo conductance peakHere we provide a simple estimate from below on the Kondo temperature based on data provided in[12]and supplemental material therein. From the supplemental material (Fig. S7)we can see at 45 mT a Kondo enhancement in the odd state (larger supercurrent amplitude in the odd state in comparison to the even state and no π phase shift), while at 40 mT we see a π phase behaviour and similar supercurrent amplitudes in odd and even sectors. As a result, we can crudely estimate the Kondo temperature from below as T K ∆ at 45 mT. NRG calculations predict a transition from a doublet ground state to a many-body Kondo singlet at T K ≈ 0.3∆[33,35,41], however, it was defined at zero phase difference, while the current-phase relation at 45 mT already shows a critical current enhancement as well as 0 phase behaviour at all the phases, for nonzero phase the transition occurs at higher values of T K /∆[36,65], which suggests that the system is already deep in the Kondo regime. The gap is suppressed at 50 mT, we thus can use a simplified formula for a thin shell of radius R[66,67]):where ξ is the superconducting coherence length, n = Φ/Φ 0 the number of phase windings in the shell (n = 0 for the case here, as the system is still in the zeroth lobe). Therefore, for an estimate we take 1 − ξ 2 R 2 − 50 120 2 = 0, as the center of the first lobe is at B = 120 mT, which corresponds exactly to one flux quantum. Then at 45 mTThe order parameter in the Al shell may be estimated from below from the differential conductance at zero magnetic field: ∆ 0 > 0.1 meV. Then, we can estimate the Kondo temperature from below: T K > 19 µeV; one can see that this estimate from below gives a value twice larger than originally assumed. We stress again that the estimate is crude and well below the real value, a more accurate evaluation of the Kondo temperature could be possible with more data around the transition from the doublet to the Kondo singlet (40 − 45 mT) or the Kondo peak analysis in the destructive regime (normal state). Data on the second device provided in the supplemental material of Ref.[12]shows similar features suggesting an enhancement of the Kondo effect. Moreover, inFig. S2the differential conductance peak at low bias voltage appears to be split in the first lobe, which may be due to Zeeman splitting of the Kondo peak.Fig. S4shows that the critical current in the ancilla junction is larger in the first lobe than in the QD-based junction (the amplitude of the supercurrent in the SQUID is changing with the QD occupation), which suggests that the tunneling amplitude in the second device could be smaller than in the first one. Therefore, we expect that the Kondo temperature in the second device is lower (or the effective g-factor can be larger) and the Kondo peak acquires a visible splitting in the magnetic field (2V Z > T K in the first lobe), which cannot be explained by MBSs. Furthermore, all three devices show supercurrent enhancement in the odd state (first lobe) in comparison to even occupation, which is also a typical feature of the Kondo effect. In conclusion, the additional data on the second and third devices in the supplemental material of Ref.[12]supports our idea of an enhancement of the Kondo effect in the first lobe.Appendix C: QD interactionsThe self-energy due to interactions can be fully covered only by elaborate numerical methods, such as NRG[33][34][35]68]. However, some approaches can give at least qualitative insights in the system behaviour at different values of T K /∆. For example, in[29]the author proposed an FRG method[69], which gives within the lowest-order static approximation flow equations from high energy cutoff Λ in the Matsubara non-interacting Green-function G 0,Λ (iω) = θ(ω − Λ)G 0 (iω) in the formThe equations are for the middle of the odd sector (which implies trivial diagonal elements) and symmet-
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| {'fraction_non_alphanumeric': 0.07192263492269997, 'fraction_numerical': 0.03994015481688675, 'mean_word_length': 3.863439839713171, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We analyze results of a recent experiment [D. Razmadze et al., Phys. Rev. Lett., 125, 116803 (2020)] on transport through a quantum dot between two full-shell nanowires and show that the observed effects are caused by the Kondo effect enhancement due to a nontrivial geometry (magnetic flux in a full-shell nanowire) rather than the presence of Majorana bound states. Moreover, we propose that such a setup presents a unique and convenient system to study the competition between superconductivity and the Kondo effect and has significant advantages in comparison to other known approaches, as the important parameter is controlled by the magnetic flux through the full-shell nanowire, which can be significantly varied with small changes of magnetic field, and does not require additional gates. This competition is of fundamental interest as it results in a quantum phase transition between an unscreened doublet and a many-body Kondo singlet ground states of the system. arXiv:2301.12442v1 [cond-mat.supr-con]', 'arxivid': '2301.12442', 'author': ['Aleksandr E Svetogorov \nDepartment of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n', 'Daniel Loss \nDepartment of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n', 'Jelena Klinovaja \nDepartment of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland\n'], 'authoraffiliation': ['Department of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland', 'Department of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland', 'Department of Physics\nUniversity of Basel\nKlingelbergstrasse 82CH-4056BaselSwitzerland'], 'corpusid': 256390315, 'doi': '10.1103/physrevb.107.134505', 'github_urls': [], 'n_tokens_mistral': 15671, 'n_tokens_neox': 13575, 'n_words': 7914, 'pdfsha': 'd347722fb170c2aa429033670441b9d87c86fb2e', 'pdfurls': ['https://export.arxiv.org/pdf/2301.12442v1.pdf'], 'title': ['Enhancement of the Kondo effect in a quantum dot formed in a full-shell nanowire', 'Enhancement of the Kondo effect in a quantum dot formed in a full-shell nanowire'], 'venue': []} |
arxiv |
∆F = 1 constraints on composite Higgs models with LR parity
7 Nov 2012
Natascia Vignaroli
Department of Physics and Astronomy
Department of Physics and Astronomy
Iowa State University
50011AmesIAUSA
Michigan State University
48824East LansingMIUSA
∆F = 1 constraints on composite Higgs models with LR parity
7 Nov 2012
We analyze the bounds on the spectrum of composite Higgs models (CHM) that come from flavor observables, by means of simple two-site effective Lagrangians, which incorporate a custodial symmetry and a Left-Right parity and which could also be adopted in further phenomenological studies on CHM. We derive, in particular, an important constraint on the masses of the (t L , b L ) partners, which does not depend on the flavor structure of the sector beyond the SM. This bound is obtained from the "infrared" contribution to b → sγ induced by the flavor-conserving effective vertex W t R b R . We find that the presence of a custodial symmetry can play a role in protecting this effective coupling and, as a consequence, in attenuating the constraint, which, however, remains of the order of 1 TeV. In addition to this bound, we calculate the constraints from the "ultraviolet" contribution to b → sγ, induced by loops of heavy fermions, and to / K .
Introduction
A possible solution to the hierarchy problem is based on an analogy with the pion mass stabilization in QCD: the Higgs, similarly to the pion, might be a composite state, generated by a new strong dynamics; as such, its mass is not sensitive to radiative corrections above the compositeness scale, assumed to be of the order of the TeV scale. A further protection, that allows the Higgs to be naturally lighter than the other resonances, exists if the composite Higgs is also the pseudo-Goldstone boson of a spontaneously broken global symmetry [1]. A pseudo-Goldstone boson Higgs is expected to be light and as such in agreement with the indication from the LEP electroweak precision data (EWPD). In this project we will reconsider the bounds on the spectrum of Composite Higgs Models (CHM) that come from flavor observables, with a special focus to b → sγ. Instead of considering a full theory we will work in an effective description valid at low energy. In particular, we will refer to a "two-site" (TS) description [2,3], where two sectors, the weakly-coupled sector of the elementary fields and the composite sector, that comprises the Higgs, are linearly coupled each other through mass mixing terms [4]. After diagonalization the elementary/composite basis rotates to the mass eigenstate one, made up of SM and heavy states that are admixture of elementary and composite modes. Heavier particles have larger degrees of compositeness: heavy SM particles, like the top, are more composite while the light ones are almost elementary. In order for composite Higgs models to be compatible with LEP precision data, the presence of a custodial symmetry in the composite sector is strongly suggested to avoid large corrections to the ρ parameter. The absence of large Flavor-Changing Neutral Currents is achieved, instead, by a sort of GIM mechanism, that naturally emerges when the connection between the elementary and the strong sector proceeds via linear couplings [8]. In absence of a symmetry protection, the LEP data also point toward a small degree of compositeness of the left-handed bottom quark (small corrections to Zb L b L ), and, by gauge invariance, of the left-handed top as well. This implies that, in order to obtain a heavy enough top quark, it is necessary to have an almost fully composite right-handed top quark. It has been shown, however, that the corrections to Zb L b L can be suppressed if the custodial symmetry of the strong sector includes a Left-Right parity [7]. This can allow for a smaller right-handed top compositeness. In order to study the phenomenology at energies lower than the compositeness scale, we derive two different models which incorporate a custodial symmetry and a Left-Right parity. We label such models as TS5 and TS10. They describe the low-energy regime of the Minimal Composite Higgs Models (MCHM) defined in Ref. [5,6], in the limit in which only the leading terms in an expansion in powers of the Higgs field are retained 1 . In MCHM the Higgs arises as the pseudo-Goldstone boson associated to the SO(5) → O(4) breaking in the composite sector; where O(4) includes SO(4) ∼ SU (2) L × SU (2) R as well as a parity P LR which exchanges SU (2) L with SU (2) R . Composite fermions can be embedded in a 5 = (2, 2) + (1, 1) representation of SO (5) in the TS5 model and in a 10 = (2, 2) + (1, 3) + (3, 1) in the TS10. TS5 and TS10 extend the two-site description of [2,3] to consider 5 and 10 SO(5) representations for composite fermions. In particular, the TS5 model extends the 'two site' model of Ref. [3] to include the composite fermions needed to give mass to the bottom quark.
We find two important bounds on the masses of the heavy fermions which do not depend on the flavor structure of the sector beyond the SM (BSM). The first comes from the measurement of the Zb LbL coupling, that we already mentioned and that can be suppressed assuming a P LR symmetry. The second is obtained from the infrared (IR) contribution to b → sγ induced by the flavor conserving effective vertex W t R b R . In composite Higgs models there are two classes of effects that lead to a shift of the b → sγ decaying rate compared to the SM prediction: loops of heavy fermion resonances from the strong sector give a ultraviolet (UV) local contribution; they generate, at the compositeness scale, the flavor-violating dipole operators O 7 and O 7 , which define the effective Hamiltonian for the b → sγ decay. The virtual exchange of heavy resonances also generates the effective V+A interaction of the W boson and the SM quarks, W t R b R , which in turn leads to a shift to b → sγ via a loop of SM particles. This latter IR contribution is enhanced by a chiral factor m t /m b and, since in this case the flavor violation comes entirely from the SM V-A current,t L γ µ s L , it gives a Minimal Flavor Violating (MFV) lower bound on the heavy fermion masses. We also discuss the role of a parity P C , which is a subgroup of the custodial SU (2) V , to protect the effective coupling W b R t R . In general, stronger bounds can be obtained from the UV CHM contribution to b → sγ and from / K [18]; however, these latter bounds are model dependent and in principle could be loosened by acting on the NP flavor structure (see, for example, [27]). The bound from the IR contribution to b → sγ, on the other hand, is robust, since it is a MFV effect.
The paper is organized as follows: in sec. 2 we introduce our two-site models; in sec. 3 we discuss the bound from b → sγ; we first calculate the MFV bounds from the infrared contribution in generic CHM, by NDA, and in the specific TS5 and TS10, we then proceed to calculate the non-MFV constraints from b → sγ and from / K ; we draw our conclusions in sec. 4.
Effective theories for composite Higgs models
The idea behind Composite Higgs Models is that the Electro Weak Symmetry Breaking may be triggered by a new strong dynamics, in analogy with the chiral symmetry breaking in QCD. In these theories a new strong sector couples to a weakly coupled sector, which coincides with that of the Standard Model without the Higgs. The Higgs, as the pion in QCD, is a composite state coming from the latter strong dynamics. Its composite nature allows for a solution to the hierarchy problem. Indeed, its mass is not sensitive to radiative corrections above the compositeness scale, assumed to be of the order of the TeV. The EWSB is transmitted to SM fermions by means of linear couplings [4] (generated by some UV physics at the UV scale Λ U V ) between elementary fermions ψ and composite fermions
∆L = λψO + h.c.(1)
This way to communicate the EWSB can give a natural explanation of the hierarchies in the quark masses (through RG evolution of the composite-elementary couplings λ i ) and avoid the tension which occurs when trying to generate large enough quark masses and, at the same time, suppressing FCNC processes 2 . As a consequence of linear couplings a scenario of Partial Compositeness of the SM particles emerges. At energies below the compositeness scale a composite operator O can excite from the vacuum a tower of composite fermions of increasing mass. Linear couplings (1) thus turn into mass mixing terms between elementary fermions and towers of composite fermions χ n
0|O|χ n = ∆ n L mix = n ∆ n ψ χ n + h.c. .(2)L = L el + L com + L mix(3)
Because of the mass mixing terms the physical eigenstates, made up of SM and (new) heavy states, are admixture of elementary and composite modes. The low-energy phenomenology of such theories can be exhaustively studied, and calculation can be made easier, by considering a truncation of each tower of composite fermions to the first resonance, while other heavy states are neglected [2]. For example, the effective Lagrangian describing one elementary chiral field ψ L and its composite partner χ is
∆L =ψ L i ∂ψ L +χ(i ∂ − m * )χ − ∆ LψL χ R + h.c. .(4)
We can rotate the fermions from the elementary/composite basis to the mass eigenstate one, the light(SM)/heavy basis, according to:
tan ϕ L = ∆ L m * |light = cos ϕ L |ψ L − sin ϕ L |χ L |heavy = sin ϕ L |ψ L + cos ϕ L |χ L(5)
Our eigenstate fields are thus a heavy fermion of mass m = m 2 * + ∆ 2 L and a light fermion, to be identified with the SM field, that will acquire a mass after the EWSB. These fields, as we see, are superpositions of elementary and composite states. The angle ϕ L parametrizes the degree of compositeness of the physical fields. In particular, the SM fermion has a sin ϕ L ≡ ∆ L √ m 2 * +∆ 2 L degree of compositeness (and a cos ϕ L ≡ m * √ m 2 * +∆ 2 L degree of elementarity); the mass mixing parameter ∆ L can be naturally much smaller than the mass m * of the composite fermion 3 , therefore, SM fermions are in general mostly elementary with a small degree of compositeness, while heavy fermions are mostly composite with a small degree of elementarity. We have a similar rotation, with angle ϕ R , in the case of right-handed fermions. SM fermions acquire a mass after the EWSB; since the origin of this breaking resides, by assumption, in the composite sector (the Higgs is a fully composite state), the SM fermion mass arises from the composite part of left-handed and right-handed SM fields:
m ψ = Y * v √ 2 sin ϕ L sin ϕ R ,(6)
where Y * is a Yukawa coupling among composites, from which the SM Yukawa y = Y * sin ϕ L sin ϕ R originates. In the following we will assume that the strong sector is flavor anarchic, so that there is no large hierarchy between elements within each matrix Y * and the hierarchy in the masses and mixings of the SM quarks comes entirely from the hierarchy in the elementary/composite mixing angles (such 'anarchic scenario' has been extensively studied in the framework of 5D warped models, see Refs. [8,[12][13][14][15] Experimental data give hints on the type of the new strong dynamics responsible for the EWSB. The LEP precision data suggest the presence of a custodial symmetry in the composite sector to avoid large corrections to the ρ parameter. In order to protect ρ (or equivalently the Peskin-Takeuchi T parameter) the composite sector must respect, minimally, a global symmetry:
SU (2) L × SU (2) R × U (1) X ,
where SU (2) L × SU (2) R is broken to the diagonal SU (2) V after the EWSB; the unbroken SU (2) V invariance acts as a custodial symmetry so that ρ = 1 at tree level.
The SM electroweak group SU (2) L ×U (1) Y can be embedded into SU (2) L ×SU (2) R ×U (1) X , so that hypercharge is realized as Y = T 3 R + X.
The Composite Higgs transforms as a bidoublet (2,2)
under SU (2) L × SU (2) R , H ≡ (H, H c ),
where H is the Composite Higgs doublet and H c = iσ 2 H * is its conjugate. The H VEV breaks the SU (2) L × SU (2) R × U (1) X group down to SU (2) V × U (1) X and leads to the EWSB. Therefore, we have the following relation among charges:
Q = T 3 L + T 3 R + X = T 3 L + Y .(7)
This scheme can also results from models where the Higgs arises as the pseudo-Goldstone boson associated to a SO(5) → SO(4) ∼ SU (2) L ×SU (2) R breaking in the composite sector; or to a SO(5) → O(4) breaking, where O(4) includes SO(4) ∼ SU (2) L × SU (2) R as well as a parity P LR which exchanges SU (2) L with SU (2) R . This enhanced custodial symmetry can suppress the corrections to the coupling Zb L b L , which are strongly constrained by LEP data [7].
P LR and P C symmetries
In MCHM [5] the Higgs arises as the pseudo-Goldstone boson associated to the SO(5) → O(4) breaking in the composite sector; where the enhanced custodial symmetry O(4) includes SO(4) ∼ SU (2) L × SU (2) R as well as a parity P LR which exchanges SU (2) L with SU (2) R . As shown in [7], this P LR parity, as well as the P C symmetry, subgroup of the custodial O(4), can protect the coupling Zb L b L against large corrections from the composite sector. Each composite operator has a definite left and right isospin quantum number, T L,R , and a 3rd component, T 3 L,R . We can also univocally assign to each SM field definite quantum numbers, T L,R , T 3 L,R , corresponding to those of the composite operator to which it couples. P LR and P C are symmetries of the composite sector, P LR exchanges SU (2) L with SU (2) R and P C is the subgroup of SU (2) V that transforms |T L , T R ;
T 3 L , T 3 R → |T L , T R ; −T 3 L , −T 3 R (SO(3)
vectors transform with P C = diag(1, −1, −1)). For P LR (P C ) to be a symmetry also of the interacting terms between SM fields and composite operators, ∆L = λψO + h.c., the SM fields ψ have to be eigenstates of P LR (P C ). This implies:
T L = T R (T 3 L = T 3 R ) (P LR invariance)(8)T 3 L = T 3 R = 0 (P C invariance) .(9)
If the above formulas hold, we can see that the coupling Zψψ,
g ψ = g cos θ W (Q 3 L − Q sin 2 θ W ) ,(10)
is protected against large corrections. Indeed, the electric charge Q is conserved and the charge of the SU (2) L 3rd component, Q 3 L , is conserved by custodial invariance plus P LR symmetry and by P C symmetry. By custodial
U (1) V invariance, δQ 3 V = δQ 3 R + δQ 3 L = 0; if there is also a P LR invariance, δQ 3 R = δQ 3 L , therefore δQ 3 L = 0.
The same conservation, δQ 3 L = 0, is obtained by P C invariance: the SM W 3 L has an odd parity under P C , W 3 L → −W 3 L ; if ψ is a P C eigenstate it must have T 3 L = T 3 R = 0, then the currentψγ µ ψ is even under P C and it cannot couple to W 3 L , which is odd. We will show (sec. 3.2.1) that the P C symmetry can also protect in a similar way the effective coupling W t R b R and, as a consequence, it can be responsible for an attenuation of the bound on heavy fermion masses, coming from the process b → sγ.
In what follows we present the two-site models, TS5 and TS10, which incorporate a custodial symmetry and a P LR parity. 4
TS5
In the TS5 model, we consider composite fermions filling the following SO(4) × U (1) X ∼ SU (2) L × SU (2) R × U (1) X representations:
Q = T T 5/3 B T 2/3 = (2, 2) 2/3T = (1, 1) 2/3 Q −1/3 = B −1/3 T B −4/3 B = (2, 2) −1/3 ,B = (1, 1) −1/3(11)
and the composite Higgs in:
H = φ † 0 φ + −φ − φ 0 = (2, 2) 0(12)
The SO(4) multiplets of composite fermions can be embedded into fundamentals [28] for a study of the same representations in a two-site description of SO(5)). We are thus introducing two classes of composite fermions, those filling a 5 2/3 representation, with X charge X = 2/3 and those in a 5 −1/3 , with X = −1/3.
5 2/3 (−1/3) of SO(5) × U (1) X , that decompose as 5 2/3 (−1/3) = (2, 2) 2/3 (−1/3) ⊕ (1, 1) 2/3 (−1/3) under SU (2) L × SU (2) R × U (1) X (see Ref.
We want to consider, indeed, the possibility that the SM quark doublet (t L , b L ) couples to two different BSM operators, Q 2/3 and Q −1/3 , the first responsible for generating the top mass, the second for generating the bottom mass. (t L , b L ) is linearly coupled to (T, B) through a mass mixing term we call ∆ L1 and to (T , B ) through a mass mixing term ∆ L2 . t R and b R couple respectively toT , through a mass mixing term ∆ R1 , and toB, through a mass mixing term ∆ R2 . The fermionic Lagrangian reads, in the elementary/composite basis:
L =q i L i ∂q i L +ū i R i ∂u i R +d i R i ∂d i R + T r Q (i ∂ − M Q * ) Q +T (i ∂ − MT * )T + Y * U T r Q H T + T r Q (i ∂ − M Q * ) Q +B (i ∂ − MB * )B + Y * D T r Q H B − ∆ L1q 3 L (T, B) − ∆ R1tRT − ∆ L2q 3 L (T , B ) − ∆ R2bRB + h.c. .(13)
where the superscript i runs over the three SM families (i = 1, 2, 3), with q 3
L ≡ (t L , b L ), u 3 ≡ t R , d 3 ≡ b R .
By construction, the elementary fields couple to the composite ones only through the mass mixing terms, shown in the last row of (13). This implies that the SM Yukawa couplings arise only through the coupling of the Higgs to the composite fermions and their mixings to the elementary fermions. We further assume that the strong sector is flavor anarchic, so that the hierarchy in the masses and mixings of the SM quarks comes from the hierarchy in the mixing parameters ∆ i L,R . In this case the mixing parameters of the light elementary quarks can be safely neglected and one can focus on just the third generation of composite fermions. 5 As a consequence of the elementary/composite mass mixings, the top and the bottom masses arise, after the EWSB, from the Yukawa terms in the Lagrangian (13), Y * U T r Q H T and Y * D T r Q H B . The top mass will be proportional to ∆ L1 ∆ R1 and the bottom mass to ∆ L2 ∆ R2 . The small ratio between the bottom and the top quark masses can be thus obtained both for ∆ L2 ∆ L1 (∆ R2 ∼ ∆ R1 ) and for ∆ R2 ∆ R1 (∆ L2 ∼ ∆ L1 ). For t R , b R and their excited states the rotation from the elementary/composite basis to the mass eigenstate one, the SM/heavy basis, is given by:
tan ϕ R = ∆ R1 MT * s R ≡ sin ϕ R c R ≡ cos ϕ R tan ϕ bR = ∆ R2 MB * s bR ≡ sin ϕ bR c bR ≡ cos ϕ bR t R = c R t el R − s RT com R T R = s R t el R + c RT com R b R = c bR b el R − s bRB com R B R = s bR b el R + c bRB com R (14) s R (s bR ) defines the degree of compositeness, ξ tR (ξ bR ), of t R (b R ); c R (c bR ) that ofT (B), ξD.
We will diagonalize analytically the mixing among q 3 L and the corresponding excited states by requiring the simplifying assumption: ∆ L2 ∆ L1 , that can naturally follow, for example, from the RG flow in the full theory [6]. The first two generations of elementary quarks do not need a field rotation from the elementary/composite basis to the mass eigenstate basis, since they do not mix with the composite fermions and can thus be directly identified with the corresponding SM states. We can see that in this model t R and b R are both P C and P LR eigenstates, since they couple to
SU (2) L × SU (2) R singlets (T L (T ,B) = T R (T ,B), T 3 L (T ,B) = T 3 R (T ,B) = 0). Instead, t L is a P LR eigenstate only in the limit (∆ L1 = 0) in which it decouples from T (T 3 L (T ) = T 3 R (T )). Similarly, b L is a P LR eigenstate only for ∆ L2 = 0, in which case it decouples from B (T 3 L (B ) = T 3 R (B )
). So far we have made field rotations to the mass eigenstate basis before the EWSB. After the EWSB, the SM top and bottom quarks acquire a mass, and the heavy masses get corrections
of order Y * v √ 2m * 2 .
In the following, we assume x ≡ Y * v √ 2m * 1 and compute all quantities at leading order in x.
2.2.1 ∆ L2 ∆ L1
In this case, since ∆ L2 ∆ L1 , b L is, approximately, a P LR eigenstate so, approximately, we have a custodial symmetry protection to Zb LbL . The small ratio between the bottom and the top quark masses is obtained for
∆ L2 ∆ L1 (∆ R2 ∼ ∆ R1 ); we have: m t = v √ 2 Y * U s 1 s R (15) m b = v √ 2 Y * D s 2 s bR ,(16)
where
s 1 = sin ϕ L1 = ∆ L1 √ M 2 Q * +∆ 2 L1
defines the (t L , b L ) degree of compositeness, ξ qL , and s 2 is a rotation angle proportional to ∆ L2 , s 2 = ∆ L2 M Q * cos ϕ L1 . The physical masses of the heavy fermions read:
MT = M 2 T * + ∆ 2 R1 MB = M 2 B * + ∆ 2 R2 M T = M B = M 2 Q * + ∆ 2 L1 M T 5/3 = M T 2/3 = M Q * = M T c 1 M T = M B = M 2 Q * + ∆ 2 L2 M Q * M B−1/3 = M B−4/3 = M Q *(17)
where c 1 ≡ cos ϕ L1 is the degree of compositeness, ξ D , of the SU (2) L doublet D = (T, B). Details can be found in App. A.1.
In order for the strong sector to respect the custodial invariance, as we have shown, composite fermions have to fill multiplets of SU (2) L × SU (2) R × U (1) X . As a consequence, the heavy partner of the SM doublet q 3 L = (t L , b L ), D = (T, B) (= 2 1/6 under the SM electroweak group), is embedded in a larger multiplet, the bidoublet Q 2/3 = (2, 2) 2/3 , that includes an other doublet of heavy fermions, (T 5/3 , T 2/3 )(= 2 7/6 ). The heavy fermions T 5/3 and T 2/3 in this latter doublet are called custodians. They share the same multiplet of the heavy partners of q 3 L but they do not mix directly with the SM fermions. This implies that their masses tend to zero in the limit in which t L becomes fully composite (see for example the discussion in [25]). This can be seen from eq. (17): M T 5/3(2/3) is zero for c 1 = 0, i.e. for a fully composite t L (s 1 = 1).
TS10
In TS10 we consider composite fermions embedded into a 10 2/3 representation of SO(5) × U (1) X , that decomposes as
10 2/3 = (2, 2) 2/3 ⊕(1, 3) 2/3 ⊕(3, 1) 2/3 under SU (2) L ×SU (2) R × U (1) X .
Therefore we refer to this field content in the composite sector:
Q 2/3 = T T 5/3 B T 2/3 = (2, 2) 2/3 Q 2/3 = T 5/3 T B = (1, 3) 2/3 ,Q 2/3 = T 5/3 T B = (3, 1) 2/3 H = φ † 0 φ + −φ − φ 0 = (2, 2) 0(18)
and to the following fermionic Lagrangian in the elementary/composite basis:
L =q 3 L i ∂q 3 L +t R i ∂t R +b R i ∂b R + T r Q (i ∂ − M Q * ) Q + T r Q i ∂ − MQ * Q + T r Q i ∂ − MQ * Q + Y * T r HQQ + Y * T r Q HQ − ∆ L1q 3 L (T, B) − ∆ R1tRT − ∆ R2bRB + h.c. .(19)
We have the following expressions for the top and bottom masses:
m t = v 2 Y * s 1 s R , m b = v √ 2 Y * s 1 s bR(20)
and for the heavy fermion physical masses:
MT = M 2 Q * + ∆ 2 R1 MB = M 2 Q * + ∆ 2 R2 = MT c R /c bR MT c R MT 5/3 = MT 5/3 = MT = MB = MT c R M T = M B = M 2 Q * + ∆ 2 L1 M T 2/3 = M T 5/3 = M T c 1 .(21)
More details can be found in App. A.2. Besides the custodians T 5/3 and T 2/3 , which are light in the case of a composite q 3 L ,T 5/3 and the fermions in theQ 2/3 triplet become light for a t R with a large degree of compositeness (alsoB becomes light in this case). In this model, both t R and b R are not P LR eigenstates and only t R is a P C eigenstate, as a consequence of the couplings toQ (
T L (T ,B) = T R (T ,B); in particular, b R is not a P C eigenstate, since T 3 R (B) = 0. b L is exactly a P LR eigenstate.
Zb LbL in the TS Models
Shifts in the Z coupling to b L , g Lb , have been extensively studied in the literature. See, for example, the studies [29] in the context of Randall-Sundrum models and [30] in two-site descriptions. The shifts arise after the EWSB because of electroweak mixings among b L and heavy fermions. There is also a contribution from the mixing among neutral gauge bosons; however this mixing is of the order ( v M * ) 2 1, where M * stands for the heavy neutral boson mass, and we will neglect it in what follows. In two-site models without P LR symmetry there is no custodial symmetry protection to Zb LbL and so the shift on g Lb is large. Naive Dimensional Analysis (NDA) [10] gives (see, for example, [16,26]
): δg Lb g Lb ∼ m 2 t M 2 Q * s 2 R ∼ Y 2 * v 2 s 2 1 M 2 Q * .(22)
This formula has been obtained by approximating
q 2 = M 2 Z 0. At q 2 = M 2 Z the shift receives O M 2 Z M 2 Q * corrections: δg Lb g Lb ∼ M 2 Z s 2 1 M 2 Q * ∼ v 2 Y 2 * s 2 1 M 2 Q * g 2 Y 2 * .(23)
When compared to (22), there is a suppression g Y *
2
(see for example [11]), so we will neglect it in the following.
LEP and SLD experiments fix an upper bound of 0.25% for the (positive) shift in the g Lb from its SM value. Therefore, from the eq. (22), we derive the following bound for the heavy fermion mass in models without custodial symmetry protection to Zb LbL :
M Q * (3.2) 1 s R TeV .(24)
In order to respect this limit without requiring too large heavy fermion masses, that would contrast with naturalness arguments, it is necessary to have a quite composite right-handed top (i.e., a not small s R ). On the contrary, in models with custodial symmetry protection to Zb LbL , there is no such restriction for the t R degree of compositeness and bounds are weaker than the one in (24). Indeed, in the TS5 with ∆ L2 ∆ L1 , where we have approximately a custodial symmetry protection to Zb LbL (the breaking is proportional to ∆ L2 and is thus small), we obtain: As expected, the shift is proportional to s 2 2 (i.e., it is proportional to ∆ 2 L2 , the size of the custodial symmetry breaking) and it is small (notice that is also smaller than the effect at non-zero momentum). In the TS10, we obtain, again, a small shift:
δg Lb g Lb = Y * v √ 2 2 s 2 c bR √ 2MB 2 [T 3 L (B) − T 3 L (b L )] = 1 2 m 2 b M 2 Q * c 4 bR s 2 bR 1 2 m 2 t M 2 Q * s 2 2 s 2 R . (25) b R b L t L t L s L γ W V tsδg Lb g Lb = Y * v √ 2 2 s 2 1 M 2 Q * c 4 bR (T 3 L (B) − T 3 L (b L )) + (T 3 L (B ) − T 3 L (b L )) = − m 2 b M 2 Q * 2 − s 2 bR 2 − m 2 b M 2 Q * .(26)
Despite b L is an exact P LR eigenstate in the TS10, there is still a small modification that comes from the coupling of b R , that explicitly breaks P LR . Notice that δg Lb = 0, if we have s bR = 0.
3 Bounds from flavor observables 3.1 Constraint from the process b → sγ
We define, following [19], the effective Hamiltonian for b → sγ:
H ef f = − G F √ 2 V * ts V tb C 7 (µ b )O 7 + C 7 (µ b )O 7 ,(27)
where
O 7 = e 8π 2 m bb σ µν F µν (1 − γ 5 )s and O 7 = e 8π 2 m bb σ µν F µν (1 + γ 5 )s.
In the SM the W boson has a purely V −A interaction to the fermions and so the contribution to the b → sγ process has to proceed through mass insertions in the external legs (see Fig. 1). The Wilson coefficient C 7 is thus negligible, because of a suppression by a factor m s /m b in respect to the Wilson coefficient C 7 , that, evaluated at the weak scale µ w is [19]
C SM 7 (µ w ) = − 1 2 − (8x 3 t + 5x 2 t − 7x t ) 12(1 − x t ) 3 + x 2 t (2 − 3x t ) 2(1 − x t ) 4 ln(x t ) ,(28)with x t = m 2 t M 2 W .
In composite Higgs models there are two classes of effects that lead to a shift of the b → sγ decaying rate compared to the Standard Model prediction. The first comes from loops of heavy fermion resonances from the strong sector that generate the flavor-violating dipole operators O 7 , O 7 at the compositeness scale. We will refer to this as the UV contribution. The second contribution comes from the tree level exchange of heavy resonances, which generates an effective V+A interaction of the W boson and the SM quarks which in turn leads to a shift to b → sγ via a loop of SM particles. This latter IR contribution is enhanced by a chiral factor m t /m b . Since in this case the flavor violation can come entirely from the SM V-A current, it gives a quite model-independent lower bound on the heavy fermion masses. By taking into account the experimental average value for the b → sγ branching ratio [20] and the theoretical calculation [21], we get, if the new physics contributions to C 7 , C CH 7 , and to C 7 , C CH 7 , are considered separately, the bounds (see Appendix B):
− 0.098 C CH 7 (m * ) 0.028 (29) |C CH 7 (m * )| 0.37 ,(30)
where m * denotes the mass of the heavy fermions in the loop (we take m * = 1 TeV).
The infrared contribution to b → sγ from the composite Higgs model is at the weak scale µ w instead of m * (we take µ W = M W ); therefore, we have to account for a scaling factor
C CH 7 (µ w ) = α s (m * ) α s (m t ) 16/21 α s (m t ) α s (µ w ) 16/23 C CH 7 (m * ) ≈ 0.79C CH 7 (m * )(31)
We get:
− 0.077 C CH 7 (µ w ) 0.023 (32) |C CH 7 (µ w )| 0.29(33)
While the infrared contribution to C 7 involves a flavor-conserving operator and brings to a MFV bound, the infrared contribution to C 7 as well as the ultraviolet contributions to C 7 and to C 7 involve flavor-violating operators. As a consequence, they will require some assumptions on the flavor structure of the NP sector.
We will now evaluate the bounds on heavy masses that come from the infrared contribution to C 7 . We will first present estimates of such bounds in generic composite Higgs models, which can be obtained by NDA. Then we will calculate the bounds in the specific two-site model TS5 and TS10, introduced in sec.s 2.2 and 2.3.
MFV bound from the infrared contribution to C 7
The infrared contribution to the process b → sγ is a one loop contribution from the W boson accompanied by top quarks, where a mass insertion in the intermediate top quark states is allowed by the presence of a (V +A) interaction of the W boson with the top and the bottom quarks (Fig. 2). This interaction originates from a term:
L ⊃ C R O R ,(34)
where O R is the dimension-6 operator: Figure 2: 1 loop Infrared contribution to C 7 . The red blob denotes the effective coupling W t R b R , generated from the composite sector.
O R ≡ H c † iD µ Ht R γ µ b R + h.c. . (35) b R t R t L s L γ W V tsW − b R t R B T φ † 0 φ † 0 Figure 3: The CHM contribution to the effective coupling W t R b R (At order Y * v √ 2m * 2 ).
At low energy, after the EWSB, the interaction in (34) gives:
L ⊃ C R v 2 2 g 2 √ 2b R γ µ t R W − µ .(36)
This interaction gives a contribution to the Wilson coefficient C 7 in the eq. (27). We find:
C CH−IR 7 (µ w ) = C R v 2 2 m t m b f RH (x t )(37)where x t = m 2 t M 2 W and f RH (x t )
is the loop function [22]:
f RH (x t ) = − 1 2 1 (1 − x t ) 3 2 3 − x 3 t 2 − 3 2 x t + 2 + 3x t log(x t ) + 1 (1 − x t ) 3 − x 3 t 2 + 6x 2 t − 15 2 x t + 2 − 3x 2 t log(x t ) .(38)W t R b R , v R ≡ C R v 2 2 .
By considering the bound in (32) and the relation in (37), we obtain:
− 0.0004 < v R < 0.0013 .(39)
This bound from b → sγ can be compared with that from the measurement of the W tb anomalous couplings at colliders. Ref. [23] reports an expected bound of −0.012 < v R < 0.024, that can be imposed by 14 TeV LHC measurements with 30 fb −1 . This latter can be obtained from studies on cross sections and top decay observables (angular distributions and asymmetries) in the single top production at the LHC. Present searches for anomalous W couplings at the 7 TeV LHC [24] fix still mild bounds on v R , −0.34 < v R < 0.39, with 0.70 fb −1 . We can see that the bound obtained from b → sγ is much stronger than that from the v R measurement at collider.
The CHM contribution to the effective coupling W t R b R is given by the exchange of heavy fermions that mix electro-weakly with t R and b R ( fig. 3). At the order x 2 , only the SU (2) L heavy doublets which are partners of (t L , b L ) contribute to C R . This latter can be easily estimated by NDA [10]:
C R ∼ Y 2 * ξ bR ξ tR ξ 2 D M 2 D ∼ y b y t M 2 D ξ 2 D ξ 2 qL .(40)
(40) implies:
C CH−IR 7 (µ w ) ∼ m 2 t M 2 D f RH (x t ) ξ 2 D ξ 2 qL .(41)
Applying the condition in (32) to this infrared contribution, we get the estimated bound:
M D 1.0(0.54) ξ qL TeV ,(42)
where the first number and the second number in parenthesis refer respectively to the case of a positive and of a negative C CH−IR 7 contribution. Notice that in the case of a positive C CH−IR 7 contribution we obtain a stronger bound on M D , since the constraint in (32) is asymmetric. We find that a subgroup of the custodial symmetry SU (2) V , the P C parity, can give a suppression to the W t R b R coupling and, as a consequence, to the CHM infrared contribution to b → sγ. The estimates we have just reported refer to generic composite Higgs models where there is not such P C protection.
Protection by P C parity
The P C protection against the generation of the W t R b R vertex acts similarly to the P LR and P C protection against large corrections to the Zb L b L coupling, which we have discussed in sec. 2.1. P C is a symmetry of the sector BSM, that is respected also by the interactions of t R and b R if these latter are P C eigenstates. Since P C acts as diag(1, −1, −1) on SO(3) vectors, the W is not a P C eigenstate (the composite partners of W 1 and W 2 have not the same P C eigenvalue). In the case in which t R and b R are both P C eigenstates, both the t R and the b R interactions must respect the P C parity. Then, the W t R b R vertex, which is P C violating, since the W is not a P C eigenstate, can arise only by paying for an additional factor, that gives a suppression. Whereas, in models where t R and b R are not both P C eigenstates and, as such, their interactions have not to respect the P C parity, the W t R b R vertex can be generated without suppressions. The TS5 falls into the class of models with P C protection, since in the TS5 both t R and b R are P C eigenstates. Considering the TS5, we can evaluate the suppression factor to W t R b R due to the P C protection. We can find it in an easy way by promoting ∆ L1 and ∆ L2 to spurions, which enforce a SU (2) L × SU (2) R invariance:
−∆ L1q 3 L (T, B) → −q 3 L Q 2/3∆L1 −∆ L2q 3 L (T , B ) → −q 3 L Q −1/3∆L2 , where∆ L1 = (∆ L1 , 0) ≡ (1, 2) 1/6 and∆ L2 = (0, ∆ L2 ) ≡ (1, 2) 1/6 . We can thus write the O R operator (35) in the SU (2) L × SU (2) R invariant way: O R = 1 f 2q 3 R∆ L1 V µ∆ † L2 q 3 R γ µ + h.c. ,(43)
where f has the dimension of a mass, q 3 R = (t R , b R ) ≡ (1, 2) 1/6 and V µ ≡ H c † iD µ H. Since P C is a subgroup of the custodial SU (2) V , the SU (2) × SU (2) invariant operator in (43) is also a P C invariant. We can notice that the P C invariance has brought to an additional factor
∆ L1 ∆ L2 f 2 compared to (35). Without P C protection, the D = (T, B) contribution to the W t R b R effective vertex in the TS5 reads s R s bR c 2 1 Y * v √ 2M D 2 = m b m t M 2 D c 2 1 s 2 1 ;
the request for P C invariance brings to the additional factor ∆ L1 ∆ L2
f 2 . For f 2 = M Q * M Q * , we obtain Y * v √ 2M D 2 s R s bR c 1 ∆ L1 M Q * c 1 ∆ L2 M Q * = Y * v √ 2M D 2 s R s bR s 1 s 2 = m b m t M 2 D ,
that is a suppression by a factor s 2 1 /c 2 1 ≡ ξ 2 qL /ξ 2 D .
We can thus return to the estimated bounds on M D from C CH−IR 7 in eq. (42), and consider the case in which there is a P C protection to the t R and b R interactions. In such case the C R contribution becomes:
C R ∼ y b y t M 2 D (with P C ) ,(44)
which implies
C CH−IR 7 (µ w ) ∼ m 2 t M 2 D f RH (x t ) (with P C )(45)
and an estimated bound:
M D 1.0(0.54) TeV (with P C ) .(46)
We will now calculate the bounds on M D from C CH−IR 7 in the specific TS5 and TS10 models. As already discussed, the TS5 belongs to the class of models with P C protection. The TS10, instead, falls in the class of models without P C protection, because in the TS10 b R is not a P C eigenstate. We thus expect that the bound in the TS10 will receive an enhancement factor c 1 /s 1 , compared to that in the TS5.
In the TS5 we have a contribution to the O R operator in (35) both from the doublet D = (T, B) in the X = 2/3 representation and from the doublet D ≡ (T , B ) in the X = −1/3. We find:
C T S5 R = − y b y t M 2 D 1 + M 2 D M 2 D .(47)
This implies:
C CH−IR−T S5 7 (µ w ) = − m 2 t M 2 D f RH (x t ) 1 + M 2 D M 2 D .(48)
Notice that the C T S5 , that gives a contribution to C R . We obtain
C T S10 R = y b y t M 2 D c 2 1 s 2 1 ,(50)
which implies:
C CH−IR−T S10 7 (µ w ) = m 2 t M 2 D f RH (x t ) c 2 1 s 2 1 .(51)
From the condition in (32) we get finally the bound:
M TS10 D (0.54) c 1 s 1 TeV .(52)
Notice that, differently from the case of the TS5 contribution, C CH−IR−T S10 7 (µ w ) is negative. As such, it is constrained less strongly by the condition in (32). As expected, we have found a c 1 /s 1 enhancement of this bound, compared to (49).
We now proceed to evaluate the bounds from the C 7 contribution and then those from the UV contributions. As we already pointed out, these are contributions that involve flavorviolating operators and require assumptions on the flavor structure of the NP sector. In what follows we will consider the case of flavor anarchy of the composite Yukawa matrices. This scenario, we remind, assumes that there is no large hierarchy between elements within each matrix Y * and the quark mass hierarchy is completely explained by the elementary/composite mixing angles. We also set, for simplicity, Y * U = Y * D = Y * .
Non-MFV constraints
Generational mixing
After the EWSB, the mass eigenstate basis is obtained, as in the SM, using unitary transformations: (D L , D R ) and (U L , U R ) for down and up-type quark respectively. We will assume that the left rotation matrix has entries of the same order as those of the Cabibbo-Kobayashi-Maskawa matrix:
C CH−IR 7 (µ w ) ∼ (ytv) 2 M 2 D ξ 2 D w/ P C ESTIMATED TS5 M D 1.0(0.54) TeV M D 1.4 TeV MFV Bounds ∼ (ytv) 2 M 2 D ξ D ξ qL 2 w/o P C ESTIMATED TS10 M D 1.0(0.54)/ξ qL TeV M D 0.54/s 1 TeV C CH−IR 7 (µ w ) ∼ (ytv) 2 M 2 D ξ 2 D ms m b V 2 ts w/ P C ESTIMATED TS5 M D 0.80 TeV M D 1.1 TeV ∼ (ytv) 2 M 2 D ξ D ξ qL 2 ms m b V 2 ts w/o P C ESTIMATED TS10 M D 0.80/ξ qL TeV M D 0.80/s 1 TeV C CH−U V 7 (m * ) ∼ (Y * v) 2 M D MD ξ D ξD ESTIMATED TS5 TS10 M D MD 1.5(0.79)Y * TeV M D MD 0.52(0.28)Y * TeV M D MB 0.75(0.40)Y * TeV C CH−U V 7 (m * ) ∼ (Y * v) 2 M D MD ξ D ξD ms m b V 2 ts ESTIMATED TS5 TS10 M D MD (1.1)Y * TeV M D MD (0.40)Y * TeV M D MB (0.58)Y * TeV(D L ) ij ∼ (V CKM ) ij .(53)
The assumption of anarchical Y * fixes the form of the rotation matrix D R to be:
(D R ) ij ∼ m i m j 1 (D L ) ij for i < j .(54)
Considering the estimates (53) and (54), we can evaluate the generational mixing factors in the composite Higgs model contributions to C 7 (UV) and C 7 . For the ultraviolet contribution to C 7 , we consider the presence of a mass insertion that can generate the operatorb L σ µν F µν s R . This mass insertion brings to a factor m b (D R ) 23 ∼ ms (D L ) 23 ∼ ms Vts ; where we have first used the estimate in (54) and then that in (53). The ultraviolet contribution to C 7 involves the operatorb R σ µν F µν s L and we obtain, from the mass insertion, a generational mixing factor m b (D L ) 23 ∼ m b V ts ; where the last similitude follows from the assumption in (53). Evaluating, similarly, the generational mixing factor for the vertex W t R s R in C CH−IR comes entirely from the SM vertex W t L s L and it is accounted by a factor V ts . Therefore, we find that the composite Higgs model contribution to the Wilson coefficient C 7 is enhanced by a factor m s m b V 2 ts ∼ 8 (55) compared to the contribution to C 7 both in the ultraviolet and in the infrared case.
Infrared contribution to C 7
Taking into account the generational mixing factor in (55), the composite Higgs model contribution to the Wilson coefficient C 7 (in Fig. 4) is given by:
C CH−IR 7 (µ w ) = C R v 2 2 m s m b V 2 ts m t m b f RH (x t ) .(56)
Considering the estimates for C R in (40) in the TS10. We can discuss how the bound on heavy masses can change in the case of a fully composite top: in the TS5 the bound on doublet heavy fermion (49) does not depend on the top degree of compositeness (this remains almost true considering the full numerical calculation) and we obtain quite strong MFV bounds both for composite t L and composite t R . In the TS10, because of the P C protection, we obtain strong bounds in the case of a fully composite t R (eq. (52)). Ref. [25] finds that corrections to S and T parameters give only weak constraints on a composite t R (both in TS5 and in TS10). The IR contribution to b → sγ, on the contrary, put a quite strong constraint, especially in the TS10, on this limit case. One can finally discuss the validity of our results, which have been obtained 'analytically' (i.e. by considering an expansion in x ≡ Y * v √ 2m * and retaining only the O(x) terms). We find that the results from the numerical calculation of the bounds, obtained by diagonalizing numerically the fermionic mass matrices, do not differ more than O(1) from those we have shown, which are obtained at order x, in the assumption x 1. This can be also found by considering that the exchange of relatively light custodians, that can give a contribution Y * v √ 2m CU ST * > 1 to the effective W t R b R vertex, has to be followed by the exchange of heavier composite fermions, that reduces the overall contribution. By definition, indeed, the custodians do not directly couple to SM fermions, therefore their contribution to W t R b R is always accompanied by the exchange of heavier composite particles.
Ultraviolet contribution
In this case the P C parity does not influence the bounds and we get contributions of the same size in the different models. The leading contribution comes from diagrams with heavy fermions and would-be Goldstone bosons in the loop 6 (Fig. 5).
C CH−U V 7 , C CH−U V 7 ∝ s Li Y * ik Y * kl Y * lj s Rj(61)
The contribution (61) is not aligned with the mass matrix m dij ∼ s Li Y * ij s Rj , therefore, after the EWSB it remains non diagonal in the flavor space. Before going on the specific TS5 and TS10 models, we can obtain estimated bounds from the UV contributions in generic composite Higgs models, by means of NDA. We obtain:
C CH−U V 7 ∼ (Y * v) 2 M D MD ξ D ξD ,(62)
whereD denotes a heavy fermion which is a SU (2) L singlet, and
C CH−U V 7 ∼ m s m b V 2 ts (Y * v) 2 M D MD ξ D ξD ,(63)
where we have taken into account the generational mixing factor in (55). By comparing these results with those from the IR contributions in (42, 46), we see that the UV contribution gives approximately a bound Y * /y t ( Y * yt ξ q L , in the case of models without P C protection) times stronger than the one from the IR contribution to C 7 . Such UV bounds, however, are not as robust as the IR one, since they require, as we already pointed out, assumptions on the flavor structure of the BSM sector. In particular, we have estimated them in the scenario of flavor anarchy in the strong sector. Notice that in this anarchic scenario much stronger bounds on the resonance masses, of the order of 20 TeV [13], come from k . In Ref. [16] the Ultraviolet contribution to b → sγ in a two-site model without a P LR protection to the t R and b R interactions is evaluated. In the following we will describe in details the contribution in the TS5 and we will report the results for TS10. We can calculate the C CH−U V 7 and C CH−U V 7 ultraviolet contributions by considering the model independent analysis of Ref. [16] and the generational mixing factor in (55). We get the following effective Hamiltonian for b → sγ with loops of heavy fermions and neutral would-be Goldstone bosons:
H ef f neutral Higgs = i e 8π 2 (2 · p) M 2 w k neutral V tsb (1 − γ 5 )s + m s m b V tsb (1 + γ 5 )s (64) where k neutral ≈ 4 i=1 |α (i) 1 | 2 + |α (i) 2 | 2 m b 1 36 M 2 w m 2 * (i) + 4 i=1 α (i) * 1 α (i) 2 m * (i) 1 6 M 2 w m 2 * (i)(65)L ⊃d (i) α (i) 1 (1 + γ 5 ) + α (i) 2 (1 − γ 5 ) bH + h.c. .(66)
After the EWSB, we find the following coefficients at O(x):
α (B) 1 = Y 2 * v 2 s bR 1 M B + M B + c bR MB M 2 B − M 2 B α (B) 2 = − Y * 2 √ 2 s 2 c bR α (B ) 1 = α (B −1/3 ) 1 = − Y * 2 √ 2 s bR α (B ) 2 = α (B −1/3 ) 2 = − Y 2 * v 4 s 2 M 2 B MB − s 2 bR M 3 B − c bR M 3 B + 2c bR M B M 2 B M B MB(M 2 B − M 2 B )(67)
the heavy fermion B gives a contribution of O(s 2 2 ) to k neutral and we neglect it. Considering the eq. (65) and the coefficients in (67), neglecting again O(x 2 ) terms, we obtain:
k neutral ≈ −m b M 2 W Y 2 * 1 8 c bR M B MB − 7 18 s 2 bR M 2 B .(68)
From this expression of k neutral we obtain the following TS5 ultraviolet contributions to the Wilson coefficient of the effective Hamiltonian in (27):
C CH−U V 7 (m * ) = 1 16 √ 2 G F Y 2 * c bR M B MB − 7 18 s 2 bR M 2 B ; C CH−U V 7 (m * ) = 1 16 √ 2 G F Y 2 * c bR M B MB − 7 18 s 2 bR M 2 B m s m b V 2 ts .(69)
Assuming s bR small, the above formulas become:
C CH−U V 7 (m * ) = 1 16 √ 2 G F Y 2 * M B MB ; C CH−U V 7 (m * ) = 1 16 √ 2 G F Y 2 * M B MB m s m b V 2 ts .(70)
Finally, the condition on C CH−U V 7 in the eq. (30) gives the bound:
M B MB (0.40) Y * TeV ;(71)
where, for simplicity, we have set s bR = 0. The condition (29) on C CH−U V 7 gives a stronger bound,
M B MB (0.52) Y * TeV ,(72)
if C CH−U V 7 (m * ) is a negative contribution. There is also a contribution to b → sγ from diagrams with heavy fermions and charged Higgs in the loop. Following a similar procedure as the one used before (C) we find, neglecting O(x 2 ) terms:
k charged ≈ m b M 2 W Y 2 * 5 48 1 M B MB + O(s 2 1 ) + O(s 2 bR ) .(73)
If we can neglect O(s 2 1 ) and O(s 2 bR ) terms, k charged gives a weaker bound than the one from k neutral . The full expression of k charged can be found in App. D, here we have just reported, for simplicity, the result for small s 1 and s bR angles. In Fig. 6
Ultraviolet contribution in the TS10
For the TS10 model, applying the same procedure as for the case of TS5, we get:
k neutral = m b M 2 W Y 2 * × 7M T M 2 T s 2 1 − 18MBM 2 B 1 − s 2 1 + M 2 B 7M B s 2 1 − 18MB 1 − s 2 1 288M 2 B M B M 2 B + O(s bR ) = −m b M 2 W Y 2 * 1 16 1 M B MB + 1 M B MB + O(s 2 1 ) + O(s bR ) (74) k charged = m b M 2 W Y 2 * 5 48 1 M B MB + 5 48 1 M B MB + 5 96 s 2 R M 2 B + O(s 2 1 ) + O(s 2 bR )(75)
If the left-handed bottom quark has a small degree of compositeness, we can neglect O(s 2 1 ) (while s bR is naturally very small in the TS10, in order to account for the ratio m b /m t 1). The charged contribution, in this case, gives a stronger bound than the one from k neutral :
M B MB (0.58) Y * TeV ,(76)
from the condition (30) on C CH−U V
7
. A stronger bound,
M B MB (0.75) Y * TeV ,(77)
comes from the condition (29) on C CH−U V 7 , if this last contribution has a negative sign. In Fig. 7 fully composite t R . This is an effect caused by the exchange of the custodiansT ,B and of theB, that are light in the limit of a composite t R . In particular, when t R is fully composite (s R = 1), MB( c R MT ) and MB = MT (= c R MT ) vanish. This causes the divergence of the bounds for s R → 1. Such divergences can be seen in the curves in Figure 7, when they approach the (grey) exclusion regions for s 1 (indeed, the minimum value of s 1 allowed by the condition s R = 2mt Y * vs 1 ≤ 1 is obviously obtained in the case s R = 1).
Tab. 1 summarizes our results. It shows the bounds on heavy fermion masses that can be obtained from the process b → sγ. We report the estimated bounds in generic Composite Higgs Models (with or without P C protection), which we have found by means of NDA, and the bounds in the specific two-site models TS5 and TS10. ξ ψ/χ denotes the degree of compositeness of a SM/Heavy fermion. In the specific TS5 and TS10 models: ξ qL ≡ s 1 , ξ D ≡ c 1 . D = (T, B),D denotes a SU (2) L singlet heavy fermion. For the estimated bounds from C CH 7 and for the bounds from C CH−U V
7
, we indicate both the values that can be obtained in the case of a positive (the first number) or a negative (the second number in parenthesis) contribution.
Constraint from / K
The bound on the mass of the heavy fermions that comes from the direct CP violating observable of the K 0 → 2π system, Re( / ), can be even stronger, in the assumption of anarchic Y * , than those obtained from b → sγ, as already found in [18]. As we pointed out, however, it is a bound that strongly depends on the assumptions made on the flavor structure of the new physics sector. As for the UV contribution to b → sγ, the custodial symmetry does not influence the bound and we obtain contributions of the same size in the different models. In what follows we describe the bound in the TS5 and in the TS10. New Physics contribution can be parametrized at low energy by chromo-magnetic operators:
O G =sσ µν T a G a µν (1 − γ 5 ) d , O G =sσ µν T a G a µν (1 + γ 5 ) d .(78)
As for the UV contribution to b → sγ, the leading contribution to / K comes from diagrams with heavy fermions and Higgs in the loop, that generate the O G and O G operators (1 loop diagrams are the same as for the UV contribution to b → sγ, Fig. 5, with the replacements γ → g, b → s and s → d).
The related coefficients C G and C G , in analogy with C 7 and C 7 of the UV contribution to b → sγ, differ by a generational mixing factor that, in the assumption of anarchic Y * , we estimate to be ∼ m d msV 2
us
. We consider only the generation mixing (1 − 3) × (2 − 3), via 3rd generation. In analogy with (64), we define:
A ef f −chromo neutral Higgs = i g s 8π 2 (2 · p) M 2 w k G neutral V uss (1 − γ 5 )d + m d m s V uss (1 + γ 5 )d ,(79)
where
k G neutral ≈ 4 i=1 |α (i) 1 | 2 + |α (i) 2 | 2 m s − 1 12 M 2 w m 2 * (i) + 4 i=1 α (i) * 1 α (i) 2 m * (i) − 1 2 M 2 w m 2 * (i)(80)
the index i runs over the four down-type heavy fermions of the model, d (i) , and the α
(i) 1 , α (i) 2
coefficients are defined by the interactions:
L ⊃d (i) α (i) 1 (1 + γ 5 ) + α (i) 2 (1 − γ 5 ) bH + h.c. .(81)
After the EWSB, neglecting O(x 2 ) terms, we find in the TS5:
k G neutral = 3 8 m s M 2 w Y 2 * M B MB + O(s 2 sR ) ,(82)
where s sR defines the degree of compositeness of the right-handed strange quark and has naturally a small value. In the limit in which s sR = 0, we obtain the same result also in the TS10. We can thus calculate the C G and C G contributions:
C G = − 1 16π 2 k G neutral M 2 w m s V us , C G = m d m s V 2 us C G .(83)Defining δ = Re( / ) CH − Re( / ) SM Re( / ) exp (84) we obtain |δ | ≈ (58 T eV ) 2 B G |C G − C G | < 1 ,(85)
where Re( / ) SM has been estimated as in Ref. [18]; B G denotes the hadronic bag-parameter, 2π I=0 |y s O G |K 0 . We take B G = 1 7 and we take into account separately the contribution from C G and C G . In the limit s sR = 0 we obtain from (85):
M B MB (1.3)Y * T eV ,(86)
which is in agreement with the result in [18]. The contribution from the charged Higgs interactions gives weaker bounds than those from the neutral Higgs contribution.
Conclusions
Composite Higgs Models are among the compelling scenarios for physics beyond the Standard Model that can give an explanation of the origin of the EWSB and that are going to be tested at the LHC.
In this project we have have built simple "two-site" models, the TS5 and the TS10, which can represent the low energy regime of Minimal Composite Higgs Models with a custodial symmetry and a P LR parity. Working in these effective descriptions, we have reconsidered the bounds on the CHM spectrum implied by flavor observables. We have found in particular that the IR contribution to b → sγ induced by the flavor conserving effective vertex W t R b R implies a robust Minimal Flavor Violating bound on the mass (m * ) of the new heavy fermions (to be more specific, on the heavy doublets, partners of q L = (t L , b L )). The relevance of shifts to W t R b R has been already pointed out in the literature (see, for example, [31,32]), even though its importance in setting a bound on heavy fermion masses was unestimated in previous studies. We have also shown how this bound can be stronger in the case of the absence of a symmetry (P C ) protection to the effective W t R b R vertex. In particular, we have found an estimated bound m * 1.0 TeV , in models with P C protection to the W t R b R vertex (where both t R and b R are P C eigenstates) and a bound m * 1.0/ξ qL TeV , where ξ qL denotes the degree of compositeness of (t L , b L ), in models without P C protection. ξ qL is naturally a small number, the bound could be thus very strong in these types of models. In the specific "two-site" models, the bounds we have found are m T S5 *
TeV
in the TS5, and m T S10 * 0.54 ξ qL TeV , in the TS10. Table 1 summarizes the results obtained for the bounds from b → sγ.
In addition to these bounds, we have calculated the constraints from the UV composite Higgs model contribution to b → sγ. Figs. 6 and 7 show the bounds in the TS5 and the TS10 as functions of the t L degree of compositeness. Our results have shown that these bounds can be stronger than those from the IR contribution but they are model dependent; in particular they strongly depend on the assumptions made on the flavor structure of the composite sector. We have obtained an estimated limit m * (0.52)Y * TeV in a specific NP flavor scenario (Y * anarchic in the flavor space). Even stronger bounds,
m * (1.3)Y * TeV ,
can be obtained from / K but, again, they are model dependent and in principle could be loosened by acting on the NP flavor structure (as done, for example, in Ref. [27]). The lower IR bounds on m * we have found from b → sγ, on the contrary, are robust MFV bounds that cannot be evaded by assuming particular conditions on the structure of the strong sector.
tan ϕ L1 = ∆ L1 M Q * ≡ s 1 c 1 , s 1 ≡ sin ϕ L1 c 1 ≡ cos ϕ L1 s 2 = ∆ L2 M Q * cos ϕ L1 s 3 = ∆ L2 M Q * ∆ 2 L1 + M 2 Q * − M 2 Q * sin ϕ L1 t L = c 1 t el L − s 1 T com L − s 2 T com L T L = s 1 t el L + c 1 T com L + s 3 T com L T L = (s 2 c 1 − s 1 s 3 ) t el L − (s 1 s 2 + c 1 s 3 ) T com L + T com L b L = c 1 b el L − s 1 B com L − s 2 B com L B L = s 1 b el L + c 1 B com L + s 3 B com L B L = (s 2 c 1 − s 1 s 3 ) b el L − (c 1 s 3 + s 1 s 2 ) B com L + B com L(87)s 4 = ∆ L2 ∆ L1 ∆ 2 L1 + M 2 Q * − M 2 Q * T R = T com R + s 4 T com R T R = T com R − s 4 T com R B R = B com R + s 4 B com R B R = B com R − s 4 B com R (88) tan ϕ R = ∆ R1 MT * s R ≡ sin ϕ R c R ≡ cos ϕ R tan ϕ bR = ∆ R2 MB * s bR ≡ sin ϕ bR c bR ≡ cos ϕ bR t R = c R t el R − s RT com R T R = s R t el R + c RT com R b R = c bR b el R − s bRB com R B R = s bR b el R + c bRB com R(89)
Physical heavy fermion masses are related to the bare ones according to:
MT = M 2 T * + ∆ 2 R1 = MT * c R MB = M 2 B * + ∆ 2 R2 = MB * c bR M T = M B = M 2 Q * + ∆ 2 L1 = M Q *
In the elementary/composite basis the Yukawa Lagrangian reads:
L Y U K = Y * U T r Q H T + Y * D T r Q H B + h.c. = Y * U T φ † 0T +T 2/3 φ 0T +T 5/3 φ +T −Bφ −T + Y * D B −1/3 φ † 0B +B φ 0B +T φ +B −B −4/3 φ −B + h.c.(91)L Y U K =Y * U c 1 c R T L φ † 0T R −B L φ −T R + Y * U c R T 2/3L φ 0TR +T 5/3L φ +T R − Y * U (s 1 s 2 + c 1 s 3 ) c R T L φ † 0T R −B L φ −T R − Y * U s 1 c R t L φ † 0T R −b L φ −T R − Y * U s R T 2/3L φ 0 t R +T 5/3L φ + t R + Y * U (s 1 s 2 + c 1 s 3 ) s R T L φ † 0 t R −B L φ − t R − Y * U c 1 s R T L φ † 0 t R −B L φ − t R + Y * U s 1 s R t L φ † 0 t R −b L φ − t R + Y * U T R φ † 0T L −B R φ −T L + Y * U T 2/3R φ 0TL +T 5/3R φ +T L − Y * U s 4 T R φ † 0T L −B R φ −T L + Y * D c bR B −1/3L φ † 0B R −B −4/3L φ −B R + Y * D c bR B L φ 0BR +T L φ +B R − Y * D s bR B −1/3L φ † 0 b R −B −4/3L φ − b R − Y * D s bR B L φ 0 b R +T L φ + b R − Y * D s 2 c bR b L φ 0BR +t L φ +B R + Y * D s 2 s bR b L φ 0 b R +t L φ + b R − Y * D s 3 s bR B L φ 0 b R +T L φ + b R + Y * D s 3 c bR B L φ 0BR +T L φ +B R + Y * D B R φ 0BL +T R φ +B L + Y * U B −1/3R φ † 0B L −B −4/3R φ −B L + Y * D s 4 B R φ 0BL +T R φ +B L + h.c.(96)
A.2 TS10
Fermions rotate from the elementary/composite basis to the 'physical' light(SM)/heavy basis as:
tan ϕ L1 = ∆ L1 M Q * ≡ s 1 c 1 t L = c 1 t el L − s 1 T com L T L = s 1 t el L + c 1 T com L b L = c 1 b el L − s 1 B com L B L = s 1 b el L + c 1 B com L (97) tan ϕ R = ∆ R1 MQ * s R ≡ sin ϕ R c R ≡ cos ϕ R tan ϕ bR = ∆ R2 MQ * s bR ≡ sin ϕ bR c bR ≡ cos ϕ bR t R = c R t el R − s RT com R T R = s R t el R + c RT com R b R = c bR b el R − s bRB com R B R = s bR b el R + c bRB com R(98)
Physical heavy fermion masses are related to the bare ones as:
MT = M 2 Q * + ∆ 2 R1 = MQ * c R MB = M 2 Q * + ∆ 2 R2 = MQ * c bR
In the elementary/composite basis the Yukawa Lagrangian reads:
L Y U K = +Y * T r HQQ + Y * T r Q HQ(100)
After field rotation to the mass eigenstate basis, before EWSB, L Y U K reads as in eq. (105).
After EWSB top and bottom masses arise as:
L Y U K = Y * c 1 c R 1 √ 2 T L φ † 0T R −B L φ −T R − Y * c R 1 √ 2 T 2/3L φ 0TR +T 5/3L φ +T R − Y * s 1 c R 1 √ 2 t L φ † 0T R −b L φ −T R + Y * s 1 s R 1 √ 2 t L φ † 0 t R −b L φ − t R + Y * s R 1 √ 2 T 2/3L φ 0 t R +T 5/3L φ + t R − Y * c 1 s R 1 √ 2 T L φ † 0 t R −B L φ − t R + Y * 1 √ 2 T R φ † 0T L −B R φ −T L − Y * 1 √ 2 T 2/3R φ 0TL +T 5/3R φ +T L + Y * T 5/3L φ † 0T 5/3R −T 2/3L φ −T 5/3R + Y * T 5/3R φ † 0T 5/3L −T 2/3R φ −T 5/3L − Y * s 1 c bR b L φ 0BR +t L φ +B R + Y * s 1 s bR b L φ 0 b R +t L φ + b R − Y * c 1 s bR B L φ 0 b R +T L φ + b R + Y * c 1 c bR B L φ 0BR +T L φ +B R + Y * B R φ 0BL +T R φ +B L + Y * B R φ † 0B L + Y * T2/3R φ +B L Y * 1 √ 2 T R φ † 0T L +B R φ −T L − Y * 1 √ 2 T 2/3R φ 0T L −T 5/3R φ +T L + Y * c 1 1 √ 2 T L φ † 0T R +B L φ −T R − Y * 1 √ 2 T 2/3L φ † 0T R −T 5/3L φ +T R − Y * s 1 1 √ 2 t L φ † 0T R +b L φ −T R + Y * T 5/3R φ 0T 5/3L −T R φ −T 5/3L + Y * c 1 B L φ † 0B R −T L φ −T 5/3R − Y * s 1 b L φ † 0B R −t L φ −T
5/3R
+ Y * T2/3L φ +B R + Y * T5/3L φ 0T 5/3R + h.c.
(105)
B BOUND derivation
The SM prediction and the experimental measurement [20] of the b → sγ branching ratio are respectively: BR th = (315 ± 23)10 −6 (106)
BR ex = (355 ± 24 ± 9)10 −6 (107)
The b → sγ decay rate is:
Γ tot ∝ |C 7 (µ b )| 2 + |C 7 (µ b )| 2 ≈ |C SM 7 (µ b ) + C N P 7 (µ b )| 2 + |C N P 7 (µ b )| 2(108)
If we consider only the C 7 contribution, we obtain:
Γ tot Γ SM = 1 + 2 Re(C SM 7 (µ b ) * C N P 7 (µ b )) |C SM 7 (µ b )| 2 + O(∆C 2 7 )(109)
For µ b = 5 GeV, µ W = M W , α S = 0.118, the SM contribution to C 7 at the scale µ b reads [19]:
C SM 7 (µ b ) = 0.695C SM 7 (µ W ) + 0.086C SM 8 (µ W ) − 0.158C SM 2 (µ W ) = −0.300 .(110)
the index i runs over the four up-type heavy fermions of the model, u (i) , m * (i) denotes the physical mass of the the u (i) heavy fermion and the α
Figure 1 :
11 loop Infrared contribution to C 7 in the SM.
f
RH = −0.777, for m t = 174 GeV and M W = 80.4 GeV.We point out that the bound on the CHM contributions to b → sγ, C CH 7 in eq. (32), can be directly translated into a bound on the effective vertex
R 7 .)
7contribution is negative. This implies a positive contribution C CH−IR−T S5 7 (f RH is negative). The condition in (32) is asymmetric and is stronger in the case of a positive C CH−IR Applying this condition to the infrared contribution in (48), we get, for r = M D M D = 1, the following bound on the D = (T, B) doublet mass: TeV, changing r to r = 0.8(1.2). In the TS10, there is only one doublet, D = (T, B)
Figure 4 :
4TS5 and TS10 at small elementary/composite mixing angles s 1 and s bR . ξ ψ/χ denotes the degree of compositeness of a SM/Heavy fermion. In the specific TS5 and TS10 models: ξ qL ≡ s 1 , ξ D ≡ c 1 . D = (T, B),D denotes a SU (2) L singlet heavy fermion. We highlight (in bold) the MFV bounds from C CH 7 . For the estimated bounds from C CH 7 and for the bounds from C CH−U V7, we indicate both the values that can be obtained in the case of a positive (the first number) or a negative (the second number in parenthesis) contribution. 1 loop Infrared contribution to C 7 .
(D L ) 23 ∼ ms m b Vts , making use, again, of the estimates (54) and (53). The flavor violation in C CH−IR7
and (44), the condition on C CH−IR 7 (µ w ), eq. (33), gives thus the estimated bounds: M D 0.80 TeV(57)in models with P C without P C symmetry. Considering the specific TS5 and TS10 models, C
Figure 5 :
51 loop CHM UV contribution to C 7 .
the index i runs over the four down-type heavy fermions of the model, d (i) =B,
we show the bound on the doublet mass M T as function of s 1 from the condition on C CH−U V 7 , for different values of the ratio k = M T MT between doublet and singlet masses, fixing Y * = 3 (Left Plot), and for different value of Y * , fixing k = 1 (Right Plot). We set MB = MT and M T = M T . These values are obtained by taking into account the strongest values between the neutral Higgs contribution and the charged Higgs one. We set s bR = s 1 .
Figure 6 :Figure 7 :
67we show the bound on the doublet mass M T as function of s 1 from the condition on C CH−U V 7 , for different values of the ratio k = M T MT between doublet andT singlet mass, fixing Y * = 3 (Left Plot), and for different Y * values, setting k = M T MT = 1 (Right Plot). The custodian singlet masses have the following relations with MT : MB c R MT , MB = MT = c R MT . All these bounds are obtained by taking into account the strongest values between the neutral Higgs contribution and the charged Higgs one. We can see that in the TS10 model, the UV bounds are particularly strong in the case of Bounds from C CH−U V 7 in the TS5. Left Plot: bounds for different values of k = M T MT and Y * = 3; Right Plot: bounds for different values of Y * and k = 1. We set MB = MT and M T = M T . Also shown is the exclusion region for s 1 , obtained from the condition s R = Bounds from C CH−U V 7 in the TS10. Left Plot: bounds for different values of k = M T MT (MB c R MT , MB = MT = c R MT ), fixing Y * = 3; Right Plot: bounds for different values of Y * , fixing k = 1. We also show the exclusion region for s 1 , obtained from the condition s R = 2mt Y * vs 1 ≤ 1.
c 1 M
1T 5/3 = M T 2/3 = M Q * M T = M B = M 2 Q * + ∆ 2 L2 M Q * = M B−1/3 = M B−4/3
MT 5/ 3 =M
3MT 5/3 = MT = MB = MQ * M T = M B = M 2 T 2/3 = M T 5/3 = M Q *
). From(6) we can see that heavier SM particles have larger degrees of compositeness: heavy SM particles, like the top, have to be quite composite while the light ones are almost elementary.
Table 1 :
1Estimated bounds from b → sγ in a generic composite Higgs model and in the specific
(1 − γ 5 ) bH + + h.c. .(i)
1 , α
(i)
2 coefficients derive from the
interactions:
L ⊃ū (i) α
(i)
1 (1 + γ 5 ) + α
(i)
2 (117)
1
M B MB
+
5
96
s 2
R
M 2
B
+ O(s 2
1 ) + O(s 2
bR )
(120)
see Ref.[9], for two-and three-site effective theories where the full Higgs non-linearities are included.
Tension that instead affects Technicolor and Extended Technicolor Models.3 As a result of RG evolution above the compositeness scale. The smallness of ∆ parameters also allows for a sort of GIM mechanism that suppresses large Flavor-Changing Neutral Currents[8].
The TS5 model has been already briefly described in[17], where it was adopted to study the phenomenology of heavy-colored vectors at the LHC.
In fact, once produced, heavy fermions of the first two generations will also decay mostly to tops and bottoms, since flavor-changing transitions are not suppressed in the strong sector, while the couplings to the light SM quarks are extremely small, see the discussion in Ref.[2].
The contribution from heavy gluon and heavy fermion exchange is suppressed. Indeed this contribution is approximately diagonal in the flavor space.
That corresponds to the estimate of the hadronic matrix element 2π I=0 |y s O G |K 0 in the chiral quark model and to the first order in the chiral expansion.
AcknowledgmentsI would like to thank Roberto Contino for having followed this work from the beginning and for comments on the manuscript.After field rotation to the mass eigenstate basis, before EWSB, L Y U K reads as in eq. (96).After the EWSB top and bottom masses arise as:We have also electroweak mixings among fermions. The fermionic mass matrices for up and down states read, in the basis t LTLT2/3LTLT L t RTR T 2/3R T R T R for the up sector and in the basis b LBLB LB−1/3LBL b RBR B R B −1/3R B R for the down-type fermions:The scaling factor of the NP contribution to C 7 from the scale µ W to the scale µ b is:By considering all the previous equations, we obtain at 95% C.L.:The scaling factor of the NP contribution to C 7 from the scale m * = 1 TeV to the scale µ W is:and we obtain at 95% C.L.:If we consider only the C 7 contribution, we obtain:We haveAfter the EWSB, we diagonalize the up-type quarks mass matrix of (94) and the downtype one (95) perturbatively in x ≡ Y * v √ 2m * , neglecting O(x 2 ). We find the following coefficients:the heavy fermion T 2/3 gives a contribution of O(x 2 ) to k charged and we can neglect it. Considering the eq.(116) and the coefficients in (118), neglecting again O(x 2 ) terms, we obtain:if we can neglect O(s 2 1 ).D Ultraviolet contributionSumming up, we find in the TS5:
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| {'fraction_non_alphanumeric': 0.0749852362913564, 'fraction_numerical': 0.04818010284183387, 'mean_word_length': 3.0421518397764324, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 194, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We analyze the bounds on the spectrum of composite Higgs models (CHM) that come from flavor observables, by means of simple two-site effective Lagrangians, which incorporate a custodial symmetry and a Left-Right parity and which could also be adopted in further phenomenological studies on CHM. We derive, in particular, an important constraint on the masses of the (t L , b L ) partners, which does not depend on the flavor structure of the sector beyond the SM. This bound is obtained from the "infrared" contribution to b → sγ induced by the flavor-conserving effective vertex W t R b R . We find that the presence of a custodial symmetry can play a role in protecting this effective coupling and, as a consequence, in attenuating the constraint, which, however, remains of the order of 1 TeV. In addition to this bound, we calculate the constraints from the "ultraviolet" contribution to b → sγ, induced by loops of heavy fermions, and to / K .', 'arxivid': '1204.0478', 'author': ['Natascia Vignaroli \nDepartment of Physics and Astronomy\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIAUSA\n\nMichigan State University\n48824East LansingMIUSA\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nDepartment of Physics and Astronomy\nIowa State University\n50011AmesIAUSA', 'Michigan State University\n48824East LansingMIUSA'], 'corpusid': 119223435, 'doi': '10.1103/physrevd.86.115011', 'github_urls': [], 'n_tokens_mistral': 26189, 'n_tokens_neox': 22540, 'n_words': 14723, 'pdfsha': 'df69e655442781a17449eaf9d21f14c84984ef64', 'pdfurls': ['https://arxiv.org/pdf/1204.0478v3.pdf'], 'title': ['∆F = 1 constraints on composite Higgs models with LR parity', '∆F = 1 constraints on composite Higgs models with LR parity'], 'venue': []} |
arxiv |
Stochastic state-transition-change process and time resolved velocity spectrometry
Jiří Procházka
Institute of Physics
Czech Academy of Sciences
Na Slovance 218221Prague 8Czech Republic
Stochastic state-transition-change process and time resolved velocity spectrometry
(Dated: April 14, 2022)stochastic state-transition-change processstochastic particle motiontime resolved velocity spectrometryparticle transport CONTENTS
Motion of particles (bodies) in presence of random effects can be considered stochastic process. However, application of widely known stochastic processes used for description of particle motion is reduced to relatively small class of particle transport phenomena. Stochastic state-transition-change (STC) process is suitable for description of many systems. In this paper it is shown under which assumptions formulae of time resolved velocity spectrometry can be derived with the help of STC process. It opens up new possibilities of unified description of particles moving in a force field in presence of random effects. It extends possibilities of applications of theory of stochastic processes in physics.
Motion of particles, or any other bodies, under various conditions has been studied in physics for a very long time. In some cases the motion can be described deterministically using, e.g., Newton's second law of motion. In some other cases it is necessary to take into account several random effects. Brownian motion (random walk) is well known example of stochastic process. Stochastic cyclotron motion [1] is another example of stochastic process. It introduces randomness to motion of a charged particle in a magnetic field by adding dissipation and fluctuation terms to the corresponding deterministic equation of motion (the terms can be added to any ordinary differential equation having time as an independent variable).
However, widely known stochastic processes are not suitable for description of all particle transport phenomena in presence of random effects. Stochastic state-transition-change * [email protected] (STC) process introduced in sect. III B in [2] extends possibilities of descriptions of motion of particles which may have random initial properties (states), may or may not reach given position, and may or may not change their properties during transport. It will be shown in this paper under which conditions (assumptions) it is possible to derive main formulae of time resolved velocity spectrometry using STC process.
This paper is structured as follows. Derivation of formulae of widely know time resolved velocity spectrometry with the help of stochastic STC process is in sect. II. The spectrometry is one of well known time-of-flight (TOF) methods which allows determination of spectrum of emitted particles from a source as a function of emission time and velocity on the basis of experimental data, see sect. III. The possibilities of generalization of the time resolved velocity spectrometry with the help of STC stochastic process are discussed in sect. IV. Concluding remarks are in sect. V.
II. PROBABILITY MODEL -TIME RESOLVED VELOCITY SPECTROMETRY
A. Stochastic process Stochastic STC process can describe particle motion of particles in a force field when initial conditions are characterized by probability (density) functions.
Consider a source emitting particles of different speeds in the same direction and at different times as an example of nontrivial particle motion. It may not be possible to detect the particles at the place where they are emitted but it may be possible to measure some quantities characterizing the particle transport at several distances x i from the source (i ∈ (0, ..., M), M > 0, x i < x i+1 and the spatial x-axis has the same orientation as the direction of the velocities of the particles). One may ask how to determine the characteristics of the emitted particles at the place where they are emitted on the basis of quantities which can be measured. Assumption II.3 (Constant speed, zero force). The speeds of individual particles did not change during transport from position x i to x i+1 . I.e., no force acted on the particles during the transport (considering only inertial motion).
Assumption II.4 (Variables). Let particle at position x i , at time x i have speed v i . Let
X i = (v i ,t i )(1)X NR i = (v min , v max ,t min 0 ,t max 0 , x 0 , . . . , x i ) .(2)
for all i ∈ (0, . . . , M − 1). I.e., v i and t i are considered random variables and the other parameters are non-random variables (only some of them may or may not be explicitly written as arguments of functions in the following).
Assumption II.5 (State spaces). Let state space S i contain states of particles, represented by variables X i , when they pass through position x i .
Remark II.1. Assumption II.5 implies that number of states of system which were in any state in state space S i is the same as number of particles which passed through x i .
Assumption II.6 (Probability of transition). It holds (for all i ∈ (0, . . . , M − 1))
P T,i (X i ) = 1 ,(3)
i.e., a particle at position x i always moved to position x i+1 , independently on value of X i .
Definition II.1. If assumption II.4 holds then the density of states (DOS) dos i (X i ) defined by eq. (45) in [2] can be written also as
dos i (x i , v i ,t i ) = dos i (X i )(4)
for all i ∈ (0, . . . , M).
Definition II.2. Let dos v i (x i ,t i ) be defined by (i ∈ (0, . . . , M)) dos v i (x i ,t i ) = v i dos i (x i , v i ,t i )dv i .(5)
Definition II.3. If assumption II.4 holds then function ρ C,i (X i , X i+1 ) defined by eq. (52) [2] can be written also as
ρ C,i (x i , v i ,t i , x i+1 , v i+1 ,t i+1 ) = ρ C,i (X i , X i+1 ) .(6)
where non-random variables x i and x i+1 have been written explicitly, and i ∈ (0, . . . , M − 1). The function ρ C,i (X i , X i+1 ) has meaning of probability function that a particle at position x i of properties characterized by random variables X i had properties characterized by random variables X i+1 at position x i+1 .
Definition II.4. Let the probability density function ρ C,i (X i , X i+1 ) satisfying the assumption of constant speed of a particle (see assumptions II.2 and II.3) be denoted as
ρ C,i (x i , v i ,t i , x i+1 ,t i+1 ), i.e., it holds (for all i ∈ (0, . . . , M)) ρ C,i (x i , v i ,t i , x i+1 ,t i+1 ) = ρ C,i (x i , v i ,t i , x i+1 , v i+1 = v i ,t i+1 ) .(7)
Definition II.5 (Minimal and maximal particle arrival time).
The last (resp. the first) particle which gets to x i+1 (i ∈ (0, . . . , M − 1)) is particle having the lowest (resp. the highest) speed v min (resp. v max ) which was at x i at time t max i (resp. t min i ). Definition II.6. Let {X i : i ∈ I} be stochastic process given by definition III.21 in [2]. I.e., it satisfies assumptions III.2 to III.4 in [2]. Let it satisfy also assumptions II.1 to II.6.
B. Derivation of several formulae
The following statements will be derived assuming stochastic process given by definition II.6, if not mentioned otherwise.
Proposition II.1. Particle of constant speed v at position x i and at time t i reached position x i+1 at time t i+1 which is equal to (for all i ∈ (0, . . . , M − 1))
t i+1 = t i + x i+1 − x i v .(8)
It holds (for all i ∈ (0, . . . , M))
t i = t 0 + x i − x 0 v .(9)
Proof. It follows from assumptions II.2 and II.3 and Newton's second law of motion.
Proposition II.2. It holds (for all i ∈ (0, . . . , M − 1))
t min i+1 = t min i + x i+1 − x i v max (10) t max i+1 = t max i + x i+1 − x i v min .(11)
and (for all i ∈ (0, . . . , M)) t min
i = t min 0 + x i − x 0 v max(12)t max i = t max 0 + x i − x 0 v min .(13)
Proof. It follows from assumptions II.1 to II.4 and eqs. (8) and (9). ∈ (0, . . . , M)).
Proposition II.3. t i ∈ [t min i ,t max i ] is equivalent to v i ∈ [v min , v max ] (for all i
Proof. It follows from eqs. (12) and (13).
Proposition II.4. It holds (for all i ∈ (0, . . . , M − 1))
ρ C,i (x i , v i ,t i , x i+1 ,t i+1 ) = δ (t i − t i+1 + x i+1 − x i v i ) .(14)
Proof. It follows from assumptions II.2 and II.3 and eq. (8).
Proposition II.5 (Transformation of DOS). It holds (for all i ∈ (0, . . . , M − 1))
dos i+1 (x i+1 , v i+1 ,t i+1 ) = dos i (x i , v i = v i+1 ,t i = t i+1 − x i+1 −x i v i+1 ) if t i+1 ∈ [t min i+1 ,t max i+1 ] and v i+1 ∈ [v min , v max ] 0 otherwise (15)
where t min i+1 and t max i+1 can be determined using proposition II.2. Equivalently, it holds (for all i ∈ (0, . . . , M))
dos i (x i , v i ,t i ) = dos 0 (x 0 , v 0 = v i ,t 0 = t i − x i −x 0 v i ) if t i ∈ [t min i ,t max i ] and v i ∈ [v min , v max ] 0 otherwise(16)
where t min i and t max i are given by eqs. (12) and (13). Remark II.2. Proposition II.5 can be derived in another way using theorem III.1 in [2] and eq. (3) and function
Proof. According to assumptions II.2 and II.3 it must hold
v i = v i+1 . The number of particles in interval (v i , v i + dv i ) × (t i ,t i + dt i ) at position x i and time t i divided by dv i dt i is equal to dos i (x i , v i ,t i ) (ρ C,i (x i , v i ,t i , x i+1
, v i+1 ,t i+1 ) expressed as a 2-dimensional delta function corresponding to assumptions II.2 to II.4. To work with n-dimensional delta functions is, however, in general more delicate than in 1-dimensional case.
Proposition II.6. It holds
dos v i (x i ,t i ) = v max v min dos i (x i , v i ,t i )dv i if t i ∈ [t min i ,t max i ] 0 otherwise(17)
where t min i and t max i are given by eqs. (12) and (13).
Proof. Insertion of dos i (x i , v i ,t i ) given by eq. (15) into eq. (5) implies eq. (17).
Proposition II.7. It holds (for all i ∈ (0, . . . , M − 1))
dos v i+1 (x i+1 ,t i+1 ) = (18) v max v min t max i t min i dos i (x i , v i ,t i )δ (t i − t i+1 + x i+1 − x i v i )dt i dv i ,(19)
if t i+1 ∈ [t min i+1 ,t max i+1 ] (see eqs. (10) and (11)), otherwise
dos v i+1 (x i+1 ,t i+1 ) = 0.
Proof. Let us consider X j = (x j , v j ,t j ) for j ∈ (0, . . . , i) and X i+1 = (x i+1 ,t i+1 ) for fixed index i ∈ (0, . . . , M − 1). Theorem III.1 in [2] and eqs. (3) and (14) imply eq. (19).
Proposition II.8. It holds (for all i ∈ (0, . . . , M − 1)) ∈ (0, . . . , M)), If we put M = 1, ) and (21) are equivalent to eq. (2) in [3] where an extension of the timeof-flight (TOF) method for determination of the time resolved velocity spectrum of particles emitted in intense bursts has been presented for the first time. Many useful comments to the time resolved velocity spectrometry method are in [3] (including numerical solutions and tests).
dos v i+1 (x i+1 ,t i+1 ) = (20) v max v min dos i (x i , v i = v i+1 ,t i = t i+1 − x i+1 − x i v i+1 )dv i+1 ,(21)if t i+1 ∈ [t min i+1 ,t max i+1 ], otherwise dos v i+1 (x i+1 ,t i+1 ) = 0. Equiva- lently (for all idos v i (x i ,t i ) = v max v min dos 0 (x 0 , v 0 = v i ,t 0 = t i − x i − x 0 v i )dv i ,(22)if t i ∈ [t min i ,t max i ], otherwise dos v i (x i ,t i ) = 0.x i = 0, x i+1 = x, t i = 0 t i+1 = t, v min = v 1 , v max = v 2 ,
In [4] this method is called extended TOF method to distinguish it from basic TOF method in which velocity spectrum of particles is determined independently on the emission time. It is shown in [4] under which conditions the former method is reduced to the later one (in the case of relatively short intense burst in comparison to the time of flight of particles from source to a detector). The basic TOF method provides less information, but it is significantly easier to use it from both the experimental and data analysis point of view (it is sufficient to use only one detector in sufficiently large distance from the source ensuring that ∆T is much smaller than the time needed by an emitted particle to travel from the source to the detector).
Both the types of TOF methods are widely known and have been adapted and successfully applied in various experiments. E.g., with the help of the extended TOF method time resolved neutron energy spectra from D(d,n) 3 He fusion reactions were determined in [4] using analog Monte Carlo reconstruction method (AMCRT). In sect. 2.2.1 in [4] several other existing reconstruction methods (algorithms) are mentioned. Efficiency of different methods depends on several factors including the dependence of the density of states dos 0 (x 0 , v 0 ,t 0 ) (i.e., also measured densities of states dos v i (x i ,t i )) which one is trying to determine.
The basic TOF method was applied, e.g., to experimental data of electrons and ions emitted by laser-produced plasmas in [5][6][7][8]; further details to TOF spectra for mapping of charge density of ions produced by laser are in [9].
IV. GENERALIZATION OF TIME-OF-FLIGHT METHODS
It has been shown in sect. II that with the help of stochastic STC process introduced in sect. III B in [2] leads to the well known time-of-flight (TOF) method for determination of time resolved velocity spectrum of emitted particles from a source (TOF-TV). There are other experimental techniques based on measurement of TOF, such as the time-of-flight mass spectrometry (TOF-MS). This method uses an electric field of known strength to determine mass-to-charge ratio of particles (ions). An electric and magnetic fields of known strength are commonly used (in various configurations) to determine mass and electric charge of charged particle by measuring and analysing trajectories of the particles with the help of the Lorenz force. All time-of-flight methods and many methods of mass spectrometry have something in common. They concern particle transport phenomena under various conditions and their aim is to determine properties of the particles.
Each of the methods is typically suitable for determination of partial spectra being functions of only some variables characterizing properties of particles emitted from a source such as mass, electric charge, velocity (energy) or time of their emission from the source. E.g., the TOF-TV method allows determination of time resolved velocity spectrum (function of "only" two random variables characterizing the emitted particles). With the help of stochastic STC process it is possible to generalize the formulae in sect. II B (TOF-TV method) by taking into account:
1. non-zero external force 2. general initial and final positions and velocities of particles 3. various properties of particles such as mass, electric charge, etc.
Application of this generalized TOF-TV method to data can be more complicated than application of the TOF-TV method to data (it may be necessary to consider more random variables). It requires more experimental information. One can take advantage of various experimental methods determining the "partial" spectra (integrated over some of the variables) to constrain the "full" spectrum, see general guidelines in sect. IV in [2]. One can then derive a more general equation than eq. (21). This allows to study forces acting on particles and their properties (some of them may be specified by random variables). Inertial mass increase in dependence on velocity may be also studied on the basis of experimental data under these conditions [10].
V. CONCLUSION
Stochastic STC process introduced in sect. III in [2] can significantly help to improve existing or develop new techniques of measurement of properties of particles moving in an external force field and being specified by random variables. The measurement is essential for characterization of various sources of emitting particles (see, e.g., laser-produced plasmas mentioned in sect. III, or development of deuterium z-pinch as a powerful source of multi-MeV ions and neutrons [11]). The sources of known properties can be used for various applications. The new types of sources of emitted particles place extra demands on particle detectors (see, e.g., design of a scintillator calorimeter for laser-plasma characterization, or magnetic electron spectrometer spectrometer [12]).
With the help of stochastic STC process it is possible to describe in a unified way motion of particles in an external force field and many other particle transport phenomena which looks very distinct at first glance, such as transmission of light through sequence of polarizers [13]. Several other applications of stochastic STC process for description of (physical) systems are discussed in [2].
Assumption II.1 (Time interval of emitted particles). The particles were emitted in a burst in time interval from t min 0 to arXiv:2204.00626v2 [physics.gen-ph] Assumption II.2 (Velocity of emitted particles). All the emitted particles emitted at position x 0 had the same direction of velocity. The particles could have different values of speeds v 0 in the interval from v min to v max (v min ≤ v max ).
see definition of DOS given by eq. (38) in[2]). The particles reach x i+1 at time t i+1 given by eq.(8).It implies . dos i+1 (x i+1 , v i+1 ,t i+1 ) = 0 in regions of t i+1and v i+1 which are outside physical region. The equivalence of eqs. (15) and (16) can be proven using proposition II.1.
Proof 1 .
1By integrating eq. (15) over v i+1 and using eq. (17) one obtains eq. (21). By integrating eq. (16) over v i and using eq. (17) one obtains eq. (22). The equivalence of eqs. (21) and (22) can be proven using propositions II.1 and II.5. Proof 2. Performing the integration over t i in eq. (19) implies eq. (21) (v i in eq. (19)and v i+1 in eq. (21) are only integration variables, they can be renamed).III. ANALYSIS OF EXPERIMENTAL DATATime dependent densities of states dos v i (x i ,t i ) can be measured at several positions x i . Unknown parameters v min , v max , t min 0 and t max 0 , and unknown function dos 0 (x 0 , v 0 ,t 0 ) (time resolved velocity spectrum of emitted particles) can be determined on the basis of the measured data with the help of formulae derived in sect. II B (see mainly propositions II.2, II.5, II.6 and II.8), and (constrained) optimization techniques as discussed in sect. IV in[2].
=
∆T then eqs. (19
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| {'fraction_non_alphanumeric': 0.07847233829182684, 'fraction_numerical': 0.039570449607220455, 'mean_word_length': 3.714004333267678, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Motion of particles (bodies) in presence of random effects can be considered stochastic process. However, application of widely known stochastic processes used for description of particle motion is reduced to relatively small class of particle transport phenomena. Stochastic state-transition-change (STC) process is suitable for description of many systems. In this paper it is shown under which assumptions formulae of time resolved velocity spectrometry can be derived with the help of STC process. It opens up new possibilities of unified description of particles moving in a force field in presence of random effects. It extends possibilities of applications of theory of stochastic processes in physics.', 'arxivid': '2204.00626', 'author': ['Jiří Procházka \nInstitute of Physics\nCzech Academy of Sciences\nNa Slovance 218221Prague 8Czech Republic\n'], 'authoraffiliation': ['Institute of Physics\nCzech Academy of Sciences\nNa Slovance 218221Prague 8Czech Republic'], 'corpusid': 248157645, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7978, 'n_tokens_neox': 7111, 'n_words': 4092, 'pdfsha': '7487db6b62fa548a5b0b076b5848df39b81bfa4e', 'pdfurls': ['https://arxiv.org/pdf/2204.00626v2.pdf'], 'title': ['Stochastic state-transition-change process and time resolved velocity spectrometry', 'Stochastic state-transition-change process and time resolved velocity spectrometry'], 'venue': []} |
arxiv |
Improved Regularization of Convolutional Neural Networks with Cutout
Terrance Devries
University of Guelph
Graham W Taylor
University of Guelph
Canadian Institute for Advanced Research and Vector Institute
Improved Regularization of Convolutional Neural Networks with Cutout
Convolutional neural networks are capable of learning powerful representational spaces, which are necessary for tackling complex learning tasks. However, due to the model capacity required to capture such representations, they are often susceptible to overfitting and therefore require proper regularization in order to generalize well.In this paper, we show that the simple regularization technique of randomly masking out square regions of input during training, which we call cutout, can be used to improve the robustness and overall performance of convolutional neural networks. Not only is this method extremely easy to implement, but we also demonstrate that it can be used in conjunction with existing forms of data augmentation and other regularizers to further improve model performance. We evaluate this method by applying it to current state-of-the-art architectures on the CIFAR-10, CIFAR-100, and SVHN datasets, yielding new state-of-the-art results with almost no additional computational cost. We also show improved performance in the low-data regime on the STL-10 dataset.
Introduction
In recent years deep learning has contributed to considerable advances in the field of computer vision, resulting in state-of-the-art performance in many challenging vision tasks such as object recognition [8], semantic segmentation [11], image captioning [19], and human pose estimation [17]. Much of these improvements can be attributed to the use of convolutional neural networks (CNNs) [9], which are capable of learning complex hierarchical feature representations of images. As the complexity of the task to be solved increases, the resource utilization of such models increases as well: memory footprint, parameters, operations count, inference time and power consumption [2]. Modern networks commonly contain on the order of tens to hundreds of millions of learned parameters which provide the necessary representational power for such tasks, but with the increased representational power also comes increased probability of overfitting, leading to poor generalization.
In order to combat the potential for overfitting, several different regularization techniques can be applied, such as data augmentation or the judicious addition of noise to activations, parameters, or data. In the domain of computer vision, data augmentation is almost ubiquitous due to its ease of implementation and effectiveness. Simple image transforms such as mirroring or cropping can be applied to create new training data which can be used to improve model robustness and increase accuracy [9]. Large models can also be regularized by adding noise during the training process, whether it be added to the input, weights, or gradients. One of the most common uses of noise for improving model accuracy is dropout [6], which stochastically drops neuron activations during training and as a result discourages the co-adaptation of feature detectors.
In this work we consider applying noise in a similar fashion to dropout, but with two important distinctions. The first difference is that units are dropped out only at the input layer of a CNN, rather than in the intermediate feature layers. The second difference is that we drop out contiguous sections of inputs rather than individual pixels, as demon-strated in Figure 1. In this fashion, dropped out regions are propagated through all subsequent feature maps, producing a final representation of the image which contains no trace of the removed input, other than what can be recovered by its context. This technique encourages the network to better utilize the full context of the image, rather than relying on the presence of a small set of specific visual features. This method, which we call cutout, can be interpreted as applying a spatial prior to dropout in input space, much in the same way that convolutional neural networks leverage information about spatial structure in order to improve performance over that of feed-forward networks.
In the remainder of this paper, we introduce cutout and demonstrate that masking out contiguous sections of the input to convolutional neural networks can improve model robustness and ultimately yield better model performance. We show that this simple method works in conjunction with other current state-of-the-art techniques such as residual networks and batch normalization, and can also be combined with most regularization techniques, including standard dropout and data augmentation. Additionally, cutout can be applied during data loading in parallel with the main training task, making it effectively computationally free. To evaluate this technique we conduct tests on several popular image recognition datasets, achieving state-of-the-art results on CIFAR-10, CIFAR-100, and SVHN. We also achieve competitive results on STL-10, demonstrating the usefulness of cutout for low data and higher resolution problems.
Related Work
Our work is most closely related to two common regularization techniques: data augmentation and dropout. Here we examine the use of both methods in the setting of training convolutional neural networks. We also discuss denoising auto-encoders and context encoders, which share some similarities with our work.
Data Augmentation for Images
Data augmentation has long been used in practice when training convolutional neural networks. When training LeNet5 [9] for optical character recognition, LeCun et al. apply various affine transforms, including horizontal and vertical translation, scaling, squeezing, and horizontal shearing to improve their model's accuracy and robustness.
In [1], Bengio et al. demonstrate that deep architectures benefit much more from data augmentation than shallow architectures. They apply a large variety of transformations to their handwritten character dataset, including local elastic deformation, motion blur, Gaussian smoothing, Gaussian noise, salt and pepper noise, pixel permutation, and adding fake scratches and other occlusions to the images, in addition to affine transformations.
To improve the performance of AlexNet [8] for the 2012 ImageNet Large Scale Visual Recognition Competition, Krizhevsky et al. apply image mirroring, cropping, as well as randomly adjusting colour and intensity values based on ranges determined using principal component analysis on the dataset.
Wu et al. take a more aggressive approach with image augmentation when training Deep Image [21] on the Ima-geNet dataset. In addition to flipping and cropping they apply a wide range of colour casting, vignetting, rotation, and lens distortion (pin cushion and barrel distortion), as well as horizontal and vertical stretching.
Lemley et al. tackle the issue of data augmentation with a learned end-to-end approach called Smart Augmentation [10] instead of relying on hard-coded transformations. In this method, a neural network is trained to intelligently combine existing samples in order to generate additional data that is useful for the training process.
Of these techniques ours is closest to the occlusions applied in [1], however their occlusions generally take the form of scratches, dots, or scribbles that overlay the target character, while we use zero-masking to completely obstruct an entire region.
Dropout in Convolutional Neural Networks
Another common regularization technique is dropout [6,15], which was first introduced by Hinton et al. Dropout is implemented by setting hidden unit activations to zero with some fixed probability during training. All activations are kept when evaluating the network, but the resulting output is scaled according to the dropout probability. This technique has the effect of approximately averaging over an exponential number of smaller sub-networks, and works well as a robust type of bagging, which discourages the co-adaptation of feature detectors within the network.
While dropout was found to be very effective at regularizing fully-connected layers, it appears to be less powerful when used with convolutional layers [16]. This reduction in potency can largely be attributed to two factors. The first is that convolutional layers already have much fewer parameters than fully-connected layers, and therefore require less regularization. The second factor is that neighbouring pixels in images share much of the same information. If any of them are dropped out then the information they contain will likely still be passed on from the neighbouring pixels that are still active. For these reasons, dropout in convolutional layers simply acts to increase robustness to noisy inputs, rather than having the same model averaging effect that is observed in fully-connected layers.
In an attempt to increase the effectiveness of dropout in convolutional layers, several variations on the original dropout formula have been proposed. Tompson et al. introduce SpatialDropout [16], which randomly discards en-tire feature maps rather than individual pixels, effectively bypassing the issue of neighbouring pixels passing similar information.
Wu and Gu propose probabilistic weighted pooling [20], wherein activations in each pooling region are dropped with some probability. This approach is similar to applying dropout before each pooling layer, except that instead of scaling the output with respect to the dropout probability at test time, the output of each pooling function is selected to be the sum of the activations weighted by the dropout probability. The authors claim that this approach approximates averaging over an exponential number of sub-networks as dropout does.
In a more targeted approach, Park and Kwak introduce max-drop [13], which drops the maximal activation across feature maps or channels with some probability. While this regularization method performed better than conventional dropout on convolutional layers in some cases, they found that when used in CNNs that utilized batch normalization, both max-drop and SpatialDropout performed worse than standard dropout.
Denoising Auto-encoders & Context Encoders
Denosing auto-encoders [18] and context encoders [14] both rely on self-supervised learning to elicit useful feature representations of images. These models work by corrupting input images and requiring the network to reconstruct them using the remaining pixels as context to determine how best to fill in the blanks. Specifically, denoising autoencoders that apply Bernoulli noise randomly erase individual pixels in the input image, while context encoders erase larger spatial regions. In order to properly fill in the missing information, the auto-encoders are forced to learn how to extract useful features from the images, rather than simply learning an identity function. As context encoders are required to fill in a larger region of the image they are required to have a better understanding of the global content of the image, and therefore they learn higher-level features compared to denoising auto-encoders [14]. These feature representations have been demonstrated to be useful for pre-training classification, detection, and semantic segmentation models.
While removing contiguous sections of the input has previously been used as an image corruption technique, like in context encoders, to our knowledge it has not previously been applied directly to the training of supervised models.
Cutout
Cutout is a simple regularization technique for convolutional neural networks that involves removing contiguous sections of input images, effectively augmenting the dataset with partially occluded versions of existing samples. This technique can be interpreted as an extension of dropout in input space, but with a spatial prior applied, much in the same way that CNNs apply a spatial prior to achieve improved performance over feed-forward networks on image data.
From the comparison between dropout and cutout, we can also draw parallels to denoising autoencoders and context encoders. While both models have the same goal, context encoders are more effective at representation learning, as they force the model to understand the content of the image in a global sense, rather than a local sense as denoising auto-encoders do. In the same way, cutout forces models to take more of the full image context into consideration, rather than focusing on a few key visual features, which may not always be present.
One of the major differences between cutout and other dropout variants is that units are dropped at the input stage of the network rather than in the intermediate layers. This approach has the effect that visual features, including objects that are removed from the input image, are correspondingly removed from all subsequent feature maps. Other dropout variants generally consider each feature map individually, and as a result, features that are randomly removed from one feature map may still be present in others. These inconsistencies produce a noisy representation of the input image, thereby forcing the network to become more robust to noisy inputs. In this sense, cutout is much closer to data augmentation than dropout, as it is not creating noise, but instead generating images that appear novel to the network.
Motivation
The main motivation for cutout comes from the problem of object occlusion, which is commonly encountered in many computer vision tasks, such as object recognition, tracking, or human pose estimation. By generating new images which simulate occluded examples, we not only better prepare the model for encounters with occlusions in the real world, but the model also learns to take more of the image context into consideration when making decisions.
We initially developed cutout as a targeted approach that specifically removed important visual features from the input of the image. This approach was similar to maxdrop [13], in that we aimed to remove maximally activated features in order to encourage the network to consider less prominent features. To accomplish this goal, we extracted and stored the maximally activated feature map for each image in the dataset at each epoch. During the next epoch we then upsampled the saved feature maps back to the input resolution, and thresholded them at the mean feature map value to obtain a binary mask, which was finally overlaid on the original image before being passed through the CNN. Figure 2 demonstrates this early version of cutout.
While this targeted cutout method performed well, we found that randomly removing regions of a fixed size per- formed just as well as the targeted approach, without requiring any manipulation of the feature maps. Due to the inherent simplicity of this alternative approach, we focus on removing fixed-size regions for all of our experiments.
Implementation Details
To implement cutout, we simply apply a fixed-size zeromask to a random location of each input image during each epoch of training, as shown in Figure 1. Unlike dropout and its variants, we do not apply any rescaling of weights at test time. For best performance, the dataset should be normalized about zero so that modified images will not have a large effect on the expected batch statistics.
In general, we found that the size of the cutout region is a more important hyperparameter than the shape, so for simplicity, we conduct all of our experiments using a square patch as the cutout region. When cutout is applied to an image, we randomly select a pixel coordinate within the image as a center point and then place the cutout mask around that location. This method allows for the possibility that not all parts of the cutout mask are contained within the image. Interestingly, we found that allowing portions of the patches to lay outside the borders of the image (rather than constraining the entire patch to be within the image) was critical to achieving good performance. Our explanation for this phenomenon is that it is important for the model to receive some examples where a large portion of the image is visible during training. An alternative approach that achieves similar performance is to randomly apply cutout constrained within the image region, but with 50% probability so that the network sometimes receives unmodified images.
The cutout operation can easily be applied on the CPU along with any other data augmentation steps during data loading. By implementing this operation on the CPU in parallel with the main GPU training task, we can hide the computation and obtain performance improvements for virtually free.
Experiments
To evaluate the performance of cutout, we apply it to a variety of natural image recognition datasets: CIFAR-10, CIFAR-100, SVHN, and STL-10.
CIFAR-10 and CIFAR-100
Both of the CIFAR datasets [7] consist of 60,000 colour images of size 32 × 32 pixels. CIFAR-10 has 10 distinct classes, such as cat, dog, car, and boat. CIFAR-100 contains 100 classes, but requires much more fine-grained recognition compared to CIFAR-10 as some classes are very visually similar. For example, it contains five different classes of trees: maple, oak, palm, pine, and willow. Each dataset is split into a training set with 50,000 images and a test set with 10,000 images.
Both datasets were normalized using per-channel mean and standard deviation. When required, we apply the standard data augmentation scheme for these datasets [5]. Images are first zero-padded with 4 pixels on each side to obtain a 40 × 40 pixel image, then a 32 × 32 crop is randomly extracted. Images are also randomly mirrored horizontally with 50% probability.
To evaluate cutout on the CIFAR datasets, we train models using two modern architectures: a deep residual network [5] with a depth of 18 (ResNet18), and a wide residual network [22] with a depth of 28, a widening factor of 10, and dropout with a drop probability of p = 0.3 in the convolutional layers (WRN-28-10). For both of these experiments, we use the same training procedure as specified in [22]. That is, we train for 200 epochs with batches of 128 images using SGD, Nesterov momentum of 0.9, and weight decay of 5e-4. The learning rate is initially set to 0.1, but is scheduled to decrease by a factor of 5x after each of the 60th, 120th, and 160th epochs. We also apply cutout to shake-shake regularization models [4] that currently achieve state-of-the-art performance on the CIFAR datasets, specifically a 26 2 × 96d "Shake-Shake-Image" ResNet for CIFAR-10 and a 29 2 × 4 × 64d "Shake-Even-Image" ResNeXt for CIFAR-100. For our tests, we use the original code and training settings provided by the author of [4], with the only change being the addition of cutout.
To find the best parameters for cutout we isolate 10% of the training set to use as a validation set and train on the remaining images. As our cutout shape is square, we perform a grid search over the side length parameter to find the optimal size. We find that model accuracy follows a parabolic trend, increasing proportionally to the cutout size until an optimal point, after which accuracy again decreases and eventually drops below that of the baseline model. This behaviour can be observed in Figure 3a and 3b, which depict the grid searches conducted on CIFAR-10 and CIFAR-100 respectively. Based on these validation results we select a cutout size of 16 × 16 pixels to use on CIFAR-10 and a Table 1: Test error rates (%) on CIFAR (C10, C100) and SVHN datasets. "+" indicates standard data augmentation (mirror + crop). Results averaged over five runs, with the exception of shake-shake regularization which only had three runs each. Baseline shake-shake regularization results taken from [4].
cutout size of 8 × 8 pixels for CIFAR-100 when training on the full datasets. Interestingly, it appears that as the number of classes increases, the optimal cutout size decreases. This makes sense, as when more fine-grained detection is required then the context of the image will be less useful for identifying the category. Instead, smaller and more nuanced details are important.
As shown in Table 1, the addition of cutout to the ResNet18 and WRN-28-10 models increased their accuracy on CIFAR-10 and CIFAR-100 by between 0.4 to 2.0 percentage points. We draw attention to the fact that cutout yields these performance improvements even when applied to complex models that already utilize batch normalization, dropout, and data augmentation. Adding cutout to the current state-of-the-art shake-shake regularization models improves performance by 0.3 and 0.6 percentage points on CIFAR-10 and CIFAR-100 respectively, yielding new stateof-the-art results of 2.56% and 15.20% test error.
SVHN
The Street View House Numbers (SVHN) dataset [12] contains a total of 630,420 colour images with a resolution of 32 × 32 pixels. Each image is centered about a number from one to ten, which needs to be identified. The official dataset split contains 73,257 training images and 26,032 test images, but there are also 531,131 additional training images available. Following standard procedure for this dataset [22], we use both available training sets when training our models, and do not apply any data augmentation. All images are normalized using per-channel mean and standard deviation.
To evalute cutout on the SVHN dataset we apply it to a WideResNet with a depth of 16, a widening factor of 8, and dropout on the convolutional layers with a dropout rate of p = 0.4 (WRN- . This particular configuration currently holds state-of-the-art performance on the SVHN dataset with a test error of 1.54% [22]. We repeat the same training procedure as specified in [22] by training for 160 epochs with batches of 128 images. The network is optimized using SGD with Nesterov momentum of 0.9 and weight decay of 5e-4. The learning rate is initially set to 0.01, but is reduced by a factor of 10x after the 80th and 120th epochs. The one change we do make to the original training procedure (for both baseline and cutout) is to normalize the data so that it is compatible with cutout (see § 3.2). The original implementation scales data to lie between 0 and 1.
To find the optimal size for the cutout region we conduct a grid search using 10% of the training set for validation and ultimately select a cutout size of 20 × 20 pixels. While this may seem like a large portion of the image to remove, it is important to remember that the cutout patches are not constrained to lie fully within the bounds of the image.
Using these settings we train the WRN-16-8 and observe an average reduction in test error of 0.3 percentage points, resulting in a new state-of-the-art performance of 1.30% test error, as shown in Table 1.
STL-10
The STL-10 dataset [3] consists of a total of 113,000 colour images with a resolution of 96 × 96 pixels. The training set only contains 5,000 images while the test set consists of 8,000 images. All training and test set images belong to one of ten classes, such as airplane, bird, or horse. The remainder of the dataset is composed of 100,000 unlabeled images belonging to the target ten classes, plus additional but visually similar classes. While the main purpose of the STL-10 dataset is to test semi-supervised learning algorithms, we use it to observe how cutout performs when applied to higher resolution images in a low data setting. For this reason, we discard the unlabeled portion of the dataset and only use the labeled training set.
The dataset was normalized by subtracting the perchannel mean and dividing by the per-channel standard deviation. Simple data augmentation was also applied in a similar fashion to the CIFAR datasets. Specifically, images were zero-padded with 12 pixels on each side and then a 96 × 96 crop was randomly extracted. Mirroring horizontally was also applied with 50% probability.
To evaluate the performance of cutout on the STL-10 dataset we use a WideResNet with a depth of 16, a widening factor of 8, and dropout with a drop rate of p = 0.3 in the convolutional layers. We train the model for 1000 epochs with batches of 128 images using SGD with Nesterov momentum of 0.9 and weight decay of 5e-4. The learning rate is initially set to 0.1 but is reduced by a factor of 5x after the 300th, 400th, 600th, and 800th epochs.
We perform a grid search over the cutout size parameter using 10% of the training images as a validation set and select a square size of 24 × 24 pixels for the no dataaugmentation case and 32 × 32 pixels for training STL-10 with data augmentation. Training the model using these values yields a reduction in test error of 2.7 percentage points in the no data augmentation case, and 1.5 percentage points when also using data augmentation, as shown in Table 2.
Model STL10 STL10+ WideResNet
23.48 ± 0.68 14.21 ± 0.29 WideResNet + cutout 20.77 ± 0.38 12.74 ± 0.23 Table 2: Test error rates on STL-10 dataset. "+" indicates standard data augmentation (mirror + crop). Results averaged over five runs on full training set.
Analysis of Cutout's Effect on Activations
In order to better understand the effect of cutout, we compare the average magnitude of feature activations in a ResNet18 when trained with and without cutout on CIFAR-10. The models were trained with data augmentation using the same settings as defined in Section 4.1, achieving scores of 3.89% and 4.94% test error respectively.
In Figure 4, we sort the activations within each layer by ascending magnitude, averaged over all samples in the test set. We observe that the shallow layers of the network experience a general increase in activation strength, while in deeper layers, we see more activations in the tail end of the distribution. The latter observation illustrates that cutout is indeed encouraging the network to take into account a wider variety of features when making predictions, rather than relying on the presence of a smaller number of features.
Conclusion
Cutout was originally conceived as a targeted method for removing visual features with high activations in later layers of a CNN. Our motivation was to encourage the network to focus more on complimentary and less prominent features, in order to generalize to situations like occlusion. However, we discovered that the conceptually and computationally simpler approach of randomly masking square sections of the image performed equivalently in the experiments we conducted. Importantly, this simple regularizer proved to be complementary to existing forms of data augmentation and regularization. Applied to modern architectures, such as wide residual networks or shake-shake regularization models, it achieves state-of-the-art performance on the CIFAR-10, CIFAR-100, and SVHN vision benchmarks. So why hasn't it been reported or analyzed to date? One reason could be the fact that using a combination of corrupted and clean images appears to be important for its success. Future work will return to our original investigation of visual feature removal informed by activations.
Figure 1 :
1Cutout applied to images from the CIFAR-10 dataset.
Figure 2 :
2An early version of cutout applied to images from the CIFAR-10 dataset. This targeted approach often occludes part-level features of the image, such as heads, legs, or wheels.
Figure 3 :
3Cutout patch length with respect to validation accuracy with 95% confidence intervals (average of five runs). Tests run on CIFAR-10 and CIFAR-100 datasets using WRN-28-10 and standard data augmentation. Baseline indicates a model trained with no cutout.
Figure 5 demonstrates similar observations for individual samples, where the effects of cutout are more pronounced.
Figure 4 :Figure 5 :
45Magnitude of feature activations, sorted by descending value, and averaged over all test samples. A standard ResNet18 is compared with a ResNet18 trained with cutout at three different depths. Magnitude of feature activations, sorted by descending value. Each row represents a different test sample. A standard ResNet18 is compared with a ResNet18 trained with cutout at three different depths.
AcknowledgementsThe authors thank Daniel Jiwoong Im for feedback on the paper and for suggesting the analysis in § 4.4. The authors also thank NVIDIA for the donation of a Titan X GPU.
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| {'fraction_non_alphanumeric': 0.034543592078306394, 'fraction_numerical': 0.021710676075574777, 'mean_word_length': 4.7454634624816086, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Convolutional neural networks are capable of learning powerful representational spaces, which are necessary for tackling complex learning tasks. However, due to the model capacity required to capture such representations, they are often susceptible to overfitting and therefore require proper regularization in order to generalize well.In this paper, we show that the simple regularization technique of randomly masking out square regions of input during training, which we call cutout, can be used to improve the robustness and overall performance of convolutional neural networks. Not only is this method extremely easy to implement, but we also demonstrate that it can be used in conjunction with existing forms of data augmentation and other regularizers to further improve model performance. We evaluate this method by applying it to current state-of-the-art architectures on the CIFAR-10, CIFAR-100, and SVHN datasets, yielding new state-of-the-art results with almost no additional computational cost. We also show improved performance in the low-data regime on the STL-10 dataset.', 'arxivid': '1708.04552', 'author': ['Terrance Devries \nUniversity of Guelph\n\n', 'Graham W Taylor \nUniversity of Guelph\n\n\nCanadian Institute for Advanced Research and Vector Institute\n\n'], 'authoraffiliation': ['University of Guelph\n', 'University of Guelph\n', 'Canadian Institute for Advanced Research and Vector Institute\n'], 'corpusid': 23714201, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8909, 'n_tokens_neox': 7833, 'n_words': 5389, 'pdfsha': '5495926e45784daada9a5f8f60f966a4eafaf54d', 'pdfurls': ['https://arxiv.org/pdf/1708.04552v1.pdf'], 'title': ['Improved Regularization of Convolutional Neural Networks with Cutout', 'Improved Regularization of Convolutional Neural Networks with Cutout'], 'venue': []} |
arxiv |
Quark-antiquark potential from a deformed AdS/QCD
20 Jun 2018
Rodrigo C L Bruni
Departamento de Física Teórica
Universidade do Estado do Rio de Janeiro
20.550-900 -Rio de Janeiro-RJBrazil
Eduardo Folco Capossoli
Departamento de Física and Mestrado Profissional em Práticas da Educação Básica (MPPEB)
Colégio Pedro II, 20.921-903 -Rio de Janeiro-RJBrazil
Instituto de Física
Universidade Federal do Rio de Janeiro
21.941-972 -Rio de Janeiro-RJBrazil
Henrique Boschi-Filho
Instituto de Física
Universidade Federal do Rio de Janeiro
21.941-972 -Rio de Janeiro-RJBrazil
Quark-antiquark potential from a deformed AdS/QCD
20 Jun 20181
In this work we calculate the static limit of the energy for a quark-antiquark pair from the Nambu-Goto action using a holographic approach with a deformed AdS space. From this energy we derive the Cornell potential for the quark-antiquark interaction. We also find a range of values for our parameters which fits exactly the Cornell potential parameters. * Eletronic address:bruni.r.c.
I. INTRODUCTION
The quark-antiquark potential has been a very useful for the investigation of strong interactions and quark confinement. This potential can be used, for example, to analyse the transition between the confined and deconfined phases of matter (see for instance [1]).
Recently, efforts have been made to obtain the quark-antiquark potential [2][3][4][5][6][7][8][9][10] using the well known AdS-CFT correspondence. For another approach using effective string theory see for instance [11]. This correspondence was originally formulated as a mapping of correlation functions of a superconformal N = 4 Yang-Mills theory defined on the boundary of the AdS space and a string theory living in its bulk. It works in such a way that a strongly coupled regime on the boundary theory is mapped into a weakly coupled one in the bulk [12][13][14][15][16].
However, since the original formulation of the correspondence is based on a conformal field theory, which has no characteristic scale, the confining behaviour of the potential is not contemplated once confinement implies a typical length scale.
In order to describe both the confining and non-confining behaviours, it becomes necessary to break the conformal invariance of the theory. There are various ways of doing so but we mention just two of them: the hardwall [17][18][19][20][21][22][23] and the softwall [24][25][26][27] models which breaks conformal invariance introducing a cut off in the action. Inspired by [5], here we break the conformal invariance modifying the background metric instead of the bulk action. So the metric is given by:
ds 2 = g mn dX m dX n = R 2 z 2 h(z) (dx i dx i + dz 2 ) ,(1)
where R is the AdS radius, m, n = 0, 1, 2, 3, 4, z is the radial coordinate while x i with i = 0, 1, 2, 3 represents an Euclidean space in four dimensions. The warp factor that we consider here in this work is given by:
h(z) = exp 1 n (κz) n ,(2)
in which κ has dimensions of inverse length and n is a dimensionless number. We will keep these constants arbitrary until sec. III . Note that, if we restrict n = 2 we reobtain [5].
The main goal of this work is to calculate the energy configuration for a quark-antiquark pair from the Nambu-Goto action using a holographic approach within the deformed metric Eq.
(1) with the warp factor given by Eq. (2). From this energy we will obtain the Cornell
potential V ( ) = − ξ + a 2 + C ,(3)
and also find a range of values for the parameters κ and n which describe h(z) in order to fit this potential.
This work is organised as follows. In Section II, using the warp factor exp{κz n /n}, we compute the separation and the energy of the quark-antiquark pair using the Wilson loop from the AdS/CFT correspondence. In section III, we discuss the matching of our parameters κ and n to fit the Cornell potential. Finally, in section IV, we present our comments and conclusions.
II. THE WILSON LOOP AND THE QUARK POTENTIAL
The starting point of our calculations involves the Wilson Loop. For convenience we choose one circuit corresponding to a rectangular spacetime loop with temporal extension T and spatial extension in the association with the area of the string worldsheet that lives in the AdS space, whose boundary is just the flat spacetime in 4 dimensions where the loop is defined [2,3].
So, following this prescription, we just have to calculate the Nambu-Goto action of a string with the endpoints (identified as the quark and antiquark) fixed at z = 0, assuming a "U-shape" equilibrium configuration in the bulk of deformed AdS.
Assuming that the string configuration is, by hypothesis, static i.e. it moves in the interior of the deformed AdS without change in its shape, one can show that the interquark separation and energy for the type of metric (1) are respectively given by:
= 2 w 1 0 h(z 0 ) h(z) z 2 z 2 0 1 1 − h(z 0 ) h(z) 2 z 4 z 4 0 dz ,(4)E = 2R 2 π w 1 0 h(z) z 2 1 1 − h(z 0 ) h(z) 2 z 4 z 4 0 dz ,(5)
in the limit that w 1 → ∞. Note that z 0 is the minimum of the z coordinate and corresponds to the extreme of the U-shape curve.
The form of (4) and (5) is very convenient because it makes explicit that the expressions of energy and separation depends only on the warp factor chosen for the metric.
It is useful to rewrite the integrals (4) and (5) in terms of a dimensionless variable. If we define v := z z 0 with v 0 being the value of v when z = z 0 , i.e. v 0 = 1, the integrals became:
= 2z 0 1 0 h(1) h(v) v 2 1 − h(1) h(v) 2 v 4 − 1 2 dv. (6) E = 2R 2 π 1 z 0 1 0 h(v)v −2 1 − h(1) h(v) 2 v 4 − 1 2 dv,(7)
which makes explicit the dimension of and E and h(v) ≡ h(z).
Now we introduce the dimensionless parameter λ := κ n z n 0 such that the equations (6) and (7) became:
= 2 λ 1 n κ 1 0 v 2 e λ n (1−v n ) 1 − e 2λ n (1−v n ) v 4 − 1 2 dv,(8)E = 2R 2 π κ λ 1 n 1 0 e λ n v n v −2 1 − e 2λ n (1−v n ) v 4 − 1 2 dv,(9)
Let us analyze the above expressions when λ ≈ 0 and λ ≈ 2, which are the interesting physical limits since for λ → 0 one has → 0, while for λ → 2 one has → ∞, as we are going to discuss below.
A. Calculation of
λ close to zero
If we express the integrand in (8) as a power series in λ centered at zero, to first order in λ and integrate it, we obtain:
I(λ, n) = − √ π 2n Γ 3 4 (λ − 2n) Γ 1 4 − λΓ n+3 4 Γ n+1 4 ,(10)
where the above result is valid only if n > −3, otherwise the integral does not converge.
Substituting this result in (8) and grouping terms proportional to λ one finds:
= 1 ρ λ 1 n κ 1 − λ 2n [1 − F (n)πρ] + O(λ 2 ) ; (λ ≈ 0).(11)
where we have defined ρ := (2π)
3 2 Γ 2 ( 1 4 ) and F (n) := 2 √ π Γ( 3+n 4 ) Γ(1+n 4 )
.
λ close to 2
If we repeat the procedure of last subsection for λ now centered at 2 we will not be able to achieve an analytic expression for the integral. We note however that the integral of (8) is dominated by v ∼ 1. We thus expand the integrand around v = 1 to first order and integrate it, obtaining:
I(λ, n) = 1 λ(2λ + n − 9) + 10 − log (−2(λ − 2)(λ(2λ + n − 9) + 10)) + 2 log λ(2λ + n − 9) + [λ(2λ + n − 11) + 14][λ(2λ + n − 9) + 10] + 10
As the first logarithm of (12) diverges when λ = 2 one would expand again around λ = 2 up to first order. However, since terms of order O(1) in the expansion will not contribute to the functional form of the Cornell potential and we are extracting just the leading behavior of (8) for λ ≈ 2, we can safely neglect contributions of order O(λ) in the aforementioned expansions, obtaining:
I(λ, n) = − 1 √ 2n log(2 − λ) + O(1) ,(13)
which, due to (8) leads to:
= 2 1 n κ − 2 n log(2 − λ) + O(1) ; (λ ≈ 2).(14)
As mentioned above, the limit λ → 2 implies → ∞. So, we choose the renormalization of (9) as:
E Ren. = 2R 2 π κ λ 1 n −1 + 1 0 e λ n v n v −2 1 − e 2λ n (1−v n ) v 4 − 1 2 − 1 dv ,(15)
such that this energy expression is finite and now we can analyse again the limits of λ close to 0 and 2.
λ close to zero
Expanding the integrand in (15) with respect to λ, centered at zero, we find:
I(λ, n) = 1 − 1 2ρ + √ π(n + 1)Γ n−1 4 8nΓ n+1 4 − 1 4n 1 ρ λ,(16)
So that, the renormalised energy is:
E Ren. = − 2R 2 π 1 2ρ κ λ 1 n 1 + λ 2n [1 − G(n)πρ] + O(λ 2 ) ,(17)
where we defined the function G(n) =
(n+1)Γ( n−1 4 ) 2 √ πΓ( n+1 4 )
.
Writing the pre factor κ/λ 1 n as a function of (c.f. (11)), substituting in (17) and keeping only linear terms in λ we get:
E Ren. = −2R 2 κ 0 + 2R 2 λ 4 G(n) − F (n) ρn + O(λ 2 ),(18)
with κ 0 := 1/(2πρ 2 ). Using (11) we can rewrite λ ≈ 0 in terms of ρ and κ:
λ ≈ (κ ρ) n 1 + λ 2 (1 − F (n)πρ) .(19)
Substituting this result in (18) and keeping in mind that λ ≈ 0 is equivalent to the regime of short distances, one can safely disregard terms proportional to 2n−1 in comparison with the terms proportional to n−1 . Then, we obtain:
E Ren. = 2R 2 − κ 0 + σ 0 (n) n−1 + O( 2n−1 ) ,(20)
with σ 0 (n) := 1 4 κ n ρ n−1 G(n)−F (n) n .
λ close to 2
In this section we are going to calculate the renormalised energy for λ close to 2. Repeating the procedure employed in subsection II A 2, i.e., rewriting all the integrand in (15) inside the square root
I(λ, v, n) = v 4 e − 2λv n n 1 − v 4 e 2λ(1−v n ) n − 1 2 − v −2 ,(21)
and expanding this integrand with respect to v centred at 1, to second order we find:
I(λ, v, n) = 2(2 − λ)(1 − v)e − 2λ n − e − 2λ n 6λ 2 − λn − 23λ + 22 (1 − v) 2 +3(1 − v) 2 + 2(1 − v) .(22)
For above expression to be real, the first two terms must be positive and the last one must be negative which implies, respectively, in λ < 2 and n+23 12 − 1 12 √ n 2 + 46n + 1 < λ < 1 12 √ n 2 + 46n + 1 + n+23 12 . Now, integrating (22) one has:
I(λ, n) = −3 + − log(4 − 2λ) e − 2λ n (−6λ 2 + λ(n + 23) − 22) + 2 log λ(−6λ + n + 21) − 18 + λ(−6λ + n + 23) − 22 e − 2λ n (−6λ 2 + λ(n + 23) − 22)
.
Keeping only terms in lowest order of λ and substituting λ = 2 in the denominator of above expression we get:
E Ren. = 2R 2 π κ 2 2 1 n − e 2 n log(2 − λ) √ 2n + O(1) = 2R 2 (σ(n) + O(1)) ,(24)
where σ(n) := κ 2 2π e 2 2 n .
III. PHENOMENOLOGY
Summarizing the results of the last section, the renormalised energies (20) and (24) in terms of the separation are given by:
E λ≈0 Ren. = 2R 2 − κ 0 + σ 0 (n) n−1 + O( 2n−1 ) ,(25)E λ≈2 Ren. = 2R 2 [σ(n) + O(1)] .(26)
Now we are going to fit the constants of our model with the phenomenological constants of the Cornell potential (3) with ξ = 0.52 and a = 2.34GeV −1 [28][29][30][31] (for excellent reviews of the Cornell potential see [32,33]).
First of all, we fix 2R 2 from the slope of the linear potential at long distances, where the stringy picture is more reliable. Since this regime is equivalent to λ ≈ 2 we compare Eq. (3) with Eq. (26), which leads to the condition 1/a 2 = 2R 2 σ(n) and therefore:
2R 2 = 4π(aκ) −2 e × 2 e 2−n n .(27)
Once we have fixed the parameter 2R 2 , we can fit with phenomenological constants imposing 2R 2 κ 0 = ξ, i.e.:
ξ = 4π(2.34κ) −2 e × 2 e 2−n n 1 2πρ 2 .(28)
The above equation can be solved graphically for given values of κ: we present some of these solutions in Figure 1, for the interval ( 0.55 ≤ κ ≤ 0.70 ) GeV −1 .
It is important to highlight that we can always choose a value of κ and a corresponding value of n such that we fit exactly the phenomenological data for the Cornell potential. With the values of parameter κ and its corresponding values of n we can investigate the energy associated to the quark-antiquark pair through numerical calculations. In Figure 2 we plot the quark-antiquark potential E Ren. in terms of the quark separation , for some values of κ.
IV. CONCLUDING REMARKS
In this work we have calculated the energy corresponding to a given separation between a quark-antiquark pair from the Nambu-Goto action using a deformed AdS space as a background. The choice of the deformed AdS space is based on the introduction of an exponential factor given by h(z) = exp{(κz) n /n}, Eq. (2). We have also shown that this configuration energy has the shape of a Cornell potential. In order to fit the Cornell potential parameters we can choose a variety of possibilities for the pair (κ, n). In Figure 1, we have shown some of these possibilities and in Figure 2, we presented some profiles for the Cornell potential. Note that if we substitute n = 2 in Eq. (27), we re-obtain the result of [5], namely, 2R 2 = 0.94.
Note that in Figure 2 we observe the transition from a confining to a non-confining regime where ∼ 0.5 fm as expected from phenomenological arguments. Another interesting feature of our model is the universal non-confining behavior for ≈ 0, already pointed out by [2].
In the context of our model, this universal behavior for short distances is due to the fact that ≈ 0 is equivalent to λ ≈ 0 ( c.f. section II A 1 ) which means that h(z) → 1 and hence we recover the geometry of pure AdS space and therefore we must obtain the non-confining term due to the conformal symmetry of the background space.
Indeed the main attractive feature of our deformation of the AdS, which is a UV deformation, is the fact that it only affects the large distance physics. This modification is encapsulated by the coefficients of the linear term in Eq. (26), which become dependent on the deformation h(z), where κ and n, are the parameters that control the deformation. It is interesting that the confining behavior is maintained despite of the choice of κ and n which is actually an explicit manifestation of the criterion discussed by [3].
Finally, note that in Figure 2, for the linear confining behaviour all the curves shown present the same slope. This is not a universal property of the deformation we have considered but rather is a choice to fit the Cornell potential parameters. In other words, if we have chosen freely the parameters κ and n without the phenomenological constraint Eq. (28) we would have found different slopes for the potentials.
B. Calculation of the energyBefore we calculate the integral in eq. (9) let us point out that it diverges as 1/v 2 when v → 0. This becomes clear if one analyzes the series expansion of the integrand in λ close to 0 and 2.
Figure 1 :
1Equation (28) solved graphically: The curves are plots of Eq. (28) for some values of κ with a = 2.34 GeV −1 . The horizontal dashed line represents the phenomenological desired value of the parameter ξ, i.e., ξ = 0.52 to fit the Cornell potential.
Figure 2 :
2E Ren. against obtained directly from Eqs. (8) and (15) through numerical integration, for three particular values of κ: 0.55 GeV −1 , 0.57 GeV −1 , 0.65 GeV −1 and their respective approximate values of n: 1.2, 1.4 , 3.5. These curves correspond to possible matches with the Cornell potential. The values n come from the Figure 1 for each curve corresponding to a given κ.
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| {'fraction_non_alphanumeric': 0.0936648632055562, 'fraction_numerical': 0.10571351828402593, 'mean_word_length': 3.717958001448226, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this work we calculate the static limit of the energy for a quark-antiquark pair from the Nambu-Goto action using a holographic approach with a deformed AdS space. From this energy we derive the Cornell potential for the quark-antiquark interaction. We also find a range of values for our parameters which fits exactly the Cornell potential parameters. * Eletronic address:bruni.r.c.', 'arxivid': '1806.05720', 'author': ['Rodrigo C L Bruni \nDepartamento de Física Teórica\nUniversidade do Estado do Rio de Janeiro\n20.550-900 -Rio de Janeiro-RJBrazil\n', 'Eduardo Folco Capossoli \nDepartamento de Física and Mestrado Profissional em Práticas da Educação Básica (MPPEB)\nColégio Pedro II, 20.921-903 -Rio de Janeiro-RJBrazil\n\nInstituto de Física\nUniversidade Federal do Rio de Janeiro\n21.941-972 -Rio de Janeiro-RJBrazil\n', 'Henrique Boschi-Filho \nInstituto de Física\nUniversidade Federal do Rio de Janeiro\n21.941-972 -Rio de Janeiro-RJBrazil\n'], 'authoraffiliation': ['Departamento de Física Teórica\nUniversidade do Estado do Rio de Janeiro\n20.550-900 -Rio de Janeiro-RJBrazil', 'Departamento de Física and Mestrado Profissional em Práticas da Educação Básica (MPPEB)\nColégio Pedro II, 20.921-903 -Rio de Janeiro-RJBrazil', 'Instituto de Física\nUniversidade Federal do Rio de Janeiro\n21.941-972 -Rio de Janeiro-RJBrazil', 'Instituto de Física\nUniversidade Federal do Rio de Janeiro\n21.941-972 -Rio de Janeiro-RJBrazil'], 'corpusid': 118831299, 'doi': '10.1155/2019/1901659', 'github_urls': [], 'n_tokens_mistral': 11181, 'n_tokens_neox': 8677, 'n_words': 4218, 'pdfsha': 'b80119ea5964f320b8cbd8b11b4404d091f094e9', 'pdfurls': ['https://arxiv.org/pdf/1806.05720v3.pdf'], 'title': ['Quark-antiquark potential from a deformed AdS/QCD', 'Quark-antiquark potential from a deformed AdS/QCD'], 'venue': []} |
arxiv |
ON 12-CONGRUENCES OF ELLIPTIC CURVES
11 Aug 2022
Sam Frengley
ON 12-CONGRUENCES OF ELLIPTIC CURVES
11 Aug 2022arXiv:2208.05842v1 [math.NT]
We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over Q with 12-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen[Che16]and Fisher [Fis20] where it is assumed that the underlying isomorphism of 12-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves.A key ingredient in the proof is to construct simple (algebraic) conditions for the 2, 3, or 4-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j-invariants of the pair of elliptic curves.Date: 11 August 2022. 1 For any N ≥ 2 and r ∈ (Z/N Z) × let Z(N, r) be the (coarse) moduli space which parametrises triples (E, E ′ , φ) where E and E ′ are elliptic curves (defined up to simultaneous quadratic twist) and φ :is an (N, r)-congruence (defined up to composition with an automorphism of E or E ′ ). Following Kani-Schanz [KS98] we call Z(N, r) a modular diagonal quotient surface. The surface Z(N, r) is naturally equipped with an involution which reverses the roles of E and E ′ . We write W (N, r) for the quotient of Z(N, r) by this involution.The main contribution of this article is to give explicit equations for the surfaces Z(12, r) for each r ∈ (Z/12Z) × . We use these equations to find infinite families of (12, r)-congruent elliptic curves which are not geometrically isogenous.The surface Z(12, 1) is an elliptic K3 surface and models were computed by Chen and Fisher (see [Che16, Chapter 7] and [Fis20]) using equations for the twist X 1 E (12) of X(12) which parametrises elliptic curves which are (12, 1)-congruent to E (see Section 2.1). Chen [Che16, Chapter 7.4] also computed equations for the twist X 7 E (12) and it may be possible to compute a simple birational model for Z(12, 7) using these. Using different techniques we prove:Theorem 1.1. Each of the surfaces W (12, r) are rational and the surfaces Z(12, r) are birational over Q to the affine surfacesF 12,11 (u, v) = − (u + 1)((v 4 + 6v 2 + 1)u − 7v 4 − 2v 2 + 1)(v 4 u 4 + 8v 4 u 3 + (−9v 6 + 24v 4 + 11v 2 )u 2 + (−54v 6 − 36v 4 − 6v 2 )u + 27v 8 + 27v 6 + 9v 4 − v 2 − 1).Moreover the double covers Z(12, r) → W (12, r) are given by (u, v, z) → (u, v).The maps Z(12, r) → X(1) × X(1) which give the moduli interpretations for the surfaces Z(12, r) are too complicated to reproduce here but may be recovered from the computations in Section 5. We record them in the electronic data corresponding to this article [Fre22].Remark 1.2. Note that the polynomials F 12,r (u, v) contain only even powers of v. In Section 4 we will show that Z(12, r) is a double cover of a surface Z + (12, r). This double cover is the quotient by the involution v → −v.Remark 1.3. The model for the surface Z(12, 1) given in Theorem 1.1 admits a genus 1 fibration over the t-line given by y 2 = F 12,1 (x, t) (in fact, this fibration is elliptic, see Example 6.3).Since the surfaces Z(12, r) are of general type when r = 1 they cannot admit genus 1 fibrations. However, they do admit genus 2 fibrations over the t-line which are given by y 2 = F 12,5 −x+42t , x 2t , y 2 = F 12,7 (x, t), and y 2 = F 12,11 (x, t) respectively.JZ JZ π where J Z and J Z are morphisms. The degree of the morphism J Z is equal to |SL 2 (Z/N Z)|/2 and the degree of J Z may be read off from the degrees of the polynomials P α , P β , and P γ . Since these degrees are equal for each (N, r) the map π is birational.Letting Z ′ (1) = Spec R we have a natural open immersion Z ′ (1) ⊂ Z(1). Let Z ′ (N, r) and Z ′ (N, r) be the preimages of Z ′ (1) under the maps J Z and J Z respectively.Explicitly Z ′ (N, r) and Z ′ (N, r) are the surfaces given by deleting the points on Z(N, r) and Z(N, r) above the loci: (N, r) = (2, 1) : J = J ′ and J, J ′ ∈ {0, 1}, (N, r) = (4, r) : J = J ′ and J, J ′ ∈ {0, 1}, (N, r) = (3, 1) : J = J ′ , J, J ′ ∈ {0, 1}, and ( 3 √ J + 1)( 3 √ J ′ + 1) = 1, (N, r) = (3, 2) : J = J ′ and J, J ′ ∈ {0, 1}.We next show that π extends to an isomorphism π : Z ′ (N, r) → Z ′ (N, r). Note that the morphism Z ′ (N, r) → Z ′ (1) is finiteétale since X(N ) → X(1) is finiteétale away from j = 0, 1728, ∞.When N = 2, 3 we see that the finite morphism Z ′ (N, r) → Z ′ (1) factors via
Introduction
Let N ≥ 1 be an integer, K be a perfect field of characteristic coprime to N , and G K := Gal(K/K) be the absolute Galois group of K. Let E/K and E ′ /K be elliptic curves and let E[N ] and E ′ [N ] be their N -torsion subgroups.
We say that E and E ′ are N -congruent (or N -congruent over K) if there exists a G K -equivariant isomorphism φ : E[N ] → E ′ [N ]. In this case the isomorphism φ is said to be an N -congruence.
Frey and Mazur conjectured that there exists a constant C 0 > 1 such that for any N ≥ C 0 there are no pairs of non-isogenous N -congruent elliptic curves over Q. It is necessary to exclude isogenies since for each integer m coprime to N the restriction of an m-isogeny to E[N ] is an N -congruence.
Note that if φ :
E[N ] → E ′ [N ]
is an N -congruence then there exists an element r ∈ (Z/N Z) × such that for all P, Q ∈ E[N ] we have e E,N (P, Q) = e E ′ ,N (φ(P ), φ(Q)) r where e E,N and e E ′ ,N denote the Weil pairings on E and E ′ respectively. In this case we say that φ is an (N, r)-congruence or an N -congruence with power r. For each integer m coprime to N composing φ with multiplication by m changes the power of φ by a factor of m 2 , so we consider the power of φ only up to multiplication by a square in (Z/N Z) × .
We are concerned with the following question: for which pairs (N, r) do there exist examples (or infinite families of examples) of pairs of non-isogenous (N, r)-congruent elliptic curves over Q?
In the case when N = p is a prime number, infinite families of geometrically non-isogenous (p, r)-congruent elliptic curves over Q have been found for every power r when p ≤ 13 (see [RS01], [RS95], [Sil97], [Fis12], [HK03], [PSS07], [KR00], [Fis14], [Kum15], [Fis20], and [Fis19]). A pair of non-isogenous (17, 3)-congruent elliptic curves appearing in the LMFDB [LMF] has been found by Cremona (see [Fis14], [Bil16], [CF22]). Fisher [Fis21] also found a pair of non-isogenous (17, 1)-congruent elliptic curves over Q, and has conjectured that these are the only 17-congruences over Q. Moreover Fisher conjectured that there exist no pairs of nonisogenous, p-congruent, elliptic curves over Q for p ≥ 19 (see [Fis21, Conjecture 1.1]).
For composite N infinite families of geometrically non-isogenous (N, r)-congruent elliptic curves have been found for every power r when N ≤ 10 and for (N, r) = (12, 1) (see [Sil97], [Fis12], [Rob99], [Pap99], [RS99], [Che16], [Fis15], [Che18], and [Fis20]).
On the other hand in [Fre21] we found examples of elliptic curves over Q admitting an N -congruence with a (non-isogenous) quadratic twist for each even N ≤ 24, N = 28, 30, 32, 36, and 48. These examples arise in infinite families except in the cases when N = 30, 32, or 48. When N = 12 we showed that there are infinite families of elliptic curves admitting (12, 7) and (12, 11)-congruences with a (non-isogenous) quadratic twist. However, in these examples the elliptic curves in question are geometrically isogenous and when N = 2, 4 their mod N Galois representations are not surjective.
It is possible to find pairs of 12-congruent elliptic curves by sieving for examples where both E and E ′ are contained in the LMFDB [LMF] (as described in [CF22,Section 3]). The power of these congruences may be determined using the equations of Rubin, Silverberg, and Fisher [RS95,Sil97,RS01,Fis12] for X r E (N ) when N = 3 and 4. For example the pairs of elliptic curves with LMFDB labels (55.a1, 1045.b1), (60450.cx2, 60450.cw2), and (735.d2, 24255.bh1) are not geometrically isogenous and are (12, 1), (12, 5), and (12, 7)-congruent respectively.
On the other hand we were not able to find any examples of (12, 11)-congruences between geometrically non-isogenous curves in the LMFDB. The rational point −19 21 , 3 7 , 1844480 2470629 on the model for Z(12, 11) in Theorem 1.1 gives rise to the pair of geometrically non-isogenous (12, 11)-congruent elliptic curves E : y 2 + xy = x 3 − 21666120x − 57035036608, E ′ : y 2 + xy = x 3 + 398520965x + 166506419482597 which have conductor 4 976 690. Moreover, among pairs of geometrically non-isogenous (12, 11)-congruent elliptic curves which we found, these curves have the smallest conductor (see Section 6.4).
By exhibiting a rational curve on the elliptic K3 surface Z(12, 1) Chen found an infinite family of (12, 1)congruent elliptic curves over Q [Che16, Proposition 1.7.11]. Fisher gave a fibration with a section of infinite order, thus giving an infinite family of (12, 1)-congruent elliptic curves over Q(t) [Fis20, Corollary 1.3].
In contrast Kani-Schanz [KS98] showed that when r = 1 the surfaces Z(12, r) are of general type. In this case we expect (by the Bombieri-Lang conjecture) that there are at most finitely many curves of geometric genus 0 or 1 on Z(12, r).
By finding curves of genus 0 or 1 on the models for Z(12, r) in Theorem 1.1 we deduce:
Theorem 1.4. For each r ∈ (Z/12Z) × there are infinitely many pairs of j-invariants of (12, r)-congruent elliptic curves E/Q and E ′ /Q which are not geometrically isogenous. Moreover there are infinite subfamilies such that the mod 12 Galois representation attached to E/Q (and E ′ /Q) is surjective.
Remark 1.5. Since a pair of elliptic curves are (12, r)-congruent if and only if they are both (3, r) and (4, r)-congruent, the families of elliptic curves in Theorem 1.4 may be checked using known formulae for the twists X r E (N ). Such equations have been computed by Rubin and Silverberg [RS95, Sil97, RS01] when r = 1, and by Fisher [Fis12] for each r.
1.1. Outline of the paper. In Section 2 we give background on the invariant theory of the modular curve X(N ) when N ≤ 4 following Klein [Kle56] and Fisher [Fis12]. We describe Z(N, r) as a quotient of X(N ) × X(N ) following Kani and Schanz [KS98].
In Section 3 we use invariant theoretic techniques to find simple (symmetric) conditions for a pair of elliptic curves to be 2, 3, or 4-congruent (note that our condition for 2-congruence was already proved by Fisher, see [Fis20, Lemma 3.5]).
We describe the relationship between the surface Z(N 1 N 2 , r) and the fibre product Z(N 1 , r)× Z(1) Z(N 2 , r) when N 1 and N 2 are coprime integers in Section 4. We employ this description to compute explicit models for the surfaces Z(12, r) and prove Theorem 1.1 in Section 5.
Finally in Section 6 we find "interesting" curves on the surfaces Z(12, r). We prove Theorem 1.4 in Section 6.2 by finding curves with infinitely many rational points (and which do not arise as Hecke correspondences) on the models for Z(12, r) in Theorem 1.1. More precisely we give explicit examples of pairs of geometrically non-isogenous (12, r)-congruent elliptic curves which are defined over Q(t) when r = 1, 7, and 11 (see Examples 6.3, 6.5, and 6.6). When r = 5 we give a family defined over Q(C) where C/Q is an elliptic curve of rank 1 (see Example 6.4).
1.2. Acknowledgements. I would like to thank my supervisor Tom Fisher for suggesting this topic and for many insightful conversations and comments on earlier versions of this article. I would also like to express my gratitude to Jef Laga for several comments on earlier versions of this paper and to Noah Porcelli, Tony Scholl, and Alice Silverberg for useful discussions and suggestions. I am grateful to the Woolf Fisher and Cambridge Trusts for their financial support.
2. The modular curve X(N ) and the surface Z(N, r) Let N ≥ 1 be an integer, and let K be a field of characteristic coprime to 6N . Consider the G K -module µ N × Z/N Z endowed with the structure of symplectic abelian group via the pairing (ζ, c), (ξ, d) = ζ d ξ −c . Let Γ N denote the group of symplectic automorphisms of µ N × Z/N Z, and note that Γ N is isomorphic to SL 2 (Z/N Z) as an abstract group.
Let Y (N )/K be the (non-compact) modular curve of full level N . For each field L/K, the L-rational points on Y (N ) parametrise pairs (E/L, ι) where E/L is an elliptic curve and ι : E[N ] → µ N × Z/N Z is a G L -equivariant isomorphism which is symplectic with respect to the Weil pairing on E[N ]. Recall that when N = 1, 2 the moduli space Y (N ) is coarse, and the pair (E/L, ι) is only determined up to twist.
An element g ∈ Γ N acts on Y (N ) via (E, ι) → (E, gι) and this action extends uniquely to an automorphism of X(N ). Elements g, g ′ ∈ Γ N induce the same action on X(N ) if and only if they differ by an automorphism of E. In particular, this action descends to an action of Γ N /{±1}.
2.1. The Twist X r E (N ). Let E/K be an elliptic curve and r be an integer coprime to N . Consider a K-isomorphism ι r : E[N ] → µ N × Z/N Z such that e N (P, Q) r = ι r (P ), ι r (Q) . By the twisting principle ι r gives rise to a Galois cohomology class [ι r ] ∈ H 1 (K, Γ N ).
The G K -equivariant homomorphism Γ N → Aut(X(N ) K ) induces a map on Galois cohomology H 1 (K, Γ N ) → H 1 (K, Aut(X(N ) K )). Again by the twisting principle we may associate to the image of [ι r ] a twist X r E (N )/K of X(N ).
For each field L/K, the L-rational points on the curve X r
E (N ) parametrise pairs (E ′ /L, φ) where φ : E[N ] → E ′ [N ] is an (N, r)-congruence.
2.2. The Surface Z(N, r). Recall that for each N ≥ 2 and r ∈ (Z/N Z) × we let Z(N, r)/K be the (coarse) moduli space of triples (E, E ′ , φ) where E/K and E ′ /K are elliptic curves (defined up to simultaneous quadratic twist) and φ :
E[N ] → E ′ [N ]
is an (N, r)-congruence. Further recall that we write W (N, r) for the quotient of Z(N, r) by the involution reversing the roles of E and E ′ .
We will write Z(1) for the surface Y (1)× Y (1) and W (1) for the quotient of Z(1) by the natural involution swapping the roles of j and j ′ . It will be useful for us to note that W (1) is birational to A 2 and that we may take the quotient Z(1) → W (1) to be given by (j, j ′ ) → (jj ′ , (j − 1728)(j ′ − 1728)).
We then have a commutative diagram:
Z(N, r) W (N, r) Z(1) W (1)
Define the group ∆ r to be the subgroup {(g, ε r (g)) : g ∈ Γ N } ⊂ Γ N × Γ N , where ε r denotes conjugation by the matrix r 0 0 1 . The following lemma is due to Kani-Schanz [KS98] and Fisher [Fis19, Lemma 3.2]. 2.3. The Action of Γ N on X(N ). We briefly describe the invariant theory of the modular curves X(N ) when N = 2, 3, and 4. In these cases the invariant theory of X(N ) has been extensively studied, dating back to Klein [Kle56]. We follow the more modern treatment of Fisher [Fis12,.
For N = 2, 3, 4 we identify the modular curve X(N ) with P 1 . We define the (finite) subgroups G N ⊂ SL 2 (K) to be the subgroups generated by
N = 2 : i 1/2 1/16 12 −1/2 and i 1 0 0 −1 N = 3 : 1 √ −3 1 1/3 6 −1 and ζ 3 1 0 0 ζ 3 N = 4 : 1 √ −2 1 1/2 2 −1 and ζ −1 8 1 0 0 i
where ζ N ∈ K is a primitive N th root of unity. By mapping the generators 0 1 −1 0 and 1 1 0 1 of Γ N to the above generators of (the projective image of) G N we obtain a G K -equivariant homomorphismρ : Γ N → PGL 2 (K) giving the action of Γ N on X(N ).
Let G N be the commutator subgroup of G N . It is a classical theorem of Klein that for each N = 2, 3, 4 the invariant ring K[x 0 , x 1 ] GN is generated by invariants c 4 , c 6 , and D of weights 4N , 6N , and 12 subject only to the relation c 3 4 − c 2 6 = 1728D N (see [Fis12,Theorem 3.3]). Explicitly we take N = 2 :
D = x 0 (64x 2 0 − x 2 1 ) c 4 = 192x 2 0 + x 2 1 c 6 = x 1 (576x 2 0 − x 2 1 ) N = 3 : D = −x 0 (27x 3 0 + x 3 1 ) c 4 = −x 1 (216x 3 0 − x 3 1 ) c 6 = 5832x 6 0 − 540x 3 0 x 3 1 − x 6 1 N = 4 : D = x 0 x 1 (16x 4 0 − x 4 1 ) c 4 = 256x 8 0 + 224x 4 0 x 4 1 + x 8 1 c 6 = 4096x 12 0 − 8448x 8 0 x 4 1 − 528x 4 0 x 8 1 + x 12 1 .
Moreover, by considering ramification points it can be shown that the map X(N ) → X(1) given by j = c 3 4 /D N is the j-map (see [Fis12,Lemma 4.4]). With the models chosen above the forgetful map X(4) → X(2) is given by [x 0 :
x 1 ] → [−x 2 0 x 2 1 : 16x 2 0 + x 4 1 ].
3. Symmetric equations for families of N -congruent elliptic curves
Several approaches have been applied to parametrising (universal) families of 2, 3, and 4-congruent elliptic curves.
Equations for X 1 E (N ) when N = 2, 3, 4, and 5 were computed by Rubin and Silverberg (see [RS95], [Sil97], [RS01]). Using a direct approach Lario-Rio computed equations for X r E (3) when r = 1 and 2 (see [LR95, Section 3]).
Fisher [Fis12] described how when N = 2, 3, 4, and 5 the curve X r E (N ) is related to the Hessian of a genus one model of degree N , and hence computed explicit equations for X r E (N ) using classical invariant theory. When N = 3 similar approaches have been employed in [Kuw12] and [ATT18]. The former considers all powers, and the latter pays careful attention to the case when r = 1 and the characteristic of K is 2.
Kumar [Kum15] (for every N ≤ 11), Kuhn [Kuh88] and Bröker-Howe-Lauter-Stevenhagen [BHLS15] Our approach is to use Lemma 2.1 to compute models for the surfaces Z(N, r) for each (N, r) = (2, 1), (3, 1), (3, 2), (4, 1), and (4, 3). An advantage of our approach is that by viewing E and E ′ symmetrically we are able to exploit natural factorisations of the maps Z(N, r) → Z(1) = Y (1) × Y (1) to provide simple conditions for E and E ′ to be 2, 3, or 4-congruent (see Theorem 3.3). Indeed, it is this observation which makes it practical for us to compute (birational) models for Z(12, r) via fibre products.
3.1. Computing quotients of X(N ) × X(N ). We compute Z(N, r) from its description as a quotient of X(N ) × X(N ) under the (twisted) diagonal action of Γ N . The method we describe has been successfully employed by Fisher to compute birational models for Z(13, r) and Z(17, r) (see [Fis19] and [Fis21]).
Suppose that we have the data of:
• an embedding X(N ) ⊂ P n for some n ≥ 1, and
• a G K -equivariant homomorphism ρ : Γ N → GL n+1 (K) such that Γ N acts on X(N ) via ρ. Let f (x 0 , ..., x n , x ′ 0 , ..., x ′ n ) be a polynomial which is homogeneous of degrees m and m ′ in the two sets of variables. Let ∆ r ⊂ Γ N × Γ N act on the polynomial ring K[x 0 , ..., x n , x ′ 0 , ..., x ′ n ] via ρ. Definition 3.1. We say that f is an r-bi-invariant if f is invariant under the action of ∆ r on the polynomial ring K[x 0 , ..., x n , x ′ 0 , ..., x ′ n ]. If f (x 0 , ..., x n , x ′ 0 , ..., x ′ n ) = f (x ′ 0 , ..., x ′ n , x 0 , ..., x n ) we say that f is symmetric. By Lemma 2.1 the surface Z(N, r) is birational to Proj A/A ∩ I where I = I(X(N ) × X(N )) is the ideal cutting out X(N ) × X(N ) in the product projective space P n × P n and A = K[x 0 , ..., x n , x ′ 0 , ..., x ′ n ] ∆r .
To compute a model for Z(N, r) it is enough to compute generators for the K-algebra K[x 0 , ..., x n , x ′ 0 , ..., x ′ n ] ∆r , their relations modulo I, and their relations (modulo I) with the rational functions j = j(x 0 , ..., x n ) and
j ′ = j(x ′ 0 , ..., x ′ n ) where j is the rational function giving the j-invariant on X(N ). Similarly if B ⊂ K[x 0 , ..., x n , x ′ 0 , ..., x ′ n ] ∆r is the subalgebra of symmetric r-bi-invariants then W (N, r) is birational to Proj B/B ∩ I.
In the cases we are interested in (when N = 2, 3, and 4) we have X(N ) ∼ = P 1 and I = 0. We do not attempt to fully describe the invariant ring K[x 0 , ..., x n , x ′ 0 , ..., x ′ n ] ∆r . Instead we use the following observation:
Consider r-bi-invariants w 0 , ..., w k such that for each i the r-bi-invariant w i is an element of the K-vector space of symmetric r-bi-invariants of homogeneous bi-degree (m, m). Let ϕ N,r be the map X(N )×X(N ) P k given by ([x 0 : ... :
x n ], [x ′ 0 : ... : x ′ n ]) → [w 0 : ... : w k ].
Since w 0 , ..., w k are symmetric and invariant under the action of ∆ r we obtain a factorisation
X(N ) × X(N ) W (N, r) im(ϕ N,r ). ψN,r ϕN,r
When N = 2, 3, and 4 we will carefully choose symmetric r-bi-invariants w 0 , ..., w k so that ψ N,r is birational.
3.2. Symmetric bi-invariants. We now give choices of r-bi-invariants such that the map ψ N,r is birational for each N = 2, 3, and 4. We consider the action of Γ N on X(N ) ∼ = P 1 via the representationρ in Section 2.3.
Remark 3.2. Note that in the case when N = 4 the actions of ∆ 1 and ∆ 3 on K[x 0 , x 1 , x ′ 0 , x ′ 1 ] via ρ have the same invariants. We therefore treat both cases simultaneously.
There is also an explanation for this phenomenon in terms of the moduli interpretation of Z(4, r). Every elliptic curve E/K is (4, 3)-congruent with its quadratic twist by its discriminant (see [BD11,Corollary 7.4] or [Fre21, Proposition 3.6] for a more general statement). In particular, if E and E ′ are (4, 1)-congruent, then the quadratic twist of E by its discriminant E ∆(E) is (4, 3)-congruent to E ′ . This induces an isomorphism Z(4, 1) ∼ = Z(4, 3) which commutes with the j-invariant maps Z(4, r) → Z(1).
Recall the forms
D = D(x 0 , x 1 ), c 4 = c 4 (x 0 , x 1 ), and c 6 = c 6 (x 0 , x 1 ) defined in Section 2.3. We let D ′ = D(x ′ 0 , x ′ 1 ), c ′ 4 = c 4 (x ′ 0 , x ′ 1 ), and c ′ 6 = c 6 (x ′ 0 , x ′ 1 ). The forgetful map X(N ) × X(N ) → Z(1) is given by ([x 0 : x 1 ], [x ′ 0 : x ′ 1 ]) → (j, j ′ ) where j = c 3 4 /D N and j ′ = c ′3 4 /D ′N . We write J = j/1728 (similarly J ′ = j ′ /1728).
Additionally, we define the following (symmetric) r-bi-invariant forms:
(N, r) = (2, 1) :
I 1,1 = 192x 0 x ′ 0 + x 1 x ′ 1 , (N, r) = (4, r) : I 1,1 = 4(x 0 x ′ 1 − x 1 x ′ 0 ), I 4,4 = 256x 4 0 x ′4 0 + 16x 4 0 x ′4 1 + 192x 2 0 x 2 1 x ′2 0 x ′2 1 + 16x 4 1 x ′4 0 + x 4 1 x ′4 1 , (N, r) = (3, 1) : I 2,2 = 3(54x 2 0 x ′2 0 + x 0 x 1 x ′2 1 + x 2 1 x ′ 0 x ′ 1 ), I 6,6 = 3 6 (x 0 x ′ 1 − x 1 x ′ 0 ) 6 , (N, r) = (3, 2) : I 1,1 = 18x 0 x ′ 0 + x 1 x ′ 1 .
We then make the following choices of symmetric r-bi-invariants: Table 1. Choice of r-bi-invariants inducing a rational map W (N, r) P k .
(N, r) r-bi-invariants (w 0 , ..., w k ) (2, 1) I 3 1,1 , 2I 1,1 c 4 c ′ 4 , 1728DD ′ (4, r) I 3 4,4 , 2I 4,4 c 4 c ′ 4 , 3I 4 1,1 I 2 4,4 , 72I 2 1,1 I 4,4 DD ′ (3, 1) 12I 6,6 , 4I 3 2,2 , c 6 c ′ 6 , 144I 2,2 DD ′ (3, 2) 3 2 (3I 4 1,1 − c 4 c ′ 4 − 144DD ′ ), c 4 c ′ 4 , 144DD ′
3.2.1. Case (N, r) = (2, 1). First note that c 6 c ′ 6 = 4w 0 − 3 2 w 1 − w 2 . In particular we have
JJ ′ = w 3 1 8w 0 w 2 2 , 6 (J − 1)(J ′ − 1) = 4w 0 − 3 2 w 1 − w 2 w 2 2 .
Let S(2, 1) = P 2 with coordinates w 0 , w 1 , w 2 . Then there is a commutative diagram
W (2, 1) S(2, 1) W (1) ψ2,1
where the map S(2, 1) W (1) is given by the rational functions JJ ′ and (J − 1)(J ′ − 1) above.
3.2.2. Case (N, r) = (4, r). We first note that the invariants w 0 , w 1 , w 2 , w 3 satisfy the single relation 192w 2 0 − 96w 0 w 1 + 128w 0 w 3 + 48w 0 w 2 − w 2 2 = 0. Therefore the image of the map ϕ N,r :
X(4) × X(4) P 3 is the quadric surface S(4, r) = {192w 2 0 − 96w 0 w 1 + 128w 0 w 3 + 48w 0 w 2 − w 2 2 } ⊂ P 3 . Let H = ker (SL 2 (Z/4Z) → SL 2 (Z/2Z)),c ′ 6 = 4w 0 − 3 2 w 1 − (w 2 3 /w 2 )
. Using the equations for the forgetful map X(4) → X(2) given in Section 2.3 we see that the diagram
W (4, r) S(4, r) W (2, 1) S(2, 1) ψ4,r ψ2,1 commutes, where the map S(4, r) S(2, 1) is given by [w 0 : w 1 : w 2 : w 3 ] → [w 0 : w 1 : w 2 3 /w 2 ].
3.2.3. Case (N, r) = (3, 1). The image of the rational map ϕ N,r :
X(3) × X(3) P 3 is the cubic surface S(3, 1) = {8w 2 0 w 1 − 60w 0 w 2 1 + 12w 0 w 1 w 2 + 36w 0 w 1 w 3 − 9w 3 1 + 27w 2 1 w 3 − 27w 1 w 2 3 + 9w 3 3 = 0}.
Additionally there are the relations:
JJ ′ = w 1 (4w 2 0 − 192w 0 w 1 + 12w 0 w 2 + 72w 0 w 3 + 603w 2 1 − 144w 1 w 2 − 918w 1 w 3 + 9w 2 2 + 108w 2 w 3 + 351w 2 3 ) 36w 3 3 , (J − 1)(J ′ − 1) = w 1 w 2 2 4w 3 3 .
We therefore have a commutative diagram
W (3, 1) S(3, 1) W (1) ψ3,1
where the morphism S(3, 1) W (1) is given by the rational functions JJ ′ and (J − 1)(J ′ − 1) above.
3.2.4. Case (N, r) = (3, 2). Let S(3, 2) = P 2 . The bi-invariants w 0 , w 1 , and w 2 are linearly independent, so the map ϕ N,r : X(3) × X(3) S(3, 2) is surjective. We have the relations
JJ ′ = w 1 w 2 3 , (J − 1)(J ′ − 1) = (w 2 0 − 3w 2 1 − 3w 1 w 2 − 3w 2 2 ) 2 4w 3 2 (2w 0 + 3w 1 + 3w 2 ) . 7
Therefore there is a commutative diagram
W (3, 2) S(3, 2) W (1) ψ3,2
where the map S(3, 2) W (1) is given by the rational functions JJ ′ and (J − 1)(J ′ − 1) above.
3.3. Symmetric conditions for 2, 3, and 4-congruences. We now prove Theorem 3.3. This provides a simple characterisation of 2, 3, and 4-congruent elliptic curves in terms of their j-invariants. It is important for our computations of the surfaces Z(12, r) that, unlike the equations of Fisher [Fis12] and of Rubin and Silverberg [RS95,Sil97,RS01], the conditions in Theorem 3.3 are symmetrical in the pair of congruent elliptic curves, E and E ′ . This will be useful since we will be able to work with the surfaces W (12, r) which have simpler birational geometry. The case when N = 2 appears in [Fis20, Lemma 3.5] with a different (computational) proof using the equations for X 1 E (2). It is also possible to verify Theorem 3.3 using the equations of Fisher and of Rubin and Silverberg for X r E (N ) (the necessary transformations are recorded in the electronic data [Fre22]). We opt for a more geometric approach which explains how we found these conditions. Theorem 3.3. Let K be a field of characteristic not equal to 2 or 3. Let E/K and E ′ /K be elliptic curves with j-invariants j and j ′ respectively. Let c 4 , c 6 , ∆ and c ′ 4 , c ′ 6 , ∆ ′ be the c-invariants and discriminant of (a Weierstrass model of ) E and E ′ respectively. Suppose that j, j ′ ∈ {0, 1728}. Let J = j/1728 and J ′ = j ′ /1728. Then we have the following:
(i) E and E ′ are 2-congruent if and only if there exist α, β ∈ K such that
α 2 − (J − 1)(J ′ − 1) = 0, β 3 − 3JJ ′ β − 2JJ ′ (α + 1) = 0.
(ii) If J = J ′ then E and E ′ are (4, r)-congruent if and only if there exist α, β, γ, τ ∈ K such that α and β satisfy the conditions of (i) and
γ 4 − 6JJ ′ βγ 2 − 16(JJ ′ ) 2 γ + 3(JJ ′ ) 2 (4JJ ′ − β 2 ) = 0, τ 2 = 3c 6 c ′ 6 δα if r ≡ 1 (mod 4), 3c 6 c ′ 6 α if r ≡ 3 (mod 4). where δ = 3(JJ ′ − (α + 1) 2 ). (iii) If J = J ′ and ( 3 √ J +1)( 3 √ J ′ +1) = 1
(for all choices of cube roots) then E and E ′ are (3, 1)-congruent if and only if there exists α, β, τ ∈ K such that
α 3 − 3JJ ′ α − JJ ′ (J + J ′ ) = 0, β 4 − 6(J − 1)(J ′ − 1)β 2 − 8(J − 1) 2 (J ′ − 1) 2 β − 3(4α + 1)(J − 1) 2 (J ′ − 1) 2 = 0, τ 2 = 3c 6 c ′ 6 δ, where δ = −6 2β 3 − (5α + 2)β 2 − 10(J − 1)(J ′ − 1)β + 3(J − 1)(J ′ − 1)(13α − 2 + 6(J + J ′ )) β 3 − 3(J − 1)(J ′ − 1)β − 2(J − 1) 2 (J ′ − 1) 2 .
(iv) If J = J ′ then E and E ′ are (3, 2)-congruent if and only if there exists α, β, τ ∈ K such that
α 3 − JJ ′ = 0, β 4 − 6(α + 1)(J − 1)(J ′ − 1)β 2 − 8(J − 1) 2 (J ′ − 1) 2 β − 3(α − 1) 2 (J − 1) 2 (J ′ − 1) 2 = 0, τ 2 = 3c 6 c ′ 6 β if β = 0, −2c 6 c ′ 6 if β = 0.
Remark 3.4. Note that in Theorem 3.3: 8 (i) The conditions on α are convenient ways of writing the conditions that ∆∆ ′ is a square when N = 2, 4, that ∆/∆ ′ is a cube when (N, r) = (3, 1), and that ∆∆ ′ is a cube when (N, r) = (3, 2). When N = 4 the term δ is equal to ∆ (and ∆ ′ ) up to a square factor (modulo squares we have δ/∆ ≡ 3δ(J − 1) ≡ (α − (J − 1)) 2 ). The conditions on τ are therefore equivalent to the fact that ∆/∆ ′ is a fourth power when (N, r) = (4, 1) and ∆∆ ′ is a fourth power when (N, r) = (4, 3). (ii) When K contains an N th root of unity, it is clear that the polynomials in α in Theorem 3.3 may have more than one root in K. Indeed, the theorem would be false if we required the existence of a β (and γ, τ ) for a fixed choice of α (and β). For example, the curves with LMFDB labels 196.b1 and 196.b2 are 3-isogenous and therefore (4, 3)-congruent. In this example the polynomial in β has no root when the negative root α is chosen. When α is chosen to be positive the polynomial in β splits, but the polynomial in γ has a root for only one of the possible choices of β. When (N, r) = (3, 1) a similar example is given over Q(ζ 3 ) by the pair 15.a1 and 15.a5 and when (N, r) = (3, 2) by the pair 11.a1 and 11.a2. Proof of Theorem 3.3. If j = j ′ then E and E ′ are 2-congruent (since an elliptic curve is 2-congruent to any quadratic twist and j, j ′ ∈ {0, 1728} by assumption). Condition (i) is always satisfied in this case, so we are free to assume that j = j ′ when (N, r) = (2, 1).
For each (N, r) let P α ∈ K[J, J ′ , α], P β ∈ K[J, J ′ , α, β] (and P γ ∈ K[J, J ′ , α, β, γ] when N = 4) be the polynomials from the statement of the theorem.
When N = 2, 3 let Z(N, r) = Spec K[J, J ′ , α, β]/(P α , P β ) and Z(4, r) = Spec K[J, J ′ , α, β, γ]/(P α , P β , P γ ). Similarly when N = 2, 3 let W(N, r) = Spec K[JJ ′ , (J − 1)(J ′ − 1), α, β]/(P α , P β ) and let W(4, r) = Spec K[JJ ′ , (J − 1)(J ′ − 1), α, β, γ]/(P α , P β , P γ ). Write, in the notation of Section 3.2:
(N, r) = (2, 1) :
α = 4w 0 − 3 2 w 1 − w 2 w 2 β = w 1 /w 2 (N, r) = (4, r) : α = 4w 0 − 3 2 w 1 − (w 2 3 /w 2 ) (w 2 3 /w 2 ) β = w 1 w 2 3 /w 2 γ = JJ ′ w 2 βw 3 (N, r) = (3, 1) : α = η 12w 0 w 2 3 β = ( 3 2 (3w 1 − w 3 )/w 3 ) 2 − 3(α + 1) 2 (N, r) = (3, 2) : α = w 1 /w 2 β = (w 0 /w 2 ) 2 − 3(α 2 + α + 1) 2 where η = (w 0 + 3w 1 − 3w 3 )(8w 0 w 1 − 3w 2 1 + 6w 1 w 3 − 3w 2 3 )
. Then these choices induce maps S(N, r) W(N, r) (where S(N, r) are the surfaces defined in Sections 3.2.1-3.2.4) and therefore maps W (N, r) → W(N, r).
In particular we have a commutative diagram
Z(N, r) Z(N, r) Z(1) Z ′ (N, r) = Spec R[α, β]/(P α , P β ) Spec R[α]/(P α ) Z ′ (1).
and similarly when N = 4 with an extra factor accounting for the polynomial in γ. In particular when N = 2, 3, to show that Z ′ (N, r) → Z ′ (1) isétale it suffices to show that ∂P α /∂α and ∂P β /∂β are units in R[α]/(P α ) and R[α, β]/(P α , P β ) respectively. When N = 4 we must also check that ∂P γ /∂γ is a unit in R[α, β, γ]/(P α , P β , P γ ). Computing successive resultants in Magma and using the fact that 2, 3, J, J ′ , J −1, J ′ −1, J −J ′ , and ξ are units in R shows that the ideals P α , ∂Pα ∂α , P α , P β , ∂P β ∂β , and P α , P β , P γ , ∂Pγ ∂γ contain 1. In particular, the morphisms Z ′ (N, r) → Z ′ (1) are finiteétale. Applying the following lemma with X = Z ′ (N, r), Y = Z ′ (N, r), and S = Z ′ (1) shows that π extends to an isomorphism π : Z ′ (N, r) → Z ′ (N, r).
Lemma 3.5. Let X, Y , and S be integral K-varieties. Suppose that S is smooth and that there exists a commutative diagram X Y S pX pY π where p X and p Y are finiteétale morphisms and π is birational. Then π extends uniquely to an isomorphism π : X → Y .
Proof. Because π is birational there exist dense open subschemes U ⊂ X and V ⊂ Y such that π| U : U → V is an isomorphism. Since p X and p Y are finiteétale we may choose U and V such that p X | U : U → S and p X | V : V → S are finiteétale. By assumption S is integral and nonsingular, so natural functor from the category of finiteétale covers of S to the category of finiteétale covers of p X (U ) is fully faithful (see [Sta22, Lemma 0BQG] noting the proof of [Sta22, Lemma 0BQI]). In particular the isomorphism π| U : U → V extends uniquely to a morphism π : X → Y . A symmetrical argument shows that the inverse of π| U extends uniquely to a morphism Y → X. Therefore by uniqueness π is an isomorphism.
The surface Z(N, r)/K acts as a fine moduli space for pairs of elliptic curves E/K and E ′ /K with j(E), j(E ′ ) ∈ {0, 1728} which are (N, r)-congruent up to a simultaneous quadratic twist (or simply up to quadratic twist if N = 2). Therefore we have only shown that elliptic curves E/K and E ′ /K satisfying the conditions on α, β (and γ is N = 4) are (N, r)-congruent up to quadratic twist. When N = 3, 4 it remains to show that the condition on τ is satisfied if and only if E and E ′ are (N, r)-congruent.
The following lemma will allow us to identify the correct quadratic twist.
Lemma 3.6 (c.f., [FK22, Proposition 13(A)]). Let N ≥ 3 be an integer and let K be the field of fractions of a complete discrete valuation ring with residue field of characteristic coprime to N . Suppose that E/K has good reduction and E ′ /K has potential good reduction. If E and E ′ are N -congruent over K then E ′ has good reduction over K.
Proof. Let K ur be the maximal unramified extension of K and let L = K ur (E[N ]) and L ′ = K ur (E ′ [N ]). Since E and E ′ are N -congruent L = L ′ .
But L (resp. L ′ ) is the smallest extension of K ur over which E (resp. E ′ ) obtains good reduction (see [ST68, §2 Corollary 3]). Since E has good reduction over K we have L ′ = L = K ur . Therefore E ′ has good reduction over K.
The fibres of the maps Z ′ (N, r) j − → X(1) have genus 0 and are birational to the curves X r E (N ) where E is an elliptic curve with j-invariant j. By parametrising the generic fibre we obtain a birational map λ : Z ′ (N, r) → X(1) × P 1 which is an isomorphism onto its image since the fibres of Z ′ (N, r) j − → X(1) are nonsingular.
Let E → Z ′ (N, r) be given by the Weierstrass equation
(1)
y 2 = x 3 − 27 j j − 1728 x − 54 j j − 1728
and let E ′ → Z ′ (N, r) be given by the Weierstrass equation
(2) y 2 = x 3 − 27d 2 j ′ j ′ − 1728 x − 54d 3 j ′ j ′ − 1728 .
where (N, r) = (4, 1) :
d = 3δαjj ′ (j − 1728)(j ′ − 1728) , (N, r) = (4, 3) : d = 3αjj ′ (j − 1728)(j ′ − 1728) , (N, r) = (3, 1) : d = 3δjj ′ (j − 1728)(j ′ − 1728) , (N, r) = (3, 2) : d = 3βjj ′ (j − 1728)(j ′ − 1728)
.
By construction the rational functions j, j ′ , (j − 1728), and (j ′ − 1728) are units in R. We check in Magma that when N = 4 the functions δ and α are units in R[α, β, γ]/(P α , P β , P γ ). Similarly when (N, r) = (3, 1) the function δ is a unit in R[α, β]/(P α , P β ). Computing the discriminants of the Weierstrass equations in (1) and (2) we see that the fibres of E , E ′ → Z ′ (N, r) are nonsingular except when (N, r) = (3, 2) and β = 0.
The generic fibres of E , E ′ → Z ′ (N, r) are elliptic curves E/Q(j, t) and E ′ /Q(j, t) and we may choose models such that ∆(E), ∆(E ′ ) ∈ Z[j, t], where j and t are coordinates on X(1) × P 1 .
Since E and E ′ are (N, r)-congruent up to quadratic twist there exists some squarefree D ∈ Z[j, t] such E is (N, r)-congruent to (E ′ ) D . If D has an irreducible factor q not dividing ∆(E)∆(E ′ ) then applying Lemma 3.6 over the completion of Z[j, t] at q we see that E and (E ′ ) D are not N -congruent, hence D divides ∆(E)∆(E ′ ). In particular, there are finitely many possibilities for D ∈ Z[j, t].
Explicitly computing the parametrisation λ in Magma, specialising at a small number of j, t ∈ Q, and comparing traces of Frobenius is enough to prove that D = 1 when N = 3, and D = 1 or δ when N = 4.
The condition on τ follows immediately when (N, r) = (3, 1) and when (N, r) = (3, 2) and β = 0. Moreover we see that elliptic curves are 4-congruent if and only if they satisfy one of the conditions on τ in (ii).
Note that 3c 6 c ′ 6 α is not a square for the pair of 5-isogenous (hence (4, 1)-congruent) elliptic curves with LMFDB labels 38.b1 and 38.b2. Therefore D = 1 when (N, r) = (4, 1). Similarly 3c 6 c ′ 6 δα is not a square for the pair of 3-isogenous elliptic curves 44.a1 and 44.a2, so D = 1 when (N, r) = (4, 3).
The curve on Z ′ (3, 2) given by the vanishing of β admits an obvious parametrisation by setting J = t, J ′ = 1/t, α = 1, and β = 0. Repeating the above calculation with d = −2jj ′ (j−1728)(j ′ −1728) completes the proof.
Remark 3.7. For N ≤ 4, rather than recovering X r E (N ) it is also possible to give a parametrisation A 2 Z(N, r) so that the involution swapping the roles of E and E ′ is given by swapping the coordinates on A 2 . In the electronic data [Fre22] we record these symmetric families of (N, r)-congruent elliptic curves over Q (a, b). These parametrisations were found by first parametrising the surfaces S(N, r) which are birational to W (N, r).
Constructing Z(N, r) via fibre products
Let N 1 , N 2 ≥ 2 be coprime integers and let N = N 1 N 2 . We define Z + (N, r) to be the fibre product
Z + (N, r) Z(N 1 , r) Z(N 2 , r) Z(1)
and similarly for W + (N, r).
Remark 4.1. Note that the surfaces Z + (N, r) and W + (N, r) may depend implicitly on the factorisation N = N 1 N 2 .
The surface Z + (N, r) has the following moduli interpretation: for each point P ∈ Z + (N, r)(K) with j(P ), j ′ (P ) ∈ {0, 1728} there exist pairs of elliptic curves (E 1 , E ′ 1 ) and (E 2 , E ′ 2 ) defined over K such that for i = 1, 2 we have • j(P ) = j(E i ) and j ′ (P ) = j(E ′ i ), and • E i and E ′ i are (N i , r)-congruent over K. The surface W + (N, r) is then obtained from Z + (N, r) by taking the quotient by the involution swapping the roles of E i and E ′ i . Remark 4.2. Suppose that N 1 = 4 and that r 1 , r 2 ∈ (Z/N Z) × are equal (up to a square) modulo N 2 . Then the surfaces Z + (N, r 1 ) and Z + (N, r 2 ) are isomorphic since Z(4, 1) and Z(4, 3) are isomorphic (see Remark 3.2) and the isomorphism respects the maps Z(4, r) → Z(1).
By taking simultaneous quadratic twists of E 2 and E ′ 2 we may assume that E 1 = E 2 . If N 1 = 2 then we immediately see that Z + (N, r) is birational to Z(N, r) -every elliptic curve is 2-congruent to any quadratic twist, so we may replace E ′ 1 by its quadratic twist E ′ 2 so that E 1 is (N, r)-congruent to E ′ 1 . Otherwise, assume that N i = 2 for i = 1, 2. By the above discussion if E/Q(Z + (N, r)) is an elliptic curve with j-invariant j then there exists a pair of elliptic curves E ′ 1 /Q(Z + (N, r)) and E ′ 2 /Q(Z + (N, r)) with j-invariant j ′ such that E and E ′ i are (N i , r)-congruent. Let d ∈ Q (Z(N, r)) be such that E ′ 1 is isomorphic to the quadratic twist of E ′ 2 by d. Then the modular diagonal quotient surface Z(N, r) is birational to the double cover of Z + (N, r) whose function field is given by adjoining the square root of d to Q (Z(N, r)).
Computing Z(12, r)
We now prove Theorem 1.1 by applying Theorem 3.3 to compute the fibre products described in Section 4. To compute the surfaces Z(12, r) for r = 1, 7 we use Theorem 3.3(i)-(ii) to construct the covers Z(12, r) → Z(3, 1). In contrast, to compute the surfaces Z(12, r) for r = 5, 11 we use Theorem 3.3(iv) to construct the covers Z(12, r) → Z(4, r).
Remark 5.1. It is also possible to compute Z(12, r) for r = 1, 7 by using Theorem 3.3(iv) to construct the covers Z(12, r) → Z(4, r), and similarly when r = 5, 11 it is possible to compute Z(12, r) by using Theorem 3.3(i)-(ii) to construct the covers Z(12, r) → Z(3, 2). W (N, r) consists of both a birational model for the surface and equations for the forgetful maps we naturally want to simplify both the equations for the surface and the maps. When the surface in question is rational we exploit the birational automorphisms of P 2 to simplify the forgetful maps.
Choosing parametrisations via Cremona transformations. Since a model for
It is well known that Bir Q (P 2 ), the birational automorphism group of P 2 over Q, contains PGL 3 (Q) and the Cremona transformations -that is, birational automorphisms of the form
[x 0 : x 1 : x 2 ] → [Q 0 (x 0 , x 1 , x 2 ) : Q 1 (x 0 , x 1 , x 2 ) : Q 2 (x 0 , x 1 , x 2 )]
where Q 0 , Q 1 , Q 2 ∈ Q[x 0 , x 1 , x 2 ] are homogeneous forms of degree 2 which vanish simultaneously at three G Q -conjugate points q 1 , q 2 , q 3 ∈ P 2 (Q) in general position.
Geometrically this operation blows up q 1 , q 2 , q 3 and blows down the three lines which contain exactly two of q 1 , q 2 , q 3 . This suggests the following approach for simplifying a model for W (N, r). Let C/Q be a curve in P 2 given by the vanishing of a factor of the numerator of JJ ′ , (J − 1)(J ′ − 1), or (J − J ′ ) 2 . Applying a generic Cremona transformation to P 2 will double the degree of JJ ′ , (J − J ′ ) 2 , and (J − 1)(J ′ − 1). If instead we apply a Cremona transformation centred at a G Q -stable triple of singular points q 1 , q 2 , q 3 ∈ C(Q) we may hope to decrease their degrees.
In the discussion that follows we often make choices of parametrisation which give rise to models for W (N, r) with j-maps of small degree. In general, the parametrisations we give were not our first choice. Once we have a birational map A 2 W (N, r) (e.g., by spotting a genus 0 fibration over P 1 ), we apply iterative Cremona transformations as described above to simplify the j-maps. Z(12, 1) and Z(12, 7). We will use Theorem 3.3 to compute a model for the fibre product W + (12, r).
Models for
As in the proof of Theorem 3.3 let W(3, 1) be the surface in A 4 defined by the polynomials in Theorem 3.3(iii). We check in Magma that the rational map A 2 W(3, 1) given by taking
JJ ′ = −(−2ab + b 2 − 4a − 4b) 3 (2b + 1) 3 (−ab + b 2 − 2a − 2b) 2 , (J − 1)(J ′ − 1) = (b + 2) 2 (a 2 b − 2ab 2 + b 3 + 2a 2 + ab − 2b 2 + a + b) 2 (2b + 1) 3 (−ab + b 2 − 2a − 2b) 2 ,
(and solving for α and β) is birational. In particular W (3, 1) is birational to A 2 with the forgetful map W (3, 1) → W (1) given by the formulae for JJ ′ and (J − 1)(J ′ − 1) above. Let ∆ = ∆(E) and ∆ ′ = ∆(E ′ ) be the discriminants of the elliptic curves E/Q(Z(3, 1)) and E ′ /Q(Z(3, 1)) with j-invariants 1728J and 1728J ′ respectively. We write W (3, 1) √ ∆∆ ′ for the double cover of W (3, 1) whose function field is given by adjoining √ ∆∆ ′ to Q(W (3, 1)). Then W (3, 1) √ ∆∆ ′ is birational the surface given by
α 2 4 − (J − 1)(J ′ − 1) = 0 in A 3 . By setting α 4 = −(b + 2)(a 2 b − 2ab 2 + b 3 + 2a 2 + ab − 2b 2 + a + b) (2b + 1) 2 (−ab + b 2 − 2a − 2b) α ′ 4
we see that W (3, 1) √ ∆∆ ′ is birational to the surface given by
β 4 = (d 4 − 4cd 2 − 10d 2 − 12c + 9) (d 2 − 2d − 3)β ′ 4 − d 4 + 2cd 2 + 2d 3 + 6d 2 + 6c − 6d − 9 2d 3 (d 4 − 2cd 2 − 6d 2 − 6c + 5) · 1 β ′ 4
we find that W (6, 1) is birational to the surface given by
(β ′ 4 ) 3 + 6(β ′ 4 ) 2 − 3(d + 1)(d − 3)β ′ 4 − (2cd 2 + 6c − d 4 + 2d 3 + 6d 2 − 6d − 9) = 0 in A 3 .
We parametrise the above surface by setting (c, d, β ′ 4 ) = p 3 q + 6p 2 q 2 + 9pq 3 + 12pq 2 − 12pq + 9q 4 + 12q 3 − 24q 2 − 16q + 16 2q 2 (3q 2 + 4) , 2 q , p q .
Hence W (6, 1) is birational to A 2 and the forgetful map W (6, 1) → W (3, 1) √ ∆∆ ′ is given by (p, q) → (c, d). By Theorem 3.3(ii) the surface W + (12, 1) (which by Remark 4.2 is isomorphic to W + (12, 7)) is birational to the surface given by
γ 4 4 − 6JJ ′ β 4 γ 2 4 − 16(JJ ′ ) 2 γ 4 + 3(JJ ′ ) 2 (4JJ ′ − β 2 4 ) = 0 in A 3 .
After making the birational transformation
γ 4 = (2p 3 q + 12p 2 q 2 + 18pq 3 + 9q 4 + 24pq 2 + 24q 3 − 24pq − 8q 2 − 32q + 16) 2 64(p + 4q − 4)(p + q + 2) 3 γ ′ 4 q = (γ ′ 4 ) 2 − 3p + 4γ ′ 4 − 8 q ′ − 2pγ ′ 4 − 4p + 4γ ′ 4 − 16 6γ ′ 4
we find that W + (12, 1) is birational to the quadric surface 3(q ′ ) 2 − 2γ ′ 4 q ′ + 8q ′ + p + 4 = 0 in A 3 . We parametrise this quadric surface by setting
(p, q ′ , γ ′ 4 ) = −4(s − 3)(s + 1)(3s − 2t − 1) 2 3(s − 1) 2 (s − 2t − 1)(3s + 2t + 1) , 8s(t − 1) 3(s − 1)(s − 2t − 1) , 4s(s − 3)(3s − 2t − 1) (s − 1)(s − 2t − 1)(3s + 2t + 1) .
The forgetful map W + (12, 1) → W (6, 1) is given by (s, t) → (p, q).
5.2.1. The Surface Z(12, 1). The α 4 , β 4 , and γ 4 are elements of Q(W + (12, 1)) which satisfy the conditions of Theorem 3.3(ii). Let δ 4 = 3(JJ ′ − (α 4 + 1) 2 ) be as in Theorem 3.3(ii). By construction there exist α 3 , β 3 , δ 3 ∈ Q(W + (12, 1)) which satisfy Theorem 3.3(iii). As in Section 4 let E/Q(Z + (12, 1)) be an elliptic curve with j-invariant 1728J. Let E ′ 1 /Q(Z + (12, 1)) and E ′ 2 /Q(Z + (12, 1)) be elliptic curves with j-invariant 1728J ′ which are (3, 1) and (4, 1)-congruent to E respectively.
It follows from Theorem 3.3(ii) and (iii) that up to square factors c 6 (E) is equal to both 3δ 3 c 6 (E ′ 1 ) and 3α 4 δ 4 c 6 (E ′ 2 ). In particular E ′ 1 is isomorphic to the quadratic twist of E ′ 2 by α 4 δ 4 δ 3 . Therefore by the discussion in Section 4 the surface W (12, 1) is birational to the double cover of A 2 given by w 2 = α 4 δ 4 δ 3 . The rational function α 4 δ 4 δ 3 is equal to 3s(t − 1)(−s + 2t + 1)(3s + 2t + 1)(st + 2s − 2t − 1) up to a square factor. Therefore the surface W (12, 1) is birational to the affine surface given by (w ′ ) 2 = 3s(t − 1)(−s + 2t + 1)(3s + 2t + 1)(st + 2s − 2t − 1). We parametrise this surface by setting
(s, t, w ′ ) = −2u 2 − uv 2 − u + 4v 2 uv 2 + u + 4v 2 , 2u 2 v 2 − 4u 2 + uv 4 − 6uv 2 − 3u − 8v 4 (2u + v 2 + 3)(uv 2 + u + 4v 2 ) , 36v(u + 2)(u + v 2 + 1)(u + 2v 2 )(2u 2 + uv 2 + u − 4v 2 ) 2 (2u + v 2 + 3) 2 (uv 2 + u + 4v 2 ) 3 .
Hence W (12, 1) is birational to A 2 and the forgetful map W (12, 1) → W + (12, 1) is given by (u, v) → (s, t).
Let F 12,1 (u, v) be the polynomial from the statement of Theorem 1.1. It follows immediately that Z(12, 1) is birational to the affine surface given by the equation . We now prove Theorem 1.4. Specifically, we identify curves C ⊂ Z(12, r) of genus 0 or 1 with infinitely many rational points. We use these to construct pairs of elliptic curves defined over Q(C) which are (12, r)-congruent. Once we have such a pair of elliptic curves the following lemma will allow us to deduce Theorem 1.4. Recall that if E/Q is an elliptic curve, then fixing a basis for E[N ] we obtain an isomorphism Aut(E[N ]) ∼ = GL 2 (Z/N Z) and hence a representationρ E,N : G Q → GL 2 (Z/N Z) which we call the mod N Galois representation attached to E/Q (for more details see [RSZ21a, Section 2.2]).
z 2 = F 12,1 (u, v) m Equation for X + 0 (m) on W (12, 1) 13 ±v + 1 25 2u + v 2 ± 2v + 1 37 2u 2 + (v 2 ± 2v + 1)u ± v 3 − 3v 2 ± 3v − 1 49 4u 2 + 2(3v 2 ± 2v + 3)u + v 4 ± 4v 3 + 6v 2 ± 4v + 1 61 4u 3 + 2(v 2 ± 2v + 5)u 2 + (−v 4 ± 6v 3 − 4v 2 ± 10v + 5)u ± v 5 − 5v 4 ± 10v 3 − 10v 2 ± 5v − 1u 2 + 2u − v 2 ± 2v − 4 41 u 3 − (±v − 2)u 2 + (−v 2 − 12)u ± v 3 − 2v 2 ± 4v + 8 53 u 4 − (2v 2 ± 4v)u 2 + (±8v − 16)u + v 4 ± 4v 3 + 8v 2 + 16u 2 − (±v − 1)u ± 2v − 2 43 −(±v − 1)u 2 + (v 2 ± 2v + 1)u + 2v 2 − 2 55 (v 2 ∓ 6v + 1)u 2 + (−2v 2 ± 4v − 2)u − v 4 ∓ 2v 3 ± 2v + 11 ± v 23 ±vu + v 2 ± v + 1 35 −(±v 3 ± 3v)u − v 4 ± 3v 3 ± v + 1 47 (±3v 3 + 2v 2 ± 3v)u + v 4 ± v 3 + 4v 2 ± v + 1 59 (−3v 4 ± 3v 3 − v 2 ± v)u ± v 5 + 8v 4 ± 7v 3 + 11v 2 ± 4v + 1 71 (±7v 5 ± 10v 3 ∓ v)u + v 6 ∓ v 5 + 15v 4 ± 10v 3 + 15v 2 ± 7v + 1
Lemma 6.2. Let C/Q be a smooth projective curve with infinitely many rational points and let E/Q(C) and E ′ /Q(C) be elliptic curves which are not both isotrivial. For a point P ∈ C(Q) let E P /Q be the specialisation of E at P (and similarly for E ′ ).
(i) Suppose there exists a point P ∈ C(Q) such that E P and E ′ P are nonsingular and do not have CM. If j(E P ) and j(E ′ P ) are not j-invariants of m-isogenous elliptic curves over Q for any integer m then for all but finitely many Q ∈ C(Q) we have that j(E Q ) and j(E ′ Q ) are not the j-invariants of m-isogenous elliptic curves for any integer m.
(ii) Suppose there exists a point P ∈ C(Q) such that E P is nonsingular and the mod N Galois representation attached to E P /Q is surjective. Then there exist infinitely many Q ∈ C(Q) such that the mod N Galois representation attached to E Q /Q is surjective.
Proof. (i) This follows from a similar argument to [Fis15, Theorem 1.5] which we reproduce here for completeness.
First suppose E and E ′ are non-CM elliptic curves defined over Q and φ : E → E ′ is an isogeny defined over Q. Then φ φ = [n] ∈ End(E ′ ) for some n ∈ Z and by comparing degrees for any σ ∈ G K we have
( φ) σ φ = χ(σ)[n] ∈ End(E) for some character χ : G Q → {±1}. Taking d ∈ Q such that Q( √ d)
is cut out by the kernel of χ we see that the quadratic twist of E by d is Q-isogenous to E ′ . That is, j(E) and j(E ′ ) are the j-invariants of Q-isogenous elliptic curves.
It therefore suffices to show that there are at most finitely many Q ∈ C(Q) such that j(E Q ) and j(E ′ Q ) are j-invariants of Q-isogenous elliptic curves.
Let C ′ be the image of the map C → X(1) × X(1) given by extending t → (j(E), j(E ′ )). Consider the curves, T m , on X(1) × X(1) which are the image of the modular curves X 0 (m) under the graph of the Hecke correspondence. Since the curves T m are geometrically irreducible and j(E P ) and j(E ′ P ) are not j-invariants of Q-isogenous elliptic curves, the curve C ′ meets each T m at finitely many points (over Q).
Mazur [Maz78] and Kenku [Ken82] showed that if an elliptic curve admits a cyclic m-isogeny over Q then m ≤ 163. In particular there are finitely many rational points on C ′ which lie on the union of all T m . The claim follows since the map j : C → C ′ has finite degree.
(ii) We define a morphism j : C → P 1 ∼ = X(1) by extending j(E). Let M be the set of all maximal subgroups of GL 2 (Z/N Z). For each H ∈ M let X(H)/Q denote the modular curve whose K-points parametrise elliptic curves E/K such thatρ E,N (G Q ) is contained in a subgroup of GL 2 (Z/N Z) conjugate to H (see e.g., [RSZ21a, Section 2]). Let C H be the fibre product
C H X(H) C X(1) φH j
We claim that the set
C(Q) \ H∈M φ H (C H (Q))
is infinite.
First suppose that C has genus 0. Note that the morphisms φ H have degree greater than > 1 because E P /Q has surjective mod N Galois representation. Therefore since M is finite H∈M φ H (C H (Q)) is a thin set in the sense of Serre. The claim follows from Hilbert's Irreducibility Theorem [Ser08, Proposition 3.4.2].
If C has genus 1 we make C into an elliptic curve by declaring P to be the identity. By Faltings' theorem any of the curves C H of genus ≥ 2 have finitely many rational points. Therefore we may assume without loss of generality that for each H the curve C H has genus 1 (and that C H (Q) is non-empty).
Since every morphism between elliptic curves is the composition of a translation and an isogeny [Sil09, Example 4.7] the images φ H (C H (Q)) are cosets of subgroups Φ H ⊂ C H (Q) which have finite index by the weak Mordell-Weil theorem. Because E P /Q has surjective mod N Galois representation the point P is not contained in φ H (C H (Q)) for any H.
Since we took P to be the identity, for each H the image φ H (C H (Q)) is a non-trivial coset of Φ H . In particular the subgroup H Φ H ⊂ C(Q) is disjoint from φ H (C H (Q)) for every H. The claim follows by noting that the intersection H Φ H has finite index in C(Q), and is therefore infinite. Example 6.3 (A family of (12, 1)-Congruences). Examples of pairs of (12, 1)-congruent elliptic curves over Q(t) have previously been given by Chen [Che16, Section 7.3] and Fisher [Fis20]. We include one here for completeness. The model z 2 = F 12,1 (u, v) has an obvious elliptic fibration over the t-line given by y 2 = F 12,1 (x, t). This model has a pair of sections with x-coordinates 0. By putting this fibration in Weierstrass form we see that Z(12, 1) is birational to the elliptic K3 surface with Weierstrass equation y 2 = x 3 − 27(t 8 + 46t 6 + 859t 4 − 186t 2 + 9)x − 54(t 12 + 69t 10 + 2082t 8 + 24731t 6 − 7848t 4 + 621t 2 + 27).
Moreover the sections with x-coordinate 3(2t 8 +37t 6 +450t 4 −27t 2 +54) In order to give a family where the 12-congruent elliptic curves have simple Weierstrass equations we instead note that the curve on Z(12, 1) given by the vanishing of 2u + v 2 + 3v has genus 0. By parametrising this curve we obtain a pair of (12, 1)-congruent elliptic curves E 12,1 /Q(t) and E ′ 12,1 /Q(t) where E 12,1 : y 2 = x 3 − 3(t 2 + 1)(4t 2 + 2t + 1)p(t)x − 2(t 2 + 1) 2 (4t 2 + 2t + 1)q(t) E ′ 12,1 : y 2 = x 3 − 3(t 2 + 1)(4t 2 + 2t + 1)p ′ (t)x + 2(t 2 + 1) 2 (4t 2 + 2t + 1)q ′ (t) and p(t) = 36t 12 + 234t 11 + 693t 10 + 1530t 9 + 2808t 8 + 4128t 7 + 4964t 6 + 4792t 5 + 3652t 4 + 2000t 3 + 736t 2 + 160t + 16, q(t) = 432t 18 + 4320t 17 + 20196t 16 + 63720t 15 + 159111t 14 + 330453t 13 + 581103t 12 + 875862t 11 + 1137762t 10 + 1270440t 9 + 1208628t 8 + 960528t 7 + 622592t 6 + 320528t 5 + 127472t 4 + 37728t 3 + 7840t 2 + 1024t + 64, p ′ (t) = 55396t 12 − 97238t 11 + 230581t 10 − 206878t 9 + 177280t 8 − 86384t 7 + 2612t 6 − 6600t 5 + 900t 4 − 1008t 3 + 288t 2 + 288t + 144, q ′ (t) = 26076112t 18 − 62137376t 17 + 165178492t 16 − 207914808t 15 + 236336753t 14 − 163049017t 13 + 71325401t 12 − 36145662t 11 + 7345854t 10 − 7544088t 9 + 2972844t 8 + 1350576t 7 + 531072t 6 − 137808t 5 − 17712t 4 − 4320t 3 + 14688t 2 + 6912t + 1728.
Example 6.4 (A family of (12, 5)-Congruences). Our model for the surface Z(12, 5) contains the genus 1 curve given by the vanishing of u 3 + 3u 2 − (v 2 + 10)u − 3v 2 + 6. This curve is birational over Q to the elliptic curve C/Q given by the Weierstrass equation C : η 2 = ξ 3 − ξ 2 − 16ξ + 16.
The elliptic curve C has LMFDB label 240.a3 and rank 1.
Using the moduli interpretation for Z(12, 5) it is possible to give an explicit pair of (12, 5)-congruent elliptic curves over Q(C) in terms of ξ and η. The Weierstrass equations are too complicated to reproduce here, but we include them in the electronic data [Fre22]. 20 Example 6.5 (A family of (12, 7)-Congruences). Our model for the surface Z(12, 7) contains the genus 0 curve given by the vanishing of u 2 + 3u − 2v 2 + 2. By parametrising this curve we obtain a pair of (12, 7)-congruent elliptic curves E 12,7 /Q(t) and E ′ 12,7 /Q(t) where E 12,7 : y 2 = x 3 − 3 (t 2 + 3)(t 4 − 3t 2 + 9)p(t) (t 4 + 3)(t 4 + 27)
x − 2 (t 2 + 3)(t 4 − 3t 2 + 9)q(t) (t 4 + 3)(t 4 + 27) E ′ 12,7 : y 2 = x 3 − 3(t 2 + 3)(t 4 − 3t 2 + 9)(t 4 + 3)(t 4 + 27)p ′ (t)x − 2(t 2 + 3)(t 4 − 3t 2 + 9)(t 4 + 3) 2 (t 4 + 27) 2 q ′ (t) and p(t) = t 18 − 18t 16 + 135t 14 − 729t 12 + 2889t 10 − 7533t 8 + 15093t 6 − 15795t 4 + 53946t 2 + 2187, q(t) = t 26 − 27t 24 + 309t 22 − 2133t 20 + 10341t 18 − 35559t 16 + 85158t 14 − 109350t 12 − 141345t 10 + 1337715t 8 − 3133971t 6 + 4183731t 4 + 3483891t 2 − 59049, p ′ (t) = t 18 + 222t 16 − 585t 14 + 5031t 12 − 22599t 10 + 78003t 8 − 177147t 6 + 295245t 4 − 354294t 2 + 177147, q ′ (t) = t 26 − 531t 24 − 5739t 22 + 38691t 20 − 148635t 18 + 141345t 16 + 984150t 14 − 6897798t 12 + 25922511t 10 − 67847301t 8 + 125951517t 6 − 164215269t 4 + 129140163t 2 − 43046721.
Example 6.6 (A family of (12, 11)-Congruences). Our model for the surface Z(12, 11) contains a pair of genus 0 curves given by the vanishing of (9v 4 + 30v 2 + 5)u − 45v 4 − 6v 2 + 7. By parametrising one of these curves we obtain a pair of (12, 11)-congruent elliptic curves E 12,11 /Q(t) and E ′ 12,11 /Q(t) where E 12,11 : y 2 = x 3 − 3 (25t 6 + 63t 4 + 27t 2 − 27)(t 6 + 15t 4 + 3t 2 − 27) (t 4 − 18t 2 − 27)(t 4 + 6t 2 − 3) x − 2 (25t 6 + 63t 4 + 27t 2 − 27)(5t 8 − 24t 6 − 78t 4 + 81) (t 4 − 18t 2 − 27)(t 4 + 6t 2 − 3) E ′ 12,11 : y 2 = x 3 − 3(t 4 − 18t 2 − 27)(t 4 + 6t 2 − 3)p ′ (t)x − 2 (t 4 − 18t 2 − 27) 2 (t 4 + 6t 2 − 3) 2 q ′ (t) (25t 6 + 63t 4 + 27t 2 − 27) and p ′ (t) = 25t 20 + 18t 18 − 1971t 16 − 16200t 14 − 138942t 12 − 367092t 10 − 500526t 8 − 332424t 6 − 13851t 4 + 1458t 2 + 6561, q ′ (t) = 3125t 32 + 30000t 30 + 94800t 28 + 878688t 26 + 25070580t 24 + 173317968t 22 + 693690912t 20
Lemma 2.1 ([KS98], [Fis19, Lemma 3.2]). The surface Z(N, r) is birational over K to the quotient of X(N ) × X(N ) by ∆ r .
(when N = 3), and Bruin-Doerksen [BD11] (when N = 4) have computed explicit equations for the surface Z(N, −1) using the description of Z(N, −1) as the moduli space of genus 2 curves with (N, N )-split Jacobians. The equations of Kumar and Bröker-Howe-Lauter-Stevenhagen are also symmetrical in E and E ′ .
so that the forgetful map X(4) → X(2) is given by taking the quotient by the action of H/{±1}. Note that the r-bi-invariants I 4,4 , c 4 c ′ 4 , and (DD ′ ) 2 are invariant under the action of H × H. In particular w 0 , w 1 and w 2 3 /w 2 are invariant under H × H and we have the relation c 6
(iii) When (N, r) = (3, 1) the condition that ( 3 √ J + 1)( 3 √ J ′ + 1) = 1 is necessary. The elliptic curves with LMFDB labels 245.a1 and 1323.s1 are not (quadratic twists of) (3, 1)-congruent elliptic curves, however there do exist α, β ∈ Q satisfying Theorem 3.3(iii). Moreover when N = 3, 4 the theorem is false when J = J ′ . Counterexamples are given by the pairs of elliptic curves 14.a1 and 126.b2 when (N, r) = (3, 1), 245.a1 and 245.b1 when (N, r) = (3, 2), 14.a6 and 112.c6 when (N, r) = (4, 1), and 14.a6 and 784.b6 when (N, r) = (4, 3).
A 3 .
3We parametrise this surface by setting (a, b, α ′ 4 ) = c, d 2 −1 2 , d . Therefore W (3, 1) √ ∆∆ ′ is birational to A 2 and the forgetful map W (3, 1) √ ∆∆ ′ → W (3, 1) is given by (c, d) → c, d 2 −1 2 . By Theorem 3.3(i) we may write W (6, 1) as a degree 3 cover of A 2 given by the vanishing of β 3 4 − 3JJ ′ β 4 − 2JJ ′ (α 4 + 1) = 0 13 in A 3 . After making the change of coordinates
order. Taking multiples of this section gives an infinite family of examples over Q(t) (a different family is given by Fisher [Fis20, Corollary 1.3]).
Table 2 .
2Modular curves on Z(12, 1).m
Equation for X +
0 (m) on W (12, 5)
17
u ± v
29
Table 3 .
3Modular curves on Z(12, 5).m
Equation for X +
0 (m) on W (12, 7)
19
1 ± v
31
Table 4 .
4Modular curves on Z(12, 7).m
Equation for X +
0 (m) on W (12, 11)
11
Table 5 .
5Modular curves on Z(12, 11).6.2. Infinite families of 12-congruent elliptic curves and examples over Q(t)
since (J − J ′ ) 2 is equal to F 12,1 (u, v) up to a square factor. 5.2.2. The Surface Z(12, 7). By Theorem 3.3 the surface W (12, 7) is birational to the double cover of W + (12, 1) given by w 2 = α 4 δ 3 .The rational function α 4 δ 3 is equal to 3(t − 1)(−9s 3 + 12s 2 t + 4st 2 + 15s 2 + 4st − 12t 2 + s − 12t − 3) up to a square factor. Therefore the surface W (12, 7) is birational to the affine surface given by (w ′ ) 2 = 3(t − 1)(−9s 3 + 12s 2 t + 4st 2 + 15s 2 + 4st − 12t 2 + s − 12t − 3). We parametrise this surface by setting (s, t, w ′ ) = (u − 1)(u 2 − v 2 + 4u + 1) u 3 + u 2 − uv 2 − 3u − v 2 + 1 , u 3 − uv 2 + 7u 2 − v 2 + 9u + 1 u 3 + u 2 − uv 2 − 3u − v 2 + 1 , 36uv(u + 2)(u 2 + 4u − v 2 + 1) (u 3 + u 2 − uv 2 − 3u − v 2 + 1) 2 .Hence W (12, 7) is birational to A 2 and the forgetful map W (12, 7) → W + (12, 1) is given by (u, v) → (s, t).Let F 12,7 (u, v) be the polynomial from the statement of Theorem 1.1. It follows immediately that Z(12, 7) is birational to the affine surface given by the equationsince (J − J ′ ) 2 is equal to F 12,7 (u, v) up to a square factor. 5.3. Models for Z(12, 5) and Z(12, 11). For each r = 1, 3 let W(4, r) be the surface in A 5 defined by the polynomials in Theorem 3.3(i)-(ii). The rational map A 2 W(4, r) given by taking(and solving for α, β, and γ) is birational. In particular W (4, r) is birational to A 2 and the forgetful maps W (4, r) → W (1) are given by the rational functions JJ ′ and (J − 1)(J ′ − 1) above. Let ∆ = ∆(E) and ∆ ′ = ∆(E ′ ) be the discriminants of elliptic curves E/Q(Z(4, 1)) and E ′ /Q(Z(4, 1)) with j-invariants 1728J and 1728J ′ respectively. We write W (4, 1) 3 √ ∆∆ ′ for the triple cover of W (4, 1) whose function field is given by adjoining 3 √ ∆∆ ′ to Q(W (4, 1)). Then W (4, 1) 3 √ ∆∆ ′ is birational to the surface given byBy making a change of coordinates α 3 = 2(−2ab+b 2 +2a+2b)We parametrise this cubic surface by settingHence W (4, 1) 3 √ ∆∆ ′ is birational to A 2 and the forgetful map W (4, 1)By Theorem 3.3(iv) the surface W + (12, 5) (which by Remark 4.2 is isomorphic to W + (12, 11)) is birational to the surface given by. This is clearly birational to the surface given byTo simplify the equations for this surface, we view the forms above as defining a quadric intersection in A 2 Q(c,d) . By using 'minimisation' techniques similar to those in [CFS10, Section 4.3], but applied over the base ring Z[c, d], we arrive at the change of coordinatesIt follows that W + (12, 5) is birational to the surface given by the pair of equationswhere h 1 (s, t) = s 3 + (3t + 6) s 2 + 3t 2 + 12t + 12 s + 3t 3 + 6t 2 + 12t + 8,h 3 (s, t) = 2s 4 + (4t + 4) s 3 + 5t 2 + 18t + 12 s 2 + 6t 3 + 20t 2 + 24t + 10 s + 3t 4 + 6t 3 + 11t 2 + 8t − 1.Therefore W + (12, 5) is birational to A 2 and the forgetful map W + (12, 5) → W (4, 1) 3 √ ∆∆ ′ is given by (s, t) → (c, d).5.3.1. The Surface Z(12, 5). Note that α 3 , β 3 ∈ Q(W + (12, 5)) satisfy the conditions of Theorem 3.3(iv). By construction there exist elements α 4 , β 4 , γ 4 of Q(W + (12, 1)) which satisfy Theorem 3.3(i)-(ii). Let δ 4 = 3(JJ ′ − (α 4 + 1) 2 ) be as in Theorem 3.3(ii) By Theorem 3.3 the surface W (12, 5) is birational to the double cover of W + (12, 5) given byThe rational function α 4 δ 4 β 3 is equal to −s 2 + t 2 + 1 up to a square factor. Therefore the surface W (12, 5) is birational to the affine surface (w ′ ) 2 = −s 2 + t 2 + 1. We parametrise this surface by settingHence W (12, 5) is birational to A 2 and the forgetful map W (12, 5) → W + (12, 5) is given by (u, v) → (s, t).Let F 12,5 (u, v) be the polynomial from the statement of Theorem 1.1. It follows immediately that Z(12, 5) is birational to the affine surface given by the equationsince (J − J ′ ) 2 is equal to F 12,5 (u, v) up to a square factor. 16 5.3.2. The Surface Z(12, 11). By Theorem 3.3 the surface W (12, 11) is birational to the double cover of W + (12, 5) given byThe rational function α 4 β 3 is equal to −(s + t)(s + t + 2) up to a square factor. Therefore the surface W (12, 11) is birational to the affine surface (w ′ ) 2 = −(s+t)(s+t+2). We parametrise this surface byHence W (12, 11) is birational to A 2 and the forgetful map W (12, 11) → W + (12, 5) is given by (u, v) → (s, t). Let F 12,11 (u, v) be the polynomial from the statement of Theorem 1.1. It follows immediately that Z(12, 11) is birational to the affine surface given by the equationsince (J − J ′ ) 2 is equal to F 12,11 (u, v) up to a square factor.Curves on Z(12, r) and examples of 12-congruencesWe now identify some curves of small genus on our models for Z(12, r) and we prove Theorem 1.4. 6.1. The modular curves X 0 (m) on Z(12, r). Let m be an integer coprime to N , and suppose that m is equal to r modulo N (up to a square factor). The (non-compact) modular curve Y 0 (m) may be naturally embedded in Z(N, r). More precisely we may regard a non-cuspidal K-point on Y 0 (m) as a triple (E, E ′ , φ) where E/K and E ′ /K are elliptic curves and φ is a cyclic m-isogeny. In particular we have a natural mapdefined over Q. A pair of points (E 1 , E ′ 1 , φ 1 ) and (E 2 , E ′ 2 , φ 2 ) are identified under this morphism if and only if the the pairs (E 1 , E ′ 1 ) and (E 2 , E ′ 2 ) are simultaneous quadratic twists, in which case they define the same point on Y 0 (m). In particular ψ is an embedding.Remark 6.1. In fact, let P = (E, E ′ , φ) be a K-point on Z(N, r). Then for each integer k such that k 2 ≡ 1 (mod N ) we obtain another K-point P a = (E, E ′ , φ • [k]) on Z(N, r) which is equal to P if and only if k is equal to either 1 or −1 modulo N .In particular when N = 12 points on Z(N, r) come in pairs, namely P 1 and P 5 in the above notation, which have the same image under Z(12, r) → Z + (12, r). Hence each modular curve Y 0 (m) as above appears twice on Z(12, m). In the notation of Kani-Schanz[KS98]these are the "Hecke curves" T m,1 and T m,5 .We identify the curves X 0 (m) on our models for Z(N, r) for each m such that X 0 (m) appears in Magma's SmallModularCurve database. The equations for these curves are recorded in Tables 2-5, the two choices of sign give the pair of curves T m,1 and T m,5 .Note that the modular curves X 0 (5) on Z(12, 5) and X 0 (7) on Z(12, 7) have been blown down (this is the case since we determined Z(12, r) only up to birational equivalence). Equations for these curves on a blowup of our models are given by z 2 = F 12,5 1 2 (3 − ε), 1 2 (±1 ± ε + ε 2 t) = 1 4 −t 2 ∓ 44t + 16 ε 4 + O(ε 5 ) and z 2 = F 12,7 (−ε, ±1 ± ε + ε 2 t) = (4t 2 ± 52t − 27)ε 4 + O(ε 5 ) respectively. + 1821351744t 18 + 3069524646t 16 + 3140007120t 14 + 1733188752t 12 + 288404064t 10 − 233440380t 8 − 163447632t 6 + 7558272t 4 − 1594323.We now complete the proof of Theorem 1.4.Proof of Theorem 1.4. Consider the pairs of elliptic curves, E 12,r /Q and E ′ 12,r /Q given by specialising the pairs in Examples 6.3, 6.5, and 6.6 at t = 2 and the pair in Example 6.4 at the point (ξ, η) = (0, 4). The hypotheses of condition (i) in Lemma 6.2 are satisfied since for each r the curve E 12,r is alone in its Q-isogeny class. The first claim then follows from Lemma 6.2(i).Note that if the mod N Galois representation attached to an elliptic curve E/Q is not surjective then there exists a maximal subgroup H ⊂ GL 2 (Z/N Z) such thatρ E,N (G Q ) is contained in a subgroup conjugate to H. In particular if p is a prime number not dividing N or the denominator of j(E) then the reduction of j(E) modulo p is the j-invariant of an F p -rational point on X(H). We check using code of Rouse, Sutherland, and Zureick-Brown[RSZ21a]that for each r and each maximal subgroup H ⊂ GL 2 (Z/12Z) there exists a prime p as above such that j(E 12,r ) is not the j-invariant of an F p -rational point on X(H) (specifically we use the intrinsic GL2jInvariantTest in [RSZ21b, gl2groups.m]). The claim follows from Lemma 6.2(ii). 21 6.3. Further points and curves on Z(12, r). Since the surfaces Z(12, r) are of general type for r = 1 we would expect by the Bombieri-Lang conjecture that they contain at most finitely many curves of genus 0 or 1. InTables 6-8we record examples of curves of genus 0 or 1 on Z(12, r) (where r = 1) which do not map to a Hecke correspondence on Z(1) and do not lie above the loci where j = 0, 1728, ∞ or j ′ = 0, 1728, ∞.InTables 6-8an irreducible curve C ⊂ Z(12, r) is recorded by its image C + under the map Z(12, r) → W (12, r). The column "genus" records the genus of C. If C is not geometrically irreducible we write g * where g is the genus of a geometrically irreducible component. The column "has Q-point" records whether a smooth projective curve birational over Q to C has a rational point.Table 7. Examples of curves C ⊂ Z(12, 7) of genus 0 or 1 without a moduli interpretation.Remark 6.7. It is natural to ask for which pairs (N, r) there exist pairs of non-isogenous (N, r)-congruent elliptic curves E/Q(t) and E ′ /Q(t)?In view of this question it would be interesting to determine whether or not there exists an example of a genus 0 curve (which is birational over Q to P 1 ) on Z(12, 5) which does not lie above j = ∞, j ′ = ∞, or a Hecke correspondence on Z(1).22Equation for C +Genus of CPrimes where C is locally insoluble Has Q-point LMFDB label of the Jacobian of C Rank (9v 4 + 30v 2 + 5)u − 45v 4 − 6v 2 + 7 0 Yes u ± 2v + 1 1 Yes 32.a4 0Table 8. Examples of curves C ⊂ Z(12, 11) of genus 0 or 1 without a moduli interpretation.Remark 6.8. It is possible to find examples of curves of genus 0 and 1 on Z(12, r) which do not arise as the image of a map Y 0 (m) → Z(12, r) as described in Section 6.1 but which nevertheless map to the graph of a Hecke correspondence on Z(1). Examples are furnished by the genus 0 curve u+2 = 0 on Z(12, 1) and the genus 1 curve uv 2 −v 2 −2 = 0 on Z(12, 11). Points on these curves give rise to pairs of 12-congruent elliptic curves which are 7 and 5-isogenous respectively.We do not attempt to classify all such examples.
r)-congruent elliptic curves which have small conductor. For each r = 1 we list all examples which we found (up to simultaneous quadratic twist) with conductor ≤ 10 10 . These examples were constructed by searching for rational points on Z(12, r). When r = 1 we were able to find too many examples to record here. Table 9 we give examples of non-isogenous. 12the 18 examples we found where both E and E ′ are contained in the LMFDB are recorded in the electronic data [Fre22of 12-congruent elliptic curves with small conductor. In Table 9 we give examples of non-isogenous (12, r)-congruent elliptic curves which have small conductor. For each r = 1 we list all examples which we found (up to simultaneous quadratic twist) with conductor ≤ 10 10 . These examples were constructed by searching for rational points on Z(12, r). When r = 1 we were able to find too many examples to record here, the 18 examples we found where both E and E ′ are contained in the LMFDB are recorded in the electronic data [Fre22].
We do not include examples where E and E ′ are quadratic twists since many such examples may be constructed for r = 7, 11 by searching for points on the modular curves in. Fre21, Lemmas 6.2 and 6.3We do not include examples where E and E ′ are quadratic twists since many such examples may be constructed for r = 7, 11 by searching for points on the modular curves in [Fre21, Lemmas 6.2 and 6.3].
The pair of (12, 5)-congruent elliptic curves 60450.cx2 and 60450. cw2 appearing in Table 9 are notableThe pair of (12, 5)-congruent elliptic curves 60450.cx2 and 60450.cw2 appearing in Table 9 are notable
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Twists of X(7) and primitive solutions to x 2 + y 3 = z 7. Bjorn Poonen, Edward F Schaefer, Michael Stoll, MR 2309145. 1371Bjorn Poonen, Edward F. Schaefer, and Michael Stoll, Twists of X(7) and primitive solutions to x 2 + y 3 = z 7 , Duke Math. J. 137 (2007), no. 1, 103-158. MR 2309145 ↑1
Explicit families of elliptic curves with prescribed mod 6 representations. Joel Philip Roberts, MR. 26995631The Ohio State UniversityPh.D. thesisJoel Philip Roberts, Explicit families of elliptic curves with prescribed mod 6 representations, Ph.D. thesis, The Ohio State University, 1999, p. 46. MR 2699563 ↑1
Families of elliptic curves with constant mod p representations, Elliptic curves, modular forms, & Fermat's last theorem. Karl Rubin, Alice Silverberg, MR 1363500 ↑1, ↑2, ↑3. Hong Kong; Cambridge, MAI, Int. Press58Ser. Number TheoryKarl Rubin and Alice Silverberg, Families of elliptic curves with constant mod p representations, Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 148-161. MR 1363500 ↑1, ↑2, ↑3, ↑5, ↑8
Mod 6 representations of elliptic curves. Automorphic forms, automorphic representations, and arithmetic. Fort Worth, TX; Providence, RIAmer. Math. Soc661MR, Mod 6 representations of elliptic curves, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 213-220. MR 1703752 ↑1
Mod 2 representations of elliptic curves. MR 1694877 ↑1, ↑2, ↑3. 1298, Mod 2 representations of elliptic curves, Proc. Amer. Math. Soc. 129 (2001), no. 1, 53-57. MR 1694877 ↑1, ↑2, ↑3, ↑5, ↑8
Jeremy Rouse, Andrew V Sutherland, David Zureick-Brown, arXiv:2106.11141ℓ-adic images of Galois for elliptic curves over Q, arXiv e-prints (2021). Sigma1921To appear in Forum of MathematicsJeremy Rouse, Andrew V. Sutherland, and David Zureick-Brown, ℓ-adic images of Galois for elliptic curves over Q, arXiv e-prints (2021), arXiv:2106.11141, To appear in Forum of Mathematics, Sigma. ↑19, ↑21
. Github Repository, Au- gust 2022. ↑21, Github repository, https://github.com/AndrewVSutherland/ell-adic-galois-images, 2021, Accessed Au- gust 2022. ↑21
Jean-Pierre Serre, ; A K Peters, Ltd Wellesley, M A , With notes by Henri Darmon. MR 2363329. 119Topics in Galois theoryJean-Pierre Serre, Topics in Galois theory, second ed., Research Notes in Mathematics, vol. 1, A K Peters, Ltd., Wellesley, MA, 2008, With notes by Henri Darmon. MR 2363329 ↑19
Explicit families of elliptic curves with prescribed mod N representations, Modular forms and Fermat's last theorem. Alice Silverberg, MR 1638488 ↑1, ↑2, ↑3. Boston, MA; New YorkSpringer58Alice Silverberg, Explicit families of elliptic curves with prescribed mod N representations, Modular forms and Fer- mat's last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 447-461. MR 1638488 ↑1, ↑2, ↑3, ↑5, ↑8
The arithmetic of elliptic curves. Joseph H Silverman, MR 2514094 ↑20Graduate Texts in Mathematics. 106Springersecond ed.Joseph H. Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 ↑20
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The Stacks project authors, The stacks project. 10The Stacks project authors, The stacks project, https://stacks.math.columbia.edu, 2022. ↑10
| {'fraction_non_alphanumeric': 0.10532494130575953, 'fraction_numerical': 0.08034866876467356, 'mean_word_length': 3.088064120890341, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 3, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 80, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over Q with 12-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen[Che16]and Fisher [Fis20] where it is assumed that the underlying isomorphism of 12-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves.A key ingredient in the proof is to construct simple (algebraic) conditions for the 2, 3, or 4-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j-invariants of the pair of elliptic curves.Date: 11 August 2022. 1 For any N ≥ 2 and r ∈ (Z/N Z) × let Z(N, r) be the (coarse) moduli space which parametrises triples (E, E ′ , φ) where E and E ′ are elliptic curves (defined up to simultaneous quadratic twist) and φ :is an (N, r)-congruence (defined up to composition with an automorphism of E or E ′ ). Following Kani-Schanz [KS98] we call Z(N, r) a modular diagonal quotient surface. The surface Z(N, r) is naturally equipped with an involution which reverses the roles of E and E ′ . We write W (N, r) for the quotient of Z(N, r) by this involution.The main contribution of this article is to give explicit equations for the surfaces Z(12, r) for each r ∈ (Z/12Z) × . We use these equations to find infinite families of (12, r)-congruent elliptic curves which are not geometrically isogenous.The surface Z(12, 1) is an elliptic K3 surface and models were computed by Chen and Fisher (see [Che16, Chapter 7] and [Fis20]) using equations for the twist X 1 E (12) of X(12) which parametrises elliptic curves which are (12, 1)-congruent to E (see Section 2.1). Chen [Che16, Chapter 7.4] also computed equations for the twist X 7 E (12) and it may be possible to compute a simple birational model for Z(12, 7) using these. Using different techniques we prove:Theorem 1.1. Each of the surfaces W (12, r) are rational and the surfaces Z(12, r) are birational over Q to the affine surfacesF 12,11 (u, v) = − (u + 1)((v 4 + 6v 2 + 1)u − 7v 4 − 2v 2 + 1)(v 4 u 4 + 8v 4 u 3 + (−9v 6 + 24v 4 + 11v 2 )u 2 + (−54v 6 − 36v 4 − 6v 2 )u + 27v 8 + 27v 6 + 9v 4 − v 2 − 1).Moreover the double covers Z(12, r) → W (12, r) are given by (u, v, z) → (u, v).The maps Z(12, r) → X(1) × X(1) which give the moduli interpretations for the surfaces Z(12, r) are too complicated to reproduce here but may be recovered from the computations in Section 5. We record them in the electronic data corresponding to this article [Fre22].Remark 1.2. Note that the polynomials F 12,r (u, v) contain only even powers of v. In Section 4 we will show that Z(12, r) is a double cover of a surface Z + (12, r). This double cover is the quotient by the involution v → −v.Remark 1.3. The model for the surface Z(12, 1) given in Theorem 1.1 admits a genus 1 fibration over the t-line given by y 2 = F 12,1 (x, t) (in fact, this fibration is elliptic, see Example 6.3).Since the surfaces Z(12, r) are of general type when r = 1 they cannot admit genus 1 fibrations. However, they do admit genus 2 fibrations over the t-line which are given by y 2 = F 12,5 −x+42t , x 2t , y 2 = F 12,7 (x, t), and y 2 = F 12,11 (x, t) respectively.JZ JZ π where J Z and J Z are morphisms. The degree of the morphism J Z is equal to |SL 2 (Z/N Z)|/2 and the degree of J Z may be read off from the degrees of the polynomials P α , P β , and P γ . Since these degrees are equal for each (N, r) the map π is birational.Letting Z ′ (1) = Spec R we have a natural open immersion Z ′ (1) ⊂ Z(1). Let Z ′ (N, r) and Z ′ (N, r) be the preimages of Z ′ (1) under the maps J Z and J Z respectively.Explicitly Z ′ (N, r) and Z ′ (N, r) are the surfaces given by deleting the points on Z(N, r) and Z(N, r) above the loci: (N, r) = (2, 1) : J = J ′ and J, J ′ ∈ {0, 1}, (N, r) = (4, r) : J = J ′ and J, J ′ ∈ {0, 1}, (N, r) = (3, 1) : J = J ′ , J, J ′ ∈ {0, 1}, and ( 3 √ J + 1)( 3 √ J ′ + 1) = 1, (N, r) = (3, 2) : J = J ′ and J, J ′ ∈ {0, 1}.We next show that π extends to an isomorphism π : Z ′ (N, r) → Z ′ (N, r). Note that the morphism Z ′ (N, r) → Z ′ (1) is finiteétale since X(N ) → X(1) is finiteétale away from j = 0, 1728, ∞.When N = 2, 3 we see that the finite morphism Z ′ (N, r) → Z ′ (1) factors via', 'arxivid': '2208.05842', 'author': ['Sam Frengley '], 'authoraffiliation': [], 'corpusid': 251492853, 'doi': None, 'github_urls': ['https://github.com/SamFrengley/12-congruences.git,', 'https://github.com/AndrewVSutherland/ell-adic-galois-images,'], 'n_tokens_mistral': 34762, 'n_tokens_neox': 28121, 'n_words': 16384, 'pdfsha': '9f24a39e61024cd8910bd1c51ee66f16b6c16558', 'pdfurls': ['https://export.arxiv.org/pdf/2208.05842v1.pdf'], 'title': ['ON 12-CONGRUENCES OF ELLIPTIC CURVES', 'ON 12-CONGRUENCES OF ELLIPTIC CURVES'], 'venue': []} |
arxiv |
Defining Least Community as a Homogeneous Group in Complex Networks
Bin Jiang
Faculty of Engineering and Sustainable Development
Division of Geomatics
University of Gävle
SE-801 76GävleSweden
Ding Ma
Faculty of Engineering and Sustainable Development
Division of Geomatics
University of Gävle
SE-801 76GävleSweden
Defining Least Community as a Homogeneous Group in Complex Networks
(Draft: June 2014, Revision: October, November 2014, and February 2015)1head/tail breaksht-indexscalingk-meansnatural breaksand classification
Highlights: A new community detection algorithm inspired by the head/tail breaks. A new way of thinking for community detection or classification in general. Far more small communities than large ones in complex networks. Simple networks like mechanical watches, while complex networks like human brains. Empirical evidence on power laws of the detected communities.AbstractThis paper introduces a new concept of least community that is as homogeneous as a random graph, and develops a new community detection algorithm from the perspective of homogeneity or heterogeneity. Based on this concept, we adopt head/tail breaks -a newly developed classification scheme for data with a heavy-tailed distribution -and rely on edge betweenness given its heavy-tailed distribution to iteratively partition a network into many heterogeneous and homogeneous communities. Surprisingly, the derived communities for any self-organized and/or self-evolved large networks demonstrate very striking power laws, implying that there are far more small communities than large ones. This notion of far more small things than large ones constitutes a new fundamental way of thinking for community detection.
particular for large or complex networks. This paper introduces a new concept of least community as a homogeneous group -as homogeneous as a random graph. Based on this concept, a heterogeneous network is partitioned into many homogeneous communities by referencing its random graph. The random graph is used as a reference because it is considered homogeneous enough and its edges are imposed with the same probability, or it contains only one community. Considering a network as a set of edges characterized by the measure edge betweenness (Girvan and Newman 2002), the issue of community detection becomes that of classification, i.e., classifying all edges into different homogeneous groups as homogeneous as a random graph or, more specifically, into inside and outside edges.
The classification relies on edge betweenness to determine different classes or communities. The edge betweenness of real-world networks demonstrates a heavy tail distribution, indicating that conventional methods such as k-means (MacQueen 1967) and natural breaks (Jenks 1967) could not effectively derive the classes that reflect the underlying scaling pattern. These conventional methods use the mean or the average to characterize individual classes, but the edge betweenness is right skewed or scale free. Given the circumstance, head/tail breaks, a newly developed classification scheme (Jiang 2013), is more appropriate and effective for data with a heavy tail distribution. Head/tail breaks partitions all the edges into the head (those edges with betweenness greater than the mean) and the tail (those edges with betweenness less than the mean), and recursively continues the partition process until the head percentage is as large as that of the random graph (c.f., the next section for illustrations). This ending condition implies that the head and tail are well balanced, and the derived classes or communities are homogeneous enough. During the recursive partition process, some heterogeneous communities are identified as well. Eventually, both homogenous and heterogeneous communities are derived at different coarse-graining levels. The central argument of this paper is that any self-organized and/or naturally evolved real world network contains far more small communities than large ones, or its communities exhibit a power law or heavy-tailed distribution in general.
Community structure or community detection has received disproportionate attention in the past years, largely because of the availability of rich data from the Internet and social media, and its far-reaching implications for a variety of disciplines (e.g., Fortunato 2010, Newman 2004. Communities could be social groupings in a social network based on interest, related papers in a citation network, related researchers in a collaboration network, functional groupings in a metabolic network such as cycles and pathways, and web pages in a website on the same or similar topics. Both community structure and community detection return large amounts of hits in Google Scholar. Despite the literature on the topic having a long history that dates back to the 1920s (Rice 1927), a vast majority of the studies was conducted in the past decade, in particular since the seminal work of Girvan and Newman (2002). The algorithm developed in this paper brings new insights into community detection or classification in general.
The next section presents the new algorithm and illustrates how a network may be partitioned into many communities, both homogeneous and heterogeneous. Section III reports on our experiments by applying the community detection algorithm to many complex networks, including social, biological, technological, and informational. Finally, Section IV concludes the paper with further discussions.
II. The new community detection algorithm based on head/tail breaks
This section illustrates the new community detection algorithm based on head/tail breaks using two sample networks. We start with a fictive social network consisting of 12 vertices and 20 edges ( Figure 1). Intuitively, the fictive network contains three communities of sizes 5, 4, and 3. We first create a random network that is the counterpart of the fictive network with the same number of vertices and edges (Panel C of Figure 1). The edge betweenness of the fictive network is very heterogeneous, with a maximum-to-minimum ratio of 19.9, whereas that of the random network is relatively homogeneous, with a maximum-to-minimum ratio of 3.5. As reflected in the corresponding rank-size plots (Panels D and E), the heterogeneity and homogeneity are indicated respectively by the steep and flat distribution curves. The red dots of the curves constitute the head, which consists of edges (or outside edges) with 2), which cess, it is unity) has ble 3) are the edge removing algorithm could find their counterparts in our method-induced communities (Table 3). The table shows that all the networks have a very low match percentage of communities. This suggests that our method is very unique, and its results cannot be compared to those of the previous method. The comparison further reinforces our belief that unlike simple networks, self-organized and/or selfevolved networks, or complex networks in general, cannot be easily decomposed into parts, for the parts tend to mutually entangled. On the one hand, they are nested, corresponding to our heterogeneous and homogeneous communities; on the other hand, they tend to be very heterogeneous in sizes. In this connection, simple networks are very much like mechanical watches that are decomposable, while complex networks like human brains that are hard to decompose. This is the fundamental thinking that differentiates our method from previous methods.
IV. Further discussions and conclusion
This work is very much inspired by the natural cities extracted from social media location data Miao 2014, Jiang 2015). Individual users' check-in locations constitute a large triangle irregular network (TIN) whose edges demonstrate a heavy-tailed distribution, i.e., far more short edges than long ones. Eventually, all short edges (shorter than an average of all the edges) constitute different clumps called natural cities. In a similar manner, there are far more small betweenness edges than large ones for complex networks, indicating that they contain many clumps called communities. The major differences between the natural cities derived from the TIN and the communities from complex networks are as follows: (1) the TIN is partitioned only once to obtain the natural cities, whereas a complex network is partitioned multiple times recursively to obtain communities; therefore, (2) the derived communities are nested, whereas the natural cities are not. However, for the natural cities, we can also recursively continue the partition process to obtain hotspots in the cities. This way the natural cities and hotspots (both as communities) would be nested as well. The nested relationships are frequently seen in reality, e.g., a country as a set of cities, a city as a set of neighborhoods, and a neighborhood as a set of families. One disadvantage of the community detection algorithm lies in the computational complexity of the edge betweenness, in particular for large networks. In our experiments, we were able to afford to use only parts of some large networks, such as Brightkite, Gowalla, and WWW.
Previous studies relied on real-world networks with known communities to verify community detection algorithms. This verification approach is questionable because the known communities could still be very heterogeneous and should be further partitioned into homogeneous ones. For example, the club network contains two known communities (Zachary 1977, Girvan andNewman 2002), whereas our algorithm leads to three heterogeneous communities and 11 homogeneous ones. Intuitively, the fictive network contains three communities; instead, our algorithm results in four communities. The reader may ask how to verify our results. We believe that the scaling pattern of far more small things than large ones is universal and applies to the communities of a network as well if the network is selforganized and/or naturally evolved. We further believe that the community detection process leading to far more small communities than large ones is very similar to dropping a piece of glass into stone, resulting in far more small pieces than large ones -the fractal or scaling nature of the broken pieces.
In other words, we use the scaling pattern to verify our results.
The notion behind the community detection algorithm is holistic, i.e., taking all edges as a whole and classifying them into the head and tail or, equivalently, the outside and inside edges, and recursively continuing the classification for the inside edges until a network and its subnetworks become homogeneous enough. From the holistic perspective, whether a family is a community is relative to the other families to which it links and to the random graph counterpart. Surprisingly, we found that the derived communities demonstrate a striking scaling property, i.e., far more small homogeneous communities than large ones. During the iterative partitioning, many heterogeneous or large communities can be identified at different coarse-graining levels. The scaling property is even more striking by taking both homogeneous and heterogeneous together, and this is shown by power laws of the communities for all large networks.
AcknowledgmentsWe would like to thank the editor and two anonymous reviewers, in particular the second, who provide some valuable comments that have significantly improved our work.
Emergence of scaling in random networks. A Barabási, R Albert, Science. 286Barabási A. and Albert R. (1999), Emergence of scaling in random networks, Science, 286, 509-512.
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R Cohen, S Havlin, Complex Networks: Structure, Robustness and Function. CambridgeCambridge University PressCohen R. and Havlin S. (2010), Complex Networks: Structure, Robustness and Function, Cambridge University Press: Cambridge.
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Community structure in social and biological networks. M Girvan, M E J Newman, Proceedings of the National Academy of Sciences. 9912Girvan M. and Newman M. E. J. (2002), Community structure in social and biological networks, Proceedings of the National Academy of Sciences, 99(12), 7821-7826.
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Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution. B Jiang, The Professional Geographer. 653Jiang B. (2013), Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution, The Professional Geographer, 65 (3), 482 -494.
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The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx. B Jiang, Y Miao, PreprintJiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx. Preprint: http://arxiv.org/abs/1401.6756
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| {'fraction_non_alphanumeric': 0.038998211091234344, 'fraction_numerical': 0.024627310673822303, 'mean_word_length': 4.998211731044349, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Highlights:\uf06c A new community detection algorithm inspired by the head/tail breaks. \uf06c A new way of thinking for community detection or classification in general. \uf06c Far more small communities than large ones in complex networks. \uf06c Simple networks like mechanical watches, while complex networks like human brains. \uf06c Empirical evidence on power laws of the detected communities.AbstractThis paper introduces a new concept of least community that is as homogeneous as a random graph, and develops a new community detection algorithm from the perspective of homogeneity or heterogeneity. Based on this concept, we adopt head/tail breaks -a newly developed classification scheme for data with a heavy-tailed distribution -and rely on edge betweenness given its heavy-tailed distribution to iteratively partition a network into many heterogeneous and homogeneous communities. Surprisingly, the derived communities for any self-organized and/or self-evolved large networks demonstrate very striking power laws, implying that there are far more small communities than large ones. This notion of far more small things than large ones constitutes a new fundamental way of thinking for community detection.', 'arxivid': '1502.00284', 'author': ['Bin Jiang \nFaculty of Engineering and Sustainable Development\nDivision of Geomatics\nUniversity of Gävle\nSE-801 76GävleSweden\n', 'Ding Ma \nFaculty of Engineering and Sustainable Development\nDivision of Geomatics\nUniversity of Gävle\nSE-801 76GävleSweden\n'], 'authoraffiliation': ['Faculty of Engineering and Sustainable Development\nDivision of Geomatics\nUniversity of Gävle\nSE-801 76GävleSweden', 'Faculty of Engineering and Sustainable Development\nDivision of Geomatics\nUniversity of Gävle\nSE-801 76GävleSweden'], 'corpusid': 2808226, 'doi': '10.1016/j.physa.2015.02.029', 'github_urls': [], 'n_tokens_mistral': 4159, 'n_tokens_neox': 3565, 'n_words': 2465, 'pdfsha': '9607de91fdaed4e3bc904c56d39010785e7e7b07', 'pdfurls': ['https://arxiv.org/pdf/1502.00284v2.pdf'], 'title': ['Defining Least Community as a Homogeneous Group in Complex Networks', 'Defining Least Community as a Homogeneous Group in Complex Networks'], 'venue': []} |
arxiv |
Scenario-based Optimization Models for Power Grid Resilience to Extreme Flooding Events
Ashutosh Shukla
Erhan Kutanoglu
John J Hasenbein
Scenario-based Optimization Models for Power Grid Resilience to Extreme Flooding Events
Graduate Program in Operations Research and Industrial Engineering The University of Texas at Austin, Austin, United Stateshurricanesstorm-surgestochastic programmingrobust optimization
We propose two scenario-based optimization models for power grid resilience decision making that integrate output from a hydrology model with a power flow model. The models are used to identify an optimal substation hardening strategy against potential flooding from storms for a given investment budget, which if implemented enhances the resilience of the power grid, minimizing the power demand that is shed. The same models can alternatively be used to determine the optimal budget that should be allocated for substation hardening when longterm forecasts of storm frequency and impact (specifically restoration times) are available. The two optimization models differ in terms of capturing risk attitude: one minimizes the average load shed for given scenario probabilities and the other minimizes the worst-case load shed without needing scenario probabilities. To demonstrate the efficacy of the proposed models, we further develop a case study for the Texas Gulf Coast using storm surge maps developed by the National Oceanic and Atmospheric Administration and a synthetic power grid for the state of Texas developed as part of an ARPA-E project. For a reasonable choice of parameters, we show that a scenario-based representation of uncertainty can offer a significant improvement in minimizing load shed as compared to using point estimates or average flood values. We further show that when the available investment budget is relatively high, solutions that minimize the worst-case load shed can offer several advantages as compared to solutions obtained from minimizing the average load shed. Lastly, we show that even for relatively low values of load loss and short post-hurricane power restoration times, it is optimal to make significant investments in substation hardening to deal with the storm surge considered in the NOAA flood scenarios.
Introduction
In the past few years, hurricanes and tropical storms have caused significant damage to critical infrastructures such as transportation systems, healthcare services, and the power grid. Hurricanes Maria, Irma, and Harvey together cost nearly $265B which was more than 85% of total weatherrelated disaster costs in the U.S. in 2017 [1]. Harvey became not only the longest-lasting hurricane with a record level of rainfall but also the costliest at $130B, part of which was due to power outages.
Harvey damaged 90+ substations, downed 800+ transmission assets, 6000+ distribution poles, and 800+ miles of power lines, with a peak power generation loss of 11GW, affecting over 2 million people. It took 2 weeks and 12,000 crew members to restore power [2].
The power grid is impacted by hurricanes and tropical storms primarily due to strong winds and flooding. To address this, a vast body of research that examines the effect of wind fields on transmission lines and towers has been developed. However, to the best of our knowledge, the literature is quite scant on the models that assess the impact of flooding. At the same time, the cost of such disasters has increased in states like Texas which is exposed to the Atlantic basin through the Gulf of Mexico. Moreover, recent studies suggest that we are likely to see more frequent and intense hurricanes in the near future [3]. In response to this, some utilities have employed on-site meteorologists which they have reported to be beneficial [2]. These meteorologists localize the predictions to obtain flood estimates for the region of interest. The estimates are then used to determine the resources needed for forecasted damage and post-storm recovery. To further improve this decision-making process, we present an end-to-end scenario-based optimization approach that integrates the output from a predictive geoscience-based flood model with a power flow model to recommend a plan for substation hardening to relieve the flood impacts of the potential storms.
While doing so, the scenario-based approach accounts for the uncertainty associated with storms and their flood forecasts.
Specifically, we propose two scenario-based optimization models (stochastic and robust) for grid resilience decision making under uncertainty. The choice of the model to be used for decisionmaking depends on the available information about the uncertain parameters which in our case are the flood levels at substations in a flood scenario. We show how the proposed models can be used to identify the substations that should be protected and to what extent. We further explain how the same models can be used for deciding the optimal budget that should be allocated for substation hardening to minimize the expected total disaster management cost. The aforementioned features of the models can help power utilities and grid operators address their concerns like the unpredictable nature of the load loss, the potential for substation flooding, and the potential reduction in generator output due to loss of load as outlined in [2].
The rest of the article is organized as follows. Section 2 presents a review of the literature on power grid resilience, particularly from a modeling and decision making viewpoint. Section 3 presents the overview of the proposed models followed by the notation, assumptions, mathematical formulation, and a brief discussion on the characteristics of the models. Section 4 is dedicated to the development of a case study for the Texas Gulf Coast and Section 5 is to the discussion of the results. We conclude with directions for future research in Section 6.
Literature Review
Power grid resilience to extreme events like cyber-attacks and natural disasters has been a topic of intense research in the past few years [4]. This includes studies focused on developing resilience metrics, methodological frameworks to enhance power grid resilience, and approaches to risk analysis [5,6,7]. In addition, several mathematical models have been developed to aid decision-making in different stages of the power grid resilience management cycle. These models can be categorized based on the planning phase they are developed for: mitigation, preparedness, response, and recovery. Using the flood as an extreme event that the resilience models are designed to respond to, the mitigation decisions are about the permanent hardening of the grid components, re-design of the grid through the introduction of new substations/transmission lines, installation of backup generation, etc. These decisions are made well before the start of the hurricane season when limited information about upcoming hurricanes is available. Similarly, before an imminent hurricane, during the preparedness phase, one has to make decisions about where to install temporary flood barriers like Tiger Dams™ , where to deploy mobile substations to quickly recover damaged substations, and what part of the grid to disconnect to avoid fatal accidents due to the collapse of power lines.
In both the mitigation and preparation phases, decision-makers face significant uncertainty about storm characteristics like path, intensity, forward speed, and precipitation. The decision-making in the response and recovery phase, on the other hand, is not plagued by weather uncertainty. Since this paper focuses on decision-making under weather uncertainty, we limit the review's focus to models that aid decision-making during the mitigation and preparedness phases.
Models for both the mitigation and preparedness phases can be broadly categorized into two groups: (1) machine learning-based and (2) optimization-based. In the case of machine learningbased models, the focus is on prediction, not decision making. For example, a machine learningbased model may predict metrics of interest such as the number of outages, outage duration, etc., for an upcoming hurricane [8,9,10]. However, decision making based on these predictions, like which substations to protect and how to reconfigure the power grid network to minimize load shed, are not typically considered within the model. Optimization-based models leverage predictions for decision making. To do so, the predictions with associated uncertainty are represented using scenarios. The decision-making model is then coupled with these scenarios. The models that we propose in this study belong to the class of optimization-based models. In the subsequent paragraphs, we survey the key characteristics of some of these models and highlight their differences from what we propose.
The optimization models generally consist of two components: uncertainty quantification and decision modeling. To quantify uncertainty about the weather, we first generate a set of scenarios using various kinds of models, such as machine learning-based models, physics-based models, and expert opinions. Then, irrespective of how the scenarios are generated, the decision-making model considers the impacts on the grid under each scenario to recommend decisions that minimize a certain risk measure. The models we review in this subsection are based on different ways the aforementioned components of the optimization model can be developed. In particular, we review various methods of generating representative scenarios and incorporating them into alternative decision making models.
Scenario generation
Scenario generation is one of the most common uncertainty quantification methods for extreme weather. We divide scenario generation techniques into four categories. The first is based on fragility curves. The curve represents the failure probability of a component as a function of some loading parameter. For example, fragility curves for transmission towers have been developed with respect to wind speeds. Such curves have been used in various power grid resilience decision making studies [11,12]. The second is based on statistical methods. For example, in [13], the authors use historical hurricane and tropical storm data for developing a baseline scenario. The alternative scenarios are then developed by altering parameters from the historical data to simulate plausible climate-induced changes to storm behavior. In [14], the path and the wind field of typhoons are simulated using Monte Carlo sampling to quantify the spatio-temporal impacts of wind speed on the transmission line status. In addition to wind and flooding, winter storms in Texas, such as Uri of 2021, have propelled research in power grid resilience to extreme cold events. For example, in [15], the authors have developed a statistical model where they incorporate historical outage data to generate scenarios of generator outages due to extreme cold events. The third set of methods is based on physics-based hydrological models. Two such models, called WRF-Hydro and SLOSH, are used in [16,17] to generate flooding scenarios. In [18], the authors use physics-based climate models to evaluate the resilience of levee-protected electric power networks with the primary focus on performance degradation. The fourth category is based on combinatorial criteria like N − k. In this case, each scenario represents a way in which k out of N components can fail. A model based on this criterion is used in [19].
Decision modeling
Several optimization models have been developed for power grid resilience decision-making against extreme weather events. These include models that can aid in decision-making about the upgrade of the power grid network through a combination of hardening existing components, adding redundant lines, switches, generators, and transformers [14,19,20]. However, hardening large parts of the power grid can be financially infeasible. In such a scenario, stockpiling power grid components in strategic locations enhances resilience by expediting network restoration after the disaster. To decide how stockpiling of components should be done, Coffrin et al. [21] developed a two-stage stochastic mixed-integer program where the first-stage discrete decisions are about stockpiling power grid components and the second-stage decisions are about how to operate the power grid to minimize load-shed. Additionally, network reconfiguration before an imminent hurricane can also enhance resilience. The models proposed in [12] make such decisions using grid islanding techniques. However, none of these models have explicitly focused on assessing the impact of flooding on the power grid. On the other hand, there are several studies that assess the impact of flooding on other critical infrastructures. For example, Kim et al. [16] present a framework and a case study using hurricane Harvey to generate physics-based (hydrological) flood scenarios. These scenarios are then used for resilience decision-making for healthcare infrastructure in [22]. Scenarios generated from physicsbased models have also been used in [23] that developed a model to estimate the overall disaster cost due to physical damage loss, income losses, and inventory losses. In comparison, our proposed models are explicitly geared towards resilience decision-making for the power grid and have a power flow model nested within a larger substation hardening model.
The models closest to ours are [17] and [24]. [17] uses a set of scenarios all based on Hurricane Harvey run on a hydrology model focusing on inland flooding. We instead consider a wider range of storms and storm characteristics and scenarios that are based on NOAA's storm surge simulations. Mohadese et al. [24] propose a stochastic optimization model for identifying and protecting substations a day before the anticipated flooding event, meaning a focus on preparedness. Here, our proposed models differ in several ways. First, we generate scenarios using outputs from physicsbased hydrological models to create flood maps for the region of interest. Our choice is based on the rationale that physics-based models represent flood levels across the region of interest such that they are correlated in space and time. Mohadese et al. [24] have not considered the impact of correlated flooding. They also assume that a substation will transmit power if it is not flooded. In reality, this may not be true due to the network effects and we embed a power flow model within our larger resilience optimization model to address such effects. Finally, our models focus on long-term decision making, highlighting mitigation-phase budgeting and decision making.
Modeling
In this section, we first present an overview of the stochastic and robust optimization models developed to assist in grid resilience decision-making against extreme flooding events. Next, we state the key assumptions, introduce the notation and provide detailed mathematical formulations. Finally, we highlight some of the key characteristics of the proposed models and explain how they can be used to address a wide variety of questions in grid resilience decision-making. We highlight that the models proposed in subsection 3.4 and 3.5 are developed to minimize a risk measure over the load shed for a single flood event. In subsection 3.6, we explain how the same models can be leveraged for multi-year planning. we represent the uncertainty in the meteorological forecasts using a set of hurricane parameters (Hurricane 1, . . . , n). In the next step, using the aforementioned parameters as input, we run a hydrological model to get the corresponding flood maps. The flood maps are then used as input to the two-stage decision-making models. The final output from the decision-making models is a plan for substation hardening. Decision making in both models occurs in two stages. Here we specify first-stage decisions that determine which substations to harden and to what extent. We assume that the substation hardening measures are taken during the mitigation phase of the power grid resilience management cycle. Consequently, the decisions made using the model are not a response to any particular imminent hurricane. Instead, they are intended to harden the grid against multiple hurricanes potentially occurring over multiple years and minimize the long-term disaster costs incurred due to flooding. One such mitigation measure for substation hardening is to build permanent protective structures like walls around the substation periphery as shown in Figure 2.
Overview of the proposed models
After we make the first-stage decisions, we assess their performance in dealing with the flood levels in the second stage. The second-stage assessment involves minimization of the load shed in multiple flood scenarios that may impact the power grid during the multi-year planning horizon.
Assumptions
In the proposed models, we make the following assumptions. The first assumption is that a DCbased power flow approximation is acceptable. For a detailed explanation and derivation of the DC power flow equations from the AC equations, we refer to [25]. This approximation has been widely used and is embedded within larger strategic decision-making problems such as long-term capacity planning and operation of wholesale electricity grids. For the kind of models proposed in this paper, a detailed discussion on the difference in the quality of solutions obtained from different power flow approximation models is given in [26]. The second assumption is that we can model the substation hardening cost with fixed and variable components. The fixed cost is incurred when a substation is chosen for hardening. It can represent the cost of building the foundation on which the protective structure is built. Furthermore, it can also include the costs associated with transporting construction resources to the substation site. The variable cost, on the other hand, is a function of the height of flooding to which the substation is made resilient. In our case, we assume that the variable cost linearly depends on height. This is reasonable when we build wall-like structures to protect substations, as shown in Figure 2. Third, we assume that each substation's hardening and flood levels are discrete and finite. In the proposed formulations, they are assumed to be nonnegative integers. Fourth, we assume that all the flooded substations within the network experience the same downtime and are recovered simultaneously. Lastly, we assume that the value of load loss can be quantified in dollars per hour.
Notation
The proposed models use the following notation. Note that all the cost parameters used in the models are in dollars and all power grid parameters are in the per-unit system. Notation not defined in this section and appearing later in the text is defined as introduced.
Stochastic Optimization Model
The two-stage stochastic optimization model (SO) is expressed as:
L * SO = min x∈X L SO (x),(1a)
where
L SO (x) = k∈K p k L(x, k). (1b)
The objective function in (1a) minimizes the expected unsatisfied power demand (load shed) over the scenarios in set K. Here, X represents the set of feasible first-stage decisions. The following constraints define the set:
i∈I f f i y i + v i x i ≤ I,(2a)x i ≤ H i y i , ∀i ∈ I f .(2b)
Note that the variables x i and y i are defined only for substations that are flooded in at least one scenario. Constraint (2a) enforces that the sum of the fixed and variable costs incurred due to substation hardening does not exceed the investment budget. Constraints (2b) place an upper bound on the extent of flooding to which the substation can be made resilient while linking variables
x i and y i for each substation i. Such constraints represent engineering and practical challenges that may arise while building protective structures that are too tall.
In (1b), L SO (x) represents the expected load shed when the first-stage decision is x. Here, p k represents the probability of scenario k and L(x, k) is the recourse function representing the minimum load shed when the first-stage decision is x and scenario k is realized. The recourse function is defined as follows:
L(x, k) = minimize j∈J D j − s j ,(3a)
subject to
(1 − z j )M ≥ ∆ θ(j)k − x θ(j) , ∀j : θ(j) ∈ I f , (3b) 2z j M ≥ 1 − 2(∆ θ(j)k − x θ(j) ), ∀j : θ(j) ∈ I f ,(3c)z j = 1, ∀j : θ(j) ∈ I \ I f , (3d) u j ≤ z j , ∀j ∈ J ,(3e)s j ≤ z j D j , ∀j ∈ J , (3f) u j G j ≤ g j ≤ u j G j , ∀j ∈ J , (3g) − z λ(r) F r ≤ e r ≤ z λ(r) F r , ∀r ∈ R, (3h) − z µ(r) F r ≤ e r ≤ z µ(r) F r , ∀r ∈ R, (3i) B r (α λ(r) − α µ(r) ) ≥ M (z λ(r) + z µ(r) ) − 2M + e r , ∀r ∈ R, (3j) B r (α λ(r) − α µ(r) ) ≤ −M (z λ(r) + z µ(r) ) + 2M + e r , ∀r ∈ R, (3k) r∈N out j e r − r∈N in j e r = g j − s j , ∀j ∈ J, (3l) − π ≤ α j ≤ π, ∀j ∈ J ,(3m)α β = 0. (3n)
The objective function in (3a) minimizes the unsatisfied power demand when the first-stage decision is x and the flood scenario realized is k. Constraints (3b) and (3c) link the first-stage substation hardening decisions to the second-stage scenario-dependent power flow decisions. For a given hardening decision at a substation, the provided protection level is compared against the flood height at that substation in a given scenario. Depending on whether the hardening level can withstand the flooding, we set the status of the corresponding bus as operational or not. This is indicated by variable z j . For the substations that are not flooded in any of the scenarios, the status of the corresponding buses is set to operational in constraints (3d).
Constraints (3e) capture generator dispatch decisions for operational generators. When z j = 0, we cannot inject power to the network through bus j and therefore set u j = 0. If z j = 1, we let the recourse problem decide if power generated at bus j should be used or not. Constraints (3f) place an upper bound on the amount of power that can be supplied at bus j (which is the demand at that bus). If bus j is flooded, then z j = 0 and no power can be supplied to the loads that are connected to the bus. Constraints (3g) place upper and lower bounds on the amount of power that can be generated at bus j. If bus j is flooded, then u j = 0 and thus g j = 0. If on the other hand bus j is not flooded, the model solves the recourse problem which is a binary linear program to determine the amount of power that should be generated at bus j. Constraints
Robust optimization model
In the two-stage robust optimization model (RO), we minimize the maximum unsatisfied power demand value across all scenarios. Mathematically, the problem can be stated as follows:
L * RO = min x∈X L RO (x),(4a)
where
L RO (x) = max k∈K L(x, k).(4b)
The expression in (4b) finds the maximum scenario-based load shed via a max(·) function. RO in (4a) can be reformulated as
min x∈X {τ : τ ≥ L(x, k) ∀ k ∈ K} ,(5)
where τ is an epigraphical variable.
Model Discussion
In this section, we highlight some of the key characteristics of the models proposed above. This leads us to a feasible solution with no power generation and maximum load shed.
Second, the proposed models can be further tightened based on some simple observations. To do so, we first compute the maximum flood level across all the scenarios for the flooded substations.
Let us represent this value using parameter W i , ∀ i ∈ I f . Next, we observe that the model need not harden any substation to flood height that is higher than W i . Therefore, in constraints (2b), we
can replace H i with min(H i , W i ), ∀i ∈ I f . Constraints (3b) and (3c) use the big-M method. Here, we need to determine the smallest value for M for each constraint. The smallest big-M value for constraints (3b) and (3c) is given by W θ(j) and min(H θ(j) , W θ(j) ) + 0.5, respectively. To verify this, recall the assumption that both the flood height and hardening level can only take non-negative integer values. Also, observe that
− min(H θ(j) , W θ(j) ) ≤ ∆ θ(j)k − x θ(j) ≤ W θ(j) .
Now, in constraints (3b), we need
∆ θ(j)k −x θ(j) M ≤ 1. The smallest value of M that ensures this is W θ(j) .
Similarly, in constraints (3c), we need
1−2(∆ θ(j)k −x θ(j) ) 2M ≤ 1.
The smallest value of M to achieve this is min(H i , W i ) + 0.5. Finally, we can also tighten constraints (3j) and (3k). For both sets of constraints, the smallest value of M is F r + 2πB r .
Third, both SO and RO can be used for hardening decisions that will provide flood mitigation over a planning horizon. To see how, notice that the objective function in the SO computes the expected load shed for a single flood event. However, substation hardening in practice is done over the planning horizon that lasts multiple hurricane seasons and provides permanent protection for multiple flood events. Therefore, to help make hardening decisions that impact performance over multiple events, we first need to compute the disaster management costs due to load shedding over the multi-year planning horizon. To do so, let us assume that the expected number of hurricanes that the study region experiences during the planning horizon is γ. We further assume that during the planning horizon, the total recovery, economic and social costs are represented by the value of load loss of $δ/megawatt-hour. Finally, we assume that it takes h hours to repair all the substations (and restore power to normal operation) starting immediately after a flood event. We believe this assumption is reasonable at the mitigation phase of the decision making process and avoids explicit and detailed modeling of the recovery process. Then, for a given investment budget, the substation hardening decisions that achieve the minimum expected total disaster management cost due to load shedding during the planning horizon is found by solving
DM SO = γhδ L * SO .(6)
In (6), the optimal substation hardening plan to minimize DM SO is the same as the plan obtained by solving SO. This is because the DM SO always equals the objective function of the SO multiplied by a positive constant. Therefore, irrespective of how the frequency of hurricanes, the restoration time, and the value of load loss change over time, the optimal substation hardening plan remains the same as the one obtained from SO. In this case, we assume that the probability distribution over the flood scenarios, and thus, the hurricanes causing them, does not significantly change over time. In practice, this is reasonable for planning horizons for which substation hardening is considered (5-10 years). Similarly, the optimal substation hardening decision that minimizes the maximum total disaster management cost due to load shedding during the planning horizon is found by solving
DM RO = γhδ L * RO .(7)
A key observation is that DM RO provides an upper-bound for DM SO . To understand why, note that the set of feasible first-stage solutions is same for both SO and RO. Further, observe that both the models are bounded below with a minimum objective value of zero and have relatively complete recourse. Now, let x R be a feasible solution to RO. Then,
L * SO = min x∈X L SO (x) (8a) ≤ L SO (x R ) (8b) = k∈K p k L(x R , k) (8c) ≤ k∈K p k max k∈K L(x R , k) (8d) = max k∈K L(x R , k) k∈K p k (8e) = max k∈K L(x R , k) (8f) = L RO (x R ).(8g)
with Equation (8b) holding at equality if and only if x R is optimal for SO and Equation (8d) Finally, in the models proposed so far, we assume that we have a predetermined budget for substation hardening. One may however be interested in determining the optimal budget allocation for minimizing a risk measure over the disaster cost incurred due to both load shedding and substation hardening. The proposed models can easily be modified to find the optimal budget for substation hardening and corresponding hardening decisions. In the case of SO, this can be done by solving
T DM SO = min ω k∈K p k L(x, k) + i∈I f f i y i + v i x i : x i ≤ H i y i , ∀i ∈ I f ,(9)
where ω = γhδ. The value of i∈I f f i y i + v i x i in the optimal solution represents the value of the optimal investment budget. Similarly, for RO, we compute the optimal investment budget by
solving T DM RO = min ωτ + i∈I f f i y i + v i x i : τ ≥ L(x, k) ∀ k ∈ K .(10)
Case Study
In this section, using a case study for the Texas coastal region, we show how the proposed models can be used for power grid resilience decision making. The two main inputs to the proposed models are a set of scenarios that represent flood profiles for different hurricane types and the network parameters for the DC power flow model. To represent flood profiles, we use storm-surge maps developed by the National Oceanic and Atmospheric Administration (NOAA) [27]. For the electric grid, we use the ACTIVSg2000 dataset developed as part of an ARPA-E project [28]. The details of each component are described in the following subsections. We further highlight that although we use the proposed models for storm surge-induced damages, they can be adopted for flooding of any kind as long as the corresponding flood scenarios are available. This could include scenarios for inland flooding as developed in [16] and used for infrastructure resilience problems in [17] and [22].
Lastly, to solve the various parameterizations of the proposed models discussed in this case study, we use the Gurobi solver with the barrier algorithm [29]. Within the solver, we set the MIP-gap threshold to 0.5 percent and limit the solve time to 6 hours. The model is solved on an Apple M1 pro machine with 16 GB of unified memory.
Flood Scenarios
We use the storm-surge maps developed by NOAA using the Sea, Lake, and Overland Surges from Hurricanes (SLOSH) model as flood scenarios. To generate these flood maps, SLOSH uses a simplified parametric wind field model that takes as input the following parameters: storm track, the radius of the maximum wind speeds, and the pressure differential between the storm's central pressure and the ambient pressure. The simulated wind fields are then used to compute surface stresses on the water beneath the hurricane. Finally, the induced stress on the surface of the water is used to determine the storm surge. For a detailed discussion on SLOSH, we refer to [30].
Simulation studies developed using SLOSH have been extensively used to assist agencies like the In addition to real-time storm-surge guidance for imminent hurricanes, NOAA has developed two composite products, Maximum Envelopes of Water (MEOW) and Maximum of the MEOWs (MOM), to provide manageable datasets for medium to long-term hurricane evacuation planning.
To develop these datasets, hurricane simulations with different combinations of intensity, forward speed, direction, and tide levels are run in parallel using SLOSH for the region of interest. Each run may yield different storm surge values for the same grid cell. A maximum overall such value is taken to represent the MEOW value of that grid cell. The same process is repeated for each grid cell within the study region to construct a MEOW map. The resolution of the grid cell is varied to balance accuracy with computation cost. It is finer in regions close to the coast and gets coarser as we move farther away in the ocean. MEOW maps are used to incorporate the uncertainties associated with a given forecast and eliminate the possibility that a critical storm track will be missed in which extreme storm surge values are generated. These maps are generated from several thousand SLOSH runs. In this study, we use the MEOW maps to represent flooding due to storm
surge. An example MEOW map is shown in Figure 3.
Within a MEOW map, it is possible that the water level for adjacent cells may come from different SLOSH runs of specific simulated storms. Nevertheless, since these are the maximum water levels over multiple tracks, we can be assured that if we have hardened a substation for a particular MEOW map, it will provide resilience towards flooding for any of the parallel runs that constitute the MEOW map. Arguably, MEOW maps are still better at representing flood uncertainty than other scenario generation methods where flooding at different nodes within a network is considered independent of each other. Moreover, these MEOW maps have been considered as scenarios in different stochastic optimization models for patient evacuation [16] and grid hardening [31]. before, we assume that in the mitigation phase, the decision-maker does not have information on any specific storm. Therefore, to model the uncertainty in the mitigation phase, we assume that all the remaining MEOW maps are representative of the flooding scenarios. In our case, they are considered equally likely for the stochastic model. This is based on the premise that the larger set of simulations that produced the MEOW maps were sampled according to some underlying distribution and therefore already reflects the underlying characteristics of the distribution implicitly used by NOAA in their development of the MEOW product. Furthermore, these MEOW maps provide us with the storm-surge flood height above ground at each of the substations (and thus for buses within) in the power grid network. In the proposed models, this is represented by ∆ ik ; the level of flooding at substation i in scenario k.
Power Grid
To model the power grid for the state of Texas, we use the synthetic grid called ACTIVSg2000 which contains 2000 buses (within 1250 substations) and 3206 branches. The grid, though synthetic, is designed such that it maintains statistical similarities with the actual Texas grid. We make two further modifications to the grid instance to make it computationally tractable while also considering the coastal part of the grid which is affected due to to storm-surge and thus is the focus of the study.
First, we perform a network reduction on the original grid instance using the electrical equivalent (EEQV) feature in PSS®E to focus on the grid components subject to storm surge flooding. The reduction is such that the buses in the inland region that are not exposed to storm-surge induced flooding are aggregated within a much smaller set of nodes. The part of the grid that is in close proximity to the Texas Gulf Coast, and therefore is prone to flooding due to storm-surge, is retained almost as is. The effect of the network reduction is detailed in Table 1. The topological changes are visualized in Figure 4. Second, we alter the locations of some of the substations. This is because, in
Results and Discussion
In this section, we first determine the expected value of perfect information from a model that can produce perfect forecasts. In the same subsection, we compute the value of the stochastic solution for the different budget levels. Next, in subsection 5.2 we show how the proposed model can be used to determine the optimal investment budget for substation hardening.
Due to dynamically changing climate conditions and ocean temperature, the probabilities of different types of hurricanes can change over time. In subsection 5.3, we show when a decisionmaker can take advantage of using solutions from RO to hedge against this uncertainty by paying a relatively insignificant premium. Finally, the last subsection is dedicated to the analysis of the distribution of load shed across scenarios for the solutions obtained from SO and RO.
The Values of Stochastic Solution and Perfect Information
In the proposed two-stage decision-making models, both the number of variables and constraints grow linearly with the number of scenarios; thus making it computationally challenging. In that case, instead of solving SO for large grid instances with many scenarios, one may be interested in solving simpler versions of the problem. One approach could be to reduce the size of the problem by constructing a single scenario problem where the flood height at each substation is the average of the flood height across all the scenarios. Another way could be to solve the problem for each scenario individually. The first-stage solutions thus obtained can then be analyzed and potentially combined using some heuristic rule. In this section, we analyze the quality of the solutions we get using such approaches. To do so, we use two widely known concepts in the stochastic programming literature: the value of the stochastic solution and the expected value of perfect information.
In Figure 5, we plot the value of L * SO as a function of the investment budget. To determine the budget levels on which the parametric study should be performed, we first compute the minimum budget such that L * SO = 0. This is computed by solving a slightly modified version of SO. Specifically, we first remove constraint (2a) from the formulation and replace the objective function with the minimization of the substation hardening expenditure (i.e., the left-hand-side of (2a)). We also force full satisfaction of demand by replacing constraint (3f) with s j = D j , ∀ j ∈ J . The optimal value to this modified version of SO is in turn the minimum hardening budget required for zero load shed. For the parameters assumed in this case study, the corresponding minimum budget turns out to be $71.35M. Any additional budget beyond this will not improve the objective function value (load shed) and therefore the corresponding optimal solution. Using $71.35M as the reference, in Figure 5, we increase the budget from $0M, in increments of $10M, until the value exceeds $71.35M.
(We note that the same budget values can be used for the RO parametric study. This is because the minimum budget required to achieve the expected load shed of zero is the same as what is required to achieve zero load shed in all the scenarios.)
In addition to computing L * SO for different budget levels, we also compute lower and upper bounds on this value as shown in Figure 5. To compute the upper bound, let us first consider a single-scenario problem called the expected value problem which is defined as
L * EV = min x∈X L(x,k).(11)
Here,k represents a scenario where the flood level at each substation is the mean of the flood heights at that particular substation across all the scenarios. The optimal solution to this problem is called Next, to evaluate the quality of the first-stage substation hardening decisions obtained by solving (11), we compute the expected load shed across all the scenarios by fixing the first-stage decisions tox, denoted by L SO (x). This value serves as an upper bound on L * SO . For a detailed explanation on this, we refer to [33]. We also observe in Figure 5 that the difference between L SO (x) and L * SO increases with an increase in the budget. We refer to this difference as the value of the stochastic solution (VSS) as it represents the value of using a scenario-based representation of the uncertainty as opposed to the average flood values, all calculated within the SO framework. We further highlight that whenx is implemented, the load shed does not strictly decrease with an increase in the budget beyond $50M. This is expected because, for the mean scenario, the model obtains zero load shed with $50.15M. As discussed before, we know that it takes a minimum of $71.35M to achieve zero load shed across all the scenarios. However, once the model achieves zero load shed in the mean scenario, there is no incentive to use additional resources. This also shows that the value of using SO over the expected value problem increases with increases in the investment budget.
To compute a lower bound on L * SO for a given budget level, we assume that the decision-maker has access to perfect information about the flood levels, and therefore can better prepare for each scenario (i.e., in a way, fine-tuning the mitigation plan according to each scenario). That is, perfect information allows the decision maker to make possibly different substation hardening decisions in each scenario to minimize the load shed in that particular scenario. These solutions are referred to as wait-and-see solutions. In this case, we compute
L * WS = k∈K p k min x∈X L(x, k) ,(12)
where the first-stage decisions are scenario-dependent and the value L * WS is referred to as the waitand-see bound. Due to the scenario-specific mitigation decisions, L * WS provides a lower bound on L * SO . The difference in the values L * SO and L * WS is referred to as the expected value of perfect information. It represents the maximum value a decision-maker would be willing to pay in exchange for complete and accurate information about the uncertainty.
The key point that we want to emphasize is that unless the flood model can make perfect predictions, which is usually not the case with the weather models, then not accounting for uncertainty and using just point estimates or mean values of the flood forecasts can lead to significantly inferior decisions. This is evident from the VSS. In fact, as it turns out in this case, the first-stage decisions obtained from SO, even with not-so-perfect forecasts, lead to a load shed performance that is close to what we would get from using a flood model that offers perfect prediction. To put it another way, accounting for flood uncertainty, even with a small number of scenarios, can help reduce the burden of getting perfect information on the decision-maker without making a significant compromise on the performance.
We also note that all bounds converge to the same expected value at the zero budget. This is because, no matter how well we represent the uncertainty or how good the predictions are, if we do not have any resources to use towards mitigation in the first stage, we cannot prevent load shed in the second stage with no protection towards flooding. Then, the expected value of the load shed is only a function of the second-stage decisions (i.e., the best power flow the grid can deliver with flooded substations). Moreover, at a sufficiently high budget value, both L * SO and L * WS converge to zero. This is expected because, despite the fact that we have poor predictions or poor uncertainty representation, we can still prevent any load shed in all the scenarios if we have enough resources to harden all substations to any desirable extent. Performing analysis with the budget level as a parameter, as described in this subsection, requires repeatedly solving SO with different parameters and can be time-consuming. To address this, the property that SO has relatively complete recourse is exploited to warm start the optimization solver and improve the solution time. As was stated in subsection 3.6, we can heuristically generate an initial feasible solution with a hundred percent load shedding for an investment budget value of zero. Once we get an optimal solution corresponding to budget level zero, we use it to warm start the solver for the next higher budget level. The process is repeated to generate high-quality feasible solutions for the next budget level. We further note that the aforementioned approach for warm-starting the solver is also applicable in the case of RO.
Determining optimal budget for substation hardening
Subsection 5.1 focuses on demonstrating how SO is used for resilience decision making for a given investment budget for substation hardening. The models can alternatively be used to decide the optimal value of the investment budget that minimizes the expected total disaster management cost over a multi-year planning horizon. To demonstrate this, we solve T DM SO as described in Section 3.6 assuming that, on average, 10 hurricanes hit the Texas Gulf Coast during the planning horizon. The values of the expected total disaster management cost for different combinations of restoration times and the value of load loss are plotted in Figure 7. As we see in the figure, when the value of load loss is low and the restoration time is short ($250 and 6 hours, respectively), the total disaster management cost is relatively low. Furthermore, the model recommends investing only a quarter of the budget required to achieve zero load shed for substation hardening. This is because the cost associated with losing power is quite low and the outage is restored relatively quickly. On the other hand, when both the load loss value and restoration time are high ($5000 and 48 hours, respectively), the model recommends investing $71.35M (equal to the investment required to achieve zero load shed) for substation hardening, avoiding any costs due to load loss. This is because, for the chosen values, the costs associated with power loss are quite high and it is better to make Texas, the value of load loss was determined to be around $6000 per MWh for Texas [34]. If we take that as given, the results in Figure 7 suggest that we must make investments to achieve close to zero load shed even if restoration time is as short as 6 hours.
We further observe in Figure 7 that the solution corresponding to the value of load loss of $1000 per MWh and restoration time of 6 hours is the same as the solution with the value of load loss of $250 per MWh and a restoration time of 24 hours. This is expected because in T DM SO , both sets of parameters lead to the same optimization problem (ω is the same). It is further apparent that the optimal investment budget for substation hardening increases monotonically from 0 to $71.35M with the increase in the value of ω. Therefore, for any investment budget value between 0 and $71.35M, there exists a unique ω for which the corresponding budget is optimal. We can use this insight to quickly approximate the optimal investment budget for any combination of the value of load loss, restoration time, and the average number of hurricanes that may hit the region of study during the planning horizon. To do so, we use only the values of L * SO for I ∈ {0, 10, 20...80} as computed in Section 5.1. For any given value of γ, h, and δ for which the optimal investment budget needs to be approximated, we compute the value of DM SO + I for each I ∈ {0, 10, 20...80}. The value of I for which DM SO + I is the smallest is the best approximation for the optimal investment budget for a chosen value ω (calculated from γ, h, and δ). In Figure 8, we show the value of DM SO + I for I ∈ {0, 10, 20...80} for different values of ω. The depicted values of ω are reasonable in the sense that they can be derived from the γ, h, and δ used in Figure 7. For example, we notice in Figure 7 that when γ = 10, h = 6, and δ = 250 (ω = 15000), the optimal investment budget is $17.05M. An approximate of this value can be quickly inferred by looking at Figure 8 for the value of ω = 15000.
Optimization in the face of uncertain probabilities
While framing the power-grid resilience decision-making problem as a two-stage stochastic program, we assume that the probability distribution over the scenarios is known. However, the probabilities that we assign to each scenario need not be constant with time. Changing climate oscillations and ocean temperatures routinely affect these probabilities. This is reflected in NOAA's annual hurricane season prediction categories: normal, above normal, and below normal. In this case, if probabilities over scenarios change as compared to what we planned for, the expected performance may deteriorate. To hedge against this, we recommend solving RO and comparing the expected performance of the corresponding decisions. If the difference in the expected performance of decisions recommended by RO and SO is not significant, we advise adopting decisions as recommended by RO. In this way, irrespective of how the probabilities evolve with time, we know that the expected total cost due to the load loss will be less than or equal to the bound obtained in Section 3.6 if the optimal first-stage decisions as recommended by RO are adopted. This can be confirmed for the parameters assumed in the case study through Figure 6. The robust solution refers to the value of the expected load shed when the first-stage hardening decisions as recommended by RO are adopted. Since these decisions are not necessarily optimal for the assumed probability distribution, they lead to a higher expected load shed as compared to the stochastic solution. However, for the RO first-stage decisions, the maximum load shed in any scenario is capped by τ as represented by the red curve. In this case, we observe that the difference in the expected value of performance is almost trivial for investment budget values of $40M and above. Therefore, in those cases, it makes sense to adopt decisions recommended by RO as opposed to what we get from SO to hedge against the change in probabilities due to factors like ocean temperature and climate oscillations. In cases when the difference between the expected performance of SO and RO is significant, the decisions depend on the risk preference of the decision-maker.
SO vs RO: Analysis of the load shed distribution
We conclude the discussion with an analysis of the distribution of the load shed across scenarios for both SO and RO. The load shed in each scenario for both models is represented by the value of the recourse function corresponding to the optimal solution. These values are used to construct the corresponding histogram for both SO and RO at different budget levels as shown in Figure 9.
As expected, the histograms shift to the left with the increase in the budget for both SO and RO.
We observe that the histograms for both SO and RO coincide when the investment budget is $0M. This is expected because there is no hardening done in either model. Consequently, the load shed in each scenario is identical. Moreover, using Figure 6 and Figure 9, we observe that the robust solutions provide an inferior performance in expectation but the RO load shed remains relatively stable across scenarios as compared to SO. We also conclude that for investment budget values of more than $40M, the robust solutions offer much better performance against extreme scenarios while also offering good expected value performance. Therefore, in this case, it is reasonable to implement RO decisions for budget values beyond $40M. In this way, Figures 6 and 9 can be used together to understand the behavior of both SO and RO in expectation and across all the individual scenarios. Figure 9: The histograms of load shed across scenarios for the SO and RO solutions at different budget levels. Notice that the x-axis for each sub-plot has a different scale for better depiction.
Conclusions
In this study, we propose an integrated framework supported by two scenario-based optimization models for power grid resilience decision making against extreme flooding events. The models recommend an optimal substation hardening plan by integrating the predictions generated from a state-of-the-art hydrological model with a DC optimal power flow model. While doing so, we account for uncertainty in hurricane predictions using a scenario-based representation. Furthermore, using a case study for the state of Texas, specifically the coastal region prone to storm surge flooding, we demonstrate how the proposed models can together be used to address a wide variety of insightseeking questions related to power grid resilience decision making. Specifically, we show that using a scenario-based representation of flood uncertainty can offer significant value over mean flood forecasts. We also explain the advantages of using flood maps generated from physics-based models as opposed to other scenario generation methods popular in the literature. Furthermore, we show how can we estimate the expected value of perfect information from near-perfect flood forecasts.
For the case study developed in the paper, we observe that by using a scenario-based representation of uncertainty, the decision-makers can reduce their burden of having access to perfect forecasts.
In addition to quantifying the value of using flood scenarios, we further show how we can use the proposed two-stage framework to determine the optimal investment budget for substation hardening.
Lastly, we explain how we can use the two-stage robust optimization model for power-grid resilience decision making when information about the probability distribution over the flood scenarios is unavailable.
For future research, we suggest four directions. First is the development of scenarios that can consider precipitation-induced inland flooding in addition to storm-surge. Second is the development of methods to account for equity while making substation-hardening decisions. Third is developing models that take into account preparedness measures while making longer-term mitigation decisions, leading to three-stage optimization models. Fourth is the development of decomposition techniques to solve such models in a reasonable time. These challenges form the basis of our ongoing research.
A two-stage stochastic optimization model is developed to address situations where the uncertainty about hurricane-induced flooding is modeled using a probability distribution. In this case, the model minimizes the expected unsatisfied power demand (also referred to as the expected load shed) due to the components' failures (i.e., flooded substations) over a set of scenarios. The two-stage robust model on the other hand requires no information about the probability distribution. The model instead minimizes the maximum load shed in any scenario within the uncertainty set. A general framework representative of the proposed models is shown inFigure 1. As shown in the figure,
A scenario in the proposed models represents the water levels at different substations obtained from the flood map of a specific hurricane type. Hurricanes with different characteristics such as direction, intensity, forward speed, etc. lead to different levels of flooding, generating different scenarios. These scenarios are representative of the flooding that the region of study can experience
Figure 1 Figure 2 :
12An example permanent hardening structure at a substation over a specific time period (typically multiple years). A distinguishing feature of the proposed models is the way we generate the scenarios. Instead of using popular techniques like fragility curves, we use flood maps for scenario representation because they capture the effects of correlated flooding. This is important because the failure of a substation within a power grid can have network effects on the other parts of the grid. To account for such details, we need not only know which substations fail more frequently but also the combination of substations that fail together. Our proposed model accounts for these details and uses them to evaluate the effects of such damages (in the form of load shed) during decision-making by solving a power flow model.The power grid network considered in the proposed models is represented by a graph where the buses are represented by the nodes and the branches interconnecting the buses are represented by the edges of the graph. The branches of the network are held using the transmission towers. We assume that these towers are well above the ground and therefore immune to flooding. It is the substations and components within them that are susceptible to flooding. In this study, we assume that the substations are outdoor open-air facilities. Therefore, when a substation is flooded, we assume that all the components within the substation and the branches connected to all the buses within the substation are out of order.
For
each of these scenarios, we overlay the power grid network on the flood map to identify parts of the network that are flooded. Given the flood height and the level of hardening at a substation, the model infers if a substation is flooded in a particular scenario. If a substation is flooded, all the buses within the substation and the branches connected to those buses are considered to be out of order. Once the damaged state of the power grid network is determined, we solve the second-stage assessment (the so-called recourse problem) which is a DC power flow model to estimate the load shed given the state of the grid. The second-stage decisions in the recourse problem determine the routing of power to minimize unsatisfied power demand. It should be noted that although both the stochastic and robust models involve the same sets of decision variables, the specific solutions suggested by them can be vastly different. The robust model gives us the flexibility to make decisions in the absence of any information about the probability distribution. These decisions can however be far more conservative than the decisions recommended by the stochastic model.
Set of substations indexed by i I f : Set of substations that are flooded in at least one scenario J : Set of buses indexed by j K: Set of scenarios indexed by k R: Set of branches indexed by r B i : Set of buses at substation i N in j : Set of branches incident on bus j with power flowing into bus j N out j : Set of branches incident on bus j with power flowing out of bus j Parameters M : An arbitrarily large constant f i : Fixed cost of hardening at substation i v i : Variable cost of hardening at substation i H i : Maximum flood height to which substation i can be hardened θ(j): Substation that contains bus j ∆ ik : Flood height at substation i in scenario k (a non-negative integer value) B r : Susceptance of branch r F r : Maximum power that can flow in branch r λ(r), µ(r): Head bus and tail bus of branch r D j : Load at bus j G j , G j : Minimum and maximum generation at bus j β: Index of the reference bus p k : Probability of scenario k I: Total investment budget for substation hardening Variables y i : Binary variable indicating whether substation i is chosen for permanent hardening x i : Non-negative integer variable indicating discrete height of hardening at substation i z j : Binary variable indicating if bus j is operational s j : Non-negative real variable indicating load satisfied at bus j g j : Non-negative real variable indicating power generated at bus j u j : Binary variable indicating if generator at bus j is used α j : Real variable indicating voltage phase angle of bus j e r : Real variable indicating power flowing in branch r
(3h) and (3i) place restrictions on the amount of power that can flow through branch r. If the bus at either end of a branch is flooded, then no power can flow through it. On the other hand, if buses at both ends of the branch are operational, then a maximum of F r power can flow through it in either direction. Constraints (3j) and (3k) enforce an approximation to Ohm's Law. If both ends of a branch are operational, then the amount of power flowing in the branch is governed by equations B r (α λ(r) − α µ(r) ) = e r , ∀r ∈ R.If the bus at either end of the branch is flooded, then the above equation need not hold. This is achieved by introducing big-M values. The formulation can be further tightened by appropriately determining the values of big-M . A discussion on this is presented in Section 3.6. Constraints (3l) represent the flow balance which states that the net power injected into the network at bus j is the difference between the power generated and consumed at the same bus. Constraints (3m) impose limits on the phase angle values at buses. Finally, constraints (3n) set the phase angle of the reference/slack bus to 0.
First, both SO and RO have relatively complete recourse. That is, no matter what first-stage decisions we make, the second-stage problem always has a feasible solution. To verify this, consider a case where irrespective of the value of z j 's, we set the value of u j = 0 for all j in the recourse function.
holding at equality if and only if load shed values are equal across all the scenarios. The above inequalities establish that the objective function value corresponding to any feasible solution to RO provides an upper bound on the optimal objective function value of SO. Since any optimal solution to RO is also feasible, it acts as a valid upper bound. In fact, it is the tightest upper bound that can be obtained in this manner.
Figure 3 :
3A sample MEOW generated using category 5 storms approaching the Texas Gulf Coast in the north-west direction with a forward speed of 5 mph
Figure 4 :
4the original dataset, some of the substations are placed in the middle of a water body and are thus flooded by default. To address this, we remap the coordinates of the 1250 substations in the dataset with the coordinates of substations obtained from the Homeland Infrastructure Foundation-Level Data (HIFLD) Electric Substations dataset [32]. The HFILD dataset contains information about real-world substations across the U.S. The remapping is done by solving an optimization problem The figure shows (a) ACTIVSg2000 Synthetic Grid for Texas, and (b) the reduced grid obtained after performing the network reduction. The red elements represent the new nodes andbranches that were introduced as the artifacts of the reduction procedure to maintain equivalence in the grid characteristics that minimizes the total displacement due to remapping. Note that this process does not change the power grid's electrical structure and makes it more realistic using the real-world substation locations (closer to the actual Texas grid) and capturing their real-world flood risks via MEOW-based flood scenarios. Lastly, the fixed cost and the variable cost for substation hardening are assumed to be $25,000 and $100,000 per foot, respectively. These values are derived from various utility reports.
Figure 5 :Figure 6 :
56The graph of the expected load shed values for the expected value solution (green), the stochastic solution (blue), and the wait-and-see solution (orange), as a function of the budget for substation hardening The objective function value for RO (i.e., the optimal maximum load shed) and the expected load shed values for the robust and the stochastic solutions as a function of budget for substation hardening the expected value solution or the mean value solution, henceforth represented byx. We note that the problem in (11) is a single scenario problem and therefore much smaller in size. However, the reduction of scenarios leads to the loss of information about the substations that flood together.
Furthermore
, for sensitivity analysis, we consider three different values of restoration time: 6, 24, and 48 hours. Similarly, for the value of load loss, we consider 5 different values: $250, $500, $1000, $3000, and $5000 per MWh.
Figure 7 :
7significant investments (in fact, everything possible) in substation hardening such that there is no loss of power. For all other combinations in between, both the total expected disaster management cost and its composition vary. Based on a study undertaken by the Electric Reliability Council Total disaster management cost as a function of the value of load loss for different restoration times
For that value, the top-left curve achieves its minimum at $20M which is closest to $17.05M in the set {0, 10, 20...80}.
Figure 8 :
8Total disaster management cost as a function of budget for different values of ω
Table 1 :
1Grid characteristics before and af-ter the electrically equivalent reduction was
performed.
Grid Characteristic
Before After
Substations (#)
1250
362
Buses (#)
2000
663
Transformers (#)
860
358
Transmission Lines (#)
2346 1151
Generators (#)
544
254
Generation Capacity (GW)
96.29 50.98
Load (GW)
67.11 39.69
The original MEOW map dataset for the Texas coastal region comprises 192 flood maps whichare constructed using eight different storm directions (west-south-west, west, west-north-west, north-
west, north-north-west, north, north-north-east, and north-east), six different intensity categories
(0-5), and four different, forward speeds (5,10,15, and 25 mph). To demonstrate the usefulness of
the proposed approach with a computationally tractable use case, we reduce the size of the problem
by eliminating a subset of less severe scenarios. We first drop all the MEOW maps corresponding to
four directions (west-south-west, north, north-north-east, and northeast) as hurricanes belonging to
these categories do not cause significant flooding in the Texas Gulf Coast. We also drop the MEOW
maps corresponding to category 0-4 hurricanes. The storms belonging to category 5 are more intense
versions of these storms. Hence, the model ends up recommending decisions to prepare for worst-case
situations (and thus implicitly prepares for category 0-4 hurricanes as well). Moreover, as discussed
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A Survey of Relaxations and Approximations of the Power Flow Equations. D K Molzahn, I A Hiskens, Foundations and Trends in Electric Energy Systems. 41-2D. K. Molzahn and I. A. Hiskens. A Survey of Relaxations and Approximations of the Power Flow Equations. Foundations and Trends in Electric Energy Systems, 4(1-2):1-221, 2019.
Impacts of Approximate Power Flow Models on Optimal Flood Mitigation in a Stochastic Program. B Austgen, J J Hasenbein, E Kutanoglu, IIE Annual Conference Proceedings. B. Austgen, J.J. Hasenbein, and E. Kutanoglu. Impacts of Approximate Power Flow Models on Optimal Flood Mitigation in a Stochastic Program. In IIE Annual Conference Proceedings, pages 518-523, 2021.
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Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual. Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual. 2021.
The role of the SLOSH model in National Weather Service storm surge forecasting. B Glahn, A Taylor, N Kurkowski, W A Shaffer, National Weather Digest. B. Glahn, A. Taylor, N. Kurkowski, and W.A. Shaffer. The role of the SLOSH model in National Weather Service storm surge forecasting. National Weather Digest, pages 1-12.
A Scenario-based Optimization Approach for Electric Grid Substation Hardening Against Storm Surge Flooding. A Shukla, J J Hasenbein, E Kutanoglu, IIE Annual Conference Proceedings. A. Shukla, J.J. Hasenbein, and E. Kutanoglu. A Scenario-based Optimization Approach for Electric Grid Substation Hardening Against Storm Surge Flooding. In IIE Annual Conference Proceedings, pages 1004-1009, 2021.
Homeland Infrastructure Foundation-Level Data. Electric Substations. Available at. Homeland Infrastructure Foundation-Level Data. Electric Substations. Available at: https://hifld-geoplatform.opendata.arcgis.com/datasets/geoplatform::substations/about (last accessed on February 13, 2023).
Introduction to Stochastic Programming. R John, François Birge, Louveaux, Springer-VerlagNew York, NY, USAJohn R. Birge and François Louveaux. Introduction to Stochastic Programming. Springer- Verlag, New York, NY, USA, 1997.
Estimating the Value of Lost Load. ERCOT Resource AdequacyReportERCOT Resource Adequacy Report. Estimating the Value of Lost Load. Available at: https://www.ercot.com/gridinfo/resource/2013/ (last accessed on February 13, 2023.), 2013.
| {'fraction_non_alphanumeric': 0.03703463231570336, 'fraction_numerical': 0.013673726447558078, 'mean_word_length': 4.57276213908387, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 3, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We propose two scenario-based optimization models for power grid resilience decision making that integrate output from a hydrology model with a power flow model. The models are used to identify an optimal substation hardening strategy against potential flooding from storms for a given investment budget, which if implemented enhances the resilience of the power grid, minimizing the power demand that is shed. The same models can alternatively be used to determine the optimal budget that should be allocated for substation hardening when longterm forecasts of storm frequency and impact (specifically restoration times) are available. The two optimization models differ in terms of capturing risk attitude: one minimizes the average load shed for given scenario probabilities and the other minimizes the worst-case load shed without needing scenario probabilities. To demonstrate the efficacy of the proposed models, we further develop a case study for the Texas Gulf Coast using storm surge maps developed by the National Oceanic and Atmospheric Administration and a synthetic power grid for the state of Texas developed as part of an ARPA-E project. For a reasonable choice of parameters, we show that a scenario-based representation of uncertainty can offer a significant improvement in minimizing load shed as compared to using point estimates or average flood values. We further show that when the available investment budget is relatively high, solutions that minimize the worst-case load shed can offer several advantages as compared to solutions obtained from minimizing the average load shed. Lastly, we show that even for relatively low values of load loss and short post-hurricane power restoration times, it is optimal to make significant investments in substation hardening to deal with the storm surge considered in the NOAA flood scenarios.', 'arxivid': '2302.10408', 'author': ['Ashutosh Shukla ', 'Erhan Kutanoglu ', 'John J Hasenbein ', 'Ashutosh Shukla ', 'Erhan Kutanoglu ', 'John J Hasenbein '], 'authoraffiliation': [], 'corpusid': 257050384, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19047, 'n_tokens_neox': 17243, 'n_words': 12283, 'pdfsha': '922c55ff7bc9c6e54f693fd26edbddb2fff577c0', 'pdfurls': ['https://export.arxiv.org/pdf/2302.10408v1.pdf'], 'title': ['Scenario-based Optimization Models for Power Grid Resilience to Extreme Flooding Events', 'Scenario-based Optimization Models for Power Grid Resilience to Extreme Flooding Events', 'Scenario-based Optimization Models for Power Grid Resilience to Extreme Flooding Events', 'Scenario-based Optimization Models for Power Grid Resilience to Extreme Flooding Events'], 'venue': []} |
arxiv |
Finite sample forecasting with estimated temporally aggregated linear processes
Lyudmila Grigoryeva
Juan-Pablo Ortega
Finite sample forecasting with estimated temporally aggregated linear processes
linear modelsARMAtemporal aggregationforecastingfinite sample forecastingflow temporal aggregationstock temporal aggregationmultistep forecasting
We propose a finite sample based predictor for estimated linear one dimensional time series models and compute the associated total forecasting error. The expression for the error that we present takes into account the estimation error. Unlike existing solutions in the literature, our formulas require neither assumptions on the second order stationarity of the sample nor Monte Carlo simulations for their evaluation. This result is used to prove the pertinence of a new hybrid scheme that we put forward for the forecast of linear temporal aggregates. This novel strategy consists of carrying out the parameter estimation based on disaggregated data and the prediction based on the corresponding aggregated model and data. We show that in some instances this scheme has a better performance than the "all-disaggregated" approach presented as optimal in the literature.
Introduction
The success of parametric time series models as a tool of choice in many research fields is due in part to their good performance when it comes to empirical forecasting based on historical samples. Once a data generating process (DGP) has been selected and estimated for the forecasting problem at hand, there is a variety of well studied forecasting procedures and algorithms available in the literature. The most widespread loss function used in the construction of predictors is the mean square forecasting error (MSFE); see the monographs [Bro06,BD02,Ham94,L05] and references therein for detailed presentations of the available MSFE minimization-based techniques. This is the approach to prediction that we follow in this work; the reader is referred to [Gra69] or Section 4.2 in [GN86] for forecasting techniques based on other optimality criteria.
The stochastic nature of the time series models that we consider implies that the forecasts produced with them, carry in their wake an error that cannot be minimized even if the parameters of the model are known with total precision; we refer to this as the characteristic error of the model. Additionally, all that is known in most applications is a historical sample of the variable that needs to be forecasted, out of which a model needs to be selected and estimated. There are well-known techniques to implement this, which are also stochastic in nature and that increase the total error committed when computing a forecast; we talk in that case of model selection error and estimation error. All these errors that one incurs in at the time of carrying out a forecasting task are of different nature and much effort has been dedicated in the literature in order to quantify them in the case of linear multivariate VARMA processes.
Most results obtained in this direction have to do with the combination of the estimation and the characteristic errors; this compound error is always studied assuming independence between the realizations of the model that are used for estimation and the ones used for prediction; we refer the reader to [Bai79,Rei80,Yam80,Yam81,Duf85,BS86,L87,SH88]. Explicit expressions for these errors in the VARMA context are available in the monograph [L05]. Indeed, if we assume that the sample out of which we want to forecast is a realization of the unique stationary solution of a VAR model, this error can be written down [L05, page 97] using the time-independent autocovariance of the process; the situation in the VARMA context is more complicated and the expression provided [L05, page 490] requires Monte Carlo simulations for its estimation.
The knowledge regarding the error associated to model selection is much more rudimentary and research in this subject seems to be in a more primitive state. A good description of the state of the art can be found in [L06, page 318] as well as in [L86b, page 89], and references therein. We do not consider this source of forecasting error in our work and hence in the sequel we will use the denomination total error to refer to the combination of the characteristic with the estimation errors.
In this paper we concentrate on one dimensional linear processes, a subclass of which is the ARMA family. The first contribution in this paper is the formulation of a MSFE based predictor that takes as ingredients a finite sample and the coefficients of a linear model estimated on it, as well as the computation of the corresponding total error. The main improvements that we provide with respect to preexisting work on this question are:
• We make no hypothesis on the second order stationarity of the sample at hand; in other words, we do not assume that the sample is a realization of the stationary solution of the recursions that define the model. Such a hypothesis is extremely difficult to test in small and finite sample contexts and it is hence of much interest to be able to avoid it.
• The expression for the total forecasting error is completely explicit and does not require the use of Monte Carlo simulations.
The interplay between the characteristic error, the estimation errors, and the forecasting horizon is highly nonlinear and can produce surprising phenomena. For example, as it is well known, the characteristic error is an increasing function of the horizon, that is, the further into the future we forecast, the more error we are likely to commit. When we take into account the estimation error, the total error may decrease with the forecast horizon! We study this finite sample related phenomenon with the total error formula introduced in Theorem 3.3 and illustrate it with an example in Section 5.1. The characterization of the total forecasting error that we described serves as a basis for the second main theme of this paper, namely, the interplay between multistep forecasting, the prediction of temporal aggregates, and the use of temporal aggregation estimation based techniques to lower the total forecasting error. In this part of the paper we work strictly in the ARMA context. The temporal aggregation of ARMA processes is a venerable topic that is by now well understood [AW72, Tia72, Bre73, TW76, Wei79, SW86, Wei06, SV08] and has been extensively studied and exploited in the context of forecasting [Abr82, L86b, L86a, L87, L89a, L89b, RS95, L06, L09, L10] mainly by H. Lütkepohl. A recurrent question in this setup consists of determining the most efficient way to compute multistep forecasts or, more generally, predictions of linear temporal aggregates of a given time series. More specifically, given a sample and an underlying model, we can imagine at least two ways to construct a h time steps ahead forecast, or in general the one that is a linear combination of the h steps ahead values for the time series. First, we can simply compute the h time steps ahead forecasts of the time series out of the original disaggregated sample and to determine the needed aggregate prediction out of them; another possibility would be to temporally aggregate the sample and the time series model in such a way that the required forecast becomes a one time step ahead forecast for the new aggregated sample and model. If we do not take into consideration estimation errors and we only consider the characteristic error, there is a general result that states that the forecast based on high frequency disaggregate data has an associated error that is smaller or equal than the one associated to the aggregate sample and model (we will recall it in Proposition 4.2). In the VARMA context, H. Lütkepohl [L86b, L87, L09] has characterized the situations in which there is no loss of forecasting efficiency when working with temporally aggregated ingredients.
When estimation errors are taken into account, the inequality that we just described becomes strict [L86b, L87], that is, forecasts based on models estimated using the dissagregated high-frequency samples perform always better than those based on models estimated using aggregated data. This is so even in the situations described in [L86b, L87, L09] for which the characteristic errors associated to the use of the aggregated and the disaggregated models are identical; this is intuitively very reasonable due to the smaller sample size associated to the aggregated situation, which automatically causes an increase in the estimation error.
In Section 4.3 we propose a forecasting scheme that is a hybrid between the two strategies that we just described. We first use the high frequency data for estimating a model. Then, we temporally aggregate the data and the model and finally forecasting is carried out based on these two aggregated ingredients. We will show that this scheme presents two major advantages:
• The model parameters are estimated using all the information available with the bigger sample size provided by the disaggregated data. Moreover, these parameters can be updated as new high frequency data becomes available.
• In some situations, the total error committed using this hybrid forecasting scheme is smaller than the one associated to the forecast based on the disaggregated data and model and hence our strategy becomes optimal. Examples in this direction for both stock and flow temporal aggregates are presented in Section 5. The increase in performance obtained with our method comes from minimizing the estimation error; given that the contribution of this error to the total one for univariate time series models is usually small for sizeable samples, the differences in forecasting performance that we will observe in practice are moderate. As we will show in a forthcoming work, this is likely to be different in the multivariate setup where in many cases, the estimation error is the main source of error.
To our knowledge, this forecasting scheme has not been previously investigated in the literature and the improvement stated in the last point seems to be the first substantial application of temporal aggregation techniques in the enhancing of forecasting efficiency.
Finite sample forecasting of linear processes
In this section we introduce notations and definitions used throughout the paper and describe the framework in which we work. Additionally, since we are interested in finite sample based forecasting, we spell out in detail the predictors as well as the information sets on which our constructions are based.
Linear processes
Let ε = {ε t } ∞ t=−∞ be a set of independent and identically distributed random variables with mean zero and variance σ 2 . We will write in short
ε = {ε t } ∼ IID 0, σ 2 .
Finite sample forecasting with estimated temporally aggregated linear processes
We say that X = {X t } ∞ t=−∞ is a linear causal process whenever it can be represented as
X t = ∞ i=0 ψ i ε t−i , for all t ∈ Z, (2.1) where {ψ i } ∞ i=0
is a set of real constants such that ∞ i=0 |ψ i | < ∞. Expression (2.1) can also be rewritten as
X = Ψ (L) ε,
where L is the backward shift operator and Ψ (z) is the power series
Ψ (z) = ∞ i=0 ψ i z i . The process X defined in (2.1) is called invertible if there exist constants {π j } ∞ j=0 such that ∞ j=0 |π j | < ∞ and ε t = ∞ j=0 π j X t−j , for all t ∈ Z, (2.2) or equivalently, ε = Π (L) X, where Π (z) is the power series Π (z) = ∞ j=0 π j z j . Ψ (L)
and Π (L) can also be referred to as causal linear filter and invertible linear filter, respectively.
Finite sample forecasting of causal and invertible ARMA processes
Consider the causal and invertible ARMA(p, q) specification determined by the equivalent relations
Φ (L) X t = Θ (L) ε t , X t = ∞ i=0 ψ i ε t−i , ε t = ∞ j=0 π j X t−j .
(2.
3)
The innovations process ε = {ε t } can be either independent and identically distributed IID(0, σ 2 ) or white noise WN(0, σ 2 ). In this subsection we focus on how to forecast out of a finite sample ξ T = {x 1 , ..., x T } that satisfies the relations (2.3) and that has been generated out of a presample {x 1−p , ..., x 0 } and a preinnovations set {ε 1−q , ..., ε 0 }. A standard way to solve this problem [Bro06,BD02] consists of assuming that ξ T is a realization of the unique stationary process X that satisfies the ARMA relations (2.3) and to use its corresponding time independent autocovariance functions to formulate a linear system of equations whose solution provides the linear projection X T +h of the random variable X T +h onto {x 1 , ..., x T } using the L 2 norm; this projection X T +h minimizes the mean square error. We recall that writing the unique stationary solution of (2.3) usually requires knowledge about the infinite past history of the process. For example, for an AR(1) model of the form X t − φX t−1 = ε t , the unique stationary solution is given by
X t = ∞ i=0 φ i ε t−i .
Given that we are concentrating in the finite sample context, we prefer for this reason to avoid the stationarity hypothesis and the use of the corresponding autocovariance functions and to exploit in the forecast only the information that is strictly available, that is:
(i) The model specification (2.3): we assume that the model parameters are known with certainty and we neglect estimation errors.
(ii) The sample ξ T = {x 1 , ..., x T }.
(iii) The presample {x 1−p , ..., x 0 } and preinnovations {ε 1−q , ..., ε 0 } that have been used in the sample generation.
We now define the preset I as
I := {x 1−p , ..., x 0 } ∪ {ε 1−q , .
.., ε 0 } , when p, q = 0 {x 1−p , ..., x 0 } , when q = 0 {ε 1−q , ..., ε 0 } , when p = 0.
Let r = max {p, q} and define the enlarged preset I * as
I * := {x 1−r , ..., x 0 } ∪ {ε 1−r , ..., ε 0 } ,
where:
• if p > q: r = p and ε t := t+p−1 j=0
π j X t−j , 1 − p ≤ t < 1 − q; • if q > p: r = q and x t = t+q−1 i=0 ψ i ε t−i , 1 − q ≤ t < 1 − p;
• if q = p: I = I * .
The enlarged preset I * is defined as a function of the elements in I. Consequently, the sigma-algebras σ(I) and σ(I * ) generated by I and I * , respectively, coincide, that is, σ(I) = σ(I * ). The following result is basically known (see for example [Ham94,L05]) but we include it in order to be explicit and self-contained about the forecasting scheme that we are using in the rest of the paper and also to spell out the peculiarities of the finite sample setup in which we are working. We include a brief proof in the appendix.
Proposition 2.1 In the conditions that we just described:
(i) The information sets (sigma algebras) σ T := σ (I, ε 1 , ..., ε T ) and σ ξ T := σ (I, x 1 , ..., x T ) generated by the preset and the past histories of the innovations T := {ε 1 , . . . , ε T } and the sample
ξ T := {x 1 , . . . , x T } coincide, that is, σ (ξ T ) = σ ( T ) . (2.4) (ii)
If the innovations process is IID (respectively WN) then the optimal multistep forecast X T +h (respectively optimal linear forecast) based on σ (ξ T ) that minimizes the mean square forecasting error (MSFE) E X T +h − X T +h 2 is:
X T +h = T +h−1+r i=h ψ i ε T +h−i = T −1+r i=0 ψ i+h ε T −i = T −1+r i=0 T −i−1+r j=0 ψ i+h π j X T −i−j . (2.5) (iii)
The MSFE associated to this forecast is
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 .
(2.6) (iv) For ARMA models, the forecasts constructed in (2.5) for different horizons with respect to the same information set F T := σ (ξ T ) = σ ( T ), satisfy the following recursive formula:
X T +h = φ 1 X T +h−1 + ... + φ p X T +h−p + θ h ε T + ... + θ q ε T +h−q , q ≥ h, φ 1 X T +h−1 + ... + φ p X T +h−p , q < h. (2.7)
Remark 2.2 Testing the stationarity of small or in general finite samples is a difficult task in practice. We emphasize that the prediction in Proposition 2.1 does not require any stationarity hypothesis. Moreover, we underline that the forecast (2.5) does not coincide in general neither with the standard linear forecast for second order stationary series that uses the corresponding time independent autocovariance function (see for example [Bro06], page 63), nor with the usual finite sample approximation to the optimal forecast (see [Ham94], page 85). The main difference with the latter consists of the fact that the innovations associated to the presample are not assumed to be equal to zero but they are reconstructed out of it so that there is no loss of information.
In the examples 2.3 and 2.4 below we show how our forecast allows us to construct a predictor that:
(i) is different from the one obtained assuming stationarity;
(ii) has a better performance in terms of characteristic forecasting error.
These statements do not generalize to arbitrary ARMA models; for example, for pure AR models, the predictor that we propose and those cited above coincide.
Example 2.3 Finite sample forecasting for the MA(1) process. We consider the MA(1) model X t = ε t + θε t−1 (2.8) and the trivial sample consisting of just one value x 1 at time t = 1; this sample is generated by the preset I = {ε 0 } and the innovation ε 1 . In this case, the enlarged preset I * = {x 0 , ε 0 } with x 0 = ε 0 . Moreover, we have
• ψ 0 = 1, ψ 1 = θ and ψ i = 0, for any integer i > 1,
• π 0 = 1, π j = (−1) j θ j , for any integer j ≥ 1.
Consequently by (2.5), the forecast X 2 based on the information set F 1 = σ ({I, x 1 }), is given by
X 2 = θε 1 = θ (x 1 − θx 0 ) = θx 1 − θ 2 ε 0 ,
and has the associated error MSFE( X 2 ) = σ 2 .
On the other hand, the forecast that assumes that x 1 is a realization of the unique stationary solution of (2.8) and that uses the corresponding autocovariance function [Bro06, page 63] is given by
X S 2 := θ 1 + θ 2 x 1 ,
and has the associated error
MSFE( X S 2 ) = σ 2 1 + θ 2 − σ 2 θ 2 1 + θ 2 . We note that MSFE( X S 2 ) = σ 2 1 + θ 2 − θ 2 1 + θ 2 > σ 2 = MSFE( X 2 ),
which shows that the forecast that we propose has a better performance than the one based on the stationarity hypothesis.
The better performance of the forecast that we propose in the preceding example can be in part due to the fact that we are using for X 2 additional information on the preinnovations that is not taken advantage of at the time of writing X S 2 . In the following example we consider an ARMA(1,1) model and we see that the statements of Remark 2.2 also hold, even though in this case, unlike in the MA(1) situation, the information sets on which the two forecasts considered are based are identical.
Example 2.4 Finite sample forecasting for the ARMA(1,1) process. Consider the model
X t − φX t−1 = ε t + θε t−1 .
Then,
• π 0 = 1, π j = (−1) j (φ + θ) θ j−1 , for any integer j ≥ 1,
• ψ 0 = 1, ψ i = (φ + θ) φ i−1 , for any integer i ≥ 1.
We consider the trivial sample x 1 generated by the preset I = {x 0 , ε 0 } = I * . Using Proposition 2.1, we have that the one-step ahead forecast X 2 based on the information set F 1 = σ ({I, x 1 }), is given by
X 2 = (φ + θ) x 1 − θ (φ + θ) x 0 , with MSFE( X 2 ) = σ 2 .
On the other hand, the forecast based on the stationarity hypothesis using the same information set, yields
X S 2 := θ 2 + φθ + 1 (θ + φ) (φθ + 1) (θ 2 + θφ + 1) 2 − θ 2 x 1 − (θ + φ) (θφ + 1) θ (θ 2 + θφ + 1) 2 − θ 2 x 0 , and MSFE( X S 2 ) = θ 2 + φθ + 1 θ 4 + θ 3 φ + θφ + 1 (θ 2 + θφ + 1) 2 − θ 2 σ 2 .
It is easy to check that the statement MSFE( X S 2 ) > MSFE( X 2 ) is equivalent to θ 4 (θ + φ) 2 > 0, which is always satisfied and shows that the forecast that we propose has a better performance than the one based on the stationarity hypothesis.
Forecasting with estimated linear processes
In Proposition 2.1 we studied forecasting when the parameters of the model are known with total precision. In this section we explore a more general situation in which the parameters are estimated out of a sample. Our goal is to quantify the joint mean square forecasting error that comes both from the stochastic nature of the model (characteristic error) and the estimation error; we will refer to this aggregation of errors as the total error. This problem has been extensively studied in the references cited in the introduction always using the following two main constituents:
• Estimation and the forecasting are carried out using independent realizations of the time series model.
• The model parameter estimator is assumed to be asymptotically normal (for example, the maximum likelihood estimator); this hypothesis is combined with the use of the so called Delta Method [Ser80] in order to come up with precise expressions for the total error.
The most detailed formulas for the total error in the VARMA context can be found in [L05] where the Delta Method is applied to the forecast considered as a smooth function of the model parameters. If we assume that the sample out of which we want to forecast is a realization of the unique stationary solution of a VAR model, an explicit expression for this error can be written down by following this approach [L05, page 97] that involves the time-independent autocovariance of the process. In the VARMA setup, the situation is more complicated [L05, page 490] and the resulting formula requires the use of a Monte Carlo estimation.
In subsection 3.1 we start by obtaining a formula for the total error using a different approach at the time of invoking the Delta Method; our strategy uses this method at a more primitive level by considering the parameters of the linear representation of the process seen as a function of the ARMA coefficients. We show that discarding higher order terms on 1/ √ T , where T is the sample size used for estimation, the resulting formula for the total error can be approximated by a completely explicit expression that involves only the model parameters and the covariance matrix associated to the asymptotically normal estimator of the ARMA coefficients.
In subsection 3.2 we rederive the total error formula by H. Lütkepohl [L05, page 490] and show that it can be rewritten as explicitly as ours without using any stationarity hypothesis or Monte Carlo simulations. Moreover, we show that this formula coincides with the approximated one obtained in subsection 3.1 by discarding higher order terms on 1/ √ T .
The total error of finite sample based forecasting
Consider the causal and invertible ARMA(p, q) process {X t } determined by the equivalent relations
Φ (L) X t = Θ (L) ε t , X t = ∞ i=0 ψ i ε t−i , ε t = ∞ j=0 π j X t−j , {ε t } ∼ IID(0, σ 2 ), (3.1)
and denote Ψ := {ψ 0 , ψ 1 , . . .}, Π := {π 0 , π 1 , . . .}. In Proposition 2.1 we studied forecasting for the process (3.1) when the parameters Ψ or Π of the model are known with total precision; in this section we suppose that these parameters are estimated by using a sample independent from the one that will be used for forecasting. A more preferable assumption would have been that the parameters Ψ are estimated based on the same sample that we intend to use for prediction, but exploiting only data up to the forecasting origin; Samaranayake [SH88] and Basu et al [BS86] have shown that many results obtained in the presence of the independence hypothesis remain valid under this more reasonable assumption. Under the independence hypothesis, the model coefficients Ψ or Π become random variables independent from the process X and the innovations ε. Moreover, we assume that these random variables are asymptotically normal, as for example in the case of maximum likelihood estimation of the ARMA coefficients.
For the sake of completeness, we start by recalling the Delta Method, that will be used profusely in the following pages. A proof and related asymptotic statements can be found in [Ser80].
Lemma 3.1 (Delta Method) Let β be an asymptotically normal estimator for the vector parameter β ∈ R n , that is, there exists a covariance matrix Σ such that
√ T β − β dist − −−− → T →∞ N (0, Σ),
where T is the sample size. Let f : R n → R m be a vector valued continuously differentiable function and let J f be its Jacobian matrix, that is,
((J f )(β)) ij := (∂f i /∂β j )(β). If J f (β) = 0, then √ T f ( β) − f (β) dist − −−− → T →∞ N (0, J f ΣJ f ).
The next ingredient needed in the formulation of the main result of this section is the covariance matrix Σ Ξ P associated to the asymptotic normal character of the estimator Ξ P for the parameters Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ), for some integer P . This is spelled out in the following lemma whose proof is a straightforward combination of the Delta Method with the results in Section 8.8 of [Bro06].
Lemma 3.2 Let {X t } be a causal and invertible ARMA(p,q) process like in (3.1). Let Φ := (φ 1 , . . . , φ p ) , Θ := (θ 1 , . . . , θ q ) , and β := Φ , Θ be the ARMA parameter vectors and let Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ) be a collection of length 2P of the parameters that provide the linear causal and invertible representations of that model. Then:
(i) The maximum likelihood estimator β of β is asymptotically normal √ T β − β dist − − → N (0, Σ β ), with Σ β = σ 2 E [U t U t ] E [U t V t ] E [V t U t ] E [V t V t ] −1 , where U t := (U t , . . . , U t+1−p ) , V t := (V t , . . . , V t+1−q ) , and {U t } and {V t } are the autoregressive processes determined by Φ(L)U t = ε t and Θ(L)V t = ε t .
(ii) Consider the elements in Ξ P as a function of β, that is, Ξ P (β) := (ψ 1 (β), . . . , ψ P (β), π 1 (β), . . . , π P (β)).
Then, by the Delta Method we have that
√ T Ξ P − Ξ P d −−→ N (0, Σ Ξ P ), where Σ Ξ P := J Ξ P Σ β J Ξ P (3.2) and (J Ξ P ) ij = ∂(Ξ P ) i ∂β j , i = 1, . . . , 2P , j = 1, . . . , p + q.
Details on how to algorithmically compute the Jacobian J Ξ P are provided in Appendix 7.2.
The next theorem is the main result in this section. Its proof can be found in the appendix.
Theorem 3.3 Let ξ T = {x 1 , . .
. , x T } be a sample obtained as a realization of the causal and invertible ARMA(p,q) model in (3.1) using a preset I. In order to forecast out of this sample, we first estimate the parameters of the model Ψ = ψ 0 , ψ 1 , . . . , Π = { π 0 , π 1 , . . . } based on another sample ξ T that we assume to be independent of ξ T , using the maximum likelihood estimator β := Φ , Θ of the ARMA parameters. If we use the forecasting scheme introduced in Proposition 2.1, then:
(i) The optimal multistep forecast X T +h for X T +h based on the information set F T generated by the sample ξ T and using the coefficients estimated on the independent sample ξ T is
X T +h = T +h−1+r i=h ψ iεT +h−i , (3.3)
where r = max{p, q} andε t := t+r−1 j=0
π j x t−j .
(ii) The mean square forecasting error (MSFE) associated to this forecast is
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 + σ 2 P i=h ψ i 2 − 2 P i=h P −i j=0 P −i−j k=0 ψ i+j+k ψ k E ψ i π j + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ i+j+k,i +j +k , (3.4) where P = T + h − 1 + r.
The first summand will be referred to as the characteristic forecasting error and the second one as the estimation based forecasting error. Notice that the characteristic error coincides with (2.6) and amounts to the forecasting error committed when the model parameters are known with the total precision.
(iii) Let Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ) with P = T + h − 1 + r. Using the notation introduced in Lemma 3.2 and discarding higher order terms in 1/ √ T , the MSFE in (3.4) can be approximated by
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 + σ 2 1 T P i=h (Σ Ξ P ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ P ) i+j+k,j+P + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P,j +P δ i+j+k,i +j +k , (3.5)
where Σ Ξ P is the covariance matrix in (3.2).
On Lütkepohl's formula for the total forecasting error
As we already pointed out, H. Lütkepohl [L05, pages 97 and 490] has proposed formulas for VARMA models similar to the ones presented in Theorem 3.3 based on a different application of the Delta Method.
In this section, we rederive Lütkepohl's result in the ARMA context and show that it is identical to the approximated formula (3.5) presented in part (iii) of Theorem 3.3. In passing, this conveys that Lütkepohl's result can be explicitly formulated and computed using neither stationarity hypotheses nor Monte Carlo simulations.
The key idea behind Lütkepohl's formula is applying the Delta Method by thinking of the forecast X T +h in question as a differentiable function X T +h (β) of the model parameters β := (Φ , Θ ) . In order to develop further this idea, consider first the information sets F T := σ(ξ T ) and F T := σ(ξ T ) generated by two independent samples ξ T and ξ T of the same size. The sample ξ T is used for forecasting and hence F T determines the forecast X T +h (β) once the model parameters β have been specified. The sample ξ T is in turn used for model estimation and hence F T determines β. Consequently, the random variable X T +h ( β) is fully determined by F T and F T . In this setup, a straightforward application of the statement in Lemma 3.1 shows that
√ T X T +h β − X T +h (β) | F T dist − −−− → T →∞ N 0, ∂ X T +h ∂β Σ β ∂ X T +h ∂β , (3.6)
which, as presented in the next result is enough to compute the total forecasting error.
Theorem 3.4 In the same setup as in Theorem 3.3, the total error associated to the forecast in (3.3) can be approximated by
MSFE X T +h = σ 2 h−1 i=0 ψ i 2 + 1 T E ∂ X T +h ∂β Σ β ∂ X T +h ∂β . (3.7)
We refer to this expression as Lütkepohl's formula for the total forecasting error. Moreover:
(i) Lutkepohl's formula coincides with the approximate expression for the total error stated in (3.5). In particular, the second summand in Lütkepohl's formula, which describes the contribution to the total error given by the estimation error, can be expressed as:
Ω(h) :=E ∂ X T +h ∂β Σ β ∂ X T +h ∂β = σ 2 P i=h (Σ Ξ P ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ P ) i+j+k,j+P + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P,j +P δ i+j+k,i +j +k .
(3.8)
(ii) If we assume that the samples used for forecasting are second order stationary realizations of the model (3.1) and γ : Z → R is the corresponding time invariant autocovariance function, then the estimation error can be expressed as:
1 T Ω(h) = 1 T P i=h P −i j=0 P i =h P −i j =0 π j π j (Σ Ξ P ) i,i + 2π j ψ i (Σ Ξ P ) i,j +P + ψ i ψ i (Σ Ξ P ) j+P,j +P γ(i+j −i −j ).
(3.9)
Finite sample forecasting of temporally aggregated linear processes
The goal of this section is proposing a forecasting scheme for temporal aggregates based on using high frequency data for estimation purposes and the corresponding temporally aggregated model and data for the forecasting task. We show, using the formulas introduced in the previous section, that in some occasions this strategy can yield forecasts of superior quality than those based exclusively on high frequency data that are presented in the literature as the best performing option [L86b, L87, L09]. We start by recalling general statements about temporal aggregation that we need in the sequel. We then proceed by using various extensions of the results in Section 3 regarding the computation of total forecasting errors with estimated series in order to compare the performances of the schemes that we just indicated.
Temporal aggregation of time series
The linear temporal aggregation of time series requires the use of the elements provided in the following definition.
Definition 4.1 Given K ∈ N, X a time series, and w = (w 1 , ..., w K ) ∈ R K , define the K-period projection p K of X as
p K (X) := X (K) j j∈Z ∈ j∈Z R K , where X (K) j := X (j−i)K+1 , .
.., X jK ∈ R K , and the corresponding temporally aggregated time series Y as
Y := I w • p K (X) , (4.1) where I w : j∈Z R K −→ j∈Z R, j∈Z v j −→ j∈Z < w, v j > .
The integer K is called the temporal aggregation period and the vector w the temporal aggregation vector. Notice that the aggregated time series Y is indexed using the time scale τ = mK, with m ∈ Z and its components are given by the X-aggregates X w t+K defined by
X w t+K := w 1 X t+1 + ... + w K X t+K = Y τ . (4.2)
Definition 4.1 can be reformulated in terms of the backward shift operator L as:
Y = Π K • K−1 i=0 w K−i L i (X) , where Π K : j∈Z R −→ j∈Z R, (X j ) j∈Z −→ (X Kj+1 ) j∈Z , (4.3)
and the indices of the components (Z j ) j∈Z of Z :=
K−1 i=0 w K−i L i (X) are uniquely determined by the choice Z 1 := w 1 X 1 + ... + w K X K .
There are four important particular cases covered by Definition 4.1, namely:
(i) Stock aggregation (also called systematic sampling, skip-sampling, point-in-time sampling): it is obtained out of (4.1) or (4.2) by setting w = (0, 0, ..., 0, 1) .
(ii) Flow aggregation: w = (1, 1, ..., 1) .
(iii) Averaging: w = (1/K, 1/K, ..., 1/K) .
(iv) Weighted averaging:
w = 1 K (ξ 1 , ..., ξ K ) such that ξ 1 + ... + ξ K = 1.
Multistep approach to the forecasting of linear temporal aggregates
Let X be a time series and w = (w 1 , ..., w K ) an K-period aggregation vector. Given a finite time realization ξ T = {x 1 , ..., x T } of X such that T = M K with M ∈ N, we aim at forecasting the aggregate w 1 X T +1 + ... + w K X T +K . There are two obvious ways to carry this out; first, we can produce a multistep forecast X T +1 , ..., X T +K for X out of which we can obtain the forecast of the aggregate by setting X w T +K := w 1 X T +1 + ... + w K X T +K . Second, we can temporally aggregate X using (4.1) into the time series Y given by
Y = I w • p K (X)
and produce a one-step forecast for Y . The following result recalls a well known comparison [AW72, L84, L86b, L89a] of the forecasting performances of the two schemes that we just described using the mean square characteristic error as an optimality criterion. In that setup, given an information set encoded as a σ-algebra, the optimal forecast is given by the conditional expectation with respect to it [Ham94, page 72]. Given a time series X, we will denote in what follows by σ X T the information set generated by a realization ξ T = {x 1 , . . . , x T } of length T of X and the preset I used to produce it; more specifically
σ X T := σ(I ∪ {x 1 , . . . , x T }).
Proposition 4.2 Let X be a time series and w = (w 1 , ..., w K ) a K-period aggregation vector. Let Y = I w • p K (X) be the corresponding temporally aggregated time series. Let T = M K with M, T ∈ N and consider F T = σ X T , F M = σ Y M the information sets associated to two histories of X and Y of length T and M , respectively, related to each other by temporal aggregation. Then:
MSFE E X w T +K |F T ≤ MSFE (E [Y M +1 |F M ]) . (4.4)
Remark 4.3 The inequality (4.4) has been studied in detail in the VARMA context by H. Lütkepohl [L86b, L87, L09] who has fully characterized the situations in which the two predictors are identical and hence have exactly the same performance. This condition is stated and exploited in Section 5, where we illustrate with specific examples the performance of the forecasting scheme that we present in the following pages.
In the next two results we spell out the characteristic and the total errors associated to a multistep approach to the forecast of linear aggregates. The characteristic error is given in Proposition 4.4 and the total error is provided in Theorem 4.7 under the same independence hypothesis between the samples used for estimation and forecasting that were already invoked in Theorem 3.3.
Proposition 4.4 Let X be a time series model as in (3.1), r = max{p, q}, K a temporal aggregation period, w = (w 1 , ..., w K ) an aggregation vector, and F T := σ X T the information set generated by
a realization ξ T = {x 1 , . . . , x T } of length T of X. Let X w T +K be the forecast of X-aggregate X w T +K := K i=1
w i X T +i based on F T using the forecasting scheme in Proposition 2.1. Then: (i) The forecast X w T +K is given by:
X w T +K = K i=1 w i T +i−1+r j=i ψ j ε T +i−j . (4.5) (ii)
The corresponding mean square forecasting characteristic error is:
MSFE X w T +K = E X w T +K − X w T +K 2 = σ 2 K i=1 w 2 i i−1 l=0 ψ 2 l + 2 K−1 i=1 K j=i+1 w i w j i−1 l=0 ψ l ψ j−i+l .
(4.6)
Example 4.5 Forecast of stock temporal aggregates. It is a particular case of the statement in Proposition 4.4 obtained by taking w = (0, ..., 0, 1) . In this case
X w T +K = X T +K = T +K−1+r j=K ψ j ε T +K−j .
This shows that the forecast of the stock temporal aggregate coincides with the K-multistep forecast of the original time series. Consequently, it is easy to see by using (4.6) and (2.6) that MSFE X w T +K = MSFE X T +K . Example 4.6 Forecast of flow temporal aggregates. It is a particular case of the statement in Proposition 4.4 obtained by taking w = (1, ..., 1) . In this case
X w T +K = K i=1 T +i−1+r j=i ψ j ε T +i−j .
Consequently,
MSFE X w T +K = σ 2 K−1 j=0 (K − j) ψ 2 j + 2 K−1 i=1 K j=i+1 i−1 l=0 ψ l ψ j−i+l = σ 2 K−1 j=0 (K − j) ψ 2 j + 2 K−1 i=1 K−1 j=i (K − j) ψ j−i ψ j .
Theorem 4.7 (Multistep forecasting of linear temporal aggregates) Consider a sample ξ T = {x 1 , . . . , x T } obtained as the realization of a model of the type (3.1) using the preset I. In order to forecast out of this sample, we first estimate the parameters of the model Ψ = ψ 0 , ψ 1 , . . . and Π = { π 0 , π 1 , . . . } based on another sample ξ T of the same size that we assume to be independent of ξ T . Let w = (w 1 , . . . , w K ) be an aggregation vector and let X w T +K be the forecast of the aggregate X w T +K := K h=1 w h X T +h based on F T := σ (I ∪ ξ T ) using Proposition 4.4 and the estimated parameters Ψ, Π. Then:
(i) The optimal forecast X w T +K for X w T +K given the samples ξ T and ξ T is
X w T +K = K h=1 w h T +h−1+r j=h ψ jεT +h−j , (4.7)
where r = max{p, q} andε t = t+r−1 j=0
π j x t−j .
(ii) The mean square forecasting error associated to this forecast is
MSFE X w T +K = σ 2 < w, (A + B + C) w >, (4.8)
where A, B, C are the matrices with components given by
A hh = P (h) i=0 P (h ) i =0 ψ i ψ i δ h−i,h −i , (4.9) B hh = −2 P (h) l=0 P (h ) i=h P (h)−i j=0 P (h)−i−j k=0 ψ l ψ k E ψ i π j δ h−l,h −i−j−k , (4.10) C hh = P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ h−i−j−k,h −i −j −k , (4.11) with P (h) = T + h − 1 + r, P (h ) = T + h − 1 + r. Notice that A hh = A char hh + A res hh , where A char hh := h−1 i=0 h −1 i =0 ψ i ψ i δ h−i,h −i , A res hh := P (h) i=h P (h ) i =h ψ i ψ i δ h−i,h −i ,
and σ 2 < w, A char w > is the characteristic forecasting error in part (ii) of Proposition 4.4.
(iii) Let Ξ P := ψ 1 , . . . , ψ P (K) , π 1 , . . . , π P (K) , with P (K) = T + K − 1 + r. Using the notation introduced in Lemmas 3.1 and 3.2 and discarding higher order terms in 1/ √ T , the MSFE in (4.8) can be approximated by
MSFE X w T +K = σ 2 < w, A char + D + F + G w >, (4.12) where A char hh := h−1 i=0 h −1 i =0 ψ i ψ i δ h−i,h −i , D hh = 1 T P (h) i=h P (h ) i =h (Σ Ξ P ) i,i δ h−i,h −i , F hh = 2 T P (h) i=h P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k (Σ Ξ P ) i,P (K)+j δ h−i,h −i −j −k , G hh = 1 T P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P (K),j +P (K) δ h−i−j−k,h −i −j −k .
Remark 4.8 In order to compute the total error in (4.12) it is necessary to determine the covariance matrix Σ Ξ P . By Lemma 3.2, it can be obtained out of the covariance Σ β matrix associated to the estimator of the ARMA parameters combined with the Jacobian J Ξ P . Details on how to algorithmically compute this Jacobian are provided in Appendix 7.2.
Remark 4.9 Notice that all the matrices involved in the statement of Theorem 4.7 are symmetric except for B and F .
A hybrid forecasting scheme using aggregated time series models
In the previous subsection we presented a forecasting method for linear temporal aggregates based exclusively on the use of high frequency data and models. The performance of this approach has been compared in the literature [L86b, L87] with the scheme that consists of using models estimated using the aggregated low frequency data; as it could be expected due to the resulting smaller sample size, this method yields a performance that is strictly inferior to the one based on working in the pure high frequency setup.
In this section, we introduce and compute the performance of a hybrid recipe that consists of estimating first the model using the high frequency data so that we can take advantage of larger sample sizes and of the possibility of updating the model as new high frequency data become available. This model and the data used to estimate it are subsequently aggregated and used for forecasting. We will refer to this approach as the hybrid forecasting scheme. The main goal in the following pages is writing down explicitly the total MSFE associated to this forecasting strategy so that we can compare it using Theorem 4.7 with the one obtained with the method based exclusively on the use of high frequency data and models. In the next section we use the resulting formulas in order to prove that there are situations in which the hybrid forecasting scheme provides optimal performance for various kinds of temporal aggregation.
The main tool at the time of computing the MSFE associated to the hybrid scheme is again the use of the Delta Method [Ser80] in order to establish the asymptotic normality of the estimation scheme resulting from the combination of high frequency data with the subsequent model temporal aggregation.
In order to make this more explicit, consider a time series model X determined by the parameters β X for which an asymptotically normal estimator β X is available, that is, there exists a covariance matrix Σ βX such that
√ T β X − β X dist − −−− → T →∞ N (0, Σ βX )
with T being the sample size on which the estimation is based. Now, let K ∈ N be an aggregation period, w ∈ R K an aggregation vector, and Y := I w • p K (X) the linear temporally aggregated process corresponding to X and w.
Proposition 4.10 In the setup that we just described, suppose that the temporally aggregated process Y is also a parametric time series model and that the parameters β Y that define it can be expressed as a C 1 function β Y (β X ) of the parameters β X that determine X. Using the estimator β X , we can construct an estimator β Y for β Y based on disaggregated X samples by setting
β Y := β Y β X . Then √ T β Y − β Y dist − −−− → T →∞ N (0, Σ βY ) , (4.13)
where T is the disaggregated sample size and
Σ βY = J βY Σ βX J T βY , (4.14) with (J βY ) ij = ∂(β Y ) i ∂(β X ) j
the Jacobian matrix corresponding to the function β Y (β X ).
Once the model temporal aggregation function and its Jacobian have been determined, this proposition can be used to formulate an analog of Theorem 4.7 for the hybrid forecasting scheme by mimicking the proof of Theorem 3.3; the only necessary modification consists of replacing the asymptotic covariance matrix Σ βX of the estimator for the disaggregated model by that of the aggregated model Σ βY obtained using Proposition 4.10.
We make this statement explicit in the following theorem and then describe how to compute the model aggregation function β Y (β X ) and its Jacobian J βY in order to make it fully functional. The construction of these objects is carried out in the ARMA context where the model aggregation question has already been fully studied. Even though all necessary details will be provided later on in the section, all we need to know at this stage in order to state the theorem is that the linear temporal aggregation of an ARMA(p,q) model is another ARMA(p, q * ) model where
q * := K (p + 1) + q − p − K * K , (4.15)
K is the temporal aggregation period, K * is the index of the first nonzero entry of the aggregation vector, and the symbol · denotes the integer part of its argument. We emphasize that if the innovations that drive the disaggregated model are independent with variance σ 2 , this is not necessarily the case for the resulting aggregated model, whose innovations may be only uncorrelated with a different variance σ 2 * , making it into a so called weak ARMA model. We forecast the value of the temporal aggregate Y M +1 = X w T +K out of the sample η M by first estimating the parameters β X of the model X using another disaggregated sample ξ T of the same size, that we assume to be independent of ξ T . Let β Y (β X ) be the function that relates the ARMA parameter values of the disaggregated and the aggregated model and let J βY be its Jacobian. Consider the ARMA(p, q * ) model, with q * as in (4.15), determined by the parameters β Y := β Y ( β X ). Then:
(i) The optimal forecast Y M +1 of the temporal aggregate Y M +1 = X w T +K based on the information set F M := σ (I K ∪ η M ) using Proposition 4.4 and the estimated parameters β Y is given by
Y M +1 = T +r * j=1 ψ jεT +h−j ,(4.
16)
where I K is the preset obtained out of the temporal aggregation of I, r * = max{p, q * }, and ε t := t+r * −1 j=0 π j y t−j , with Ψ = ψ 0 , ψ 1 , . . . and Π = { π 0 , π 1 , . . . } the parameters corresponding to the causal and invertible representations of the temporally aggregated ARMA model with parameters β Y .
(ii) Let Ξ P := (ψ 1 , . . . , ψ P , π 1 , . . . , π P ) with P = T + r * . Discarding higher order terms in 1/ √ T , the MSFE corresponding to the forecast (4.16) can be approximated by
MSFE Y M +1 = σ 2 * + σ 2 * 1 T P i=h (Σ Ξ P ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ P ) i+j+k,j+P + P i=1 P −i j=0 P −i−j k=0 P i =1 P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P,j +P δ i+j+k,i +j +k , (4.17)
where σ 2 * is the variance of the innovations of the aggregated ARMA(p, q * ) model Y and Σ Ξ P is the covariance matrix given by
Σ Ξ P = J Ξ P Σ βY J Ξ P = J Ξ P J βY Σ βX J T βY J Ξ P ,
with J βY the Jacobian matrix corresponding to the function β Y (β X ) and J Ξ P the Jacobian of
Ξ P (β) := (ψ 1 (β Y ), . . . , ψ P (β Y ), π 1 (β Y ), . . . , π P (β Y )) .
As we announced above, we conclude this section by describing in detail the parameters aggregation function β Y (β X ) and its Jacobian J βY , so that all the ingredients necessary to apply formula (4.17) are available to the reader. In order to provide explicit expressions regarding these two elements, we provide a brief review containing strictly the results on the temporal aggregation of ARMA processes that are necessary for our discussion; for more ample discussions about this topic we refer the reader to [AW72, Tia72, Bre73, TW76, Wei79, SW86, Wei06, SV08] and references therein.
The temporal aggregation function β Y (β X ). Consider the ARMA(p,q) model Φ (L) X = Θ (L) ε, where Φ (L) = 1 − p i=1 φ i L i and Θ (L) = 1 + q i=1 θ i L i .
In order to simplify the discussion and to avoid hidden periodicity phenomena, we place ourselves in a generic situation in which the autoregressive and moving average polynomials of the model that we want to temporally aggregate have no common roots and all roots are different (see [Wei06] for the general case). Consider now a K-period aggregation vector w = (w 1 , ..., w K ) . Our first goal is to find polynomials T (z) and Φ * (z) that satisfy
T (L) Φ (L) = Φ * L K • Π K • K i=1 w i L i ,(4.18)
with Π K : j∈Z R −→ j∈Z R as in (4.3). The intuition behind (4.18) is that for any time series X, its temporal aggregation
Y = Π K • K i=1 w i L i (X) satisfies T (L) Φ (L) X = Φ * L K Y. (4.19)
In other words, the polynomial T (L), that we will call the temporal aggregation polynomial transforms the AR polynomial for X into an AR polynomial for Y in the aggregated time scale units. Let
T (L) = t 0 + t 1 L + ... + t n L n and Φ * L K = 1 − φ * 1 L K − ... − φ * c L Kc be the unknown polynomials. Equation (4.18) can be written in matrix form as [Bre73]:
t 0 0 0 . . . 0 t 1 t 0 0 . . . 0 t 2 t 1 t 0 . . . 0 . . . . . . . . . . . . . . . t p t p−1 t p−2 . . . t 0 . . . . . . . . . . . . . . . t n t n−1 t n−2 . . . t n−p 0 t n t n−1 . . . t n−p−1 0 0 t n . . . t n−p−2 . . . . . . . . . . . . . . . 0 0 0 . . . t n A 1 −φ 1 . . . −φ p Z = w −φ * 1 w . . . . . . . . . . . . . . . . . . −φ * c w D (4.20)
where w = (w K , w K−1 , ..., w 1 ) is the reflection of w. We start by determining the unknown values n and c using the two following dimensional restrictions:
• Since A is a matrix of size (n + p + 1) × (p + 1), Z and D are vectors of size p + 1 and cK + K, respectively, and we have AZ = D then, necessarily n + p + 1 = cK + K.
(4.21)
• The system AZ = D contains n + p + 1 equations that need to coincide with the number of unknowns, that is, the n + 1 + c coefficients (t 0 , t 1 , . . . , t n ) and (φ * 1 , . . . , φ * c ) of the polynomials T (z) and Φ * (z), respectively. Consequently, n + p + 1 = n + c + 1.
(4.22)
The conditions (4.21) and (4.22) yield c = p, n = pK + K − p − 1 = (p + 1) (K − 1) .
(4.23)
The first condition in (4.23) shows that the autoregressive order does not change under temporal aggregation.
Let now K * ≤ K be the index of the first nonzero component in the vector w. This implies that w is a vector of the form w = (w K , w K−1 , ..., w K * , 0, ..., 0) with w K , w K−1 , . . . , w K * = 0. Since (4.20) is a matrix representation of the polynomial equality in (4.18), we hence have that deg (T (L) Φ (L)) = deg (D (L)), where D (L) is the polynomial associated to the vector in the right hand side of (4.20). It is clear that deg (D (L)) = K (p + 1) − K * ; as deg (T (L) Φ (L)) = n + p, the degree n of T (L) is therefore n = K (p + 1) − p − K * .
(4.24)
Solving the polynomial equalities (4.20), we have found polynomials T (L) and Φ * L K such that the temporally aggregated time series Y satisfies
Φ * L K Y = T (L) Φ (L) X (4.25)
Our goal now is showing the existence of a polynomial Θ * L K and a white noise {ε * } ∼ WN 0, σ 2 * such that T (L) Θ (L) ε lK = Θ * L K ε * lK for any l ∈ Z.
(4.26)
This equality, together with (4.25) shows that the temporally aggregated process Y out of the ARMA process X is a weak ARMA process, as it satisfies the relation
Φ * (B) Y = Θ * (B) ε * , {ε * } ∼ WN 0, σ 2 * , (4.27) where B = L K , Φ * (B)
is a polynomial of degree p, and Θ * (B) a polynomial of degree n + q K whose coefficients will be determined in the following paragraphs. We recall that the symbol · denotes the integer part of its argument. Indeed, by (4.24) we have that deg (T (L) Θ (L)) = K (p + 1)−p−K * +q = n + q. Additionally, by (4.25) The coefficients of the polynomial Θ * (B) are obtained by equating the autocovariance functions of the processes on both sides of (4.30) at lags 0, K, 2K, ..., q * K, which provide q * + 1 nonlinear equations that determine uniquely the q * + 1 unknowns corresponding to the coefficients θ * 1 , . . . , θ * q * of the polynomial Θ * (B) and the variance σ 2 * of the white noise of the aggregated model. In order to explicitly write down the equations that we just described, let us denote C (L) := T (L) Θ (L) as in (4.30) and set C (L) = n+q i=0 c i L i . Let now γ and Γ be the autocovariance functions of the MA (n + q) and MA (q * ) processes Consequently, the coefficients of the polynomial Θ * (B) and the variance σ 2 * are uniquely determined by the q * + 1 equations The equations (4.34) can be written down in matrix form, which is convenient later on at the time of spelling out the Jacobian of the aggregation function. Indeed, we can write:
T (L) Θ (L) ε = T (L) Φ (L) X = Φ * L K Y.V t = C (L) ε t and U τ = Θ * (B) ε * τ , respectively,(4.γ (i) = σ 2 ε T (L) Θ (L) S i T (L) Θ (L) = σ 2 ε C (L) S i C (L) , (4.35) Γ (i) = σ 2 * Θ * (B) S i Θ * (B) ,(4.36)
where the bars over the polynomials in the previous expressions denote the corresponding coefficient vectors, that is, given a polynomial q (x) = n i=1 a i x i , then q (x) = (a 0 , a 1 , . . . , a n ) . Additionally S i is the lower ith-shift matrix, that is, In conclusion, if we denote β X = (Φ, Θ) and β Y = (Φ * , Θ * ), the construction that we just examined shows that
(S i ) jl = δ j−l,i .β Y (β X ) = (Φ * (Φ) , Θ * (Φ, Θ)) . (4.38)
The function Φ * (Φ) is given by the solution of the polynomial equalities (4.20) and Θ * (Φ, Θ) by the coefficients (t 0 , t 1 , . . . , t n ) determined by (4.20) and the solutions of (4.37).
Example 4.12 Stock temporal aggregation of an ARMA(p,q) model. In this case, w = (0, . . . , 0, 1) and hence K * = K, n = p (K − 1), and q * = p (K − 1) + q K .
Example 4.13 Flow temporal aggregation of an ARMA(p,q) model. In this case, w = (1, . . . , 1) and
hence K * = 1, n = (p + 1) (K − 1), q * = (p + 1) (K − 1) + q K .
The Jacobian J βY of the temporal aggregation function β Y (β X ). The goal of the following paragraphs is the computation of the Jacobian J βY of the function β Y (β X ) = (Φ * (Φ) , Θ * (Φ, Θ)) in (4.38). We first compute the Jacobian of the function Φ * (Φ) by taking derivatives with respect to the components of the vector Φ on both sides of the equations (4.20) that determine Φ * (Φ). This results in the following p matrix equations that will be needed later on in the computation of the remaining blocks of the Jacobian. Given that there is no Θ dependence on the function Φ * (Φ), the (1, 2)-block of the Jacobian is a zero matrix of size p × q * . The remaining two blocks are computed by using the function Θ * (Φ, Θ) uniquely determined by the equations (4.37). Its derivatives are obtained out of a new set of equations resulting from the differentiation of both sides of this relation, namely,
∂t 0 ∂φ i 0 . . . 0 ∂t 1 ∂φ i ∂t 0 ∂φ i . . . 0 ∂t 2 ∂φ i ∂t 1 ∂φ i . . . 0 . . . . . . . . . . . . ∂t p ∂φ i ∂t p−1 ∂φ i . . . ∂t 0 ∂φ i . . . . . . . . . . . . ∂t n ∂φ i ∂t n−1 ∂φ i . . . ∂t n−p ∂φ i 0 ∂t n ∂φ i . . . ∂t n−p−1 ∂φ i 0 0 . . . ∂t n−2 ∂φ i . . . . . . . . . . . . 0 0 . . . ∂t n ∂φ i 1 −φ 1 −φ 2 . . . −φ p + t 0 0 . . . 0 t 1 t 0 . . . 0 t 2 t 1 . . . 0 . . . . . . . . . . . . t p t p−1 . . . t 0 . . . . . . . . . . . . t n t n−1 . . . t n−p 0 t n . . . t n−p−1 0 0 . . . t n−p−2 . . . . . . . . . . . . 0 0 . . . t n 0 − ∂φ 1 ∂φ i − ∂φ 2 ∂φ i . . . − ∂φ p ∂φ i = 0 −w ∂φ * 1 ∂φ i −w ∂φ * 2 ∂φ i . . . −w ∂φ * p ∂φ i ,σ 2 ε ∂T (L) ∂(β X ) i Θ (L) + T (L) ∂Θ (L) ∂(β X ) i S jK T (L) Θ (L) + T (L) Θ (L) S jK ∂T (L) ∂(β X ) i Θ (L) + T (L) ∂Θ (L) ∂(β X ) i = ∂σ 2 * ∂(β X ) i Θ * (B) S j Θ * (B) + σ 2 * ∂Θ * (B) ∂(β X ) i S jK Θ * (B) + Θ * (B) S j ∂Θ * (B) ∂(β X ) i ,
(4.41) j = 0, 1, . . . , q * , i = 1, . . . , p + q.
We recall that the entries of the vector ∂T (L) ∂φ i correspond to the values previously obtained in (4.40)
and that ∂T (L) ∂θ i = 0. Expression (4.41) provides (q * + 1) (p + q) equations that allow us to find the values of the (q * + 1) (p + q) unknowns
∂ Θ * (B) j ∂φ r , ∂ Θ * (B) j ∂θ s , ∂σ 2 * ∂φ r , ∂σ 2 * ∂θ s , j = 1, .
. . , q * , r = 1, . . . , p, s = 1, . . . , q.
(4.42)
5 Comparison of forecasting efficiencies. Examples.
In the previous section we proposed a new hybrid scheme for the forecasting of temporal aggregates coming from ARMA processes. We recall that this strategy consists of first using high frequency disaggregated data for estimating a model; then we temporally aggregate both the data and the model, and finally we forecast using these two ingredients. As we announced in the introduction, there are situations in which our strategy is optimal with respect to the total error, that is, the predictor constructed following this procedure performs better than the one based exclusively on high frequency data and the underlying disaggregated model. In this section we give a few examples of specific models for which our scheme provides optimal efficiency of prediction. Before we proceed, we introduce abbreviations for the various predictors that we will be working with:
(i) Temporally aggregated multistep predictor (TMS predictor): this is the denomination that we use for the forecast of the aggregate that is constructed out of the disaggregated data and the underlying disaggregated model estimated on them.
(ii) Temporally aggregated predictor (TA predictor): this is the forecast based on use of the temporally aggregated sample and a model estimated on it.
(iii) Hybrid predictor (H predictor): this is the predictor introduced in Section 4.3 whose performance is spelled out in Theorem 4.11. In this scheme, a first model is estimated on the disaggregated high frequency data sample, then the data and the model are temporally aggregated with an aggregation period that coincides with the forecasting horizon; finally, both the temporally aggregated model and the sample are used to produce a one-step ahead forecast that amounts to a prediction of the aggregate we are interested in.
(iv) Optimal hybrid predictor (OH predictor): this predictor is constructed by taking the multistep implementation of the H predictor that yields the smallest total error. More explicitly, suppose that the aggregate that we want to forecast involves K time steps; let {K 1 , . . . , K r } be the positive divisors of K and {C 1 , . . . , C r } the corresponding quotients, that is, K = K i C i for each i ∈ {1, . . . , r}. There are aggregation schemes (stock and flow for example) for which a K-temporal aggregate can be obtained as the aggregation of C i K i -temporal aggregates, for all i ∈ {1, . . . , r}. The total error associated to the forecasting of these aggregates using a multistep version of the H predictor obviously depends on the factor K i used. The OH predictor is the one associated to the factor K i that minimizes the total error.
As we already mentioned, the forecasting performance of the TMS predictor is always superior or equal than that of the TA predictor when we take into account only the characteristic error, and it is strictly superior when the total error is considered. In view of these results and given that the H and the OH predictors carry out the forecasting with temporally aggregated data, they are going to produce worse characteristic errors than their TMS counterpart; hence, the only way in which the H and OH predictors can be competitive in terms of total error performance is by sufficiently lowering the estimation error. In order to check that they indeed do so, we will place ourselves in situations that are particularly advantageous in this respect and will choose models for which the TMS and the TA predictors have identical characteristic errors and hence it is only the estimation error that makes a difference as to the total error. The linear models for which this coincidence of characteristic errors takes place have been identified in the works of H. Lütkepohl [L86b, L87, L09] via the following statement.
Theorem 5.1 (Lütkepohl) Let X t = ∞ i=0 ψ i ε t−i be a linear causal process and let w = (w 1 , . . . , w K ) ∈ R K be a K-period aggregation vector. Then the TMS and TA predictors for the K-temporal aggregate determined by w have identical associated characteristic errors if and only if the following identity holds:
K−1 i=0 w K−i L i Ψ (L) = ∞ j=0 K−1 i=0 w K−i ψ jK−i L jK K−1 j=0 j i=0 w K−i ψ j−i L j .
(5.1)
The equality (5.1) is satisfied for both stock (w = (0, . . . , 0, 1) ) and flow aggregation (w = (1, . . . , 1) ) if {X t } is a purely seasonal process with period K, that is,
X t = ∞ i=0 ψ iK ε t−iK . (5.2)
Given a specific model we want to compare the performances of the H and the TMS predictors for a variety of forecasting horizons. Given that condition (5.1) is different for each aggregation period K and cannot be solved simultaneously for several of them, we will content ourselves either with approximate solutions that are likely to produce very close H and TMS characteristic errors for several periods K or with exact solutions that provide exactly equal errors for only a prescribed aggregation period. The following points describe how we have constructed examples following the lines that we just indicated:
• We first choose the orders p and q of the disaggregated ARMA(p,q) model that we want to use as the basis for the example.
• We fix an aggregation period K and a number n of parameters ψ i for which the equation (5.1) will be solved. The choice of p and q imposes a minimal number n min = q − p + 1.
• We determine a vector Ψ * = (ψ 0 , ψ 1 , ..., ψ n−1 ) that consists of the n first components of the set Ψ = {ψ 0 , ψ 1 , ...} that satisfies condition (5.1). We emphasize that in general this condition does not determine uniquely the vector Ψ * and that arbitrary choices need to be made. The vector Ψ * is a truncation at order n − 1 of the MA representation of the ARMA process that we want to construct.
• We conclude the construction of the ARMA(p,q) model that we are after by designing either an AR(p) polynomial Φ consistent with causality or a MA(q) polynomial Θ consistent with invertibility. Then:
-In the first case, the required model is given by
Φ(L)X = Θ * (L)ε, with Θ * = Ψ * · Φ. (5.3)
-In the second case, the required model is given by
Φ * (L)X = Θ(L)ε, with Φ * = (Ψ * ) −1 · Θ. (5.4)
In both cases, the MA and AR polynomials that are obtained in this way have to be checked regarding invertibility and causality, respectively. Additionally, the finite truncation of Ψ is likely to give rise to common roots between the AR and MA polynomials in (5.3) or in (5.4) which may make necessary a slight perturbation of the coefficients in order to be avoided.
• We emphasize that the resulting ARMA model satisfies (5.1) only approximately and hence the characteristic errors of the two predictors will be not identical but just close to each other for the specific aggregation period K used. For pure MA models no truncation is necessary and hence exact equality can be achieved.
Stock aggregation examples
In the particular case of stock temporal aggregation, condition (5.1) is written as:
Ψ (L) = ∞ j=0 ψ jK L jK K−1 j=0 ψ j L j . (5.5)
We now consider the truncated vector Ψ * with n components, that is, Ψ * = (ψ 0 , . . . , ψ n−1 ) . Then, the truncated version of (5.5) is:
n−1 j=0 ψ j L j = (n−K+1)/K j=0 ψ jK L jK K−1 j=0 ψ j L j , (5.6)
where the symbol · denotes the integer part of its argument. We now provide a few examples of models whose specification is obtained following the approach proposed in the previous subsection and the relation (5.6).
Example 5.2 MA(10) model.
Let p = 0, q = 10, n = n min = 11 and let K = 2. Equation (5.6) becomes
10 j=0 ψ j L j = 5 j=0 ψ 2j L 2j 1 j=0 ψ j L j ,
which imposes the following relations:
ψ 0 = 1, ψ 1 ψ 2i = ψ 2i+1 , i = 0, . . . , 5; ψ i = 0 for i ≥ n.
This system of nonlinear equations has many solutions. We choose one of them by setting ψ j = 0, for j = 1, . . . , 9, and ψ 10 = 0.3. This way we can trivially determine a MA(10) model which satisfies exactly the relation (5.5) by taking θ j = 0 for j = 1, . . . , 9 and θ 10 = 0.3. Figure 1 shows the values of the characteristic errors for different values of the forecasting horizon for the TMS predictor, the H predictor, and the OH predictor. For the horizon h = 2, the values of the characteristic errors of all the predictors coincide, which is a consequence of the fact that the model has been constructed using the relation (5.5) with K = 2. Moreover, it is easy to see by looking at (5.2), that the particular choice of MA coefficients that we have adopted ensures that the resulting model is seasonal for the periods 2, 5, and 10; this guarantees that (5.5) is also satisfied for the corresponding values of K and hence there is coincidence for the characteristic errors at those horizons too.
The total errors for a sample size of T = 50 are then computed using the formulas presented in sections 3 and 4. The corresponding results are also plotted in Figure 1 for the different forecasting schemes. This plot shows that for several forecasting horizons both the H and the OH predictors perform better than the TMS predictor.
A quick inspection of this plot reveals another interesting phenomenon consisting on the decrease of the total error associated to the three predictors as the forecasting horizon increases; this feature is due to the decrease of the estimation error using these forecasting schemes as the horizon becomes longer. The characteristic errors for the H and OH predictors do not increase monotonically with the forecasting horizon either; however, in this case, this is due to the fact that for each value of the forecasting horizon, these predictors are constructed using a different model since the aggregation period changes and hence so does the aggregated model used for forecasting.
In conclusion, in this particular example, both the H and the OH predictors exhibit a better forecasting performance than the TMS predictor and, additionally, the results regarding the OH predictor help in making a decision on what is the best possible aggregation period to work with in order to minimize the associated total forecasting error. Figure 2 shows the errors with respect to the forecasting horizon for all the predictors as in the previous example. The H and the OH predictors perform better than the TMS predictor for h = 3, 6, 9, 10. Additionally the OH predictor performs better than the H predictor for h = 4. Taking into account the initial choice of K = 3 when constructing the example, it becomes clear why the characteristic errors associated to the H and the OH predictors are very close to those associated to the TMS predictor for horizons h that are multiples of 3. Figure 3 shows that the H and the OH predictors have equal associated total errors and exhibit a better forecasting efficiency than the TMS predictor for all forecasting horizons except at h = 2. The initial choice of K = 4 at the model construction stage results in the fact that for h = 4 the values of the characteristic errors associated to the H and the OH predictors are very close to the one committed by the TMS predictor.
Flow aggregation examples
In the particular case of flow temporal aggregation, condition (5.1) can be written as:
Ψ (L) K−1 i=0 L i = ∞ j=0 L jK K−1 i=0 ψ jK−i K−1 j=0 L j j i=0 ψ j−i . (5.7)
We now consider the truncation Ψ * of Ψ with n components, that is, Ψ * = (ψ 0 , . . . , ψ n−1 ). Then, the truncated version of (5.7) can be expressed as:
K−1 i=0 L i n−1 j=0 ψ j L j = (n−K+1)/K j=0 L jK K−1 i=0 ψ jK−i K−1 j=0 L j j i=0 ψ j−i , (5.8)
where symbol · denotes the integer part of its argument. We now provide a few examples of models whose specification is obtained following the approach described in the beginning of the section and the relation (5.8).
Example 5.5 MA(10) model.
Let p = 0, q = 10, n = n min = 11, and let K = 2. Then the expression (5.8) reads
1 i=0 L i 10 j=0 ψ j L j = 5 j=0 L 2j 1 i=0 ψ 2j−i 1 j=0 L j j i=0 ψ j−i ,
and consequently ψ 0 = 1, (1 + ψ 1 )(ψ i + ψ i+1 ) = ψ i+1 + ψ i+2 , for i = 1, . . . , 9, and ψ i = 0 for i ≥ n.
(5.9)
A solution for these equations is given by the choice ψ j = 0, for j = 1, . . . , 9, and ψ 10 = 0.3. Since the order of the AR polynomial is zero, the method that we proposed determines uniquely in this case the MA(10) polynomial that we are after with Θ = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0.3) . The evolution of the forecasting errors versus the forecasting horizon in plotted in the Figure 4. Both the H and the OH predictors perform better than the TMS predictor. For h = 4 the OH predictor has the smallest total error among the three predictors. Figure 5 shows the corresponding characteristic and total errors for the three predictors. The H and the OH predictors exhibit better performance than the TMS predictor for horizons h = 2, 4, 5, 6, 7. Remark 5.7 The model that we just presented can be used to illustrate the fact that the construction method that we presented in this sections is not the unique source of examples in which the H and the OH predictors perform better than the TMS scheme. Indeed, as it can be seen in Figure 6, the very same ARMA(3,10) model prescription used in the context of stock aggregation also shows this feature even though it has not been obtained by finding a solution of the equation (5.5).
Remark 5.8 Notice that when the forecasting horizon h equals one all predictors coincide because there is no temporal aggregation and hence they obviously have the same errors associated.
Conclusions
In this work we have carried out a detailed study of the total error committed when forecasting with one dimensional linear models by minimizing the mean square error. We have introduced a new hybrid scheme for the forecasting of linear temporal aggregates that in some situations shows optimal performance in comparison with other prediction strategies proposed in the literature. We work in a finite sample context. More specifically, the forecasting is based on the information set generated by a sample and a model whose parameters have been estimated on it and we avoid the use of second order stationarity hypotheses or the use of time independent autocovariance functions.
In this setup, we provide explicit expressions for the forecasting error that incorporate both the error incurred in due to the stochastic nature of the model (we call it characteristic error) as well as the one associated to the sample based estimation of the model parameters (estimation error). In order to derive these expressions we use certain independence and asymptotic normality hypotheses that are customary in the literature; our main contribution consists of providing expressions for the total error that do not require neither stationarity on the samples used nor Monte Carlo simulations to be evaluated. We subsequently use these formulas to evaluate the performance of a new forecasting strategy that we propose for the prediction of linear temporal aggregates; we call it hybrid scheme. This approach consists of using high frequency data for estimation purposes and the corresponding temporally aggregated data and model for forecasting. This scheme uses all the information available at the time of estimation by using the bigger sample size provided by the disaggregated data, and allows these parameters to be updated as new high frequency data become available. More importantly, as we illustrate with various examples, in some situations the total error committed using this scheme is smaller than the one associated to the forecast based on the disaggregated data and model; in those cases our strategy becomes optimal. As the increase in performance obtained with our method comes from minimizing the estimation error, we are persuaded that this approach to forecasting may prove very relevant in the multivariate setup where in many cases the estimation error is the main source of error. (ii) Suppose first that the innovations {ε t } are IID(0, σ 2 ). Then the forecast X T +h that minimizes the mean square forecasting error E X T +h − X T +h 2 is given by the conditional expectation (see for example [Ham94], page 72):
X T +h = E X T +h |σ ξ T = E X T +h |σ T = T +h−1+r i=0 ψ i E ε T +h−i |σ T = T +h−1+r i=h ψ i ε T +h−i = T −1+r i=0 ψ i+h ε T −i = T −1+r i=0 T −i−1+r j=0 ψ i+h π j X T −i−j ,
as required.
When {ε t } is WN(0, σ 2 ) our goal is finding the linear combination
T −1+r j=0 a j X T −j that minimizes E X T +h − T −1+r i=0 a i X T −i 2 .
Given that by (7.1), the elements X T −i can be written as a linear combination of the elements in T −i , this task is equivalent to finding the vector b = (b 0 , ..., b T −1+r ) that minimizes the function
S (b 0 , ..., b T −1+r ) = E X T +h − T −1+r i=0 b i ε T −i 2 = E T +h−1+r i=0 ψ i ε T +h−i − T −1+r i=0 b i ε T −i 2 = E h−1 i=0 ψ i ε T +h−i + T −1+r i=0 (ψ i+h − b i ) ε T −i 2 = σ 2 h−1 i=0 ψ 2 i + T −1+r i=0 (ψ i+h − b i ) 2 .
Hence, in order to minimize the function S (b 0 , ..., b T −1+r ) we compute the partial derivatives ∂S/∂b i and we set them to zero, which shows that the optimal values are attained when b i = ψ i+h . Consequently, the optimal linear forecast is given by X T +h = T −1+r i=0 ψ i+h ε T −i , as required.
(iii) We first compute X T +h − X T +h . By (2.5) and (7.1) we have
X T +h − X T +h = h−1 i=0 ψ i ε T +h−i . Therefore MSFE X T +h = E h−1 i=0 ψ i ε T +h−i 2 = σ 2 h−1 i=0 ψ 2 i .
(iv) Given the model Φ (L) X = Θ (L) ε, we have
X T +h − φ 1 X T +h−1 + ... + φ p X T +h−p = ε T +h + θ 1 ε T +h−1 + ... + θ q ε T +h−q .
We first recall that by (2.4) we have that σ ξ T = σ T =: F T . We now project both sides of this equality onto the information set F T by thinking of this σ-algebra as σ ξ T for the left hand side projection and as σ T for the right hand side. We obtain:
X T +h − φ 1 X T +h−1 + ... + φ p X T +h−p = E [ε T +h + θ 1 ε T +h−1 + ... + θ q ε T +h−q |F T )] = θ h ε T + ... + θ q ε T +h−q , q ≥ h 0, otherwise.
In the presence of white noise innovations, the conditional expectation in the previous equality should be replaced by a linear projection.
7.2 Computation of the Jacobian J Ξ P In this section we provide a simple algorithmic construction for the computation of the Jacobian J Ξ P when the elements in Ξ P are considered as a function of β, that is, Ξ P (β) := (ψ 1 (β), . . . , ψ P (β), π 1 (β), . . . , π P (β)).
We will separately compute the components ∂ψ i ∂β k and ∂π i ∂β k , i = 1, . . . , P , k = 1, . . . , p + q.
The causality and invertibility hypotheses on the ARMA process we are working with, guarantee that for any P ≥ max {p, q} there exist polynomials Ψ P (z), Π P (z) of order P uniquely determined by the relations:
Φ(z)Ψ P (z) = Θ(z), (7.2) Φ(z) = Π P (z)Θ(z), (7.3)
which are equivalent to
Ψ P (z) = Φ −1 (z)Θ(z), Π P (z) = Φ(z)Θ −1 (z).
These polynomial relations determine the functions Ψ P (Φ, Θ), Π P (Φ, Θ) needed in the computation of the Jacobian. We now rewrite (7.2) and (7.3) as Φ(z)Ψ P (Φ, Θ)(z) = Θ(z), (7.4) Φ(z) = Π P (Φ, Θ)(z)Θ(z).
(7.5)
If we take derivatives with respect to θ j and φ i , j ∈ {1, . . . , q}, i ∈ {1, . . . , p} on both sides of (7.4), we obtain a set of p + q polynomial equations:
Φ(z) ∂Ψ P (φ, θ) ∂θ j (z) = z j , j ∈ {1, . . . , q} , z i Ψ P (φ, θ) + Φ(z) ∂Ψ P (φ, θ) ∂φ i = 0, i ∈ {1, . . . , p} ,
that determine uniquely the corresponding entries of the Jacobian due to the invertibility of Φ(z). At the same time, taking derivatives on both the right and left hand sides of (7.5) with respect to θ j and φ i , we obtain another set of p + q polynomial equations ∂Φ ∂θ j (z) = ∂Π P (φ, θ) ∂θ j (z) + Π P z j , j ∈ {1, . . . , q} ,
z i = ∂Π P (φ, θ) ∂φ i , i ∈ {1, . . . , p} ,
that determine uniquely the corresponding entries of the Jacobian due to the invertibility of Θ(z).
Proof of Theorem 3.3
(i) It is a straightforward consequence of the independence hypothesis between the samples ξ T and the ξ T , and part (ii) of Proposition 2.1.
(ii) By (3.1) and part (i) of Proposition 2.1 we have that
MSFE X T +h = E X t+h − X t+h 2 = E P i=0 ψ i ε T +h−i − P i=h ψ iεT +h−i 2 .
In order to compute this error notice thatε T +h−i can be rewritten in terms of the original innovations asε
T +h−i = P −i j=0 π j x T +h−i−j = P −i j=0 P −i−j k=0 π j ψ k ε T +h−i−j−k . (7.6)
Hence,
MSFE X T +h = E P i=0 ψ i ε T +h−i − P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = E P i=0 ψ i ε T +h−i 2 − 2 P l=0 P i=h P −i j=0 P −i−j k=0 ψ l ψ i π j ψ k ε T +h−l ε T +h−i−j−k + P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = σ 2 P i=h ψ i 2 − 2 P l=0 P i=h P −i j=0 P −i−j k=0 ψ l ψ k E ψ i π j δ l,i+j+k + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ i+j+k,i +j +k + σ 2 h−1 i=0 ψ i 2 .
(iii) By part (i) the forecast X T +h is given by
X T +h = P i=h ψ iεT +h−i . (7.7)
According to the statement (3.2), both ψ i and π j can be asymptotically written as
ψ i = ψ i + r i √ T , π j = π j + t j √ T ,
with r i and t j as Gaussian random variables of mean 0 and variances (Σ Ξ ) i,i and (Σ Ξ ) j+P,j+P , respectively. Consequently by (7.7) and (7.6)
X T +h = P i=h ψ iεT +h−i = P i=h P −i j=0 P −i−j k=0 ψ i + r i √ T π j + t j √ T ψ k ε T +h−i−j−k = P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k + ψ i t j ψ k √ T + r i π j ψ k √ T + r i t j ψ k T ε T +h−i−j−k .
We now recall that
P −i j=0 P −i−j k=0 π j ψ k ε T +h−i−j−k = ε T +h−i ,
and we eliminate in this expression the term that decays as 1/T; we hence approximate X T +h as
X T +h P i=h ψ i + r i √ T ε T +h−i + P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k = P i=h ψ i ε T +h−i + P i=h P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k .
Using this approximation we compute now the MSFE:
MSFE X T +h = E X t+h − X t+h 2 = E P i=0 ψ i ε T +h−i − P i=h ψ i ε T +h−i − P i=h P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = E h−1 i=0 ψ i ε T +h−i + P i=h ψ i − ψ i ε T +h−i − P i=h P −i j=0 P −i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = σ 2 h−1 i=0 ψ i 2 + σ 2 1 T P i=h (Σ Ξ ) i,i + 2 P i=h P −i j=0 P −i−j k=0 ψ i ψ k (Σ Ξ ) i+j+k,j+P + P i=h P −i j=0 P −i−j k=0 P i =h P −i j =0 P −i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ ) j+P,j +P δ i+j+k,i +j +k .
Proof of Theorem 3.4
The mean square error associated to the forecast X T +h ( β) carried out using estimated parameters β is given by:
MSFE X T +h ( β) = E X T +h − X T +h ( β) 2 = E X T +h − X T +h (β) + X T +h (β) − X T +h ( β) 2 = E X T +h − X T +h (β) 2 + 2E X T +h − X T +h (β) X T +h (β) − X T +h ( β) + E X T +h (β) − X T +h ( β) 2 .
(7.8)
We now recall that
X T +h (β) = P i=h ψ i ε T +h−i , with P = T + h − 1 + r,
and notice that
E X T +h − X T +h (β) X T +h (β) − X T +h ( β) = E h−1 j=0 ψ j ε T +h−j P i=h ψ i ε T +h−i − ψ iεT +h−i = E h−1 j=0 ψ j ε T +h−j P i=h ψ i ε T +h−i − P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k = 0,
since the first term in the product involves the innovations {ε T +1 , . . . , ε T +h } and the second one {ε 1−r , . . . , ε T }; these two sets are disjoint and hence independent. Consequently by (2.6) and (7.8) we have
MSFE X T +h ( β) = σ 2 h−1 i=0 ψ 2 i + E X T +h (β) − X T +h ( β) 2 .
The second summand of this expression can be asymptotically evaluated using (3.6). Indeed, ∂ψ k ∂β j π l + ψ k ∂π l ∂β j ∂ψ m ∂β i π n + ψ m ∂π n ∂β i (Σ β ) ij E [X T +h−k−l X T +h−m−n ] . ((Σ Ξ P ) k,m π l π n + (Σ Ξ P ) k,n+p π l ψ m +(Σ Ξ P ) m,l+p ψ k π n + (Σ Ξ P ) l+p,n+p ψ k ψ m ) ψ u ψ v δ k+l+u,m+n+v .
E X T +h (β) − X T +h ( β) 2 = E E X T +h (β) − X T +h ( β) 2 |F T = 1 T E [Ω(h)] = 1 T E p+q i,j=1 J i J j (Σ β ) ij ,(7.
(7.12)
The required identity (3.8) follows directly from (7.12) by noticing that π n ψ v δ k,m+n+v = δ k,m .
Proof of Proposition 4.4
First, we notice that by (4.2) we have F M ⊂ F T .
The same relation guarantees that X w T +k = Y M +1 . Hence the result is a consequence of the following general fact:
Lemma 7.1 Let z be a random variable in the probability space (Ω, P, F). Let F * be a sub-sigma algebra of F , that is, F * ⊂ F. Then
E (z − E [z|F * ]) 2 ≥ E (z − E [
Proof of Proposition 4.4
Part (i) is a straightforward consequence of (2.5). Regarding (ii), we first have that
X w T +K − X w T +K = K i=1 w i i−1 j=0 ψ j ε T +i−j .
Consequently,
MSFE X w T +K = E K i=1 K j=1 w i w j i−1 l=0 j−1 m=0 ψ l ψ m ε T +i−l ε T +j−m = σ 2 K i=1 w 2 i i−1 l=0 ψ 2 l + 2 K−1 i=1 K j=i+1 w i w j i−1 l=0 ψ l ψ j−i+l .
7.7 Proof of Theorem 4.7
(i) It is a straightforward consequence of part (i) in Theorem 3.3.
(ii) By (4.7) we have that
MSFE X w T +K = E X w t+K − X w t+K 2 = E K h=1 w h P (h) i=0 ψ i ε T +h−i − P (h) i=h ψ iεT +h−i 2 ,
where P (h) = T + h − 1 + r. We now notice that
ε T +h−i = P (h)−i j=0 P (h)−i−j k=0
π j ψ k ε T +h−i−j−k .
(7.14)
Hence,
MSFE X w T +K = E K h=1 w h P (h) i=0 ψ i ε T +h−i − K h=1 w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = E K h=1 w h P (h) i=0 ψ i ε T +h−i 2 − 2 K h=1 K h =1 w h w h P (h ) l=0 P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ l ψ i π j ψ k ε T +h −l ε T +h−i−j−k + K h=1 w h P i=h P −i j=0 P −i−j k=0 ψ i π j ψ k ε T +h−i−j−k 2 = E K h=1 K h =1 w h w h P (h) i=0 P (h ) i =0 ψ i ψ i ε T +h−i ε T +h −i 2 (7.15) − 2σ 2 K h=1 K h =1 w h w h P (h) l=0 P (h ) i=h P (h)−i j=0 P (h)−i−j k=0 ψ l ψ k E ψ i π j δ h−l,h −i−j−k + E K h=1 K h =1 w h w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0
ψ k ψ k ψ i π j ψ i π j ε T +h−i−j−k ε T +h −i −j −k = σ 2 < w, (A + B + C)w >, (7.16)
where A, B, C are the matrices with components given by
A hh = P (h) i=0 P (h ) i =0 ψ i ψ i δ h−i,h −i , (7.17) B hh = −2 P (h) l=0 P (h ) i=h P (h)−i j=0 P (h)−i−j k=0
ψ l ψ k E ψ i π j δ h−l,h −i−j−k , (7.18)
C hh = P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ k ψ k E ψ i π j ψ i π j δ h−i−j−k,h −i −j −k ,(7.19)
and P (h) = T + h − 1 + r, P (h ) = T + h − 1 + r.
(iii) We recall that by (4.7), the forecast X w T +K is given by
X w T +K = K h =1 w h P (h) i=h ψ iεT +h−i .
According to (3.2), both ψ i and π j can be asymptotically written as
ψ i = ψ i + r i √ T and π j = π j + t j √ T ,
with r i and t j Gaussian random variables of mean 0 and variances (Σ Ξ P ) i,i and (Σ Ξ P ) j+P (K),j+P (K) , respectively. Consequently,
X w T +K = K h=1 w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i + r i √ T π j + t j √ T ψ k ε T +h−i−j−k = K h=1 w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i π j ψ k + ψ i t j ψ k √ T + r i π j ψ k √ T + r i t j ψ k T ε T +h−i−j−k .
(7.20)
We now recall that
P (h)−i j=0 P (h)−i−j k=0 π j ψ k ε T +h−i−j−k = ε T +h−i ,
and we eliminate in (7.20) the term that decays as 1/T. We hence approximate X w T +K as
X w T +K K h=1 w h P (h) i=h ψ i + r i √ T ε T +h−i + P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k = K h=1 w h P (h) i=h ψ i ε T +h−i + P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k .
(7.21)
Using this approximation we now compute the MSFE:
MSFE X w T +K = E X w T +K − X w T +K 2 = E K h=1 w h P (h) i=0 ψ i ε T +h−i − P (h) i=h ψ i ε T +h−i − P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = E K h=1 w h h−1 i=0 ψ i ε T +h−i + P (h) i=h ψ i − ψ i ε T +h−i − P (h) i=h P (h)−i j=0 P (h)−i−j k=0 ψ i ψ k t j √ T ε T +h−i−j−k 2 = E K h=1 K h =1 w h w h h−1 i=0 h −1 i =0 ψ i ψ i ε T +h−i ε T +h −i + E K h=1 K h =1 w h w h P (h) i=h P (h ) i =h ψ i − ψ i ψ i − ψ i ε T +h−i ε T +h −i + E K h=1 K h =1 w h w h P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ k t j t j T ε T +h−i−j−k ε T +h −i −j −k + 2E K h=1 K h =1 w h w h P (h) i=h P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ i − ψ i ( π j − π j ) ε T +h−i ε T +h −i −j −k .
Using Lemma 3.2, this expression can be asymptotically approximated by:
MSFE X w T +K = σ 2 < w, A char + D + F + G w >,
where A char , D, F , G are matrices whose components are given by
A char hh := h−1 i=0 h −1 i =0 ψ i ψ i δ h−i,h −i , D hh = 1 T P (h) i=h P (h ) i =h (Σ Ξ P ) i,i δ h−i,h −i , F hh = 2 T P (h) i=h P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k (Σ Ξ P ) i,P (K)+j δ h−i,h −i −j −k , G hh = 1 T P (h) i=h P (h)−i j=0 P (h)−i−j k=0 P (h ) i =h P (h )−i j =0 P (h )−i −j k =0 ψ i ψ k ψ i ψ k (Σ Ξ P ) j+P (K),j +P (K) δ h−i−j−k,h −i −j −k .
Theorem 4.11 (Hybrid forecasting of linear temporal aggregates) Let ξ T = {x 1 , . . . , x T } be a sample obtained as a realization of a causal and invertible ARMA(p,q) model X as in (3.1). Let w = (w 1 , . . . , w K ) be a temporal aggregation vector such that T = M K, for some M ∈ N, and let Y = I w • p K (X) be the temporal aggregation of the model X and η M := {y 1 , . . . , y M } the temporal aggregated sample obtained out of ξ T .
right hand side of this expression only involves time steps that are integer multiples of K, the relation (4.30) only imposes requirements on the left hand side at those time steps. Moreover, it is easy to see thatE [(T (L) Θ (L) ε lK ) (T (L) Θ (L) ε lK+jK )] = 0, (4.29) for any Kj > K (p + 1) + q − p − K * . This implies that the process is {T (L) Θ (L) ε lK } l∈Z is (K (p + 1) + q − p − K * )-correlated,which guarantees in turn by [Bro06, Section 3.2] the existence of a weak MA(q * ) representation T (L) Θ (L) ε lK = Θ * (B) ε * lK , l ∈ Z, (4.30) where deg (Θ * (B)) = K (p + 1) + q − p − K * K := q * . (4.31)
l c l+jK = σ 2 * q * −j l=0θ * l θ * l+j , j = 0, 1, . . . , q * .(4.34)
For any given vectorv = (v 1 , . . . , v n ) , S i v = (0, . . . , 0 i , v 1 , .. . , v n−i ) . With this notation, the equations (4.34) can be rewritten asσ 2 ε T (L) Θ (L) S jK T (L) Θ (L) = σ 2 * Θ * (B) S j Θ * (B) , j = 0, 1, . . . , q * .(4.37)
= 1, ..., p. These equations uniquely determine the (1, 1)-block ∂Φ * ∂Φ = ∂φ * i ∂φ j i,j of the Jacobian J βY , as well as the derivatives d ij := ∂t i ∂φ j (4.40)
Figure 1 :
1Characteristic and total errors associated to the forecast of the temporal stock aggregate of the MA(10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5. Example 5.3 ARMA(3,11) model.Let p = 3, q = 11, n = 10 and let K = 3. In this case, the relation (5.ψ j ψ 3i = ψ 3i+1 , i = 0, . . . , 2, j = 1, 2, ψ i = 0 for i ≥ n − 1.We choose a solution for these relations of the form Ψ * = (1, −0.9, 0.8, 0, 0, 0, −0.7, 0.63, −0.56, 0) . We now introduce an AR(3) polynomial of the form Φ = (−0.9, 0.8, −0.4) . We then determine the MA(11) part of the model by using(5.3), which yields the coefficients Θ =(−1.8, 2.41, −1.84, 1, −0.32, −0.7, 1.26, −1.687, 1.288, −0.7, 0.224). In order to avoid the common roots between the AR and the MA polynomials that are obtained when the coefficients of the MA part are derived in this manner, we slightly perturb the values of some of the components of the vector Θ that we now set to be Θ = (−1.8, 2.4102, −1.8403, 1, −0.32, −0.7, 1.26, −1.687, 1.288, −0.7, 0.224) .
Figure 2 :
2Characteristic and total errors associated to the forecast of the temporal stock aggregate of the ARMA(3,11) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5. Example 5.4 ARMA(1,4) model.Let p = 1, q = 4, n = 5 and let K = 4. In this setup, relation (5.L j , and consequently ψ 0 = 1 and ψ 4 = 0, necessarily, while the values of the coefficients ψ 1 , ψ 2 , and ψ 3 are not subjected to any constraint. We hence set Ψ *= (1, 0.3, −0.3, 0.3, 0). We now introduce the AR(1) polynomial determined by the coefficient Φ = 0.8. We then determine the MA(4) part of the model by using (5.3) which yields Θ = (−0.5, −0.54, 0.54, −0.24) . Again in order to avoid common roots between the AR and the MA polynomials, we perturb the polynomial Θ by setting: Θ = (−0.5, −0.5403, 0.54, −0.24) .
Figure 3 :
3Characteristic and total errors associated to the forecast of the temporal stock aggregate of the ARMA(1,4) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5.
Figure 4 :
4Characteristic and total errors associated to the forecast of the temporal flow aggregate of the MA(10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5. Example 5.6 ARMA(3,10) model. In the previous example we chose ψ 10 = 0. Let us now use another solution of the system (5.9) in order to obtain another model with target orders p = 3 and q = 10. If we set ψ 10 = 0, then a possible solution is Ψ * = (1, −0.5, 0.45, −0.475, 0.3, −0.3875, 0.1, −0.2438, 0, 0, 0) . Let now Φ = (0.21, 0.207, 0.0162) be a causal AR(3) polynomial which determines via (5.3) the MA(10) polynomial Θ = (−0.71, 0.348, −0.4822, 0.3147, −0.3595, 0.1270, −0.1894, 0.0368, 0.0488, 0.0039) . In order to avoid common roots for the AR and MA polynomials, we perturb the MA coefficients and set Θ = (−0.71, 0.3481, −0.4823, 0.3148, −0.3595, 0.1270, −0.1894, 0.0368, 0.0488, 0.0039) .
Figure 5 :
5Characteristic and total errors associated to the forecast of the temporal flow aggregate of the ARMA(3,10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5.
Figure 6 :
6Characteristic and total errors associated to the forecast of the temporal stock aggregate of the ARMA(3,10) model. The sample size used for estimation is T = 50. The innovations of the model have variance σ 2 = 5.
It is a straightforward consequence of the causality and invertibility hypotheses on the ARMA model that we are dealing with. Indeed, we can writeε t = t−1+r j=0 π j X t−j and X t = t−1+r i=0 ψ i ε t−i ,(7.1) which proves (2.4).
presence of the stationarity hypothesis in part (ii) of the theorem we have thatE [X T +h−k−l X T +h−m−n ] = γ(k + l − m − n)and hence (3.9) follows. Otherwise, since we have in general thatE [X t X s ] ψ k ψ m (Σ β ) ij ψ u ψ v δ k+l+u,m+n+v
l ψ u δ k+l+u,m+n+v ) = δ k,m+n+v , l ψ u δ k+l+u,m+n+v ) π n ψ v =
Proof of Lemma 7.1E (z − E [z|F * ]) 2 = E (z − E [z|F * ] − E [z|F] + E [z|F]) 2 = E (z − E [z|F]) + E (E [z|F] − E [z|F * ]) 2 + 2E [(z − E [z|F]) (E [z|F] − E [z|F * ])] . that E (z − E [z|F * ]) 2 ≥ 0,the inequality (7.13) follows if we show that E [(z − E [z|F]) (E [z|F] − E [z|F * ])] = 0. E [(z − E [z|F]) (E [z|F] − E [z|F * ]) |F] = E zE [z|F] − zE [z|F * ] − E [z|F] 2 + E [z|F] E [z|F * ] |F = E [z|F] 2 − E [z|F] E [z|F * ] − E [z|F] 2 + E [z|F] E [z|F * ] = 0.z|F])
2 .
(7.13)
2
Given Indeed,
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| {'fraction_non_alphanumeric': 0.08836605022228537, 'fraction_numerical': 0.027486246726619434, 'mean_word_length': 3.2028410545175325, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 261, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We propose a finite sample based predictor for estimated linear one dimensional time series models and compute the associated total forecasting error. The expression for the error that we present takes into account the estimation error. Unlike existing solutions in the literature, our formulas require neither assumptions on the second order stationarity of the sample nor Monte Carlo simulations for their evaluation. This result is used to prove the pertinence of a new hybrid scheme that we put forward for the forecast of linear temporal aggregates. This novel strategy consists of carrying out the parameter estimation based on disaggregated data and the prediction based on the corresponding aggregated model and data. We show that in some instances this scheme has a better performance than the "all-disaggregated" approach presented as optimal in the literature.', 'arxivid': '1209.4188', 'author': ['Lyudmila Grigoryeva ', 'Juan-Pablo Ortega '], 'authoraffiliation': [], 'corpusid': 88512739, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 35907, 'n_tokens_neox': 32278, 'n_words': 19279, 'pdfsha': 'eebcd0032850fc81359683c3bd17f815fa056f95', 'pdfurls': ['https://arxiv.org/pdf/1209.4188v1.pdf'], 'title': ['Finite sample forecasting with estimated temporally aggregated linear processes', 'Finite sample forecasting with estimated temporally aggregated linear processes'], 'venue': []} |
arxiv |
A Robust Circle-criterion Observer-based Estimator for Discrete-time Nonlinear Systems in the Presence of Sensor Attacks and Measurement Noise
19 Sep 2018
Tianci Yang
Carlos Murguia
Margreta Kuijper
Dragan Nešić
A Robust Circle-criterion Observer-based Estimator for Discrete-time Nonlinear Systems in the Presence of Sensor Attacks and Measurement Noise
19 Sep 2018
We address the problem of robust state estimation and attack isolation for a class of discrete-time nonlinear systems with positive-slope nonlinearities under (potentially unbounded) sensor attacks and measurement noise. We consider the case when a subset of sensors is subject to additive false data injection attacks. Using a bank of circle-criterion observers, each observer leading to an Input-to-State Stable (ISS) estimation error, we propose a estimator that provides robust estimates of the system state in spite of sensor attacks and measurement noise; and an algorithm for detecting and isolating sensor attacks. Our results make use of the ISS property of the observers to check whether the trajectories of observers are consistent with the attack-free trajectories of the system. Simulations results are presented to illustrate the performance of the results.{1, · · · , p}\J(i) is the set of sensors potentially under attack in the i-th time window, which is Algorithm 1.Algorithm 1 SENSOR ATTACKS ISOLATION Input: N , q ⋆ 1: Design a circle-criterion observer for each subset J ⊂ {1, · · · , p} with card(J) = p − q ⋆ and for each subset S ⊂ {1, · · · , p} with card(S) = p − 2q ⋆ . 2: We intialize the counter variable n J (i) = 0 for each J with card(J) = p − q ⋆ and for all i ∈ Z >0 . 3: for each i ∈ Z >0 do 4:for each k ∈ [1 + (i − 1)N, iN ] do 5: calculate π ⋆ J (k) for all J with card(J) = p − q ⋆ as follows:
I. INTRODUCTION
Networked Control Systems (NCSs) have emerged as a technology that combines control, communication, and computation and offers the necessary flexibility to meet new demands in distributed and large scale systems. Recently, security of NCSs has become an important issue as wireless communication networks might serve as new access points for attackers to adversely affect the operation of the system dynamics. Cyber-physical attacks on NCSs have caused substantial damage to a number of physical processes. One of the most well-known examples is the attack on Maroochy Shire Councils sewage control system in Queensland, Australia that happened in January 2000. The attacker hacked into the controllers that activate and deactivate valves and caused flooding of the grounds of a hotel, a park, and a river with a million liters of sewage. Another incident is the very recent SuxNet virus that targeted Siemens supervisory control and data acquisition systems which are used in many industrial processes. It follows that strategic mechanisms to identify and deal with attacks on NCSs are strongly needed.
In [1]- [22], a range of topics related to security of control systems have been discussed. In general, they provide analysis tools for quantifying the performance degradation induced by different classes of attacks; and propose reaction strategies to identify and counter their effect on the system dynamics. Most of the existing work, however, has considered control systems with linear dynamics, although in most engineering applications the dynamics of the plants being monitored [email protected] and controlled is highly nonlinear. There are some results addressing the nonlinear case though. In [23], exploiting sensor redundancy, the authors address the problem of sensor attack detection and state estimation for uniformly observable continuous-time nonlinear systems. Similarly, in [24], the authors provide an algorithm for isolating sensor attacks for a class of discrete-time nonlinear systems with bounded measurement noise.
In this manuscript, we consider the case when the system has p sensors, all of which are subject to measurement noise and up to q < p/2 of them are attacked. Following the results in [25] for linear systems, using a bank of circle criterion observers [26]- [29], each observer leading to an ISS estimation error, we construct an estimator that provides robust estimates of the system state in spite of sensor attacks. In particular, the proposed estimator leads to estimation errors satisfying an ISS property with respect to measurement noise but independent of attack signals. Next, we propose an algorithm for detecting and isolating false data injection sensor attacks. Our results make use of the ISS property of the observers to check whether the trajectories of observers are consistent with the attack-free trajectories of the system. The main idea behind our results is the following. Each observer in the bank is driven by a different subset of sensors. Thus, without attacks, the observers produce ISS estimation errors with respect to measurement noise only. For every pair of observers in the bank, we compute the largest difference between their estimates. If a pair of observers is driven by a subset of attack-free sensors, then the largest difference between their estimates is also ISS with respect to measurement noise only. However, if there are attacks on some of the sensors, the observers driven by those sensors might produce larger differences than the attack-free ones. These ideas work well under the assumption that less than p/2 sensors are attacked, i.e, q < p/2. To design the observers in the bank, we give an extension to the result in [28] for designing robust discrete-time circle-criterion observers. In particular, we use the incremental multiplier technique introduced in [29] to cast the observer design as the solution of a semidefinite program. We minimize the ISSgain from the measurement noise to the estimation error.
The paper is organized as follows. In Section II, we present preliminary results needed for the subsequent sections. In Section III, we provide tools for designing optimal robust circle criterion observers in the attack-free case. In Section IV, assuming that a sufficiently small number of sensors are subject to attacks, we propose an estimation scheme using a bank of robust circle criterion observers. In Section V, an algorithm for isolating sensor attacks is given. Finally, in Section VI, we give concluding remarks.
II. PRELIMINARIES
A. Notation
We denote the set of real numbers by R, the set of natural numbers by N , the set of integers by Z, and R n×m the set of n×m matrices for any m, n ∈ N. For any vector v ∈ R n , we denote v J the stacking of all v i , i ∈ J and J ⊂ {1, · · · , n},
|v| = √ v ⊤ v and supp(v) = {i ∈ {1, · · · , n} |v i = 0}. For a sequence of vectors {v(k)} ∞ k=0 , we denote v [0,k] a sequence of vectors v(i), i = 0, · · · , k, ||v|| ∞ sup k≥0 |v(k)| and ||v|| T sup 0≤k≤T |v(k)|.
We say a sequence {v(k)} ∈ l ∞ if ||v|| ∞ < ∞. We denote the cardinality of a set S as card(S). We denote matrix P to be positive definite as P > 0. The identity matrix is denoted by
I. A function β : R ≥0 × R ≥0 → R ≥0 is said to be of class exp − KL if there exist c > 0, λ ∈ (0, 1), such that β(s, k) = cλ k · s.
The binomial coefficient is denoted as a b , where a, b are nonnegative integers. We denote a variable m uniformly distributed in the interval (a, b) as m ∼ U(a, b).
B. Definitions and lemmas
Several definitions and lemmas that are important in this paper are introduced here.
Definition 1: [29] (Incremental Multiplier Matrices). Suppose f : R nq → R n f . A symmetric matrix M ∈ R (nq+n f )×(nq+n f ) is an incremental multiplier matrix (δM M ) for f if the following incremental quadratic constraint (δQC) is satisfied for all q 1 , q 2 ∈ R nq :
△q △f T M △q △f ≥ 0,(1)
where △q = q 1 − q 2 and △f = f (q 1 ) − f (q 2 ). Definition 2: Consider a discrete-time system
e + = F (e, m),(2)
with state e ∈ R n , the input m ∈ R p with {m(k)} ∈ l ∞ . The system is said to be input-to-state stable (ISS) with a linear gain γ and an exp − KL function if there exist c > 0, λ ∈ (0, 1), and γ ≥ 0 such that the following condition is satisfied:
|e(k)| ≤ cλ k |e(0)| + γ||m|| k ,(3)
for all e(0) ∈ R n , k ≥ 0, and {m(k)} ∈ l ∞ . The next lemma provides sufficient conditions for system (2) to be ISS with a linear gain γ and an exp − KL function. It is a special case of the lemma in [29]. Lemma 1: [29] Let c 1 , c 2 > 0, c 3 ∈ (0, 1) and µ 1 > 0. Suppose there exists V : R n → R ≥0 such that
c 1 |e| 2 ≤ V (e) ≤ c 2 |e| 2 ,(4)V (F (e, m)) − V (e) ≤ −c 3 V (e) + c 3 µ 1 |m| 2 ,(5)
for all k ≥ 0, e ∈ R n , m ∈ R p . Then the system (2) is ISS with a linear gain and an exp − KL function with respect to the bounded sequence {m(k)}, and
|e(k)| ≤ cλ k |e(0)| + γ||m|| k ,(6)
for all k ≥ 0, e(0) ∈ R n , and {m(k)} ∈ l ∞ , where c = c2 c1 , λ = √ 1 − c 3 , and γ = µ1 c1 .
III. A CIRCLE-CRITERION OBSERVER ROBUST TO
MEASUREMENT NOISE
In [28], Ibrir designs a discrete-time nonlinear observers through circle criterion, but no disturbance in the system is considered. Our goal in this section is to give an extension of the result given in [28] by taking measurement noise into consideration and propose a design method of a robust circle-criterion observer with respect to measurement noise in the absence of attack signals. The design method uses a similar idea as the one in [29] by characterizing the nonlinearity with an incremental multiplier matrix, but we present an extension of the results given in [29] by solving an optimization problem with more degrees of freedom, which leads to a less conservative ISS gain. More specifically, we show that in some circumstances our circle-criterion observer provides state estimates less sensitive to measurement noise (see Example 1). We consider a discrete-time nonlinear system formulated as:
x + =Ax + Gf (Hx) + ρ(u, y), y =Cx + m,(7)
where x ∈ R n is the state, y ∈ R ny is the sensor measurements, m ∈ R ny is the measurement noise with {m(k)} ∈ l ∞ and G ∈ R n×r , H ∈ R r×n . The term ρ(u, y) is a known arbitrary real-valued vector that depends on the system inputs and outputs. The state-dependent nonlinearity f (Hx) is an rdimensional vector where each entry is a function of a linear combination of the states
f i = f i n j=1 H ij x j , i = 1, · · · , r(8)
where H ij are the entries of matrix H.
Assumption 1:
For any i ∈ {1, · · · , r}, the following holds,
f i (v i ) − f i (w i ) v i − w i ≥ 0, ∀v i , w i ∈ R with v i = w i (9)
We consider a circle-criterion observer with the following structure:
x + = Ax + Gf (Hx + K(Cx − y)) + L(Cx − y) + ρ(u, y)(10)
wherex denotes the estimate of the state x, and K ∈ R r×ny , and L ∈ R n×ny are the observer gains to be designed. Then the error e =x − x has the following dynamics
e + = (A + LC)e − Lm + G△f,(11)
where
△f = f (q) − f (q),(12)whereq = Hx andq = Hx + K(ŷ − y) andŷ = Cx △q =q −q = (H + KC)e − Km.(13)
Our objective is to design the gains K and L, such that a quadratic Lyapunov function V (e) satisfies (4) and (5). Then we can show that the error dynamics of the observer is ISS with a linear gain and an exp − KL function with respect to the bounded measurement noise and (6) holds.
Proposition 1: Consider system (7), for given c 3 ∈ (0, 1), suppose there exist matrix P ∈ R n×n and P > 0, K ∈ R r×ny and Y ∈ R n×ny , an incremental multiplier matrix M for the nonlinearity f , and scalars µ > 0 and µ 1 > 0 that satisfy the matrix inequalities:
−P ⋆ Ξ 21 Ξ 22 + 0 0 0 Γ T M Γ ≤0, P I I µI ≥0,(14)
where
Ξ T 21 = P A + Y C −Y P G , Ξ 22 = (c 3 − 1)P 0 0 0 −c 3 µ 1 I 0 0 0 0 ,(15)
and
Γ = H + KC −K 0 0 0 I ,(16)
then the observer (10) characterized by gains L = P −1 Y and K has ISS error dynamics with a linear gain γ = √ µµ 1 and an exp − KL function with respect to m.
Proof: The proof of Proposition 1 can be obtained from the proof of Theorem 1 in [29] by letting H = I, B = 0, D = I, D q = 0 and adding a new variable µ 1 in Ξ 22 .
From Proposition 1 , we see that if we could solve (14) while minimizing √ µµ 1 , then the designed observer is robust to measurement noise. We take advantage of the results in [29] by using an incremental multiplier matrix to characterize the nonlinearity f in the design of a robust circle-criterion observer, but we do not fix µ 1 = 1 as a constant as [29] does. Hence, our observer could provide estimates more robust to measurement noise in some circumstances. From (9), we have
(q −q) ⊤ (f (q) − f (q)) ≥ 0.(17)
Recalling (12) and (13), we know △q ⊤ △f ≥ 0 ∀q ∈ R r and ∀q ∈ R r . Hence, any matrix
M = κ 0 1 1 0 ,(18)
with κ > 0 is an incremental multiplier matrix for f . The following linear matrix inequality is equivalent to (14).
Lemma 2: [29] For some matrix Y 2 ∈ R r×ny , consider the linear matrix inequality
−P ⋆ Ξ 21 Ξ 22 + 0 0 0 Γ T 1 M Γ 1 + Γ T 1 Γ 2 + Γ T 2 Γ 1 ≤ 0,(19)
where Ξ 21 , Ξ 22 are described in (15), and (19) and (14) are equivalent.
Γ 1 = H 0 0 0 0 I , Γ 2 = 0 0 0 Y 2 C −Y 2 0 , then with L = P −1 Y, K = Y 2 κ(20)Proof: Recalling (16), we let Γ = Γ 1 +Γ 2 whereΓ 2 = KC −K 0 0 0 0 . Note that MΓ 2 = 0 0 0 κKC κK 0 , with
K given by (20), we see that κK = Y 2 . Therefore,
MΓ 2 = 0 0 0 Y 2 C Y 2 0 = Γ 2 . As we haveΓ ⊤ 2 MΓ 2 = 0, thus Γ ⊤ M Γ =(Γ 1 +Γ 2 ) ⊤ M (Γ 1 +Γ 2 ) =Γ ⊤ 1 M Γ 1 + Γ ⊤ 1 MΓ 2 +Γ ⊤ 2 M Γ 1 +Γ ⊤ 2 MΓ 2 =Γ ⊤ 1 M Γ 1 + Γ ⊤ 1 Γ 2 + Γ ⊤ 2 Γ 1(21)
which implies that (19) and (14) are equivalent. The proof is complete. By replacing (14) with (19) in Proposition 1, we obtain the following result. Theorem 1: Consider the system (7), for given c 3 ∈ (0, 1), suppose there exist matrix P ∈ R n×n and P > 0, matrix
Y ∈ R n×ny , matrix Y 2 ∈ R r×ny , scalars κ > 0, µ > 0, µ 1 > 0 that satisfy the linear matrix inequalities −P ⋆ Ξ 21 Ξ 22 + 0 0 0 Γ T 1 M Γ 1 + Γ T 1 Γ 2 + Γ T 2 Γ 1 ≤0, P I I µI ≥0,(22)
where Ξ 21 , Ξ 22 are described in (15), M is given by (18),
and Γ 1 = H 0 0 0 0 I , Γ 2 = 0 0 0 Y 2 C −Y 2 0
, then the observer (10) characterized by gains given in (20) has ISS error dynamics with a linear gain γ = √ µµ 1 and an exp − KL function, which means there exist c > 0, λ ∈ (0, 1) such that
|e(k)| ≤ cλ k |e(0)| + γ||m|| k ,(23)
for all e(0) ∈ R n , k ≥ 0 and m ∈ R ny with {m(k)} ∈ l ∞ . Corollary 1: A circle-criterion observer robust to measurement noise can be obtained by solving (22) while minimizing µ + µ 1 .
Theorem 1 provides a way to design the observer for (7). If we solve (22) while minimizing √ µµ 1 , we obtain an observer robust to measurement noise. To make the objective function convex, we consider using µ + µ 1 instead as our objective function. We know that (µ + µ 1 ) 2 ≥ 4 · µµ 1 , which yields γ = √ µµ 1 ≤ 1 2 (µ + µ 1 ) as µ, µ 1 are positive. Therefore, we can minimize the upper bounds of γ by minimizing µ+µ 1 . By solving (22) while minimizing µ+µ 1 , we can obtain an observer that attenuates measurement noise.
Since c 3 ∈ (0, 1) is in a bounded set, we do a grid-search over c 3 , i.e. we make a grid in (0, 1) and for each grid point we solve (22) while minimizing µ + µ 1 and then we choose the c 3 that minimizes √ µµ 1 . In our design method, besides regarding µ 1 as a variable, we also do not assume c 3 is a fixed constant with a given value as [29] does, which makes our LMIs less conservative than those in [29] and further improves the robustness of our observer to measurement noise in some circumstances. We use the model used in Example 1 in [28] and compare their performance by introducing measurement noise m. All LMIs were solved using PENLAB [30] in MATLAB.
Example 1 Consider the discrete-time nonlinear system subject to measurement noise:
x + = 1 δ 0 1 x + 1 2 δα sin(x 1 + x 2 ) δα sin(x 1 + x 2 ) + δu δu , y = 3 0.3 3 0.6 6 0.9 1.2 12 x + m.(24)
We let δ = 0.1 and α = 1. (24) can be rewritten in the form of (7) with Assumption 1 holding, see [28] for more details. We solve (22) while minimizing µ + µ 1 and doing a grid search over c 3 to obtain observer matrices K, L, c 3 = 0.900, and γ = 0.924. We obtain K, L via solving the LMIs in [28].
We solve the LMIs in [29] by letting c 3 = 0.500, M given as
IV. A CIRCLE-CRITERION OBSERVER-BASED ESTIMATOR
ROBUST TO MEASUREMENT NOISE AND SENSOR ATTACKS
In this section, we introduce a circle-criterion observerbased estimator for the same class of discrete-time nonlinear systems as in Section III, but we assume a small number of sensors are also subject to sensor attacks:
x + =Ax + Gf (Hx) + ρ(u, y),
y =Cx + a +m,(25)
where x ∈ R n is the state,ỹ ∈ R p is the sensor measurement, m ∈ R p is the measurement noise with {m(k)} ∈ l ∞ , a ∈ R p is the vector of attacks: if sensor i ∈ {1, · · · , p} is not attacked, then the ith component of the vector a(k), a i (k) = 0, ∀k ≥ 0; otherwise sensor i is attacked and a i (k) is arbitrary and possibly unbounded. We denote W ⊆ {1, · · · , p} the set of attacked sensors, then we have supp(a) = W . We assume the set W is unknown to us. We denoteỹ k; x(0), a [0,k] ,m [0,k] as the output of the system at time k when the initial state is x(0) and the outputs are subject to measurement noisem and sensor attacks a. (8), (9) still hold. Suppose some of the sensors are corrupted by attack signals. Having received the measured output sequences ỹ(k; x(0), a [0,k] ,m [0,k] ) ∀k ≥ 0, a circle-criterion observer-based estimator is used to estimate the states of the system. Our objective is to present a design method of a robust circle-criterion observer-based estimator that provides exponential convergence of the estimatesx(k) to a neighborhood of the true states x(k) under some assumptions, and the error e(k) =x(k) − x(k) is ISS with a linear gain and exp − KL function with respect to the measurement noise only, which means there existc > 0,λ ∈ (0, 1),γ y ≥ 0 such that, |e(k)| ≤cλ k |e(0)| +γ y ||m|| k ,
for all e(0) ∈ R n , k ≥ 0. We now outline our estimation strategy which is inspired by the method for the linear case from [25] . For (25), let 0 < q < p 2 be the largest integer such that for each subset J ⊂ {1, · · · , p} of sensors with card(J) ≥ p−2q, the circlecriterion observer of the form
x + J =Ax J + Gf (Hx J + K J (C JxJ −ỹ J )) + L J (C JxJ −ỹ J ) + ρ(u, y),(27)
exists forỹ J .x J denotes the estimate of the state x from y J , and K J , L J are the observer gains.C J is the stacking of allC i , i ∈ J whereC i is the ith row ofC. When we say the observer exists forỹ J , we mean when a J (k) = 0 for all k ≥ 0 the error of each observer e J (k) = x J (k) − x(k) with the following dynamics
e + J = (A + L JCJ )e J − L JmJ + G△f J ,(28)
where
△f J = f (q) − f (q J ),q = Hx andq J = Hx J + K J (ŷ J −ỹ J ) andŷ J =C JxJ ,ỹ J =C J x +m J is ISS with
a linear gain γ J and an exp − KL function with respect to measurement noisem J . This implies that there exist c J > 0,
λ J ∈ (0, 1), γ J ≥ 0 such that |e J (k)| ≤ c J λ k J |e J (0)| + γ J ||m J || k ,(29)
for all e J (0) ∈ R n , k ≥ 0 andm J ∈ R card(J) with {m J (k)} ∈ l ∞ . We assume that:
Assumption 2: There are at most q sensors attacked,
card(W ) ≤ q.(30)
By using the design method proposed in Section III, we construct a robust observer for each subset J ⊂ {1, 2, · · · , p} with card(J) = p − q and for each subset S ⊂ {1, · · · , p} with card(S) = p − 2q. For each subset J with card(J) = p−q, we define π J (k) for all k ≥ 0 to be the largest deviation between the estimatex J (k) and the estimatex S (k) that is given by any subset S ⊂ J with card(S) = p − 2q.
π J (k) := max S⊂J:card(S)=p−2q |x J (k) −x S (k)|.(31)
Recalling that among the total p sensors, there is at least one subsetĪ ⊂ {1, · · · , p} of sensors with card(Ī) = p − q thatỹĪ = CĪ x +mĪ as aĪ = 0, then in general all of the estimates that appear in the definition of πĪ (k) are more consistent than all the subsets J with card(J) = p − q and y J =C J x + a J +m J with a J = 0. This motivates the following state estimation: for all k ≥ 0,
σ(k) = arg min J⊂{1,2,··· ,p}:card(J)=p−q π J (k),(32)
and then we say for all k ≥ 0, the estimate given by the subset σ(k) is a good estimate,
x(k) =x σ(k) (k),(33)
wherex σ(k) (k) represents the estimates given by the subset σ(k). The following result states that the proposed estimator is robust with respect to sensor attacks and measurement noise. For simplicity, we initialize all the observers to the same conditionx(0). Theorem 2: For the system (25), suppose Assumptions 1-2 hold, recalling from (31)-(33), denoting e(k) =x(k) − x(k), there exist positive constantsc > 0,λ ∈ (0, 1),γ y ≥ 0 such that the following inequality holds:
|e(k)| ≤cλ k |e(0)| +γ y ||m|| k ,(34)
for all e(0) ∈ R n , k ≥ 0, andm ∈ R p with {m(k)} ∈ l ∞ . Proof: From the result of Section III, we know for each subset J ⊂ {1, · · · , p} with card(J) ≥ p − 2q, the observation error dynamics satisfies (29). Since a i (k) = 0 for all i ∈ {1, · · · , p} \ supp(a) and ∀k ≥ 0, we conclude for J =Ī ⊆ {1, · · · , p} \ supp(a) with card(Ī) = p − q, there exist cĪ > 0, λĪ ∈ (0, 1) and γĪ ≥ 0, such that
|eĪ (k)| ≤ cĪ λ k I |e(0)| + γĪ ||mĪ || k ,(35)
for all e(0) ∈ R n and k ≥ 0. Also for any set S ⊂Ī with card(S) = p − 2q, we have a S (k) = 0 ∀k ≥ 0, hence there exist c S > 0, λ S ∈ (0, 1) and γ S ≥ 0 such that
|e S (k)| ≤ c S λ k S |e(0)| + γ S ||m S || k ,(36)
for all e(0) ∈ R n and k ≥ 0. Recalling the definition of πĪ from (31), we have that
πĪ (k) =max S⊂Ī |xĪ (k) −x S (k)| =max S⊂Ī |xĪ (k) − x(k) + x(k) −x S (k)| ≤|eĪ (k)| + max S⊂Ī |e S (k)|(37)
for all k ≥ 0. From (35) and (36), we obtain
πĪ (k) ≤ 2c ′Ī λ ′ k I |e(0)| + 2γ ′Ī ||mĪ || k ,(38)
for all e(0) ∈ R n and k ≥ 0, where c ′Ī := max π J (k), hence π σ(k) (k) ≤ πĪ (k). We know that there exist at least one setS ⊂ σ(k) with card(S) = p − 2q such that aS(k) = 0 ∀k ≥ 0, and there exist cS > 0, λS ∈ (0, 1) and γS ≥ 0 such that
|eS(k)| ≤ cSλ k S |e(0)| + γS||mS|| k ,(39)
for all e(0) ∈ R n and k ≥ 0. From (31), there is a fact that π σ(k) (k) = max
S⊂σ(k):card(S)=p−2q |x σ(k) (k) −x S (k)| ≥ |x σ(k) (k) −xS(k)|.
From the triangle inequality we have that
|e σ(k) (k)| =|x σ(k) (k) − x(k)| =|x σ(k) (k) −xS(k) +xS(k) − x(k)| ≤|x σ(k) (k) −xS(k)| + |eS(k)| ≤π σ(k) (k) + |eS(k)| ≤πĪ (k) + |eS(k)|(40)
for all k ≥ 0. From (38) and (39), we have
|e σ(k) (k)| ≤cλ k |e(0)| +γ y · max {||mS|| k , ||mĪ || k } ,(41)
for all e(0) ∈ R n and k ≥ 0, wherec = 3 · max cS, c ′Ī , λ = max λS, λ ′Ī ,γ y = 3 · max γS, γ ′Ī . Since ||m|| k ≥ max {||mS|| k , ||mĪ || k }, we can see (41) satisfies (34). The proof is complete.
We still use the model in Example 1, but here we assume sensor attacks and measurement noise both occur to test the performance of our designed estimator.
Example 2 Consider the discrete-time nonlinear system subject to measurement noise and sensor attacks:
x + = 1 δ 0 1 x + 1 2 δα sin(x 1 + x 2 ) δα sin(x 1 + x 2 ) + δu δu , y = 3 0.3 3 0.6 6 0.9 1.2 12 x + a +m.(42)
We still let δ = 0.1 and α = 1,m ∼ U(−0.5, 0.5). We find that the circle-criterion observer of the form (27) In this section, we still consider system (25). We let q be the largest integer such that a circle-criterion observer exists for each subset J ⊂ {1, · · · , p} with card(J) ≥ p − 2q. We propose an algorithm for isolating attacked sensors when we know how many sensors are attacked, which is denoted as q ⋆ (q ⋆ ≤ q).
Assumption 3:
There are q ⋆ ≤ q attacked sensors, i.e.,
card(W ) = q ⋆ ,(43)
and q ⋆ ≤ q is a known positive integer.
We construct a circle-criterion observer for each subset J ⊂ {1, · · · , p) with card(J) = p − q ⋆ and for each subset S ⊂ {1, · · · , p} with card(S) = p − 2q ⋆ . For each subset J with card(J) = p − q ⋆ and for all k ≥ 0, we define π ⋆ J (k) as
π ⋆ J (k) := max S⊂J:card(S)=p−2q ⋆ |x J (k) −x S (k)|.(44)
Since there are q ⋆ sensors under attack, we know there is one subsetĪ ⊂ {1, · · · , p} of sensors with card(Ī) = p − q ⋆ thatỹĪ =CĪ x +mĪ as aĪ = 0, then all of the estimates that appear in the definition of πĪ (k) are very likely to be more consistent than all the subsets J with card(J) = p − q ⋆ and y J =C J x + a J +m J with a J = 0. For all k > 0, if we denoteJ(k) as the set of attack-free sensors at time k, then J(k) is given as
J(k) = arg min J⊂{1,2,··· ,p}:card(J)=p−q ⋆ π J (k).(45)
Then the set {1, · · · , p} \J(k) is isolated as the set of attacked sensors at time k. We make our decision in every N time steps, where N ∈ Z >0 is the window size we choose, i.e. in each N time steps we keep obtainingJ(k) from (45) for each k, and we choose the subset J(i) that is equal toJ(k) most often in the i-th window. Then we claim if for some J with card(J) = p − q ⋆ we havē J(k) == J then 8: update n J (i) as follows:
π ⋆ J (k) = max S⊂J:card(S)=p−2q ⋆ |x J (k) −x S (k)|.
n J (i) = n J (i) + 1. Select the subset J that is equal toJ(k) most often J(i) = arg max J∈{1,··· ,p}:card(J)=p−q ⋆ n J (i).
12:
The set of sensors potentially under attack is given as:
A(i) = {1
, · · · , p} \ J(i).
13:
ReturnÃ(i). 14: end for Example 3 We still consider model (42) in Example 2, with δ = 0.1. We consider two cases where α is equal to 1 and 0 respectively. In each case, we letm ∼ U(−0.5, 0.5), q ⋆ = 1 and W = {3}. We let a 3 ∼ U(−b, b), and b given by 1, 2.5. In each case, we choose the window size N to be 50, 100, 200 respectively. The major advantage of Algorithm 1 is that it can be applied to isolate attacked sensors when sensor attacks and measurement noise both occur as long as measurement noise is bounded.
VI. CONCLUSION
Following the way of [29], a design method of a discretetime circle-criterion observer robust to measurement noise is given as a series of linear matrix inequalities in the absence of attack signals. An less conservative ISS gain is obtained by solving an optimization problem with more degrees of freedom. Then a circle-criterion observer-based estimation strategy is proposed in the presence of measurement noise and sensor attacks. We show that the designed circle-criterion observer-based estimator provides ISS estimation errors with a linear gain and an exp − KL function with respect to measurement noise when a sufficiently small subset of sensors are corrupted by (potentially unbounded) attack signals and all sensors are affected by bounded measurement noise. This work can be seen as an extension of the existing observerbased estimator for linear systems [25]. An algorithm for isolating attacked sensors is also proposed when we know how many sensors are attacked.
This work was supported by the Australian Research Council under the Discovery Project DP170104099.The authors are with the Department of Electrical and Electronics Engineering, the University of Melbourne, Australia.
( 18 )
18, H = I and minimizing µ to obtain observer with γ = 22.4. We let m ∼ U(−0.5, 0.5). x(0) is randomly selected from a normal distribution andx(0) = 0. The performance of these observers are compared respectively inFigures 1-2.
Fig. 1 .Fig. 2 .
12Estimated statesx converges to a neighbourhood of the true states x. Legend: Observer obtained via Theorem 1 (red), Observer from[29] (grey), true states (black) Estimated statesx converges to a neighbourhood of the true statesx. Legend: Observer obtained via Theorem 1 (red), Observer from[28] (grey), true states (black)
{cĪ , c S }, λ ′Ī := max S⊂Ī {λĪ , λ S }, and γ ′Ī := max S⊂Ī {γĪ , γ S }. Observe that since S ⊂Ī with card(S) = p − 2q. Recall from (31)-(33) thatx(k) =x σ(k) (k) where σ(k) = arg min J⊂{1,2,··· ,p}:card(J)=p−q
Fig. 3 .Fig. 4 .
34exists for each subset of J ⊂ {1, 2, 3, 4} with card(J) ≥ 1 and p = 4, we have q = 1. We let W = {3}, which means the 3-rd sensor is under attack. The estimator knows there is at most one sensor under attack, but does not know which. By using the design method proposed in Section III, we design an observer for each J ⊂ {1, 2, 3, 4} with card(J) = 3 and each S ⊂ {1, 2, 3, 4} with card(S) = 2. Therefore, totally 4 3 + 4 2 = 10 observers are designed, and they are all initialized atx(0) = [0, 0] ⊤ . x 1 (0), x 2 (0) are randomly selected from a standard normal distribution. We let a 3 ∼ U(−b, b) with b given by 1, 10. For all k ∈ [0, 500], (31)-(33) is used to constructx(k). The performance of the designed estimator is shown in Figures 3Estimated statesx converges to a neighbourhood of the true states x when a 3 ∼ U (−1, 1). Legend:x (grey), true states (black) Estimated statesx converges to a neighbourhood of the true states x when a 3 ∼ U (−10, 10). Legend:x (grey), true states (black) V. ISOLATION OF SENSOR ATTACKS
Fig. 5 .Fig. 6 .
5610 circle-criterion observers which are all initialized withx(0) = [0, 0] ⊤ and x 1 (0), x 2 (0) are randomly selected from a standard normal distribution. We follow the steps in Algorithm 1. We check in 1000 time steps which sensor is isolated in each time window, which is shown in Figures 5-6. Case 2. α = 0, (42) becomes a discrete-time linear systems subject to measurement noise and sensor attacks. We construct observers via solving (22) by letting G = 0 and minimizing µ + µ 1 . We apply Algorithm 1 in a similar way as what we do when α = 1 and Figures 7-8 show the performance of Algorithm 1 when α = 0. The sensor isolated by Algorithm 1, α = 1, a 3 ∼ U (The sensor isolated by Algorithm 1, α = 1, a 3 ∼ U (−2.5, 2.5).
Fig. 7 .Fig. 8 .
78The sensor isolated by Algorithm 1, α = 0, a 3 ∼ U (The sensor isolated by Algorithm 1, α = 0, a 3 ∼ U (−2.5, 2.5).
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| {'fraction_non_alphanumeric': 0.08945226600611683, 'fraction_numerical': 0.03298536511386902, 'mean_word_length': 3.6595218466611708, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 24, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 29, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We address the problem of robust state estimation and attack isolation for a class of discrete-time nonlinear systems with positive-slope nonlinearities under (potentially unbounded) sensor attacks and measurement noise. We consider the case when a subset of sensors is subject to additive false data injection attacks. Using a bank of circle-criterion observers, each observer leading to an Input-to-State Stable (ISS) estimation error, we propose a estimator that provides robust estimates of the system state in spite of sensor attacks and measurement noise; and an algorithm for detecting and isolating sensor attacks. Our results make use of the ISS property of the observers to check whether the trajectories of observers are consistent with the attack-free trajectories of the system. Simulations results are presented to illustrate the performance of the results.{1, · · · , p}\\J(i) is the set of sensors potentially under attack in the i-th time window, which is Algorithm 1.Algorithm 1 SENSOR ATTACKS ISOLATION Input: N , q ⋆ 1: Design a circle-criterion observer for each subset J ⊂ {1, · · · , p} with card(J) = p − q ⋆ and for each subset S ⊂ {1, · · · , p} with card(S) = p − 2q ⋆ . 2: We intialize the counter variable n J (i) = 0 for each J with card(J) = p − q ⋆ and for all i ∈ Z >0 . 3: for each i ∈ Z >0 do 4:for each k ∈ [1 + (i − 1)N, iN ] do 5: calculate π ⋆ J (k) for all J with card(J) = p − q ⋆ as follows:', 'arxivid': '1805.04242', 'author': ['Tianci Yang ', 'Carlos Murguia ', 'Margreta Kuijper ', 'Dragan Nešić '], 'authoraffiliation': [], 'corpusid': 21681198, 'doi': '10.1109/cdc.2018.8619724', 'github_urls': [], 'n_tokens_mistral': 14043, 'n_tokens_neox': 12337, 'n_words': 7345, 'pdfsha': '4f9e7ca0958604ab2cbb8c0b3ea5b66b528f0906', 'pdfurls': ['https://arxiv.org/pdf/1805.04242v2.pdf'], 'title': ['A Robust Circle-criterion Observer-based Estimator for Discrete-time Nonlinear Systems in the Presence of Sensor Attacks and Measurement Noise', 'A Robust Circle-criterion Observer-based Estimator for Discrete-time Nonlinear Systems in the Presence of Sensor Attacks and Measurement Noise'], 'venue': []} |
arxiv |
Partitioning the Bags of a Tree Decomposition Into Cliques
Thomas Bläsius
Karlsruhe Institute of Technology
Germany
Maximilian Katzmann
Karlsruhe Institute of Technology
Germany
Marcus Wilhelm
Karlsruhe Institute of Technology
Germany
Partitioning the Bags of a Tree Decomposition Into Cliques
2012 ACM Subject Classification Mathematics of computing → Graph algorithms; Theory of computation → Graph algorithms analysis Keywords and phrases treewidthweighted treewidthalgorithm engineeringcliquesclusteringcomplex networks
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions.Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.
Introduction
The treewidth is a measure for how treelike a graph is in terms of its separators. It is defined via a tree decomposition, a collection of vertex separators called bags that are arranged in a tree structure. The size of the largest bag determines the width of the decomposition and the treewidth of a graph is the minimum width over all tree decompositions. The concept of treewidth has its origins in graph theory with some deep structural insights [22,24]. Additionally, there are algorithmic implications. Intuitively speaking, the separators of a tree decomposition split the graph into pieces that can be solved independently except for minor dependencies at the separators. This is often formalized using a dynamic program over the tree decomposition, yielding an FPT-algorithm (fixed-parameter tractable) with the treewidth as parameter [9]. As this is a versatile framework that can be applied to many problems, it comes to no surprise that there has been quite a bit of effort to develop algorithms for computing low-width tree decompositions (see, e.g., [15,16]).
A major obstruction for low treewidth are large cliques, which inevitably lead to large separators. This is particularly true for so-called complex networks, i.e., graphs with strong community structure and heterogeneous degree distribution, which appear in various domains such as communication networks, social networks, or webgraphs. One could, however, hope for two aspects that together mitigate this negative effect of large cliques. First, though some separators need to be large, these separators are structurally simple, e.g., they form a clique or can be covered with few cliques. Second, separators that are large but structurally simple still let us solve the separated pieces individually with low dependence between them. The first hope is supported by the fact that the treewidth is asymptotically equal to the clique number in hyperbolic random graphs [5]; a popular model for complex networks [21]. This indicates that cliques are indeed the main obstruction for low treewidth in these kinds of networks. The second hope is supported by the results of de Berg et al. [14], who introduced the concept P-flattened tree decompositions. There, the graph is partitioned into cliques and the width of the tree decomposition is measured in terms of the (weighted) number of cliques in a bag. Thus, the width does measure the complexity of separators rather than their size. Based on this definition, the authors then show that these structurally simple separators help to solve various graph problems efficiently.
To the best of our knowledge, these extended concepts have not yet been studied from a practical perspective. With this paper, we want to initiate this line of research by addressing two questions. First, can such clique-partitioned tree decompositions lead to substantially smaller width values than classical tree decompositions? Second, how can such tree decompositions be computed? For the second question, we design and evaluate different algorithmic strategies for computing a novel yet closely related variant of tree decompositions. Our experiments yield some interesting algorithmic insights and provide a good starting point for further development. On networks that do exhibit clique structures, the constructed tree decompositions indeed have sufficiently low width to answer the first question affirmatively. We believe that there is plenty of room for improvement in our approaches, which may yield even better insights into the applicability of the new parameter. In the following, we discuss related work before stating our contribution more precisely.
Related Work
There are multiple lines of research that investigate variants of treewidth where additional structural properties are taken into account. As mentioned above, De Berg et al. [14] propose a variant of tree decompositions where the initial graph is partitioned into cliques (or unions of constantly many connected cliques) that are contracted into weighted vertices. The weight of a clique of size s is log(s + 1) and the weight of a bag of the tree decomposition is the sum of its weights. Using this technique, they give subexponential algorithms for a range of problems on geometric intersection graphs, including Independent Set, Steiner Tree and Feedback Vertex Set. For some of these problems, the algorithms are also representation agnostic, while for most others, the geometric representation is required. They also prove that the running time of the algorithms is tight under the exponential-time-hypothesis (ETH). Kisfaludi-Bak [20] applied the same algorithmic framework to intersection graphs of constantly sized objects in the hyperbolic plane.
A similar parameter called tree clique width has been proposed by Aronis [2]. Here, the idea is to consider tree decompositions where each bag is annotated with an edge clique cover (ecc) and where the size of the cover determines the width of a bag. The paper shows several hardness results and adapts common treewidth algorithms to the newly proposed parameter.
Another approach to capture graph structures that lead to high treewidth despite being structurally simple has been proposed by Dallard, Milanič, and Štorgel. They define the independence number of a tree decomposition as the size of the largest independent set of any of its bags and the tree-independence number of a graph as the minimum independence number of any tree decomposition [13]. This parameter connects to the more theoretical study of (tw, ω)-bounded graphs, i.e., graph classes in which the treewidth depends only on the clique number [11,12]. This line of research is mostly concerned with the classification and characterization of the considered graph classes both in terms of graph theory and algorithmic exploitability. However, apart from a factor 8 approximation with running time 2 O(k 2 ) · n O(k) due to Dallard, Fomin, Golovach, Korhonen, and Milanič [10], we are not aware of any work that tries to actually build algorithms for this or similar parameters.
Contribution
In this paper, we propose clique-partitioned treewidth as a parameter that captures structurally simple separators in graphs. It can be seen as a close adaptation of P-flattened treewidth [14], where we first compute a tree decomposition and then determine clique partitions of the subgraphs induced by the bags. Thus, instead of using a global clique partition of the whole graph, we consider clique partitions that are local to a single bag.
The remainder of this paper is structured as follows. In Section 2, we formalize our definition for clique-partitioned treewidth and prove several statements comparing it with P-flattened treewidth. In Section 3, we present multiple approaches to compute low-weight clique partitions for the bags of a tree decomposition. They include various heuristic methods, as well as an exact branch-and-bound algorithm for which we propose several adjustments with the potential to improve its running time in practice. Afterwards, in Section 4 we combine an implementation of our approaches with existing methods for computing tree decompositions and study the upper bounds on the clique-partitioned treewidth of real-world networks. Furthermore, we evaluate the performance of the exact and heuristic clique partition solvers proposed in Section 3.
Clique-partitioned treewidth
We first introduce some basic notation and give the definition for traditional tree decompositions. We write [n] = {1, . . . , n} for the first n natural numbers. Throughout the paper, we assume graphs G = (V, E) to be simple and undirected and write V (G) and E(G) for the sets of vertices and edges, respectively. For a subset X ⊆ V we write G[X] for the subgraph of G induced by X.
A tree decomposition of G is a pair (T, B), for a tree T and a function B mapping vertices of T to subsets of V called bags such that T and B have the following three properties: (1) every vertex of G is contained in some bag, (2) for every edge, there is a bag containing both endpoints, and (3) for any vertex v of G, the set of bags containing v forms a connected subtree of T . The width of a tree decomposition is the size of the largest bag minus 1. The treewidth tw(G) is the smallest width obtainable by any tree decomposition of G.
We define a clique-partitioned tree decomposition of G as a tree decomposition where for every t ∈ V (T ) we have a partition P t of the subgraph induced by the corresponding bag (i.e., the graph G[B(t)]) into cliques. Following de Berg et al. [14], we define the weight of a clique C as log(|C| + 1) and the weight of a bag B(t) as the sum of weights of the cliques in its partition P t . Throughout this paper we assume 2 to be the default base of logarithms. The weight of a clique-partitioned tree decomposition is the maximum weight of any of its bags and the clique-partitioned treewidth (short: cp-treewidth) of G, denoted by cptw(G), is the minimum weight of any clique-partitioned tree decomposition.
As mentioned before, the clique-partitioned treewidth is closely related to the parameter defined by de Berg et al. [14]. For a clique partition P of the whole graph G, we say that a P-flattened tree decomposition is a clique-partitioned tree decomposition of G where the partition into cliques within a bag is induced by the global partition P. As before, the weight of a P-flattened tree decomposition is the maximum total weight of the cliques in any of its bags. In reference to the authors [14], we call the minimum weight over all P the BBKMZ-treewidth.
We note that our parameter can also be seen as an adaptation of tree clique width [2], where instead of considering the size of an edge clique cover of each bag, we consider the logarithmically weighted sum of clique sizes of a clique partition. That is, we are using the weight function of the P-flattened treewidth to define a parameter which considers individual clique partitions, similar to tree clique width.
In the following, we compare the clique-partitioned treewidth to the more closely related BBKMZ-treewidth. First, as a global partition P can also be used locally in each bag of a clique-partitioned tree decomposition, we obtain that the clique-partitioned treewidth of a graph is at most its BBKMZ-treewidth. Additionally, the clique-partitioned treewidth can also be substantially smaller than the BBKMZ-treewidth, as shown in the following lemma.
Lemma 1.
There is an infinite family of graphs G such that a graph G ∈ G with n vertices, has clique-partitioned treewidth in O (log log n) and BBKMZ-treewidth in Ω (log n).
Proof. The family G contains for every h ∈ N one graph G h . The Graph G h is a complete binary tree of height h, where additionally for every leaf we connect all h vertices that lie on a path between the root r and into a clique. Note that we have h ∈ Θ(log n).
Let P h be a clique partition of G h . Then, via a simple induction over h, it is easy to see that in G h there is a path between the root r and some leaf of G h such that every vertex on the path belongs to a different partition class. These vertices form a clique in G h that has to be prosent in some bag of any P h -flattened tree decomposition of G h . This bag thus contains all h partition classes on the path and has weight h · log(1 + 1) ∈ Ω(log n).
At the same time we can construct a clique-partitioned tree decomposition (T, σ), that has one bag for every path between the root r and each leaf . Then, T forms a path. As every bag consists of a single clique on h vertices, there is a clique partition of this tree decomposition with weighted width log(h + 1) ∈ O(log log n).
Finally, we show the algorithmic usefulness of clique-partitioned treewidth in the following lemma, which is an extension of the one proposed by de Berg et al. [14].
Lemma 2.
Let G be a graph with a clique-partitioned tree decomposition (T, σ) of weight τ . Then a smallest independent set of G can be found in O(2 τ · poly(n)) time.
Proof. We use a standard dynamic programming approach on tree decompositions based on introduce, forget, and join nodes (see for example Cygan et. al [9]). For each node t ∈ V (T ), we store a number of partial solutions for the subgraph of G induced by the bags of nodes in the subtree below t.
A partial solution consists of a subset of the vertices in the current bag as well as the size of the total partial independent set for the subgraph induced by the subtree below the current bag. This makes it easy to initialize partial solutions for leaf nodes in the tree decomposition.
In an introduce node, two new partial solutions are created, one where the new vertex is in the independent set and one where it is not. In a forget node, the removed vertex is removed from each partial solution. In a join node, the partial solutions from the child-nodes are combined by taking their union.
In a traditional tree decomposition of width k, this leads to at most 2 k partial solutions per bag. In a clique-partitioned tree decomposition, this is even smaller, as there are only k + 1 ways an independent set can intersect a clique of size k. Thus, assuming {P t | t ∈ V (T )} denotes the clique partition of weight τ , the number of partial solutions that need to be considered per bag t are at most
C∈Pt (|C| + 1) = 2 C∈P t log(|C|+1) = 2 τ .
As the number of bags and time spent per bag is polynomial, this concludes the proof.
By the above argumentation, it follows that the clique-partitioned treewidth introduced in this paper is upper bounded by the version of de Berg et al. and can be exponentially lower. Additionally, it retains some power in solving NP-hard problems in FPT-time.
3
The weighted clique partition problem
We split the task of computing a clique-partitioned tree decomposition in two phases. First, we compute a tree decomposition, minimizing the traditional tree width. Secondly, fixing the structure and bags of this decomposition, we compute a clique partition for every bag. We note that we already lose optimality by this separation, i.e., the result may be suboptimal even if we get optimal solutions in each of the two phases. However, we expect that small bags should also allow for low-weight clique partitions.
In the first phase, we use established algorithms for the computation of tree decompositions. Consequently, we focus on the second step in this section. To this end, we define the Weighted Clique Partition problem, short Clique Partition. For a given graph G and an integer w, decide if there is a partition of V (G) into cliques P 1 , . . . , P k such that i∈[k] (|P i | + 1) ≤ w. Note that this function differs from the one in the definition of clique-partitioned treewidth, but is equivalent, as i∈ [k] log(|P i | + 1) = log( 1≤i≤k (|P i | + 1)) and the logarithm is monotonic.
In the following, we prove some technical lemmas that are useful throughout the section, before showing that Weighted Clique Partition is NP-complete (Section 3.1). Afterwards, we give different heuristic approaches (Section 3.2) and an optimal branch-and-bound algorithm in (Section 3.3). We start with following lemma, which intuitively states that the weight of a partition is smaller the more imbalanced the individual weights are, i.e., moving a vertex from a smaller to a larger clique reduces the total weight.
Lemma 3. Let a, b, c, d ∈ N 0 such that a + b = c + d and a ≥ b, c ≥ d, d > b. Then (a + 1)(b + 1) < (c + 1)(d + 1). Proof. There is an x > 0 such that c = a − x and d = b + x. As c ≥ d, x can be at most (a − b)/2. We derive (c + 1)(d + 1) = (a − x + 1)(b + x + 1) = ab − bx + b + ax − x 2 + x + a − x + 1 = (ab + a + b + 1) + ax − bx − x 2 = (a + 1)(b + 1) + x(a − b − x). We have x(a − b − x) > 0, as 0 < x ≤ a−b
2 and thus the claimed strict inequality follows.
With the above lemma (i.e., repeated applications thereof) we can compare the weight of two partitions.
. , r be different non-increasing sequences of natural numbers such that
2 ≤ k ≤ , i∈[k] s i = i∈[ ] r i , and s i ≥ r i for all i ∈ [k − 1]. Then i∈[k] (s i + 1) < i∈[ ] (r i + 1).
Proof. This follows from repeatedly applying Lemma 3 to go from R = r 1 , . . . , r to S = s 1 , . . . , s k while reducing the product in each step. To make this precise let i be the first index where s i > r i . We adjust R by adding 1 to r i ans subtracting 1 from r . Note that this maintains the sum. We apply Lemma 3 with a = r i + 1, b = r − 1, c = r i , and d = r . Then, we have (a + 1)(b + 1) < (c + 1)(d + 1), i.e., the product of the adjusted sequence is smaller than that of the original sequence R. Moreover, after a finite number of steps, we reach S and thus the product for S is smaller than the product for R.
Hardness
To prove that Weighted Clique Partition is NP-complete, we perform a reduction in two steps. We start with the NP-hard problem 3-Coloring. It asks for a given graph whether each vertex can be colored with one of three colors such that no two neighbors have the same color. As an intermediate problem in the reduction, we introduce Weighted Independent Set Partition. It is defined equivalently to Weighted Clique Partition, but instead of partitioning the graph into cliques, we partition it into independent sets, i.e., sets of pairwise non-adjacent vertices. Note that independent sets are cliques in the complement graph and vice versa. Thus, Weighted Independent Set Partition and Weighted Clique Partition are computationally equivalent. Thus, to obtain the following theorem, it remains to reduce 3-Coloring to Weighted Independent Set Partition.
Theorem 5. Weighted Clique Partition is NP-complete.
Proof. Membership in NP is easy to see as polynomial time verification of a solution is straightforward. For hardness, we reduce from 3-Coloring to Weighted Independent Set Partition. Thus, we now assume that we are given a graph G and need to transform it into a graph G and integer w such that such that G can be colored with three colors if and only if G has a partition into independent sets of weight at most w. We construct G as follows. For every vertex v of G, we add two new vertices v 1 and v 2 that form a triangle together with v, but have no other edges. We denote n = |V (G)| and set w = (n + 1) 3 . Note that any independent set in G can contain at most n vertices, because every appended triangle admits only one independent vertex.
Assume that G admits a proper three-coloring. This coloring directly translates to a three-coloring of G as follows. Every vertex v of G keeps its color in G . moreover, v 1 and v 2 each get one of the two other colors. Thus, the coloring classes in G have size exactly n each and form an independent set partition with weight (n + 1) 3 .
If otherwise G does not admit a proper three-coloring, then neither does G and there is no partition of G into at most three independent sets. Any partition of V (G ) into more than three independent sets has a weight larger than (n + 1) 3 by Lemma 4, as no independent set in G can have more than n vertices. Consequently, G is three-colorable if and only if there is a partition of V (G ) into independent sets with weight at most (n + 1) 3 .
Heuristic approaches
We now explain different approaches to solving the optimization variant of Weighted Clique Partition both optimally and heuristically.
Throughout this section we make use of the fact that enumerating all maximal cliques of a graph is not only output polynomial [19], but also highly feasible in practice as shown by Eppstein, Löffler, and Strash [17]. We use an implementation of their algorithm from the igraph 1 library.
Maximal clique heuristic. Recall from Lemma 3 that the weight function favors imbalanced clique sizes over more balanced ones. It therefore makes sense to try to find few large cliques that cover all vertices. A basic greedy heuristic that tries to achieve this works as follows. First, we enumerate all maximal cliques C of the graph. Then we iteratively add one clique to the partition by greedily selecting the clique with the largest number of remaining uncovered vertices. We call this the maximal clique heuristic.
In order to efficiently implement this heuristic, we use a priority queue to fetch the largest clique and keep track of the cliques C v ⊆ C that a vertex v is part of. This way, after choosing the remaining vertices of a clique C ∈ C as a partition, we have to update the sizes of O( v∈C |C v |) cliques. The total number of such updates throughout the whole algorithm is at most the sum of clique sizes in C. Thus, using a Fibonacci Heap, a total running time of O(|V | log |C| + C∈C |C|) can be achieved. In our implementation we use a binary heap due to it being faster in practice. This costs an additional factor of log |C| for the second term.
Repeated maximal clique heuristic. Note that the MC heuristic does not recompute the maximal cliques of the remaining graph after selecting a clique. As deleting the vertices of one clique can have the effect that a non-maximal clique becomes maximal, the MC heuristic might miss a clique we would want to select. The repeated maximal clique heuristic recomputes the set of maximal cliques after each decision, i.e., it selects a maximum clique of the remaining graph in each step.
Set Cover heuristics.
Observe that for the Weighted Clique Partition problem, we have to choose a set of cliques of minimum weight that cover all vertices. Thus, we essentially have to solve a weighted Set Cover problem. As there are reasonably efficient solvers for Set Cover (or the equivalent Hitting Set problem), it seems like a promising approach to use those. However, this has the disadvantage, that we would need to list all cliques and not only the maximal cliques. Nonetheless, it seems like a good heuristic to just consider maximal cliques and find a minimum set cover (unweighted or weighted).
The heuristic consists of two steps. First, we compute a minimum set cover, using the maximum cliques as sets and the vertices as elements. We consider two variants for this steps; weighted (a set of size k has weight log(k + 1)) and unweighted (each set has weight 1). Afterwards, in the second step, we convert the cover into a partition by assigning the overlap between selected cliques to only one clique. We call the resulting two approaches the (maximal clique) set cover and (maximal clique) weighted set cover heuristics.
For the first step, i.e., solving Set Cover, we use a state of the art branch-and-bound solver [6] for the unweighted case. Additionally, for the weighted case, we use the straightforward formulation of set cover as an ILP and solve it with Gurobi [18]. To the best of our knowledge, ILP solvers are currently the state-of-the-art for weighted set cover.
For the second step, we have to compute clique partitions from the resulting set covers by assigning each vertex that is covered by multiple cliques to a single one of these cliques.
The goal is to minimize the weight of the resulting cliques, i.e., by Lemma 3, we want to distribute them as unevenly as possible. We employ a simple greedy heuristic, assigning each vertex to the largest clique it is part of and braking ties arbitrarily in case of ambiguity.
At a first glance it seems possible that doing both steps optimally (solving set cover and resolving the overlaps) could yield an overall optimal solution. However, this is not the case, as briefly discussed in Appendix A.
Exact branch-and-bound solver
Our branch-and-bound branches on which clique to select next. How to branch is described in Section 3.3.1 where we show that we can, in each step, select a maximal clique and that the cliques of the resulting sequence are non-increasing in size. In Section 3.3.2 and Section 3.3.3, we describe lower bounds for pruning the search space, i.e., if the best solution found so far is better than the lower bound in the current branch, we can prune that branch.
Branching
The following structural insight enables us to branch on the maximal cliques.
Lemma 6.
Let P be a minimum weight clique partition of a graph G and let C ∈ P be the largest clique of P. Then C is maximal clique in G.
Proof. Assume that C is a non-maximal clique. That is, there is a vertex v ∈ V (G) \ C with C ⊆ N (v)
. Let C ∈ P be the clique containing v. We construct a clique partition P by removing v from C and adding it to C. As C was the largest clique in P, via Lemma 3 we have (|C| + 2)(|C |) < (|C| + 1)(|C | + 1), contradicting the optimality of P.
Thus, even though not all cliques of an optimal solution might be maximal, we at least know that the largest one is. We can use the decision of which maximal clique to select as the largest one as the branching decision of our algorithm. This way, we can solve the optimization variant of Weighted Clique Partition, i.e., the algorithm takes a graph G and finds a minimum weight clique partitioning.
After a clique C has been selected as the largest one, the remaining problem is to find a clique partition of G[V \ C] that does not use any clique larger than C. This means that we can view our algorithm as a simple recursive subroutine that solves the same problem at every node of the recursion tree. As input it gets the graph G and the cliques C 1 , . . . , C i that have already been selected by previous recursive calls. It then tries to compute an optimal clique partition of the remaining graph G := G \ j∈[i] C j . This is done by either returning a trivial solution if G can be covered with a single clique or by branching on the decision of which maximal clique to select as the largest one for the partition of G . Note that for this decision, only maximal cliques that are at most as large as any of the previously selected cliques C 1 , . . . , C i need to be considered. The result of the subroutine call is then the cheapest solution found in any of the branches.
In order to quickly obtain a good upper bound, we explore branches corresponding to larger cliques first. This way, the first leaf of the search tree constructs the same solution as the repeated maximal clique heuristic.
Size lower bound
We call the lower bound given by the following lemma the size lower bound.
Lemma 7.
Let G be a graph with n vertices and P be a clique partition of G consisting of cliques of size at most s. Then P has weight at least (s + 1) n/s · ((n mod s) + 1).
Proof. The stated minimum weight is achieved by a partitioning P that uses as many cliques of size s as possible and one clique with all remaining vertices. Any other partitioning P using only cliques of size at most s is at least as expensive, as it can be transformed into P by of Lemma 4.
Note that the size lower bound can trivially be evaluated in constant time. Even though it is rather basic, we expect this lower bound to be effective at pruning branches in which very small cliques are selected early on.
Valuable sequence lower bound
Note that the size lower bound optimistically assumes that there are n/s non-overlapping cliques of size s. This yields a bad lower bound if, e.g., there is only one clique of size s while all other cliques are much smaller. In the following, we describe an improved bound based on this observation. We note that we have to be careful when considering what clique sizes are available for the following reason. Assume the branching has already picked a clique of size s, i.e., subsequent selected cliques have to have size at most s. Then it seems natural to derive a lower bound by summing over the sizes of all maximal cliques of size at most s. However, we have to account for the fact that selecting (and deleting) one clique can shrink a maximal clique that was larger than s to become a clique of size s. Thus, there might me more cliques of size s available than initially thought. In order to formalize this, we first introduce a different problem that considers only sizes of the cliques without making any assumptions on the overlap between the cliques.
In the Valuable Sequence problem, we are given a multiset A of natural numbers and a natural number n. The task is to construct a sequence of total value n and minimum weight. Such a sequence S = s 1 , s 2 , . . . , s k consists of elements s i ∈ A such that each number is repeated at most as often as it appears in A. In the following, we define value and weight of a sequence and give additional restrictions to what constitutes a valid sequence. To this end, let S i = s 1 , . . . , s i for i ≤ k denote a prefix of S. We define a value val(s i ) for each s i in the sequence as follows. The first element s 1 has value val(s 1 ) = s 1 . For subsequent elements s i+1 , we have val(s i+1 ) = min{s i+1 , val(s i ), n − val(S i )}, where val(S i ) = j∈[i] val(s j ) is the total value of the prefix S i . 2 If val(s i ) = s i , we say that the element contributes fully to the sequence. Otherwise, it contributes partially. The weight of S is i∈[k] (val(s i ) + 1). For the subsequence S i , we call the next element s i+1 eligible if s i+1 − val(S i ) ≤ val(s i ); s 1 is always eligible. The sequence S is valid if each element is eligible.
To make the connection back to Weighted Clique Partition, interpret the numbers in A as the clique sizes. The total value n corresponds to the number of vertices that have to be covered. The value val(s i ) corresponds to the number of vertices from the maximal clique of size s i in G that have not been covered by previous cliques, i.e., the number of vertices that are newly covered in step i. Note that in step i + 1, at least s i − val(S i ) new vertices are covered as only val(S i ) have been covered previously. Thus, the eligibility requirement ensures that the number of vertices covered in step i + 1 is not larger than the number of vertices covered in step i (recall, that we can assume the chosen cliques to form a non-increasing sequence). Moreover, for the definition of val(s i+1 ), note that the minimum with val(s i ) ensures that the sequence of values is non-increasing and the minimum with n − val(S i ) ensures that the total value is n.
The following two lemmas formalize this connection between Valuable Sequence and Weighted Clique Partition. Afterwards, we discuss how Valuable Sequence can be solved optimally.
Lemma 8.
Let P be a minimum weight clique partition of a graph G and let C be the set of maximal cliques in G. Then, any mapping f : P → C with P ⊆ f (P ) for each P ∈ P is injective and there exists at least one such mapping.
Proof. There are mappings from P to C, because each clique P ∈ P is either a maximal clique or a subset of a larger maximal clique. Assume that a mapping f : P → C is not injective. Then, there are two partition classes P and P that are mapped to the same clique C ∈ C. Thus, these partition classes could be merged, contradicting the optimality of P. Proof. We now show that for a minimum clique partition P of G, we find a solution S of (A, n) whose weight is at most the weight of P.
Let P 1 , . . . , P k be the cliques of P sorted by size in decreasing order. We can think of P as constructed iteratively in that order, so that each P i is a maximal clique in
G[V \( j∈[i−1] P j )].
We construct S iteratively until val(S) = n. For each element s i in the sequence, we prove by induction that val(s i ) ≥ |P i | except for the last element. This then lets us use Lemma 4 to obtain that the weight of S is at most the weight of P. For i = 1, we simply choose s 1 = |P 1 | ∈ A. Since the first element always contributes fully, we have val(s 1 ) = |P 1 |.
Assuming we constructed the sequence until i, we continue with step i + 1 as follows. If P i+1 is one of the initial maximal cliques in C, then we can simply choose s i+1 = |P i+1 | ∈ A. Note that s i+1 is eligible, as s i+1 = |P i+1 | ≤ |P i | ≤ val(s i ), which in particular implies s i+1 − val(S i ) ≤ val(s i ). In this case, s i+1 contributes fully, i.e., val(s i+1 ) = |P i+1 |, which implies the claim.
Otherwise, if P i+1 is not in C, it is at least a subset of some clique C ∈ C such that P i+1 = C \ j∈[i] P j . We choose s i+1 = |C|. The eligibility of s i+1 follows from the facts that the cliques in P are ordered non-increasingly, i.e. |P i | ≥ |P i+1 |, and that val(s j ) ≥ |P j | holds by induction for all j < i + 1:
val(s i ) ≥ |P i | ≥ |P i+1 | = C \ j∈[i] P j ≥ |C|− j∈[i] |P j | ≥ s i+1 − j∈[i] val(s j ) = s i+1 −val(S i ).
Note that s i+1 contributes partially, i.e., val(s i+1 ) = val(s i ) unless this is the last item in S. As we just argued, we have val(s i ) ≥ |P i+1 | and thus val(s i+1 ) ≥ |P i+1 |, proving the claim.
To conclude, observe that our construction of S implicitly defines a mapping from P to C as in Lemma 8. As such a mapping is injective, no number in A is chosen twice. Moreover as we have val(s i ) ≥ |P i | for i < k, but both sum to n, the weight of P is at least the weight of S by Lemma 4.
Valuable Sequence can be solved optimally with a simple greedy algorithm. We call the resulting lower bound the valuable sequence bound. Proof. We construct a solution S = s 1 , . . . , s i by iteratively choosing an eligible and not yet chosen number a ∈ A with maximum value, until the value of the sum reaches n.
We note that this greedy strategy maximizes how many numbers in A are eligible, as the corresponding upper bound val(S) + val(s i ) decreases as slowly as possible. The optimality of the produced sequence S = s 1 , . . . , s i follows, again, via Lemma 4 as for j ∈ [i], the value val(s j ) is at least as large as the value of any other number that can be chosen in round j.
Regarding the running time, the greedy strategy can be implemented by sorting the numbers in A (in O(|A| + n) time) and keeping track of the largest unchosen number that is eligible and contributes fully, as well as the smallest unchosen number that can contribute partially (which is larger than the ones that can contribute fully). Both of these values can be updated in constant time each time a number has been chosen.
Sufficient weight reduction
To speed-up the computation of clique partitions for all bags of a tree decomposition, we additionally apply the following reduction rule. In the sufficient weight reduction, we immediately accept the first solution that is lighter or equally light as the largest weight of any of the already considered bags.
Evaluation
With our evaluation, we aim to answer the following questions.
1.
How do the different algorithms compare in regards to run time and quality? 2. How do the algorithms scale? 3. What is the impact of the lower bounds and the reduction rules on the performance of the exact branch-and-bound solver? 4. How do different network properties influence the performance of the algorithms? 5. How do the resulting upper bounds on the clique-partitioned treewidth compare to traditional treewidth?
Experimental setup. Our implementation is written in Python. The source code along with all evaluation scripts and results is available on our public GitHub repository 3 . The experiments were run with Python 3.10.1 on a Gigabyte R282-Z93 (rev. 100) server (2250MHz) with 1024GB DDR4 (3200MHz) memory. For each input graph, we perform the following two steps. First, we compute a tree decomposition using the heuristics implemented in the HTD library [1]. Specifically, we use the min-fill-in heuristic, which is known to provide a good tradeoff between run time and solution quality [23]. Secondly, we solve the Weighted Clique Partition problem for each bag of the tree decomposition using all algorithms proposed in Section 3.
We use a time limit of five minutes for the heuristic computation of low-weight tree decompositions with the HTD library. For the Weighted Clique Partition algorithms, we set a time limit of three minutes per bag and five minutes in total.
To discern the different solvers from Sections 3.2 and 3.3, in our plots, we use the following abbreviations: branch and bound solver (B&B), maximal clique set cover heuristic (SC), maximal clique weighted set cover heuristic (WSC), maximal clique heuristic (MC), and repeated maximal clique heuristic (RMC).
(a) Number of solved instances with 500 (bright) 5 k (medium) and 50 k (dark) vertices.
(b) Distribution of run time relative to fastest solver within time limit on a given graph.
(c) Distribution of obtained width relative to best found solution on a given graph.
Input instances.
For the input, we use a large collection of real-world networks as well as generated networks. For the latter, we use geometric inhomogeneous random graphs (GIRGs) [8], which resemble real-world networks in regards to important properties and have been shown to be well suited for the evaluation of algorithms [3]. GIRGs can be generated efficiently [4] and allow to vary the power-law exponent (ple) of the degree distribution controlling its heterogeneity, as well as a parameter α controlling the locality by either strengthening the influence of the geometry (high values of α) or increasing the probability for random edges not based on the geometry (low values of α). We mainly use the following two datasets, where each graph has been reduced to its largest connected component.
A collection of 2967 real-world networks [7] that essentially consists of all networks with at most 1 M edges from Network Repository [25]; see [3] for details. GIRGs with n ∈ {500, 5000, 50000} vertices, expected average degree 10, dimension 1, ple ∈ {2.1, 2.3, 2.5, 2.7, 2.9}, and α ∈ {1.25, 2.5, 5, ∞}. For each parameter configuration, we generate ten networks with different random seeds, to smooth out random variations.
Performance comparison
Here, we evaluate the performance of our Clique Partition approaches on the two datasets.
Generated instances. In Figure 1, we compare the run times as well as the solution quality of the different considered Clique Partition algorithms on the dataset of generated networks. In Figure 1a, we show how many of the 600 instances were solved within the time limit by each solver. While the greedy heuristics are able to finish on almost all instances, the set cover heuristic and the branch-and-bound solver get timed on some of the larger networks with 5 k and 50 k vertices. The weighted set cover heuristic performs much worse, finishing only on few networks. We therefore exclude it from the other comparisons.
In Figures 1b and 1c, we compare the performance for all instances that were solved within the time limit by all other algorithms. Figure 1b shows the run time of each algorithm relative to the fastest one on each instance. Figure 1c shows the obtained upper bound on the cp-treewidth relative to the optimal solution computed by the branch-and-bound solver.
Our findings are as follows. The branch-and-bound solver solves the fewest instances of the four considered algorithms, but is quick on most of the instances it is able to solve within the time limit. Both greedy heuristics (MC and RMC) are similarly fast, significantly outcompeting the other approaches. In terms of quality, all three heuristics perform well, achieving solutions within few percent of the optimum. The set cover heuristic slightly outperforms the greedy heuristics in terms of quality, but pays for this with substantially higher running time.
Real-world networks. We complement the above evaluation of our Clique Partition algorithms, by comparing their performance on the collection of real-world networks. As above, we exclude the weighted set cover heuristic. The other four approaches were able to finish on 1243 (B&B), 2619 (MC), 2622 (RMC), and 2204 (SC) of the 2967 networks within the time limit. We compare our algorithms on the 1237 networks that were solved by all four approaches. Figure 2 shows the run time of each solver relative to the fastest solver on each instance. In Table 1 we describe the distribution of the obtained upper bounds on the cp-treewidth relative to the optimal solution found by the branch-and-bound solver.
Our results are the following. In general, our observations on generated networks are replicated on the real-world networks. Even though the branch-and-bound algorithm solved fewer instances than the set cover heuristic, it is comparatively faster on the networks it is able to solve. Both approaches are, however, considerably slower than the greedy heuristics and this difference is more pronounced than on the generated networks. Regarding the solution quality, all three heuristic solvers perform even better than on the generated networks, with only a tiny fraction of instances not being solved almost optimally.
Discussion. We find that the proposed algorithms show good performance both on generated and real-world instances. Although, the branch-and-bound solver was only able to solve about half of the considered networks, it's run time typically beats the set cover heuristic on the networks it can solve. In addition, it is a valuable tool for evaluating the solution quality of the other approaches. We find that especially the set cover heuristic, but also the greedy heuristics (MC and RMC) often find close to optimal clique partitions. Due to their excellent trade-off between speed and solution quality, the greedy heuristics are probably the best approach in most practical settings. In general, we do not expect that there is substantial room for improvement in the engineering of Clique Partition solvers for the computation of cp-treewidth. Instead, in order to achieve better upper bounds, we suggest future research to optimize the tree decomposition and the partition into cliques at the same time.
Run time scaling
Next, we consider the scaling behavior of our solvers. For this, we generated GIRGs of varying sizes up to around 50 k vertices for various parameters. As in Section 4.1, we did not evaluate the weighted set cover heuristic. Figure 3 shows the run times for GIRGs with two different parameter configurations. On the networks with high locality (α = ∞), all four approaches seem to have close to linear run time, despite enumerating all maximal cliques present in each bag of the tree decomposition. However, as we decrease the locality (α = 5) the performance of the branch-and-bound solver deteriorates while the greedy heuristics and especially the set cover heuristic are only slightly affected. In the logarithmic plot, we observe clearly super-polynomial scaling behavior only for the branch-and-bound solver. Further experiments on a larger grid of parameter settings confirm the above findings.
Branch-and-bound: lower bounds and reduction rule
In the following, we evaluate the effectiveness of the lower bounds and the reduction rule in speeding up our branch-and-bound solver. For this, we use the dataset of generated networks. As the performance without lower bounds does not allow for the timely evaluation on larger instances, we consider only graphs generated with 5 k vertices. Figure 4 shows the average run time without lower bounds (none), with only the size lower bound (S) and with the valuable sequence bound in addition to the size bound (S+V) as well as with and without the sufficient weight reduction for different network parameters. We only show α ∈ {5, ∞}, as for lower values the variant without lower bounds did not finish within the time limit.
We find that especially for smaller power-law exponents, the lower bounds bring large speed-ups of up to multiple orders of magnitude. The additional gain of using the size lower bound is much larger than that of the much simpler valuable sequence bound. The sufficient weight reduction yields similar speed-ups for all settings. Overall, we conclude that the lower bounds are effective in speeding up the branch-and-bound solver. On a more general note, it is striking how strongly all variants of the solver are affected by lower values of α, especially also below the values shown in Figure 4. In additional experiments we found that the above observations also apply to the remainder of the dataset, even though for 50 k vertices the time limit is reached even more frequently.
Impact of network properties
At multiple points throughout the last sections, we found that, especially for the branch-andbound algorithm, the performance strongly depended on the parameter α controlling the locality of the generated networks.
In order to better understand this, we study the structure of cliques in the generated networks depending on their parameters. Specifically, for each network we count the number of maximal cliques in the graph, we count the number of maximal cliques in each bag of the tree decomposition and take the maximum, we count the number of cliques used per bag in the clique-partitioned tree decomposition and take the maximum, and consider the width of the clique-partitioned tree decomposition. The clique-partitioned tree decompositions are obtained using the MC and MCR heuristic. Figure 5 shows these values for GIRGs with varying power-law exponent and α.
We see that with decreasing values of α, all considered measures increase. However, while the total number of maximal cliques in the network only increases by a factor of roughly 4 to 10, the highest number of cliques intersecting some bag of the tree decomposition as well as the highest number of cliques in a lowest-weight clique partition increase by multiple orders of magnitude. Intuitively, this can be explained by cliques starting to fray if the locality is too low. This explains, why the Clique Partition problem is harder on GIRGs with lower (a) Dependency between clustering coefficient and heuristic upper bounds on clique-partitioned treewidth and treewidth.
(b) Dependency between clustering coefficient and relative difference between clique-partitioned treewidth and treewidth. values of α, which slows down the branch-and-bound algorithm. We also observe, that the obtained upper bounds on the cp-treewidth are not much lower than the highest number of cliques per bag of a solution, explaining the good performance of the set cover heuristic.
Clique-partitioned treewidth compared to traditional treewidth
Here we consider the data set of real-world networks. As we have seen in Section 4.1, the maximal clique and repeated maximal clique heuristics are efficient and tend to perform well in terms of quality. Thus, we use these two heuristics to find an upper bound on the clique-partitioned treewidth.
In Figure 6a we compare the obtained upper bounds for the weighted treewidth and the treewidth. Even though the parameter does not decrease much for the majority networks, there are some networks on which substantial reductions are achieved. This is particularly true for networks with high clustering coefficient, where for some instances our clique-partitioned tree decomposition has width 10 while the corresponding traditional tree decomposition has width above 100. This correspondence with the clustering coefficient fits well to the observations in Section 4.4. For the networks for which we do not yet see a big improvement, it would be interesting to see whether adjusting the computation of the initial tree decomposition can yield better bounds; see also the discussion in Section 4.1.
(a) Unweighted set cover.
(b) Weighted set cover.
Figure 7
Counter-examples for the optimality of the set cover heuristics.
A Limits of the set cover heuristics
We want to briefly discuss why the set cover solutions are not always optimal clique partitions. First, we give an instance on which the unweighted set cover approach fails.
Observation 11. There are graphs on which the minimum size clique cover cannot give an optimal clique partition.
Proof. Consider a clique on k vertices for even k where half of the vertices are connected to one additional vertex and the other half to another additional vertex, as illustrated in Figure 7a. Then, for k ≥ 6 the partition into three cliques of sizes 1, 1, and k has lower weight than the partition into two cliques of size k 2 + 1, which corresponds to the optimal solution of the set cover instance.
For the minimum weight set cover, we can use the fact that the weights of the set cover instance correspond to the size of the whole clique and do not reflect the potential overlap between multiple selected cliques.
Observation 12.
There are graphs on which the minimum size clique cover cannot give an optimal clique partition.
Proof. For the weighted approach, consider the instance depicted in Figure 7b. The small circles represent the vertices of a graph and the regions mark maximal cliques. The optimal clique partitioning uses cliques of sizes 6, 2, and 1 (the dotted clique plus the remainders of the two solid cliques). In the set cover instance these cliques have (partly overlapping) sizes 6, 4, and 4, which is more expensive than the set cover solution with sizes 5, 4, and 4 (using the dashed clique instead of the dotted one), which results in a solution with sizes 5, 3, 1.
The above problem could be avoided by extending the set cover instance to also include all non-maximal subsets of each clique that can be obtained by removing vertices shared with some subset of overlapping cliques. This would, however, lead to an exponential blowup of the set cover instances, which is not feasible even with state of the art solvers.
Lemma 4 .
4Let s 1 , . . . , s k and r 1 , . .
Lemma 9 .
9Let G be a graph with maximal cliques C = {C 1 , . . . , C k } and let (A, n) with A = {|C 1 |, . . . , |C k |} and n = |V (G)| be an instance of Valuable Sequence. The weight of a minimum solution of (A, n) is a lower bound for the weight of every clique partition of G.
Theorem 10 .
10An instance (A, n) of Valuable Sequence can be solved in O(|A| + n) time.
Figure 1
1Comparison of run time and solution quality of the different exact (red), greedy (green, blue) and set cover based (violet) solvers for the Weighted Clique Partition problem on GIRGs.
Figure 2
2Distribution of run time relative to fastest solver within time limit on our set of real-world networks.
Figure 3
3Scaling behavior of Clique Partition algorithms on GIRGs with different parameters.
Figure 4
4Run time of different variants of the branch-and-bound solver on GIRGs with 5 k vertices and different values for the power-law exponent (left to right) and α (top / bottom).
Figure 5
5Total clique count (number of maximal cliques) per network, and highest clique count in any bag of a greedy tree decomposition as well in the lowest weight clique partition of any bag, and clique-partitioned treewidth (lowest upper bound) of the entire instance on GIRGs with 5 k vertices and varying parameters. Note the logarithmic y-axes on all except the first plot.
Figure 6
6Upper bounds for clique-partitioned treewidth on large real-world networks.
Table 1
1Distribution of obtained cp-treewidth relative to optimum clique partition on our set of real-world networks.Measure MC
RMC SC
Mean 1.008 1.009
1.002
Median 1
1
1
90th percentile 1.035 1.036
1.000
99th percentile 1.108 1.121
1.046
Maximum 1.254 1.192
1.113
https://igraph.org/
Note that val(s i+1 ) only depends on values of previous elements in S, i.e., the definition is not cyclic.
https://github.com/marcwil/cptw_code
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| {'fraction_non_alphanumeric': 0.04725372638446321, 'fraction_numerical': 0.029358019843846745, 'mean_word_length': 4.3221347050313454, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 4, 'https://': 5, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With this paper, we take a first step towards computing clique-partitioned tree decompositions.Our focus lies on the subproblem of computing clique partitions, i.e., for each bag of a given tree decomposition, we compute an optimal partition of the induced subgraph into cliques. The goal here is to minimize the product of the clique sizes (plus 1). We show that this problem is NP-hard. We also describe four heuristic approaches as well as an exact branch-and-bound algorithm. Our evaluation shows that the branch-and-bound solver is sufficiently efficient to serve as a good baseline. Moreover, our heuristics yield solutions close to the optimum. As a bonus, our algorithms allow us to compute first upper bounds for the clique-partitioned treewidth of real-world networks. A comparison to traditional treewidth indicates that clique-partitioned treewidth is a promising parameter for graphs with high clustering.', 'arxivid': '2302.08870', 'author': ['Thomas Bläsius \nKarlsruhe Institute of Technology\nGermany\n', 'Maximilian Katzmann \nKarlsruhe Institute of Technology\nGermany\n', 'Marcus Wilhelm \nKarlsruhe Institute of Technology\nGermany\n'], 'authoraffiliation': ['Karlsruhe Institute of Technology\nGermany', 'Karlsruhe Institute of Technology\nGermany', 'Karlsruhe Institute of Technology\nGermany'], 'corpusid': 257019639, 'doi': '10.48550/arxiv.2302.08870', 'github_urls': ['https://github.com/marcwil/cptw_code'], 'n_tokens_mistral': 18934, 'n_tokens_neox': 16735, 'n_words': 10533, 'pdfsha': '61a1d37ebbd5c9d383aa08a385169a5f3264e32b', 'pdfurls': ['https://export.arxiv.org/pdf/2302.08870v1.pdf'], 'title': ['Partitioning the Bags of a Tree Decomposition Into Cliques', 'Partitioning the Bags of a Tree Decomposition Into Cliques'], 'venue': []} |
arxiv |
Forecasting Sequential Data using Consistent Koopman Autoencoders
Omri Azencot [email protected]
N Benjamin Erichson [email protected]
Vanessa Lin [email protected]
Michael W Mahoney [email protected]
UC
Los Angeles
UC Berkeley
ICSI
UC Berkeley
ICSI
UC Berkeley
ICSI
Forecasting Sequential Data using Consistent Koopman Autoencoders
Recurrent neural networks are widely used on time series data, yet such models often ignore the underlying physical structures in such sequences. A new class of physically-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. In this work, we propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis that unravels the interplay between consistent dynamics and their associated Koopman operators. Our network is interpretable from a physical viewpoint and its computational requirements are comparable to other baselines. We evaluate our method on a wide range of high-dimensional and short-term dependent problems. The datasets include nonlinear oscillators, sea surface temperature data, and fluid flows on a curved domain. The results show that our model yields accurate estimates for significant prediction horizons, while being robust to noise. * Equal contribution.
INTRODUCTION
Sequential data processing and forecasting is a fundamental problem in the engineering and physical sciences. Recurrent Neural Networks (RNNs) provide a powerful class of models for these tasks, designed to learn long-term dependencies via their hidden state variables. However, training RNNs over long time horizons is notoriously hard (Pascanu et al., 2013) due to the problem of exploding and vanishing gradients (Bengio et al., 1994). Several approaches have been proposed to mitigate this issue using unitary hidden-to-hidden weight matrices (Arjovsky et al., 2016) or analyzing stability properties (Miller & Hardt, 2019), among other solutions. Still, attaining long-term memory remains to be a challenge and, moreover, short-term dependencies might be affected due to limited expressivity of unitary RNNs (Kerg et al., 2019).
Another shortcoming of traditional RNNs is their lack of interpretability. In this context, physicallybased methods have been proposed, relating RNNs to dynamical systems (Sussillo & Barak, 2013) or differential equations (Chang et al., 2019). This point of view allows to construct models which enjoy high-level properties such as time invertibility via Hamiltonian (Greydanus et al., 2019) or Symplectic (Chen et al., 2020;Zhong et al., 2020) networks. Other techniques suggested reversible RNNs (MacKay et al., 2018;Chen et al., 2018) to alleviate the large memory footprints RNNs induce during training. In this work, we advocate that modeling time-series data which exhibit strong short-term dependencies can benefit from relaxing the strict stability and time invertibility requirements.
An interesting physically motivated alternative for analyzing time series data has been introduced in Koopman-based models (Takeishi et al., 2017;Morton et al., 2018Morton et al., , 2019Li et al., 2020). Koopman theory is based on the insight that a nonlinear dynamical system can be fully encoded using an operator that describes how scalar functions propagate in time. The Koopman operator is linear, and thus preferable to work with in practice as tools from linear algebra and spectral theory can be directly applied. While Koopman's theory (Koopman, 1931) was established almost a century ago, significant advances have been recently accomplished in the theory and methodology with applications in the fluid mechanics (Mezić, 2005) and geometry processing (Roufosse et al., 2019) communities.
The Koopman operator maps between function spaces and thus it is infinite-dimensional and can not be represented on a computer. Nevertheless, most machine learning approaches hypothesize that there exists a data transformation under which an approximate finite-dimensional Koopman operator is available. Typically, this map is represented via an autoencoder network, embedding the input onto a low-dimensional encode decode Nonlinear Linear ϕ(z k ) Cq k Figure 1: Illustration of a (non-linear) data transformation that maps the high-dimensional states z k (evolving on a nonlinear trajectory) to a new latent space in which the dynamics are linearized. latent space. In that space, the Koopman operator is approximated using a linear layer that encodes the dynamics (Takeishi et al., 2017). The main advantage of this framework is that the resulting models are highly interpretable and allow for accurate prediction of short-term dependent data. Specifically, predicting forward or backward in time can be attained via subsequent matrix-vector products between the Koopman matrix and the latent observation. Similarly, stability features can be analyzed and constrained via the operator spectrum. Based on the Koopman theory, we propose a new model for forecasting high-dimensional time series data. In contrast to previous approaches, we assume that the backward map exists. That is, the system from a future to current time can be properly defined. While not all systems exhibit this feature (e.g., diffusive systems), there are many practical cases where this assumption holds. We investigate the interplay between the forward and backward maps and their consistency in the discrete and continuous space settings. Our work can be viewed as relaxing both reversibility and stability requirements, leading to higher expressivity and improved forecasts. Ideas close to ours have been applied to training Generative Adversarial Networks (Zhu et al., 2017;Hoffman et al., 2018). However, to the best of our knowledge, our work is first to establish the link between consistency of latent variables and dynamical systems.
Background and Problem Setup
In what follows, we focus on dynamical systems that can be described by a time-invariant model
z k+1 = ϕ(z k ) , z ∈ M ⊂ R m ,(1)
where z k denotes the state of the system at time k ∈ N. The map ϕ : M → M is a (potentially non-linear) update rule on a finite dimensional manifold M, pushing states from time k to time k + 1. The above model assumes that future states depend only on the current state z k and not on information from a sequence of previous states. In order to predict future states, one might be tempted to train a neural network which learns an approximation of the map ϕ. However, the resulting model ignores prior knowledge about the problem and is potentially difficult to interpret and analyze. As an alternative, one could seek a data transformation for the states z k so that the corresponding latent variables q k evolve on a linear path, as illustrated in Figure 1. In turn, the dynamics could be approximated by a linear model, which improves interpretability and facilitates the integration of prior physical knowledge into the training process. Intriguingly, Koopman theory suggests that there exists a data transformation for any non-linear dynamical system so that the states can be pushed from time k to time k + 1 by a linear map. In this paper, we advocate the use of Koopman's perspective on data which are high-dimensional or exhibit strong short-term dependencies.
More concretely, the dynamics ϕ induces an operator K ϕ that acts on scalar functions f : M → R ∈ F , with F being some function space on M (Koopman, 1931). Formally, the Koopman operator is given by i.e., the function f is composed with the map ϕ. Intuitively, the operator details the evolution of a scalar function by pulling-back its values from a future time. In other words, K ϕ f at z is the value of f evaluated at the future state z k+1 . Hence, the Koopman operator is also commonly known as the pull-back operator. Also, it is easy to show that K ϕ is linear for any α, β ∈ R
K ϕ f (z) = f • ϕ(z) ,(2)K ϕ (α f + βg) = (α f + βg) • ϕ = α f • ϕ + βg • ϕ = αK ϕ f + βK ϕ g .
Finally, we assume that the backward dynamics ψ exists, and we denote by U ψ the associated Koopman operator. In Fig. 2, we show an illustration of our setup. Unfortunately, K ϕ is infinite-dimensional. Nevertheless, the key assumption in most of the practical approaches is that there exists a transformation χ whose conjugation with K ϕ leads to a finite-dimensional approximation which encodes "most" of the dynamics. Formally,
C = χ • K ϕ • χ −1 , C ∈ R κ×κ ,(3)
i.e., χ and its inverse extract the crucial structures from K ϕ , yielding an approximate Koopman matrix C. Similarly, we denote by D = χ • U ψ • χ −1 the approximate backward system. The main focus in this work is to find the matrices C and D, and a nonlinear transformation χ such that the underlying dynamical system is recovered well. We assume to be given scalar observations of the dynamics { f k : M → R} n k=1 such that
f k+1 = f k • ϕ + r k ,
where the function r k ∈ F represents deviation from the true dynamics due to e.g., measurement errors or missing values. We focus on the case where ϕ and ψ are generally unknown, and our goal is to predict future observations from the given ones. Namely,
f k+l = f k • ϕ l , l = 1, 2, ... ,(4)
where ϕ l means we repeatedly apply the dynamics. In practice, as C approximates the system, we exploit the relation χ −1 • C l • χ( f k ) ≈ f k • ϕ l to produce further predictions. That is, the matrix C fully determines the forward evolution of the input observation f k .
Main Contributions
Our main contributions are as follows.
• We develop a Physically Constrained Learning (PCL) framework based on Koopman theory and consistent dynamics for processing complex time series data.
• Our model is effective and interpretable and its features include accurate predictions, time reversibility and a stable behavior even over long time horizons.
• We evaluate on high-dimensional clean and noisy systems including the pendulum, cylinder flow, vortex flow on a curved domain, and climate data, and we achieve exceptionally good results with our model.
RELATED WORK
Modeling dynamical systems from Koopman's point-of-view has gained increasing popularity in the last few years (Mezić, 2005). An approximation of the Koopman operator can be computed via the Dynamic Mode Decomposition (DMD) algorithm (Schmid, 2010). While many extensions of the original algorithm have been proposed, most related to our approach is the work of Azencot et al. (2019) where the authors consider the forward and backward dynamics in a non-neural optimization setting. A network design similar to ours was proposed by Lusch et al. (2018), but without our analysis, back prediction and consistency terms. Other techniques minimize the residual sum of squares (Takeishi et al., 2017;Morton et al., 2018), promote stability (Erichson et al., 2019b;Pan & Duraisamy, 2020), or use graph convolutional networks (Li et al., 2020). Sequential data are commonly processed using RNNs (Elman, 1990;Graves, 2012). The main difference between standard neural networks and RNNs is that the latter networks maintain a hidden state which uses the current input and previous inner states. Variants of RNNs such as Long Short Term Memory (Hochreiter & Schmidhuber, 1997) and Gated Recurrent Unit (Cho et al., 2014) have achieved groundbreaking results on various tasks including language modeling and machine translation, among others. Still, training RNNs involves many challenges and a recent trend in machine learning focuses on finding new interpretations of RNNs based on dynamical systems theory (Laurent & von Brecht, 2016;Miller & Hardt, 2019).
Several physically motivated models have been recently proposed. Based on Lagrangian mechanics, Lutter et al. (2019) encoded the Euler-Lagrange equations into their network to attain physical plausibility and to alleviate poor generalization of deep models. Other methods attempt to learn conservation laws from data and their associated Hamiltonian representation, leading to exact preservation of energy (Greydanus et al., 2019) and better handling of stiff problems (Chen et al., 2020). To deal with the limited expressivity of unitary RNNs, Kerg et al. (2019) suggested to employ the Schur decomposition to their connectivity matrices. By considering the normal and nonnormal components, their network allows for transient expansion and compression, leading to improved results on tasks which require continued computations across timescales.
METHOD
In what follows, we describe our PCL framework which we use to handle time series data. Similar to other methods, we model the transformation χ above via an encoder χ e and a decoder χ d . Our approach differs from other Koopman-based techniques (Lusch et al., 2018;Takeishi et al., 2017;Morton et al., 2018) in two key components. First, in addition to modeling the forward dynamics, our network also takes into account the backward system. Second, we require that the resulting forward and backward Koopman operators are consistent.
Autoencoding Observations
Given a set of observations F = { f k } n k=1 as defined in Sec. 1.1, we design an autoencoder (AE) to embed our inputs in a low-dimensional latent space using a nonlinear map χ e . The decoder χ d map allows to reconstruct latent variables in the spatial domain. To train the AE, we definẽ
C D f k f k+1 χ e |χ d χ d |χ ef = χ d • χ e ( f ) ,(5)E id = 1 2n n ∑ k=1 f k −f k 2 2 ,(6)
i.e.,f is the reconstructed version of f , and E id derives the optimization to obtain an AE such that
χ d • χ e ≈ id.
We note that the specific requirements from χ e and χ d are problem dependent, and we detail the particular design we used in Appendix A.
Backward Dynamics
In general, a dynamical system ϕ prescribes a rule to move forward in time. There are numerous practical scenarios where it makes sense to consider the backward system, i.e., ψ : z k → z k−1 . For instance, the Euler equation, which describes the motion of an inviscid fluid, is invariant to sign changes in its time parameter (see Fig. 11 for an example). Previous approaches incorporated the backward dynamics into their model as in bi-directional RNNs (Schuster & Paliwal, 1997). However, the inherent nonlinearities of a typical neural network make it difficult to constrain the forward and backward models. To this end, a few approaches were recently proposed (Greydanus et al., 2019;Chen et al., 2020) where the obtained dynamics are reversible by construction due to the leapfrog integration. In comparison, most existing Koopman-based techniques do not consider the backward system in their modeling or training.
To account for the forward as well as backward dynamics, we incorporate two linear layers with no biases into our network to represent the approximate Koopman operators. As we assume that χ e transforms our data into a latent space where the dynamics are linear, we can directly evolve the dynamics in that space. We introduce the following notation
f k+1 = χ d • C • χ e ( f k ) ,(7)f k−1 = χ d • D • χ e ( f k ) ,(8)
for every admissible k. Namely, the Koopman operators C, D ∈ R κ×κ allow to obtain forward estimateŝ f k+1 , and backward forecastsf k−1 .
In practice, we noticed that our models predict as well as generalize better if instead of computing one step forward and backward in time, we employ a multistep forecasting. Given a choice of λ s ∈ N, the total number of prediction steps, we define the following loss terms
E fwd = 1 2λ s n λ s ∑ l=1 n ∑ k=1 f k+l −f k+l 2 2 ,(9)E bwd = 1 2λ s n λ s ∑ l=1 n ∑ k=1 f k−l −f k−l 2 2 ,(10)
where we assume that f k+l and f k−l are provided during training for any l ≤ λ s , see Eq. (4). Also,f k+1 anď f k−l are obtained by taking powers l of C, respectively D, in Eqs. (7), (8). We show in Fig. 3 a schematic illustration of our network design including the encoder and decoder components as well as the Koopman matrices C and D. Notice that all the connections are bi-directional, that is, data can flow from left to right and right to left.
Backward Prediction.
Koopman operators and their approximating matrices are linear objects that allow for greater flexibility when compared to other models for time series processing. One consequence of this linearity is that while existing Koopman-based nets (Lusch et al., 2018;Takeishi et al., 2017;Morton et al., 2018) are geared towards forward prediction, their evolution matrix C can be exploited for backward prediction as well. This computation is obtained via the inverse of C, i.e.,
f k−1 = χ d • C −1 • χ e ( f k ) .(11)
However, models that were trained for forward prediction typically produce poor backward predictions as we show in Appendix B. In contrast, we note that our model allows for the direct back prediction using the D operator and Eq. (8). Thus, while other techniques can technically produce backward predictions, our model supports it by construction.
Consistent Dynamics. The backward prediction penalty E bwd in itself only affects D, and it is completely independent of C. That is, C will not change due to backpropagating the error in Eq. (10). To link between the forward and backward evolution matrices, we need to introduce an additional penalty that promotes consistent dynamics. Formally, we say that the maps ϕ and ψ are consistent if ψ • ϕ( f ) = f for any f ∈ F . In the Koopman setting, we will show below that this condition is related to requiring that DC = I κ , where I κ is the identity matrix of size κ. However, our analysis shows that in fact the continuous space and discrete space settings differ, yielding related but different penalties. In this work, we will incorporate the following loss to promote consistency
E con = κ ∑ k=1 1 2k D k * C * k − I k 2 F + 1 2k C k * D * k − I k 2 F ,(12)
where D k * and C * k are the upper k rows of D and leftmost k columns of the matrix C, and · F is the Frobenius norm.
Stability.
Recently, stability has emerged as an important component for analyzing neural nets (Miller & Hardt, 2019). Intuitively, a dynamical system is stable if nearby points stay close under the dynamics. Mathematically, the eigenvalues of a linear system fully determine its behavior, providing a powerful tool for stability analysis. Indeed, the challenging problem of vanishing and exploding gradients can be elegantly explained by bounding the modulus of the weight matrices' eigenvalues (Arjovsky et al., 2016).
To overcome these challenges, one can design networks that are stable by construction, see e.g., Chang et al. (2019), among others. We, on the other hand, relax the stability constraint and allow for quasi-stable models. In practice, our loss term (12) regularizes the nonconvex minimization by promoting the eigenvalues to get closer to the unit circle. The comparison in Fig. 4 highlights the stability features our model attains, whereas a non regularized network obtains unstable modes. From an empirical viewpoint, unstable behavior leads to rapidly diverging forecasts, as we show in Sec. 5.
A Consistent Dynamic Koopman Autoencoder
Combining all the pieces together, we obtain our PCL model for processing time series data. Our model is trained by minimizing a loss function whose minimizers guarantee that we achieve a good AE, that predictions through time are accurate, and that the forward and backward dynamics are consistent. We define our loss
E = λ id E id + λ fwd E fwd + λ bwd E bwd + λ con E con ,(13)
where λ id , λ fwd , λ bwd , λ con ∈ R + are user-defined positive parameters that balance between reconstruction, prediction and consistency. Finally, E id , E fwd , E bwd , E con are defined in Eqs. (6), (9), (10), and (12), respectively.
CONSISTENT DYNAMICS VIA KOOPMAN
We now turn to prove a necessary and sufficient condition for a dynamical system to be invertible from a Koopman viewpoint in the continuous space setting. We then show a similar result in the spatial discrete case, yielding a more elaborate condition which we use in practice. We recall that Koopman operators take inputs and return outputs from a function space F . Thus, many properties of the underlying dynamics can be related to the action of K ϕ on every function in F . A natural approach in this case is to consider a spectral representation of the associated objects. Specifically, we choose an orthogonal basis for F which we denote by {ξ k } ∞ k=0 where for any i, j we have
ξ i , ξ j M = M ξ i (z) ξ j (z) d z = δ ij ,(14)
with δ ij being the Kronecker delta function. Under this choice of basis, any function f ∈ F can be represented by f = ∑ k f , ξ k M ξ k . Moreover, due to the linearity of K ϕ we also have that K ij = ξ i , K ϕ ξ j M . In the following proposition we characterize invertibility in the continuous-space case, which is a known result in Ergodic theory (Eisner et al., 2015).
Proposition 1 Given a manifold M, the dynamical system ϕ is invertible if and only if for every i and j the Koopman operators K ϕ and U ψ satisfy ξ i , U ψ K ϕ ξ j M = δ ij .
Proof. If ϕ is invertible then the composition ψ • ϕ = id for every z ∈ M. Thus, Conversely, we assume that ξ i , U K ξ j M = δ ij for all i, j. It follows that for every k we have U K ξ k = ξ k since ξ k is orthogonal to every ξ l , l = k. Let f be some scalar function, then f (ψ • ϕ(z)) = U K f (z) = f (z) and thus ψ • ϕ = id. The main advantage of Prop.
ξ i , U K ξ j M = M ξ i (z) ξ j (ψ • ϕ(z)) d z = ξ i , ξ j M .
(1) is that it can be used directly in a computational pipeline. In particular, if we denote by C and D the κ × κ matrices that approximate K and U , respectively, then the above condition takes the form
1 2 D C − I κ 2 F = 0 ,(15)
where I κ is an identity matrix of size κ. This loss term was recently used in (Azencot et al., 2019) to construct a robust scheme for computing DMD operators. However, we prove below that in the discrete-space setting, a more elaborate condition is required. To discretize the above objects, we assume that our manifold M is represented by the domain M ⊂ R d which is sampled using m vertices. In this setup, scalar functions f : M → R are vectors f ∈ R m storing values on vertices with in-between values obtained via interpolation. A map ϕ : M → M can be encoded using a matrix P ϕ ∈ R m×m defined by
P ϕ δ z = h ϕ(z) ,(16)
where h x is a function that stores the vertices' coefficients such that h T x X = x T , with X ∈ R m×d being the spatial coordinates of M. Similarly, we denote by Q ψ the matrix associated with ψ, i.e., Q ψ δ z = h ψ(z) . Note that P and Q are in fact discrete Koopman operators represented in the canonical basis. We show in Fig. 5 an illustration of our spatial discrete setup including some of the notations.
Similar to the continuous setting, we can choose a basis for the function space on M. We denote B ∈ R m×m as the matrix that contains the orthogonal basis elements in its columns, i.e., b i , b j M = b T i b j = δ ij for every i, j. We use this basis to define the matrices C and D by
C = B T P ϕ B , D = B T Q ψ B .(17)
Finally, instead of invertible maps, we consider consistent maps. That is, a discrete map ϕ is consistent if for every z, we have that ψ • ϕ(z) = z. Using the above constructions and notations, we are ready to state our main result.
Proposition 2 Given a domain M, the map ϕ is consistent if and only if for every i and j the matrices C and D satisfy ∑ k 1 2k D k * C * k − I k 2 F = 0, where D k * and C * k are the upper k rows of D and leftmost k columns of the matrix C. Figure 5: The Kronecker delta δ z centered at z is mapped via P ϕ to the function h ϕ(z) which holds the coefficients of the vertices of M whose combination generates ϕ(z).
0 1 P ϕ z ϕ(z) δ z h ϕ(z)
Proof. If ϕ is a consistent map, then Q Pδ z = δ z for every z and thus Q P = I. In addition, for every k we have that
D k * C * k = B T k QBB T PB k = B T k QPB k = B T k B k = I k ,
where B k are the first k basis elements of B and BB T = I. Conversely, we assume that C, D are related to some maps ϕ and ψ and constructed via Eq (17). In addition, the condition ∑ k 1 2k D k * C * k − I k 2 F = 0 holds. Then B T k Q PB k = I k for every k. By induction on k it follows that Q P b k = b k for every k where b k is the kth column of B and thus Q P = I as B spans the space of scalar functions on M.
EXPERIMENTS
To evaluate our proposed consistent dynamic Koopman AE, we perform a comprehensive study using various datasets and compare to state-of-the-art Koopman-based approaches as well as other baseline sequential models. Our network minimizes Eq. (13) with a decaying learning rate initially set to 0.01. We fix the loss weights to λ id = λ fwd = 1, λ bwd = 0.1, and λ con = 0.01, for the AE, forward forecast, backward prediction and consistency, respectively. We use λ s = 6 prediction steps forward and backward in time. We provide additional details in Appendix A.
Baselines
Our comparison is mainly performed against the state-of-the-art method of Lusch et al. (2018), henceforth referred to as the Dynamic AE (DAE) model. Their approach may be viewed as a special case of our network by setting λ bwd = λ con = 0. While we use this work as a baseline, other models such as (Takeishi et al., 2017;Morton et al., 2018) could be also considered. The main difference between DAE and the latter techniques is the least squares solution for the evolution matrix C per training iteration. In our experience, this change leads to delicate training procedures and thus it is less favorable. Unless said otherwise, both models are trained using the same parameters, where DAE does not have the regularizing penalties.
We additionally compare against a feed-forward model and a recurrent neural network. The feedforward network simply learns a nonlinear function ζ : f k → f k+1 , where during inference we takef k+1 as input for predictingf k+2 , and so on. The RNN adds an hidden state h k such that h k = σ(U f k + Wh k−1 + b), and the prediction is obtained viaf k = Vh k + c. We performed a parameter search when comparing with these baselines.
Nonlinear Pendulum with no Friction
The nonlinear (undamped) pendulum (Hirsch et al., 1974) is a classic textbook example for dynamical systems, which is also used for benchmarking deep models (e.g., Greydanus et al., 2019;Bertalan et al., 2019; Chen et al., 2020). This problem can be modeled as a second order ODE by d 2 θ dt 2 + g sin θ = 0, where the angular displacement from an equilibrium is denoted by θ ∈ [0, 2π). We use l and g to denote the length and gravity, respectively, with l = 1 and g = 9.8 in practice. We consider the following initial conditions θ(0) = θ 0 andθ(0) = 0. The motion of the pendulum is approximately harmonic for a small amplitude of the oscillation θ 0 1. However, the problem becomes inherently nonlinear for large amplitudes of the oscillations.
We experiment with oscillation angles θ 0 = 0.8 and θ 0 = 2.4 on the time interval t = [0, 51]. The data are generated using a time step ∆t = 0.03, yielding T = 1700 equally spaced points x 1 , ..., x T ∈ R 2 . In addition, we map the sequence {x t } to a high-dimensional space via a random orthogonal transformation to obtain the training snapshots, i.e., P ∈ R 64×2 such that f t = P x t for any t. Finally, we split the new sequence into a training set of 600 points and leave the rest for the test set. We show in Fig. 6 examples of the clean and noisy trajectories for these data.
Experimental results. Fig. 7 shows the pendulum results for initial conditions θ 0 = 0.8 (top row) and θ 0 = 2.4 (bottom row). We used a bottleneck κ = 6 and α = 0.5 for the DAE and our models. (The parameter α controls the width of the network, i.e., the number of neurons used for the hidden layers.) The relative forecasting error is computed at each time step via f t −f t 2 / f t 2 , wheref t is the high-dimensional estimated prediction, see also Eq. (7). We forecast over a time horizon of 1000 steps, and we average the error over 30 different initial observations f t , where the shaded areas represent the ±1 standard deviations. Overall, our model yields the best or second best results in all the cases we explored. The RNNs obtains good measures in the clean case and for short prediction times, but its performance deteriorates in the noisy setup and when forecasting is required for long horizons. The DAE model (Lusch et al., 2018) recovers the pendulum dynamics in the linear regime but struggles when the nonlinearity increases.
High-dimensional Fluid Flows
Next, we consider two challenging fluid flow examples. The first instance is a periodic flow past a cylinder that exhibits vortex shedding from boundary layers. This flow is commonly used in physically-based machine learning studies (Takeishi et al., 2017;Morton et al., 2018). The data are generated by numerically solving the Navier-Stokes equations given here in their vorticity form ∂ t ω = − v, ∇ω + 1 Re ∆ω, where ω is the vorticity taken as the curl of the velocity, ω = curl(v). We employ an immersed boundary projection solver (Taira & Colonius, 2007) with Re = 100. Our simulation yields 300 snapshots of 192 × 199 grid points, sampled at regular intervals in time, spanning five periods of vortex shedding. We split the data in half for training and testing.
Our second example is an inviscid flow, i.e., Re = ∞, of a vortex pair travelling over a curved domain of a sphere given as a triangle mesh with 2562 nodes. This flow is characterized by a continuous motion along the great center geodesic of the sphere, and it has been used previously to assess the stability of numerical algorithms (Azencot et al., 2014). We facilitate the intrinsic solver (Azencot et al., 2014) by producing 600 snapshots of which we use 550 for training.
Experimental results for the flow past a cylinder. It can be shown that the cylinder dynamics evolve on a low-dimensional attractor (Noack et al., 2003), which can be viewed as a nonlinear oscillator with a state-dependent damping (Loiseau & Brunton, 2018). As the dynamics are within the linear regime, we expect that on clean data, both DAE and ours will obtain good prediction results. We compare both Figure 11: Vortices that rotate in opposite directions travel along geodesics of the domain. We show the input data (top row) as well as our forecast results (bottom row) for several different times. models using width α = 2 and bottleneck κ = 10. The error plots for this experiment are provided in Fig. 8. In general, our model outperforms DAE when the data are noisy and the prediction horizon is long, illustrating the regularizing effect of our loss terms. Fig. 10 shows the obtained predictions for t = 100. Indeed, for the clean data (top) the results are similar in nature, whereas for the noisy case (bottom), our model provides a more robust profile.
Experimental results for the inviscid flow. Next, we consider the inviscid flow as an input to our network as well as to the DAE model. In general, our method yields extremely good results for this flow, even for long time predictions, as can be seen in Fig. 9. This example highlights the benefits of our approach with respect to DAE, as for some of the times we obtain a gain which is five times better than DAE. Moreover, we note that the particular choice of fully connected layers in Tab 2 is important in this case, since computing convolutions on curved domains is still considered a difficult problem whose solutions are not as robust as in the structured setting (Bronstein et al., 2017). In Fig. 11, we show a few estimated predictions for the flow over a sphere.
Initial condition DAE
Ours Ground truth Figure 12: Starting from the same initial conditions, we use our model and DAE to forecast the sea surface temperature for the 120th day (top row) and 175th day (bottom row). The results can be visually compared to the ground truth data (right column). Our model generally attains predictions that are much closer to the ground truth compared to the results obtained by DAE.
Sea Surface Temperature Data
Our last dataset includes complex climate data representing the daily average sea surface temperature measurements around the Gulf of Mexico. This dataset is used in climate sciences to study the intricate dynamics between the oceans and the atmosphere. Climate prediction is generally a hard task involving challenges such as irregular heat radiation and flux as well as uncertainty in the wind behavior. Nevertheless, the input dynamics exhibit non-stationary periodic structures and are empirically low-dimensional, suggesting that Koopman-based methods can be employed. We extract a subset of the NOAA OI SST V2 High Resolution Dataset hereafter SST, and we refer to (Reynolds et al., 2007) for additional details. In our experiments, we use data with a spatial resolution of 100 × 180 spanning a time horizon of 1, 305 days, of which 1, 095 snapshots are used for training.
Experimental results. Fig. 13 shows the prediction error over a time horizon of 180 days. Here we are using width α = 6 and bottleneck κ = 10. It can be seen, that our model provides good estimations over a long time horizon, whereas the DAE diverges rapidly. Fig. 12 provides a visual comparison of the estimated predictions as obtained from the DAE model and ours. In both cases, forecasting for the 120th day (top) and 175th day (bottom), our results are closer to the ground truth by a large margin.
Recall that changes in sea surface temperature causes far-reaching effects on global climate and lead to climatic phenomenons, such as storms and floods. Thus, models that provide a more accurate forecast can improve response rate to the potential effects and damages. However, predicting climate data is in general a notoriously difficult problem. Nevertheless, the prediction results of our model are of practical significance.
ABLATION STUDY
To support our empirical results, we have also conducted an ablation study to quantify the effect of our additional loss terms when weighted differently. To this end, we revisit the noisy pendulum flow with an initial condition θ 0 = 0.8. Our results are summarized in Tab. 1 where we explore various values for λ bwd and λ con which balance the back prediction and consistency penalties, respectively. Our model generally outperforms other baselines for all the parameters we checked, measured via the average error (fifth column) and most distant prediction error (sixth column).
Type #par λ bwd λ con
1 T ∑ T t=1 |f t − f t |/| f t | |f T − f T |/| f T | DAE
DISCUSSION
In this paper, we proposed a novel PCL framework for processing high-dimensional time series data. Our method is based on Koopman theory as we approximate dynamical systems via linear evolution matrices. Key to our approach is that we consider the backward dynamics during prediction, and we promote the consistency of the forward and backward systems. These modifications may be viewed as relaxing strict reversibility and stability constraints, while still regularizing the parameter space. We evaluate our method on several challenging datasets and compare with a state-of-the-art Koopman-based network as well as other baselines. Our approach notably outperforms the other models on noisy data and for long time predictions. We believe that our work can be extended in many ways, and in the future, we plan on considering our setup within a recurrent neural network design.
A NETWORK ARCHITECTURE
In our evaluation, we employ an autoencoding architecture where the encoder and decoder are shallow (Erichson et al., 2019a) and contain only three layers each. Using a simple design allows us to focus our comparison on the differences between the DAE model (Lusch et al., 2018) and ours. Specifically, we list the network structure in Tab. 2 including the specific sizes we used as well as the different activation functions. We recall that m represents the spatial dimension of the input signals, whereas κ is the bottleneck of our approximated Koopman operators. Thus, p = 32 · α is the main parameter with which we control the width and expressiveness of the autoencoder. We facilitate fully connected layers as some of our datasets are represented on unstructured grids. Finally, we note that the only difference between our net architecture and the DAE model is the additional backward linear layer.
B BACKWARD PREDICTION OF DYNAMICAL SYSTEMS
One of the key features of our model is that it allows for the direct backward prediction of dynamics. Namely, given an observation f t , our network yields the forward prediction viaf t+1 = χ d • C • χ e ( f t ), as well as the backward estimate usingf t−1 = χ d • D • χ e ( f t ). Time reversibility may be important in various contexts (Greydanus et al., 2019). For instance, given two different poses of a person, we can consider the trajectory from the first pose to the second or the other way around. Typically, neural networks require that we re-train the model in the reverse direction to be able to predict backwards. In contrast, Koopman-based methods can be used for this task as the Koopman matrix is linear and thus back forecasting can be obtained simply viaf t−1 = χ d • C −1 • χ e ( f t ). We show in Fig. 14 the backward prediction error computed withf t−1 for the cylinder flow data using our model and the DAE (blue and red curves). In addition, as our model computes the matrix D, we also show the errors obtained forf t−1 . The solid lines correspond to the clean version of the data, whereas the dashed lines are related to its noisy version. Overall, our model clearly outperforms DAE by an order of magnitude difference.
C COMPUTATIONAL REQUIREMENTS
The models used in this work are relatively shallow. The amount of parameters per model can be computed as follows 2(m + 32 + κ) · 32α + (4 · 32α + κ + m) + 2 · κ 2 , corresponding to the number of weights, biases and Koopman operators, respectively. Notice that DAE is different than our model by having κ 2 less parameters. In general, our model is almost two times slower than DAE during training.
We recorded the average training time per epoch, and we show the results for DAE and our models in Fig. 15 for many of our test cases. Specifically, the figure shows from left to right the run times for cylinder flow, noisy cylinder flow, sphere flow, linear pendulum, noisy linear pendulum, nonlinear pendulum, noisy nonlinear pendulum, SST and noisy SST. The behavior of our model is consistent in comparison to DAE for the different tests. On average, if DAE takes x milliseconds per epoch, than our model needs ≈ 1.8x time.
This difference in time is due to the additional penalty terms, and it can be asymptotically bounded via O(E bwd ) + O(E con ) = O(λ s nm) + O(κ 4 ) .
We note that the asymptotics for the forward prediction are equal to the backward component, i.e., O(E fwd ) = O(λ s nm). The consistency term E con is composed of a sum of sequence of cubes (matrix products) which can be bounded by κ 4 , assuming matrix multiplication is O(κ 3 ) and thus it is a non tight bound. Moreover, while the quartic bound is extremely high, we note that a cheaper version of the constraint can be used in practice, i.e., |CD − I| 2 F . Also, since our models are loaded to the GPU where matrix multiplication computations are usually done in parallel, the practical bound may be much lower. Finally, the inference time is insignificant (≈ 1 ms) and it is the same for DAE and ours and thus we do not provide a comparison.
Figure 2 :
2Our analysis and computational pipeline take into account the forward and backward dynamical systems.
Figure 3 :
3Our network takes observations f k , and it learns a latent representation of them via an autoencoder architecture χ e and χ d . Then, the latent variables are propagated forward and backward in time using the linear layers C and D, respectively. We emphasize that all the above connections are bi-directional and so information can flow freely from left to right or right to left.
Figure 4 :
4Koopman operators are linear and thus their spectrum can be investigated. We visualize the eigenvalues of an unregularized model (a) and ours (b) in the complex plane. Indeed, our PCL framework promotes stability, whereas the other model exhibits an eigenvalue with modulus greater than one.
Figure 6 :Figure 7 :
67We show the pendulum's trajectories for the initial conditions of amplitude oscillations θ 0 = 0.8 (left) and θ 0 = 2.4 (right). The discrete sampled data points are slightly perturbed. Prediction errors, over a time horizon of 1000 steps, for clean and noisy observations from a pendulum with initial conditions θ 0 = 0.8 (top row) and θ 0 = 2.4 (bottom). Our model outperforms the DAE in all settings.
Figure 8 :
8We compare the behavior of our approach vs. DAE on the cylinder flow on clean (a) and noisy (b) inputs. Overall, our model achieves consistent results even in the presence of noise, whereas DAE struggles with noise and over long range forecasts.
Figure 9 :
9Prediction errors for the vortex pair flow over the surface of a sphere for a time horizon of 100 steps on clean (a) and noisy (b) inputs. Again, our model achieves consistent results, whereas DAE struggles with noise and over long range forecasts.
Figure 10 :
10We demonstrate a few estimation examples for the flow past a cylinder, showing the predicted fluctuations around the mean for clean (top row) and noisy (bottom row) inputs. Our model recovers more flow structures over a time horizon of 130 steps, where the advantage is pronounced for the noisy inputs. Ground truth Ours
Figure 13 :
13Prediction errors for SST over a time horizon of 180 days. Our results clearly outperform DAE across various times.
Figure 14 :Figure 15 :
1415The cylinder flow is used for backward prediction with our model (red) and DAE (blue). Our results hint that DAE overfits in the forward direction, whereas our network generalizes well when the time is reversed. We show above the average run time for an epoch in milliseconds for several of the test cases in this work.
Table 1: Summary of ablation study for the pendulum. The star indicates the setting that is used during our experiments above.10k
0.0
0.0
0.20
0.36
RNN
20k
0.0
0.0
0.19
0.33
Ours
10k
1e-1
0.0
0.15
0.27
Ours (*) 10k
1e-1 1e-2
0.09
0.15
Ours
10k
1e-1 1e-1
0.11
0.18
Ours
10k
2e-1 1e-2
0.07
0.11
Ours
10k
2e-1 1e-1
0.11
0.18
Table 2 :
2Our network architecture, where p = 32 · α with α controlling the width per encoder and decoder layer.
ACKNOWLEDGEMENTSOA would like to acknowledge the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 793800. MWM would like to acknowledge ARO, DARPA, NSF and ONR for providing partial support of this work. We would like to thank J. Nathan Kutz, Steven L. Brunton, Lionel Mathelin and Alejandro Queiruga for valuable discussions about dynamical systems and Koopman theory. Further, we would like to acknowledge the NOAA for providing the SST data (https://www.esrl.noaa.gov/psd/).
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arxiv |
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J Boucrot
O Callot
M Davier
L Duflot
PhJ.-F Grivaz
A Heusse
Jacholkowska
C Loomis
L Serin
J.-J Veillet
J.-B De Vivie De Régie
Now at Centre de Physique des Particules de Marseille
Univ Méditerranée
F-13288MarseilleFrance
C Yuan
G Bagliesi
T Boccali
L Foà
A Giammanco
A Giassi
F Ligabue
A Messineo
F Palla
G Sanguinetti
A Sciabà
R Tenchini
Also at CERN
1211Geneva 23Switzerland
A Venturi
Also at CERN
1211Geneva 23Switzerland
P G Verdini
O Awunor
G A Blair
J Coles
G Cowan
A Garcia-Bellido
M G Green
L T Jones
T Medcalf
A Misiejuk
J A Strong
P Teixeira-Dias
B Bloch-Devaux
D Boumediene
P Colas
B Fabbro
E Lançon
M.-C Lemaire
E Locci
P Perez
J Rander
J.-F Renardy
A Rosowsky
P Seager
A Trabelsi
Now at Département de Physique
Faculté des Sciences de Tunis
24 Also at Dipartimento di Fisica e Tecnologie Relative
Università di Palermo
1060Le Belvédère, PalermoTunisia., Italy
B Tuchming
B Vallage
N Konstantinidis
A M Litke
G Taylor
C N Booth
S Cartwright
F Combley
P N Hodgson
M Lehto
L F Thompson
K Affholderbach
A Böhrer
S Brandt
C Grupen
J Hess
A Ngac
G Prange
U Sieler
C Borean
G Giannini
H He
J Putz
J Rothberg
S R Armstrong
K Berkelman
K Cranmer
D P S Ferguson
Y Gao
Also at Department of Physics
30 Now at SLAC
U.S.A. 31 Deceased. 32 Also at Groupe d' Astroparticules de Montpellier
Tsinghua University
94309Beijing, StanfordCAThe People's Republic of China
Université de Montpellier II
34095MontpellierFrance
S González
O J Hayes
H Hu
S Jin
J Kile
IIIP A Mcnamara
J Nielsen
Y B Pan
J H Von Wimmersperg-Toeller
W Wiedenmann
J Wu
Sau Lan Wu
X Wu
G Zobernig
G Dissertori
Physikalisches Institut das RWTH-Aachen
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
D-52056AachenGermany
Institut de Física d'Altes Energies
Dipartimento di Fisica, INFN Sezione di Bari
Institute of High Energy Physics
Laboratoire de Physique des Particules (LAPP)
Universitat Autònoma de Barcelona
IN 2 P 3 -CNRSF-74019, E-08193, I-70126Annecy-le-Vieux Cedex, Bellaterra, BariFrance, Italy
Academia Sinica
BeijingThe People's Republic of China
European Laboratory for Particle Physics (CERN)
CH-1211Geneva 23Switzerland
Laboratoire de Physique Corpusculaire
Université Blaise Pascal
IN 2 P 3 -CNRSF-63177Clermont-Ferrand, AubièreFrance
Niels Bohr Institute
2100CopenhagenDenmark
Nuclear Research Center Demokritos (NRCD)
GR-15310AttikiGreece
Dipartimento di Fisica, Università di Firenze, INFN Sezione di Firenze
Laboratoire de Physique Nucléaire et des Hautes Energies, Ecole Polytechnique
IN 2 P 3 -CNRSF-91128, I-50125Palaiseau Cedex, FirenzeFrance, Italy
Department of Physics and Astronomy
Laboratori Nazionali dell'INFN (LNF-INFN)
University of Glasgow
I-00044, G12 8QQFrascati, GlasgowItaly, United Kingdom
Department of Physics
Kirchhoff-Institut für Physik
Haverford College
19041-1392HaverfordPAU.S.A
Universität Heidelberg
D-69120HeidelbergGermany
Department of Physics, Imperial College
SW7 2BZLondonUnited Kingdom
Institut für Experimentalphysik
Universität Innsbruck
A-6020InnsbruckAustria
Department of Physics
University of Lancaster
LA1 4YBLancasterUnited Kingdom
Département de Physique
Institut für Physik
O. van der Aa, C. Delaere, V. Lemaitre Institut de Physique Nucléaire
Université Catholique de Louvain
1348Louvain-la-NeuveBelgium
Universität Mainz
D-55099MainzGermany
Centre de Physique des Particules de Marseille
Dipartimento di Fisica, Università di Milano e INFN Sezione di Milano
Max-Planck-Institut für Physik
Univ Méditerranée
IN 2 P 3 -CNRS, Werner-Heisenberg-InstitutF-13288, I-20133, D-80805Marseille, Milano, MünchenFrance, Italy., Germany
Dipartimento di Fisica dell'Università, INFN Sezione di Pisa, e Scuola Normale Superiore
Department of Physics, Royal Holloway & Bedford New College
Laboratoire de l'Accélérateur Linéaire
Université de Paris-Sud
IN 2 P 3 -CNRSF-91898, I-56010Orsay Cedex, PisaFrance, Italy
University of London
TW20 OEXEghamSurreyUnited Kingdom
R.W. Clifft, T.R. Edgecock, P.R. Norton, I.R. Tomalin Particle Physics Dept., Rutherford Appleton Laboratory, Chilton
OX11 OQXDidcotOxonUnited Kingdom
DAPNIA/Service de Physique des Particules
CEA
CE-Saclay, F91191Gif-sur-Yvette CedexFrance
Institute for Particle Physics
University of California at Santa Cruz
95064Santa CruzCAUSA
Department of Physics
University of Sheffield
S3 7RHSheffieldUnited Kingdom
Fachbereich Physik
Universität Siegen
D-57068SiegenGermany
Dipartimento di Fisica, Università di Trieste e INFN Sezione di Trieste
Experimental Elementary Particle Physics
Department of Physics
University of Washington
I-34127, 98195Trieste, SeattleWAItaly, U.S.A
University of Wisconsin
53706MadisonWIUSA
Institute for Particle Physics, ETH Hönggerberg
8093ZürichSwitzerland
23 Now at Skyguide, Swissair Navigation Services
Geneva, Switzerland4 Feb 2002 February 1, 2002The ALEPH Collaboration * ) Submitted to Physics Letters B * ) See next pages for the list of authors
A search for the pseudoscalar meson η b is performed in two-photon interactions at LEP 2 with an integrated luminosity of 699 pb −1 collected at e + e − centre-of-mass energies from 181 GeV to 209 GeV. One candidate event is found in the six-chargedparticle final state and none in the four-charged-particle final state, in agreement with the total expected background of about one event. Upper limits of Γ γγ (η b )×BR(η b → 4 charged particles) < 48 eV Γ γγ (η b )×BR(η b → 6 charged particles) < 132 eV are obtained at 95% confidence level, which correspond to upper limits of 9.0% and 25% on these branching ratios.
Introduction
The bb ground state, the η b meson, has not yet been observed. Because of their initial state, two-photon collisions are well suited for the study of pseudoscalar mesons, for which J P C = 0 −+ . The high γγ cross section and the high LEP luminosity and energy, as well as the low background from other processes, make LEP 2 a good environment to search for this meson.
Theoretical estimates (from pertubative QCD and lattice nonrelativistic QCD) of the mass difference, ∆m, between the η b and the Υ (m Υ = 9.46 GeV/c 2 ) are summarized in Table 1 and those of the partial decay width of the η b into two photons, Γ γγ (η b ), in Table 2. For the former, values ranging from ∆m = 34 MeV/c 2 to 141 MeV/c 2 are obtained. For the latter, a value of Γ γγ (η b ) = 557 ± 85 eV, chosen in this letter, is obtained from the average of the first order estimates (488 eV) shifted by 69 eV at the second order in α s . It yields an exclusive η b production cross section of 0.304 ± 0.046 pb in e + e − collisions at √ s = 197 GeV. The branching ratios of the η b into four and six charged particles are estimated as in Ref. [1] to be 2.7% and 3.3% respectively. (The same estimate gives 9.9% for the η c decay branching fraction into four charged particles, in agreement with the measured value of 9.3 ± 1.8% [2].) Six and seven exclusive η b are therefore expected to be produced in the 699 pb −1 of data collected by ALEPH above the WW threshold, in the four-and six-charged-particle final states, respectively. A measurement of the η b mass and of its decay modes would therefore provide a test of pQCD and NRQCD [3,4,5]. Searches have already been conducted by the CUSB and CLEO Collaborations in the cascade decay of the Υ(3S): the CUSB Collaboration finds for the product of the branching ratios BR(Υ(3S) → ππh b ) × BR(h b → γη b ) < 0.45% at 90% C.L. for an Υ-η b splitting between 50 MeV/c 2 and 110 MeV/c 2 [6]. The CLEO Collaboration has published a 90% C.L. upper limit on the product of the branching ratios BR(Υ(3S) → π + π − h b ) × BR(h b → γη b ) of about 0.1% for the η b mass range from 9.32 GeV/c 2 to 9.46 GeV/c 2 with a photon energy ranging from 434 MeV to 466 MeV and the h b mass restricted to 9.900 ± 0.003 GeV/c 2 [7].
In this letter, a search is presented for the η b meson via its decay into four and six charged particles. The search is performed in quasi-real two-photon interactions where the meson is produced exclusively. This letter is organized as follows. A description of the ALEPH detector is given in Section 2. The data analysis with event selection, efficiency calculation, background estimate and systematic uncertainty determination is described in Section 3. The results of the search are presented in Section 4. Finally, in Section 5 a summary is given.
ALEPH Detector
A detailed description of the ALEPH detector and its performance can be found in Ref. [19]. The central part of the ALEPH detector is dedicated to the reconstruction of the trajectories of charged particles. The trajectory of a charged particle emerging from the interaction point is measured by a two-layer silicon strip vertex detector 1 (VDET), a cylindrical drift chamber (ITC) and a large time projection chamber (TPC). The three tracking detectors are immersed in a 1.5 T axial magnetic field provided by a superconducting solenoidal coil. Together they measure charged particle transverse momenta with a resolution of δp t /p t = 6 × 10 −4 p t ⊕ 0.005 (p t in GeV/c). The TPC also provides a measurement of the specific ionization dE/dx meas . An estimator may be formed to test a particle hypothesis, χ h = (dE/dx meas −dE/dx exp,h )/σ exp,h , where dE/dx exp,h and σ exp,h denote the expected specific ionization and the estimated uncertainty for the particle hypothesis h, respectively. Photons are identified in the electromagnetic calorimeter (ECAL), situated between the TPC and the coil. The ECAL is a lead/proportional-tube sampling calorimeter segmented in 0.9 • ×0.9 • projective towers read out in three sections in depth. It has a total thickness of 22 radiation lengths and yields a relative energy resolution of 0.18/ √ E+0.009, with E in GeV, for isolated photons. Electrons are identified by their transverse and longitudinal shower profiles in ECAL and their specific ionization in the TPC.
The iron return yoke is instrumented with 23 layers of streamer tubes and forms the hadron calorimeter (HCAL). The latter provides a relative energy resolution of charged and neutral hadrons of 0.85/ √ E, with E in GeV. Muons are distinguished from hadrons by their characteristic pattern in HCAL and by the muon chambers, composed of two double-layers of streamer tubes outside HCAL.
The information from the tracking detectors and the calorimeters are combined in an energy-flow algorithm [19]. For each event, the algorithm provides a set of charged and neutral reconstructed particles, called energy-flow objects in the following.
Analysis
Event Selection
The search is performed in the four-and six-charged-particle modes, where four (or six) charged energy-flow objects with a net charge zero are required. In order to keep the efficiency high, loose selection cuts are chosen. No attempt is made to reconstruct K S mesons at this stage. The dE/dx measurement, when available, must be consistent with the pion or kaon hypothesis (χ 2 π,K < 9); the more likely hypothesis is used for mass assignment. When no dE/dx information is available the pion mass is assigned to the particle. No neutral energy-flow object with E > 1 GeV must be present within 20 • of the beam axis. No muon and no electron (as defined by the ECAL) must be observed. Events are also excluded if a photon conversion is detected, where the electron and positron are identified by requiring χ 2 e < 9, and the pair invariant mass is smaller than 25 MeV/c 2 . The total transverse momentum of charged particles in the event ( p t,i ) must be smaller than 250 MeV/c. The energy-flow objects in the event are boosted into their centre-of-mass frame and the thrust is computed in this frame. The thrust axis must form an angle θ thrust larger than 45 • with respect to the beam axis to reject events from the γγ continuum background. The γγ → τ + τ − background is reduced to a negligible fraction by the rejection of events in which both hemispheres, as defined by the thrust axis, have a net charge of ±1 and an invariant mass less than 1.8 GeV/c 2 .
Signal Efficiency
Selection and reconstruction efficiencies are studied with events generated with PHOT02 [20] in which the η b mass is set to 9.4 GeV/c 2 and the total width to 7 MeV/c 2 . The width is calculated under the assumption that the two-gluon decay is dominant [2,21,22]. Four samples of 2500 events each are generated for the final state with four charged particles (2(π + π − ), π + π − K + K − , 2(K + K − ), K S K + π − ). Four other samples of 2500 events each are generated for the final state with six charged particles (3(π + π − ), 2(π + π − )K + K − , π + π − 2(K + K − ), 3(K + K − )). For the decays, it is assumed that the momenta are distributed according to phase space. The event samples are passed through the detector simulation and reconstruction programs. The mass resolution of the selected events is about 0.14 GeV/c 2 and is dominated by wrong mass assignment from π-K misidentification. A signal region between 9.0 GeV/c 2 and 9.8 GeV/c 2 is chosen. The event selection efficiencies averaged over the four decay channels are found to be 16.7% and 9.3% for the four-and six-charged-track channels, respectively.
Systematic Uncertainties
The lack of knowledge of the decay modes and kinematics of the η b meson is the source of the dominant systematic uncertainties in the analysis. The uncertainty on the selection efficiency due to the unknown decay mode of the η b meson is estimated from the spread of the efficiencies of the four simulated decay modes. The relative uncertainty is 7.5% and 20.4% for the four-and six-charged-particle final states. In order to check the effect of the selection efficiency due to the assumption of phase space decays, the η b is forced to decay into a pair of φ mesons, each giving two charged kaons. In this case a relative increase of 10% in the detection efficiency is found.
Further studies are performed without the final cut on neutral energy or with modified cuts on p t,i , θ thrust , and hemisphere mass. An uncertainty of 5.5% is estimated. The limited statistics of simulated events contribute an uncertainty of 2.4% and 3.2% for the two decay modes, respectively.
A total relative uncertainty of 9.7% (21.4%) on the selection efficiency is calculated for the four-(six-) charged-track channel.
Background Estimate
The background estimate suffers from the low statistics of the simulated events selected and from the poor description of the shape of the invariant mass spectra. The background, dominated by γγ continuum production, is therefore estimated from data by means of a fit to the ratio of the mass spectra after all cuts are applied and before the final cuts on p t,i , θ thrust , and hemisphere mass are applied. The ratio is fitted with an exponential function up to m = 6 GeV/c 2 (m = 7 GeV/c 2 ) for the four-(six-) charged-particle topology. The average of the values of this function at m = 6 GeV/c 2 (m = 7 GeV/c 2 ) and at m = 9.4 GeV/c 2 is then multiplied by the number of events in the signal region before the final cuts to obtain the background estimate. Half of the difference between these two values is taken as the systematic uncertainty on the estimate. The background in the signal region is determined to be 0.30 ± 0.25 (0.70 ± 0.34) events for the four-(six-) charged-particle topology.
Results
Invariant mass spectra of the selected events are shown in Fig. 1. A total of 33727 (3432) events is selected in the four-(six-) charged-particle final states. In the signal region, only one event is found in the six-prong topology.
Cross Section Upper Limit
From the knowledge of the background b and the efficiency ε, the observed number of events n is converted [23] into an upper limit on the signal events µ into a confidence level α given by
1 − α = g(b)f (ε) n i=0 P (i | µε + b)dεdb g(b) n i=0 P (i | b)db ,
where P (j | x) is the Poisson probability that j events be observed, when x are expected. The probability density functions for the background g(b) and the efficiency f (ε) are assumed to be Gaussian, but restricted to the range where b and ε are positive. Upper limits of 3.06 (4.69) events at 95% confidence level are calculated for the four-(six-) prong topology. This translates into the upper limits Γ γγ (η b )×BR(η b → 4 charged particles) < 48 eV Γ γγ (η b )×BR(η b → 6 charged particles) < 132 eV . With a two-photon width of 557 ± 85 eV, upper limits on the branching ratios BR(η b → 4 charged particles) < 9.0% and BR(η b → 6 charged particles) < 25% are derived.
Mass of the Candidate
The raw reconstructed mass of the candidate, as obtained from the measured momenta of the six charged particles and with masses assigned according to the dE/dx measurement, is 9.45 GeV/c 2 . The mass estimate can be refined with additional information visible from the event display shown in Fig. 2. Two of the six tracks form a secondary vertex compatible with the decay of a K S into π + π − . This hypothesis is supported by the presence of a third track compatible with a K − (χ 2 π = 6.0 and χ 2 K = 3.8 × 10 −5 ). The secondary vertex is therefore fitted to this hypothesis, and the five particles (three charged pions, one charged kaon and one K S ) are forced to originate from a common primary vertex. A mass of 9.30 ± 0.02 ± 0.02 GeV/c 2 is derived from these constraints.
A control sample of η c mesons is selected in the K S K + π − decay mode, without the final cuts but that on the total transverse momentum, which is relaxed to p t,i < 500 MeV/c. The analysis is repeated with this control sample for the study of the systematic uncertainty on the mass determination. The mass of the η c meson is fitted and is found consistent with the world average value [2] within its statistical accuracy of 4.7 MeV/c 2 . A systematic uncertainty of the same size is assigned. The total uncertainty is then rescaled with the mass ratio m(candidate)/m(η c ) and a systematic uncertainty of 21 MeV/c 2 is obtained for the mass estimate of the η b candidate. The η c signal is shown in Fig. 3 together with the D 0 signal as observed in its K − π + decay mode. The fitted D 0 mass agrees with the world average value [2] within its statistical accuracy of 0.9 MeV/c 2 . The number of observed η c mesons is consistent with previous measurements [2,22,24].
Summary
With an integrated luminosity of 699 pb −1 collected at e + e − centre-of-mass energies between 181 GeV and 209 GeV, the pseudoscalar meson η b is searched for in its decays to four and six charged particles. One candidate is retained in the decay mode into six charged particles, while no candidate is found in the four-charged-particle decay mode. The candidate η b has a reconstructed invariant mass of 9.30 ± 0.02 ± 0.02 GeV/c 2 . The observation of one event is consistent with the number of events expected from background.
Upper limits on Γ γγ (η b )×BR of 48 eV and 132 eV, corresponding to limits on the branching ratios BR(η b → 4 charged particles)< 9.0% and BR(η b → 6 charged 5 particles)< 25%, are obtained at a confidence level of 95%. ρ π + π + π − π + K − π − K s K π π π π π π − − − − + + + + Figure 2: An rφ view of the η b → K S K − π + π − π + candidate event with the reconstructed mass of 9.30 ± 0.02 ± 0.02 GeV/c 2 , selected in the signal region. The track coordinates recorded in the VDET and the ITC are shown. The tracks are appropriately labeled. The plot to the right shows an rz view of the ALEPH apparatus. Information is given for each track: particle type, momentum (GeV/c), momentum error (GeV/c), azimuthal and polar angle (degrees), transverse and longitudinal impact parameter (cm).
Figure 1 :
1Invariant mass distribution of selected events for four-and six-charged-particle final states (solid line: data). The dashed line represents the expected signal for a 100% branching ratio into the mode under consideration. The signal region is indicated by the vertical dashed lines.
Figure 3 :
3(a) Invariant mass distribution of the selected events of the K S K + π − control sample showing the signal of the η c meson. (b) The D 0 signal reconstructed in its K − π + decay mode.
Table 1 :
1Estimates for the mass splitting ∆m = m(Υ) − m(η b ) from QCD calculations.∆m [ MeV/c 2 ] Ref.
lattice NRQCD
45 − 100
[3, 8, 9]
lattice potential
60 − 110
[10]
pQCD
36 − 55
[11]
1/m expansion
34 − 114
[12]
potential model
60 − 141
[13, 14, 15]
Table 2 :
2Estimates for the two-photon width Γ γγ (η b ).Γ γγ (η b ) [ eV] Ref.
estimates O(α s )
potential model
500 ± 30
[16]
potential model, Γ e + e − (Υ)
490 ± 40
[16]
NRQCD
460
[17]
NRQCD, Γ e + e − (Υ)
501
[18]
estimates O(α 2
s )
NRQCD, Γ e + e − (Υ)
570 ± 50
[18]
used in this letter
557 ± 85
AcknowledgementsWe wish to thank our colleagues in the CERN accelerator divisions for the successful operation of LEP. We are indebted to the engineers and technicians in all our institutions for their contribution to the excellent performance of ALEPH. Those of us from nonmember countries thank CERN for its hospitality. We would like to thank Ted Barnes and Gunnar Bali for discussions.
Also at CERN, 1211 Geneva 23. SwitzerlandAlso at CERN, 1211 Geneva 23, Switzerland.
. Now at LAPP. 74019Now at LAPP, 74019 Annecy-le-Vieux, France
Also at Dipartimento di Fisica di Catania and INFN Sezione di Catania. 95129Catania, ItalyAlso at Dipartimento di Fisica di Catania and INFN Sezione di Catania, 95129 Catania, Italy.
. Now at LBNL. Now at LBNL, Berkeley, CA 94720, U.S.A.
Supported by the National Science Foundation of China. Supported by the National Science Foundation of China.
Supported by the Danish Natural Science Research Council. Supported by the Danish Natural Science Research Council.
Supported by the UK Particle Physics and Astronomy Research Council. Supported by the UK Particle Physics and Astronomy Research Council.
17 Supported by the Direction des Sciences de la Matière, C.E.A. 18 Supported by the Austrian Ministry for Science and Transport. 19 Now at SAP AG, 69185 Walldorf, Germany 20 Now at Groupe d' Astroparticules de Montpellier. 2234095Germany; Montpellier, France; Tunisia; Geneva, SwitzerlandSupported by the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie ; Université de Montpellier II ; US Department of Energy21 Now at Département de Physique, Faculté des Sciences de Tunis, 1060 Le Belvédère. grant DE-FG03-92ER40689. 23 Now at Skyguide, Swissair Navigation ServicesSupported by the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie, Germany. 17 Supported by the Direction des Sciences de la Matière, C.E.A. 18 Supported by the Austrian Ministry for Science and Transport. 19 Now at SAP AG, 69185 Walldorf, Germany 20 Now at Groupe d' Astroparticules de Montpellier, Université de Montpellier II, 34095 Montpellier, France. 21 Now at Département de Physique, Faculté des Sciences de Tunis, 1060 Le Belvédère, Tunisia. 22 Supported by the US Department of Energy, grant DE-FG03-92ER40689. 23 Now at Skyguide, Swissair Navigation Services, Geneva, Switzerland.
27 Now at INFN Sezione di Roma II, Dipartimento di Fisica. Dipartimento di Fisica e Tecnologie Relative. Palermo, Italy; Geneva, Switzerland; Honeywell, Phoenix AZ, U.S.A26133Università di Palermo ; Università di Roma Tor Vergata25 Now at McKinsey and Compagny, Avenue Louis Casal 18Also at Dipartimento di Fisica e Tecnologie Relative, Università di Palermo, Palermo, Italy. 25 Now at McKinsey and Compagny, Avenue Louis Casal 18, 1203 Geneva, Switzerland. 26 Now at Honeywell, Phoenix AZ, U.S.A. 27 Now at INFN Sezione di Roma II, Dipartimento di Fisica, Università di Roma Tor Vergata, 00133
28 Now at Centre de Physique des Particules de Marseille. Italy Roma, Marseille, FranceUniv MéditerranéeRoma, Italy. 28 Now at Centre de Physique des Particules de Marseille, Univ Méditerranée, F-13288 Marseille, France.
The People's Republic of China. 30 Now at SLAC. Beijing; Stanford, CA 94309, U.S.A. 31 DeceasedAlso at Department of Physics, Tsinghua UniversityAlso at Department of Physics, Tsinghua University, Beijing, The People's Republic of China. 30 Now at SLAC, Stanford, CA 94309, U.S.A. 31 Deceased.
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| {'fraction_non_alphanumeric': 0.04835944428022465, 'fraction_numerical': 0.04026012415016258, 'mean_word_length': 3.9180113388573923, 'pattern_counts': {'":': 0, '<': 12, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'A search for the pseudoscalar meson η b is performed in two-photon interactions at LEP 2 with an integrated luminosity of 699 pb −1 collected at e + e − centre-of-mass energies from 181 GeV to 209 GeV. One candidate event is found in the six-chargedparticle final state and none in the four-charged-particle final state, in agreement with the total expected background of about one event. Upper limits of Γ γγ (η b )×BR(η b → 4 charged particles) < 48 eV Γ γγ (η b )×BR(η b → 6 charged particles) < 132 eV are obtained at 95% confidence level, which correspond to upper limits of 9.0% and 25% on these branching ratios.', 'arxivid': 'hep-ex/0202011', 'author': ['A Heister ', 'S Schael ', 'R Barate ', 'R Brunelière ', 'I De Bonis ', 'D Decamp ', 'C Goy ', 'S Jezequel ', 'J.-P Lees ', 'F Martin ', 'E Merle ', 'M.-N Minard ', 'B Pietrzyk ', 'B Trocmé ', 'G Boix \nNow at McKinsey and Compagny\nAvenue Louis Casal 181203GenevaSwitzerland\n', 'S Bravo ', 'M P Casado ', 'M Chmeissani ', 'J M Crespo ', 'E Fernandez ', 'M Fernandez-Bosman ', 'Ll Garrido ', 'E Graugés ', 'J Lopez ', 'M Martinez ', 'G Merino ', 'R Miquel \nNow at LBNL\n94720BerkeleyCAU.S.A\n', 'Ll M Mir \nNow at LBNL\n94720BerkeleyCAU.S.A\n', 'A Pacheco ', 'D Paneque ', 'H Ruiz ', 'Barcelona Spain ', 'A Colaleo ', 'D Creanza ', 'N De Filippis ', 'M De Palma ', 'G Iaselli ', 'G Maggi ', 'M Maggi ', 'S Nuzzo ', 'A Ranieri ', 'G Raso ', 'F Ruggieri ', 'G Selvaggi ', 'L Silvestris ', 'P Tempesta ', 'A Tricomi \nAlso at Dipartimento di Fisica di Catania and INFN Sezione di Catania\n95129CataniaItaly\n', 'G Zito ', 'X Huang ', 'J Lin ', 'Q Ouyang ', 'T Wang ', 'Y Xie ', 'R Xu ', 'S Xue ', 'J Zhang ', 'L Zhang ', 'W Zhao ', 'D Abbaneo ', 'P Azzurri ', 'T Barklow ', 'O Buchmüller ', 'M Cattaneo ', 'F Cerutti ', 'B Clerbaux ', 'H Drevermann ', 'R W Forty ', 'M Frank ', 'F Gianotti ', 'T C Greening \nDipartimento di Fisica\nNow at Honeywell, Phoenix AZ, U.S.A. 27 Now at INFN Sezione di Roma II\nUniversità di Roma Tor Vergata\n00133RomaItaly\n', 'J B Hansen ', 'J Harvey ', 'D E Hutchcroft ', 'P Janot ', 'B Jost ', 'M Kado \nNow at LBNL\n94720BerkeleyCAU.S.A\n', 'P Mato ', 'A Moutoussi ', 'F Ranjard ', 'L Rolandi ', 'D Schlatter ', 'G Sguazzoni ', 'W Tejessy ', 'F Teubert ', 'A Valassi ', 'I Videau ', 'J J Ward ', 'F Badaud ', 'S Dessagne ', "A Falvard \nNow at Groupe d' Astroparticules de Montpellier\nUniversité de Montpellier II\n34095MontpellierFrance\n", 'D Fayolle ', 'P Gay ', 'J Jousset ', 'B Michel ', 'S Monteil ', 'D Pallin ', 'J M Pascolo ', 'P Perret ', 'J D Hansen ', 'J R Hansen ', 'P H Hansen ', 'B S Nilsson ', 'A Kyriakis ', 'C Markou ', 'E Simopoulou ', 'A Vayaki ', 'K Zachariadou ', 'A Blondel \nNow at Departement de Physique Corpusculaire\nUniversité de Genève\n1211 Genève 4Switzerland\n', 'J.-C Brient ', 'F Machefert ', 'A Rougé ', 'M Swynghedauw ', 'R Tanaka ', 'H Videau ', 'V Ciulli ', 'E Focardi ', 'G Parrini ', 'A Antonelli ', 'M Antonelli ', 'G Bencivenni ', 'F Bossi ', 'P Campana ', 'G Capon ', 'V Chiarella ', 'P Laurelli ', 'G Mannocchi \nAlso Istituto di Cosmo-Geofisica del C.N.R\nTorinoItaly\n', 'F Murtas ', 'G P Murtas ', 'L Passalacqua ', 'A Halley ', 'J Kennedy ', 'J G Lynch ', 'P Negus ', 'V O'shea ', 'A S Thompson ', 'S Wasserbaech ', 'R Cavanaugh ', 'S Dhamotharan ', 'C Geweniger ', 'P Hanke ', 'V Hepp ', 'E E Kluge ', 'G Leibenguth ', 'A Putzer ', 'H Stenzel ', 'K Tittel ', 'S Werner ', 'M Wunsch ', 'R Beuselinck ', 'D M Binnie ', 'W Cameron ', 'G Davies ', 'P J Dornan ', 'M Girone \nAlso at CERN\n1211Geneva 23Switzerland\n', 'R D Hill ', 'N Marinelli ', 'J Nowell ', 'H Przysiezniak \nNow at LAPP\n74019Annecy-le-VieuxFrance\n', 'S A Rutherford ', 'J K Sedgbeer ', 'J C Thompson \n19 Now at SAP AG\nAlso at Rutherford Appleton Laboratory, Chilton\n15 Permanent address: Universitat de Barcelona08208, 69185Barcelona, WalldorfDidcot, UKSpain., Germany\n', 'R White ', 'V M Ghete ', 'P Girtler ', 'E Kneringer ', 'D Kuhn ', 'G Rudolph ', 'E Bouhova-Thacker ', 'C K Bowdery ', 'D P Clarke ', 'G Ellis ', 'A J Finch ', 'F Foster ', 'G Hughes ', 'R W L Jones ', 'M R Pearson ', 'N A Robertson ', 'M Smizanska ', 'U Blumenschein ', 'F Hölldorfer ', 'K Jakobs ', 'F Kayser ', 'K Kleinknecht ', 'A.-S Müller ', 'G Quast \nNow at Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\n76128KarlsruheGermany\n', 'B Renk ', 'H.-G Sander ', 'S Schmeling ', 'H Wachsmuth ', 'C Zeitnitz ', 'T Ziegler ', 'A Bonissent ', 'P Coyle ', 'C Curtil ', 'A Ealet ', 'D Fouchez ', 'P Payre ', 'A Tilquin ', 'F Ragusa ', 'A David ', 'H Dietl ', 'G Ganis ', 'K Hüttmann ', 'G Lütjens ', 'W Männer ', 'H.-G Moser ', 'R Settles ', 'G Wolf ', 'J Boucrot ', 'O Callot ', 'M Davier ', 'L Duflot ', 'PhJ.-F Grivaz ', 'A Heusse ', 'Jacholkowska ', 'C Loomis ', 'L Serin ', 'J.-J Veillet ', 'J.-B De Vivie De Régie \nNow at Centre de Physique des Particules de Marseille\nUniv Méditerranée\nF-13288MarseilleFrance\n', 'C Yuan ', 'G Bagliesi ', 'T Boccali ', 'L Foà ', 'A Giammanco ', 'A Giassi ', 'F Ligabue ', 'A Messineo ', 'F Palla ', 'G Sanguinetti ', 'A Sciabà ', 'R Tenchini \nAlso at CERN\n1211Geneva 23Switzerland\n', 'A Venturi \nAlso at CERN\n1211Geneva 23Switzerland\n', 'P G Verdini ', 'O Awunor ', 'G A Blair ', 'J Coles ', 'G Cowan ', 'A Garcia-Bellido ', 'M G Green ', 'L T Jones ', 'T Medcalf ', 'A Misiejuk ', 'J A Strong ', 'P Teixeira-Dias ', 'B Bloch-Devaux ', 'D Boumediene ', 'P Colas ', 'B Fabbro ', 'E Lançon ', 'M.-C Lemaire ', 'E Locci ', 'P Perez ', 'J Rander ', 'J.-F Renardy ', 'A Rosowsky ', 'P Seager ', 'A Trabelsi \nNow at Département de Physique\nFaculté des Sciences de Tunis\n24 Also at Dipartimento di Fisica e Tecnologie Relative\nUniversità di Palermo\n1060Le Belvédère, PalermoTunisia., Italy\n', 'B Tuchming ', 'B Vallage ', 'N Konstantinidis ', 'A M Litke ', 'G Taylor ', 'C N Booth ', 'S Cartwright ', 'F Combley ', 'P N Hodgson ', 'M Lehto ', 'L F Thompson ', 'K Affholderbach ', 'A Böhrer ', 'S Brandt ', 'C Grupen ', 'J Hess ', 'A Ngac ', 'G Prange ', 'U Sieler ', 'C Borean ', 'G Giannini ', 'H He ', 'J Putz ', 'J Rothberg ', 'S R Armstrong ', 'K Berkelman ', 'K Cranmer ', 'D P S Ferguson ', "Y Gao \nAlso at Department of Physics\n30 Now at SLAC\nU.S.A. 31 Deceased. 32 Also at Groupe d' Astroparticules de Montpellier\nTsinghua University\n94309Beijing, StanfordCAThe People's Republic of China\n\nUniversité de Montpellier II\n34095MontpellierFrance\n", 'S González ', 'O J Hayes ', 'H Hu ', 'S Jin ', 'J Kile ', 'IIIP A Mcnamara ', 'J Nielsen ', 'Y B Pan ', 'J H Von Wimmersperg-Toeller ', 'W Wiedenmann ', 'J Wu ', 'Sau Lan Wu ', 'X Wu ', 'G Zobernig ', 'G Dissertori ', '\nPhysikalisches Institut das RWTH-Aachen\nEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)\nD-52056AachenGermany\n', "\nInstitut de Física d'Altes Energies\nDipartimento di Fisica, INFN Sezione di Bari\nInstitute of High Energy Physics\nLaboratoire de Physique des Particules (LAPP)\nUniversitat Autònoma de Barcelona\nIN 2 P 3 -CNRSF-74019, E-08193, I-70126Annecy-le-Vieux Cedex, Bellaterra, BariFrance, Italy\n", "\nAcademia Sinica\nBeijingThe People's Republic of China\n", '\nEuropean Laboratory for Particle Physics (CERN)\nCH-1211Geneva 23Switzerland\n', '\nLaboratoire de Physique Corpusculaire\nUniversité Blaise Pascal\nIN 2 P 3 -CNRSF-63177Clermont-Ferrand, AubièreFrance\n', '\nNiels Bohr Institute\n2100CopenhagenDenmark\n', '\nNuclear Research Center Demokritos (NRCD)\nGR-15310AttikiGreece\n', '\nDipartimento di Fisica, Università di Firenze, INFN Sezione di Firenze\nLaboratoire de Physique Nucléaire et des Hautes Energies, Ecole Polytechnique\nIN 2 P 3 -CNRSF-91128, I-50125Palaiseau Cedex, FirenzeFrance, Italy\n', "\nDepartment of Physics and Astronomy\nLaboratori Nazionali dell'INFN (LNF-INFN)\nUniversity of Glasgow\nI-00044, G12 8QQFrascati, GlasgowItaly, United Kingdom\n", '\nDepartment of Physics\nKirchhoff-Institut für Physik\nHaverford College\n19041-1392HaverfordPAU.S.A\n', '\nUniversität Heidelberg\nD-69120HeidelbergGermany\n', '\nDepartment of Physics, Imperial College\nSW7 2BZLondonUnited Kingdom\n', '\nInstitut für Experimentalphysik\nUniversität Innsbruck\nA-6020InnsbruckAustria\n', '\nDepartment of Physics\nUniversity of Lancaster\nLA1 4YBLancasterUnited Kingdom\n', '\nDépartement de Physique\nInstitut für Physik\nO. van der Aa, C. Delaere, V. Lemaitre Institut de Physique Nucléaire\nUniversité Catholique de Louvain\n1348Louvain-la-NeuveBelgium\n', '\nUniversität Mainz\nD-55099MainzGermany\n', '\nCentre de Physique des Particules de Marseille\nDipartimento di Fisica, Università di Milano e INFN Sezione di Milano\nMax-Planck-Institut für Physik\nUniv Méditerranée\nIN 2 P 3 -CNRS, Werner-Heisenberg-InstitutF-13288, I-20133, D-80805Marseille, Milano, MünchenFrance, Italy., Germany\n', "\nDipartimento di Fisica dell'Università, INFN Sezione di Pisa, e Scuola Normale Superiore\nDepartment of Physics, Royal Holloway & Bedford New College\nLaboratoire de l'Accélérateur Linéaire\nUniversité de Paris-Sud\nIN 2 P 3 -CNRSF-91898, I-56010Orsay Cedex, PisaFrance, Italy\n", '\nUniversity of London\nTW20 OEXEghamSurreyUnited Kingdom\n', '\nR.W. Clifft, T.R. Edgecock, P.R. Norton, I.R. Tomalin Particle Physics Dept., Rutherford Appleton Laboratory, Chilton\nOX11 OQXDidcotOxonUnited Kingdom\n', '\nDAPNIA/Service de Physique des Particules\nCEA\nCE-Saclay, F91191Gif-sur-Yvette CedexFrance\n', '\nInstitute for Particle Physics\nUniversity of California at Santa Cruz\n95064Santa CruzCAUSA\n', '\nDepartment of Physics\nUniversity of Sheffield\nS3 7RHSheffieldUnited Kingdom\n', '\nFachbereich Physik\nUniversität Siegen\nD-57068SiegenGermany\n', '\nDipartimento di Fisica, Università di Trieste e INFN Sezione di Trieste\nExperimental Elementary Particle Physics\nDepartment of Physics\nUniversity of Washington\nI-34127, 98195Trieste, SeattleWAItaly, U.S.A\n', '\nUniversity of Wisconsin\n53706MadisonWIUSA\n', '\nInstitute for Particle Physics, ETH Hönggerberg\n8093ZürichSwitzerland\n'], 'authoraffiliation': ['Now at McKinsey and Compagny\nAvenue Louis Casal 181203GenevaSwitzerland', 'Now at LBNL\n94720BerkeleyCAU.S.A', 'Now at LBNL\n94720BerkeleyCAU.S.A', 'Also at Dipartimento di Fisica di Catania and INFN Sezione di Catania\n95129CataniaItaly', 'Dipartimento di Fisica\nNow at Honeywell, Phoenix AZ, U.S.A. 27 Now at INFN Sezione di Roma II\nUniversità di Roma Tor Vergata\n00133RomaItaly', 'Now at LBNL\n94720BerkeleyCAU.S.A', "Now at Groupe d' Astroparticules de Montpellier\nUniversité de Montpellier II\n34095MontpellierFrance", 'Now at Departement de Physique Corpusculaire\nUniversité de Genève\n1211 Genève 4Switzerland', 'Also Istituto di Cosmo-Geofisica del C.N.R\nTorinoItaly', 'Also at CERN\n1211Geneva 23Switzerland', 'Now at LAPP\n74019Annecy-le-VieuxFrance', '19 Now at SAP AG\nAlso at Rutherford Appleton Laboratory, Chilton\n15 Permanent address: Universitat de Barcelona08208, 69185Barcelona, WalldorfDidcot, UKSpain., Germany', 'Now at Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\n76128KarlsruheGermany', 'Now at Centre de Physique des Particules de Marseille\nUniv Méditerranée\nF-13288MarseilleFrance', 'Also at CERN\n1211Geneva 23Switzerland', 'Also at CERN\n1211Geneva 23Switzerland', 'Now at Département de Physique\nFaculté des Sciences de Tunis\n24 Also at Dipartimento di Fisica e Tecnologie Relative\nUniversità di Palermo\n1060Le Belvédère, PalermoTunisia., Italy', "Also at Department of Physics\n30 Now at SLAC\nU.S.A. 31 Deceased. 32 Also at Groupe d' Astroparticules de Montpellier\nTsinghua University\n94309Beijing, StanfordCAThe People's Republic of China", 'Université de Montpellier II\n34095MontpellierFrance', 'Physikalisches Institut das RWTH-Aachen\nEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)\nD-52056AachenGermany', "Institut de Física d'Altes Energies\nDipartimento di Fisica, INFN Sezione di Bari\nInstitute of High Energy Physics\nLaboratoire de Physique des Particules (LAPP)\nUniversitat Autònoma de Barcelona\nIN 2 P 3 -CNRSF-74019, E-08193, I-70126Annecy-le-Vieux Cedex, Bellaterra, BariFrance, Italy", "Academia Sinica\nBeijingThe People's Republic of China", 'European Laboratory for Particle Physics (CERN)\nCH-1211Geneva 23Switzerland', 'Laboratoire de Physique Corpusculaire\nUniversité Blaise Pascal\nIN 2 P 3 -CNRSF-63177Clermont-Ferrand, AubièreFrance', 'Niels Bohr Institute\n2100CopenhagenDenmark', 'Nuclear Research Center Demokritos (NRCD)\nGR-15310AttikiGreece', 'Dipartimento di Fisica, Università di Firenze, INFN Sezione di Firenze\nLaboratoire de Physique Nucléaire et des Hautes Energies, Ecole Polytechnique\nIN 2 P 3 -CNRSF-91128, I-50125Palaiseau Cedex, FirenzeFrance, Italy', "Department of Physics and Astronomy\nLaboratori Nazionali dell'INFN (LNF-INFN)\nUniversity of Glasgow\nI-00044, G12 8QQFrascati, GlasgowItaly, United Kingdom", 'Department of Physics\nKirchhoff-Institut für Physik\nHaverford College\n19041-1392HaverfordPAU.S.A', 'Universität Heidelberg\nD-69120HeidelbergGermany', 'Department of Physics, Imperial College\nSW7 2BZLondonUnited Kingdom', 'Institut für Experimentalphysik\nUniversität Innsbruck\nA-6020InnsbruckAustria', 'Department of Physics\nUniversity of Lancaster\nLA1 4YBLancasterUnited Kingdom', 'Département de Physique\nInstitut für Physik\nO. van der Aa, C. Delaere, V. Lemaitre Institut de Physique Nucléaire\nUniversité Catholique de Louvain\n1348Louvain-la-NeuveBelgium', 'Universität Mainz\nD-55099MainzGermany', 'Centre de Physique des Particules de Marseille\nDipartimento di Fisica, Università di Milano e INFN Sezione di Milano\nMax-Planck-Institut für Physik\nUniv Méditerranée\nIN 2 P 3 -CNRS, Werner-Heisenberg-InstitutF-13288, I-20133, D-80805Marseille, Milano, MünchenFrance, Italy., Germany', "Dipartimento di Fisica dell'Università, INFN Sezione di Pisa, e Scuola Normale Superiore\nDepartment of Physics, Royal Holloway & Bedford New College\nLaboratoire de l'Accélérateur Linéaire\nUniversité de Paris-Sud\nIN 2 P 3 -CNRSF-91898, I-56010Orsay Cedex, PisaFrance, Italy", 'University of London\nTW20 OEXEghamSurreyUnited Kingdom', 'R.W. Clifft, T.R. Edgecock, P.R. Norton, I.R. Tomalin Particle Physics Dept., Rutherford Appleton Laboratory, Chilton\nOX11 OQXDidcotOxonUnited Kingdom', 'DAPNIA/Service de Physique des Particules\nCEA\nCE-Saclay, F91191Gif-sur-Yvette CedexFrance', 'Institute for Particle Physics\nUniversity of California at Santa Cruz\n95064Santa CruzCAUSA', 'Department of Physics\nUniversity of Sheffield\nS3 7RHSheffieldUnited Kingdom', 'Fachbereich Physik\nUniversität Siegen\nD-57068SiegenGermany', 'Dipartimento di Fisica, Università di Trieste e INFN Sezione di Trieste\nExperimental Elementary Particle Physics\nDepartment of Physics\nUniversity of Washington\nI-34127, 98195Trieste, SeattleWAItaly, U.S.A', 'University of Wisconsin\n53706MadisonWIUSA', 'Institute for Particle Physics, ETH Hönggerberg\n8093ZürichSwitzerland'], 'corpusid': 119381572, 'doi': '10.1016/s0370-2693(02)01329-1', 'github_urls': [], 'n_tokens_mistral': 12149, 'n_tokens_neox': 10357, 'n_words': 5531, 'pdfsha': '9d4c15f010c3046665da29f1943d8124fa979381', 'pdfurls': ['https://export.arxiv.org/pdf/hep-ex/0202011v1.pdf'], 'title': ['23 Now at Skyguide, Swissair Navigation Services', '23 Now at Skyguide, Swissair Navigation Services'], 'venue': []} |
arxiv |
An Informative Path Planning Framework for Active Learning in UAV-based Semantic Mapping
Julius Rückin
Federico Magistri
Cyrill Stachniss
Marija Popović
An Informative Path Planning Framework for Active Learning in UAV-based Semantic Mapping
SUBMITTED TO IEEE TRANSACTIONS ON ROBOTICS, EVOLVED PAPER 1Index Terms-Informative Path PlanningActive LearningBayesian Deep LearningSemantic Segmentation and Mapping
Unmanned aerial vehicles (UAVs) are frequently used for aerial mapping and general monitoring tasks. Recent progress in deep learning enabled automated semantic segmentation of imagery to facilitate the interpretation of large-scale complex environments. Commonly used supervised deep learning for segmentation relies on large amounts of pixel-wise labelled data, which is tedious and costly to annotate. The domain-specific visual appearance of aerial environments often prevents the usage of models pre-trained on publicly available datasets. To address this, we propose a novel general planning framework for UAVs to autonomously acquire informative training images for model re-training. We leverage multiple acquisition functions and fuse them into probabilistic terrain maps. Our framework combines the mapped acquisition function information into the UAV's planning objectives. In this way, the UAV adaptively acquires informative aerial images to be manually labelled for model re-training. Experimental results on real-world data and in a photorealistic simulation show that our framework maximises model performance and drastically reduces labelling efforts. Our map-based planners outperform state-of-the-art local planning.
I. INTRODUCTION
Unmanned aerial vehicles (UAVs) enable highly agile, lowcost operations in various aerial imaging applications [1,2], such as precision agriculture [3,4], wildlife conservation [2], and urban planning [5][6][7][8]. Combined with advances in deep learning for semantic segmentation through fully convolutional neural networks (FCNs) [9,10], deploying UAVs accelerates automated scene understanding in large-scale and complex aerial environments [11]. Classical deep learning-based semantic segmentation models often used in this context are usually trained on a static curated dataset in a supervised fashion only once before deployment. This leads to two major drawbacks. First, training a semantic segmentation model requires enormous amounts of pixel-wise labelled images, which is a repetitive and time-consuming process often executed by costly domain experts. Second, visual appearance can differ significantly between environments or change over time. Thus, a critical requirement for robot autonomy is the ability to learn about an environment by continuously improving the robot's semantic perception with minimal expert guidance.
In this work, we examine the problem of active learning (AL) in UAV-based semantic mapping. Our goal is to Fig. 1. Our general planning framework for active learning in UAV-based semantic mapping deployed in a photo-realistic simulator [12] (top). We compute an acquisition function, e.g. model uncertainty, and predict semantic segmentation online (centre-right) and fuse both in terrain maps (bottomright). Our map-based planners replan a UAV's path (orange, bottom-left) to collect the most informative, e.g. most uncertain (yellow), images for network re-training. Our approach reduces the number of images that must be manually labelled to maximise semantic segmentation performance.
improve the robot's vision capabilities in initially unknown environments while minimising the total amount of humanlabelled data. To this end, our approach exploits ideas from AL research and incorporates them into a new informative path planning (IPP) framework. The framework replans the UAV's path online as new observations are collected to actively target regions of informative training data. The newly gathered images are labelled by a human annotator and used to re-train an FCN, maximising its semantic segmentation performance.
Various AL methods for machine learning effectively reduce the requirements for human-labelled training data [13][14][15][16][17][18][19]. Recently, AL approaches for deep learning models are gaining attention [20][21][22][23][24][25]. These works develop acquisition functions for selecting to-be-labelled training data to maximise model performance. However, they cannot be directly applied to robotic missions as they assume access to large pre-recorded unlabelled in-domain data pools. An open problem is how to leverage AL to improve robot perception with minimal expert guidance when operating in initially unknown environments. More recent AL works for aerial imagery consider the UAV to be a passive data collection device to record static data pools [2,6]. In contrast, we aim to utilise the UAV's decisionmaking capabilities to improve its perception and, thus, its arXiv:2302.03347v2 [cs.RO] 24 May 2023 reasoning about the environment for downstream tasks.
The main contribution of this work is a novel and generally applicable active planning framework linking ideas from AL to robotic planning objectives, as illustrated in Fig. 1. The key benefit of our approach is that it reduces human labelling effort for continuous robotic perception improvement. We exploit various model uncertainty and training data novelty estimation techniques for deep learning models [26][27][28] and apply them to semantic segmentation with a pre-trained FCN [29]. The inferred pixel-wise semantic labels and estimated model uncertainty and novelty scores are fused sequentially into a probabilistic terrain map as new observations are acquired. As a key feature, our new IPP framework iteratively replans the UAV's path to collect the most informative, i.e. the most uncertain or novel, images for labelling and model re-training in a targeted fashion.
This article builds upon our previous conference paper [30]. In our previous work [30], we proposed an IPP approach linking globally mapped model uncertainties to a robotic planning objective for AL in UAV-based terrain monitoring. This submission extends our previous method and generalises experimental findings in the following ways. First, we introduce a general IPP framework for AL in UAV-based semantic mapping by linking various uncertainty-and representationdriven acquisition functions to planning objectives, as opposed to just utilising model uncertainties computed via Monte-Carlo dropout. Second, we propose new terrain mapping features to improve map-based planning compared to our conference version. Third, we systematically evaluate mapping, planners, and AL planning objectives on new datasets from different domains and in a photorealistic simulator. This way, we present a thorough empirical analysis of combining AL acquisition functions and IPP approaches, giving new insights into how to connect AL and autonomous robotic decision-making.
In sum, we make the following four claims. First, our active planning framework for AL in UAV-based semantic mapping reduces the number of labelled images needed to maximise segmentation performance compared to both traditionally used coverage and random walk data collection. Second, probabilistic global mapping of gathered information enhances map-based planning performance for AL. Third, our map-based planners outperform state-of-the-art local planning for AL [31]. Fourth, we demonstrate the generality of our approach, showing that it significantly reduces labelling effort in largely varying domains irrespective of the used uncertainty estimation methods, planning strategies, and AL-based objectives. We open-source our code for usage by the community at: https://github.com/dmar-bonn/ipp-al-framework.
II. RELATED WORK
Our goal is to collect the most informative images to train a semantic segmentation model with a minimal amount of labelled data using UAVs in aerial mapping missions. Our approach combines advances in AL with IPP. This section overviews how our work is placed within these research areas.
A. Active Learning
Active learning aims to maximise model performance while minimising labelled training data. It assumes the existence of a large unlabelled data pool, then iteratively selects a data point from the pool by maximising an acquisition function until a labelling budget is exceeded [13][14][15][16]. Settles et al. [32] provide a comprehensive overview of AL approaches for low-dimensional machine learning problems. Recent AL approaches focus on training deep learning models from highdimensional inputs, e.g. images, where a single data point has a negligible effect on model performance. AL methods for deep learning collect a batch of data from the pool instead of single data points, called batch-mode AL [20,[33][34][35]. However, these strategies are not applicable in robotic settings since they reason about which images from an existing large data pool should be labelled. In contrast, we propose an IPP framework for AL collecting new batches of to-be-labelled data directly during a mission in initially unknown environments. We link the AL acquisition function to an IPP objective, adaptively guiding the UAV towards regions of informative training data.
Further, we answer the following two open research questions. First, how to incorporate recently proposed acquisition functions [20,31,36] into our IPP framework and second, in which ways planners, planning objectives, and terrain mapping influence AL performance.
Uncertainty-based AL methods select data with the highest model uncertainty [17,20,34,36]. Early methods use Gaussian processes [17] or support-vector machines [34] to quantify model uncertainty in tasks with low-dimensional inputs. Measuring model uncertainty in deep neural networks is computationally challenging due to their parameter space dimensionality. One approach aims at estimating the model uncertainty deterministically in a single forward pass. Although computationally efficient, these methods are often not wellcalibrated in real-world vision tasks [28]. Alternatively, Gal et al. [26] propose using dropout at test time, called Monte-Carlo (MC) dropout, to efficiently approximate the Bayesian posterior over the network parameters. They utilise MC dropout in acquisition functions applied to image classification maximising model uncertainty [20]. Other works use neural network ensembles for uncertainty estimation [27,37]. Each network is independently initialised and trained. Ensembles achieve higher prediction performance and better calibration than MC dropout [36,38]. Further, recent advances make ensemble training computationally more efficient [38,39]. In this work, we study the applicability of different uncertaintybased AL objectives in a robotic planning context.
Representation-based AL methods maximise training data diversity by selecting data points with novel representations in feature space [21][22][23]. Generative adversarial networkinspired approaches use a generator learning the joint data representation, while the discriminator distinguishes labelled and unlabelled data [22,23]. Sener et al. [21] select a number of data points, called a core-set, geometrically covering a data pool in the model's latent space with a minimal number of data points. However, both approaches require large in-domain data pools to learn rich representations of the data-generating distribution. These methods are impractical in our scenario as autonomous robots operate in unknown and visually varying environments. In contrast, Blum et al. [31] propose a method for quantifying data novelty in semantic segmentation tasks without access to large in-domain data pools by using kerneldensity estimation of unlabelled images in the network's latent space. They use this novelty estimation in a local planning objective and apply it for AL in aerial semantic mapping. We integrate their novelty estimation into new global map-based planning objectives. We rigorously analyse its AL performance using various global planning schemes and datasets, outperforming their local planning strategy.
B. Informative Path Planning
Informative path planning enables autonomous robots to efficiently and actively explore initially unknown environments subject to platform constraints, such as battery capacity [40]. IPP methods have been applied to various environmental monitoring scenarios, including lake monitoring [41], underwater inspection [42], infrastructure surface inspection [43], and agricultural monitoring [44]. We distinguish between nonadaptive approaches, which precompute paths before a mission starts, e.g. coverage planning [45], and adaptive approaches, which replan paths online as new data is collected [3,41,42]. We focus on adaptive methods as our goal is to collect informative training data on-the-fly.
Combinatorial approaches solve IPP problems in a nearoptimal fashion [46][47][48][49]. However, they exhaustively query the search space scaling exponentially in problem size, which makes most of them impractical for online replanning. In contrast, sampling approaches break the curse of dimensionality to increase the computational efficiency of online IPP [50,51]. Hollinger et al. [50] propose receding-horizon rapidly exploring information gathering algorithms to sample motion plans. Choudhury et al. [51] combine MC planning with cost-benefit rollouts to increase sampling efficiency.
Similarly, optimisation approaches directly optimise IPP objectives [3,41,52]. Vivaldini et al. [52] utilise Bayesian optimisation to choose a sequence of informative measurement positions for UAV-based tree disease monitoring. In a similar problem setup, Hitz et al. [41] leverage the covariance matrix adaptation evolution strategy (CMA-ES) to optimise a sequence of measurement positions. Popović et al. [3] extend this approach by introducing a greedily optimised initial sequence of measurement positions, then using the CMA-ES to fine-tune the initial solution resulting in more informative paths.
Geometric approaches collect candidate measurement positions for efficient exploration. The candidate maximising an objective function is chosen as the most informative one [53][54][55]. Gonzalez. et al. [55] choose the position maximising the potentially visible unexplored space. Ghaffari et al. [53] generate candidate positions along probabilistic frontiers of explored space, greedily selecting the one which maximises the expected information gain. Similarly, Cheng et al. [54] train an agent choosing frontiers to minimise localisation uncertainty and maximise information gain.
The above-mentioned works consider adaptive IPP for mapping environmental phenomena. In contrast, our framework applies planning algorithms to the problem setting of improving robot vision with minimal human labelling effort. We design new IPP objective functions to replan paths towards informative training data as new observations are collected and demonstrate their integration into map-based planners.
C. Informative Path Planning for Active Learning
Using autonomous robots to reduce manual labelling effort for training deep learning models is a relatively unexplored research area. Georgakis et al. [56] propose a framework for active semantic goal navigation which uses ensembles to estimate model uncertainty in their planning objective. In contrast, Frey et al. [57] introduce a self-improving continual learning framework for semantic segmentation in indoor scenes without manual labelling by generating pseudo-labels from 3D maps. Zurbrügg et al. [58] extend this approach to an embodied agent autonomously navigating towards viewpoints of high training data novelty. Chaplot et al. [59] suggest a similar self-supervised approach for semantic segmentation in indoor scenes training an exploration policy targeting uncertain 3D map parts. This approach introduces several sim2real assumptions. As discussed by Chaplot et al. [59], the approaches [57][58][59] rely on large labelled indoor datasets for pre-training a semantic segmentation model to produce highquality pseudo-labels in new indoor scenes. If the pre-trained model misclassifies objects, these errors not only prevent learning semantics but could even be reinforced in the case of over-confident predictions. Aerial mapping missions, as in our problem setting, present much more visual variability, with very little and often small pre-training datasets being available, further exacerbating these issues. Thus, these purely selfsupervised methods are not directly applicable in our scenario.
Most similar to our work is the local planning approach of Blum et al. [31] for AL in semantic mapping. Their planning objective aims to promote training data novelty in semantic prediction tasks. We combine their ideas on novelty estimation for AL with our previous work on IPP for AL [30]. In contrast to Blum et al. [31], we propose a general and unified IPP framework supporting probabilistic semantic mapping, various acquisition functions, planning objectives, and map-based planning algorithms. Further, we provide in-depth empirical analyses and show that our map-based planners outperforms existing methods [30,31].
III. OUR APPROACH
We present our general IPP framework for AL in UAVbased semantic mapping. Our setup considers a UAV collecting images of a flat terrain using a downwards-facing RGB camera. Assuming no further prior knowledge about the terrain, the goal is to autonomously collect informative training data to improve the robot's perception with minimal human labelling effort. As shown in Fig. 2, our framework links AL with planning objectives guiding the UAV to regions of informative training data. As new data is collected, we utilise a lightweight FCN to predict pixel-wise semantics. Further, we estimate the pixel-wise model uncertainty associated with the prediction and training data novelty of the collected image Fig. 2. Overview of our approach. We start with a pre-trained semantic segmentation network deployed on a UAV. During a mission, the network processes RGB images to predict pixel-wise semantic labels, model uncertainties (Sec. III-A1), and novelty scores (Sec. III-A2), which are projected onto the terrain to build global maps capturing these variables (Sec. III-B). Based on the current UAV position, budget, and posterior map state, our algorithm plans paths for the UAV to collect informative training data for improving the network performance (Sec. III-C). After the mission, the collected images are labelled by an annotator and used for network re-training. By guiding the UAV to collect informative training data, our pipeline reduces the human labelling effort. Fig. 3. ERFNet architecture proposed by Romera et al. [29]. The network takes an RGB image (left) as input and outputs semantic labels (right). We utilise the network in our ensemble method to predict model uncertainty.
Conv Dropout
and then fuse them into probabilistic terrain maps. The UAV position, its remaining budget, and the current map state are combined into new AL-based information objectives used to replan the future path towards informative training data. A key feature of our framework is its general applicability, as it is agnostic to the chosen network and supports different uncertainty estimation techniques, mapping methods, and map-based planners. The following subsections detail the framework's individual modules and the specific methods we investigate in this work.
A. Active Learning Acquisition Functions
We first derive measures for an image's information value when a network is re-trained on this data. To this end, AL works propose two main paradigms, uncertainty-based and representation-based acquisition functions. We demonstrate the generality of our approach using either paradigm.
We adapt the ERFNet encoder-decoder architecture proposed by Romera et al. [29] depicted in Fig. 3 to our AL use case. Although our framework is agnostic to the chosen network architecture, the lightweight ERFNet is particularly suitable for online robot deployment with limited computational resources. In the following, the model f W (·) is parameterised by weights W and outputs a probability tensor
p(y | f W (z)) = softmax(f W (z)) ∈ [0, 1] K×w×h , where z
is the input RGB image with width w and height h, and y is the pixel-wise semantic label over the K classes. The training set contains N images Z = {z 1 , . . . , z N } and semantic labels Y = {y 1 , . . . , y N }. Our network is trained to minimise crossentropy with weight decay regularisation factor λ:
L(θ) = − 1 N N i=1 log p(y i | f W (z i )) + λ∥W ∥ 2 2 .(1)
The following subsections describe different methods to estimate the information value of a candidate image for AL.
1) Bayesian Uncertainty-based Methods:
We estimate pixel-wise model uncertainty over the prediction p(y | f W (z)) as a measure for the informativeness of image z for retraining [20,26,36,60,61]. We leverage advances in Bayesian deep learning, transforming the deterministic ERFNet into a probabilistic version [30]. We consider using two alternative methods: Monte-Carlo (MC) dropout [26] and ensembles [36]. To measure model uncertainty, we utilise Bayesian active learning by disagreement [61], which computes the mutual information between the unknown labels y and the posterior distribution over weights p(W | Z, Y ). However, the weights' posterior is intractable for FCNs [26]. Thus, we approximate the true posterior prediction [60]:
p(y | z, Z, Y ) = 1 T T i=1 softmax(fŴ i (z)) ,(2)
where we independently sample T weightsŴ i from a prior weight distribution q(W ) performing MC integration. MC dropout and ensemble methods provide two alternative approaches to construct the prior q(W ). In MC dropout, dropout is applied independently to the weights W before each of the T forward passes at test time. In the ensemble method, we train T independently randomly initialised ERFNet models with stochastic mini-batch gradient descent. For further details on MC dropout and ensembles, we refer to [26,30] and [36], respectively. Following Gal et al. [20], Fig. 4. Representation-based image novelty score [31]. An RGB image z is passed through our ERFNet encoder and its latent vectors r z i,j ∈ r z are extracted along the channel dimension. We compute the cosine distance between each r z i,j and its k-nearest neighbors from the training representation vector database. The resulting novelty image is upsampled to the spatial dimensions of z. Last, we add all r z i,j to the representation vector database. Lighter colours indicate higher novelty, i.e. higher informativeness for AL.
we approximate the mutual information using Eq. (2):
I(y, W | z, Z, Y ) ≈ −p(y | z, Z, Y ) T log p(y | z, Z, Y ) + 1 T T i=1 p(y | z,Ŵ i ) T log p(y | z,Ŵ i ) ,(3)
where log(·) is applied element-wise. Intuitively, model uncertainty is high whenever the posterior prediction entropy is high, while single prediction entropy is low, but disagreeing with each other. We exploit this measure to guide the UAV towards more informative areas, i.e. regions of high model uncertainty. Note that our framework is agnostic to both, the model uncertainty estimation method and the chosen network.
2) Representation-based Method: Inspired by recent AL works [21][22][23], we study a representation-based planning objective as an alternative to uncertainty-based objectives. We deterministically quantify the network's confidence in its prediction by estimating the image's novelty to the network f W given training images and labels Z, Y [28,31,62,63]. Intuitively, the image's novelty is high whenever the network's latent representation of a new image z and training images Z is dissimilar. Although confidence measures for classification are well-known [62,63], they are not directly applicable for semantic segmentation as they do not provide pixel-wise scores and are not invariant to object locations. Thus, we utilise the novelty measure for semantic segmentation proposed by Blum et al. [31].
We perform kernel-density estimation in the network latent space by computing the average cosine distance between the latent representations of image z and its k-nearest latent representations of training images Z. We exploit the FCN's architecture, where the network f W (·) = d W d (e We (·)) consists of an encoder e We parameterised by W e , a decoder d W d parameterised by W d , and W = {W e , W d }. Specifically, we extract representations e We (z) = r z ∈ R w 8 × h 8 ×C after the encoder's last convolutional layer with spatial dimensions downsampled by a factor of 8 compared to the image, and C channel dimensions as induced by the ERFNet architecture.
Hence, r z i,j is a C-dimensional latent vector of the (i, j)th 8 × 8 pixels patch of image z. After model training, we generate a database R = {r z1
1,1 , ..., r z N w 8 , h 8 } of w 8 · h 8 · N
patchwise representations of the training images Z. Given an image z at inference time, its (i, j)-th novelty score is:
r(z) i,j = 1 k r∈N N (r z i,j ) 1 − r ⊤ r z i,j ∥r z i,j ∥ 2 ∥r∥ 2 ,(4)
where
N N (r z i,j ) is the set of k-nearest neighbors of r z i,j
in R with respect to the cosine distance. Intuitively, higher novelty indicates higher informativeness of image z for retraining. Fig. 4 provides a schematic of an image's novelty score computation. For more details, we refer to Blum et al. [31].
A key feature of our framework is that it can easily be adapted to other acquisition functions and FCNs. This work shows its generality using the uncertainty-and representationbased objectives with ERFNet, as described above.
B. Probabilistic Semantic Mapping
An important basis for our new planning objective functions is our 2D multi-layer terrain map. This map captures global semantics, model uncertainties, representation novelties, and training data statistics to provide different sources of information for informative planning. We propose a probabilistic mapping module updating this information online as the UAV collects new images of the terrain. To achieve this, we utilise sequential probabilistic occupancy grid mapping [64] to update each map layer when a new measurement arrives. We discretise the terrain into three 2D maps
G S : G → {0, 1} K×W ×L , G U : G → [0, 1] W ×L , G R : G → [0, 1] W ×L
defined over a grid lattice G with W ×L spatially independent cells capturing the discrete semantic classes, continuous model uncertainties, and continuous novelty scores.
The semantic map G S consists of K independent layers G Si : G → {0, 1} W ×L to map i ∈ [K] > 2 classes. Each grid cell's G m,n random state follows a uniform prior distribution G m,n Si ∼ p(G m,n Si = 1) = 1 K . When a new image z t arrives at time step t, the semantic predictionsp(y | f W (z t )), see Eq. (2), are projected to the flat terrain given the UAV position p t ∈ R 3 and camera intrinsics. We utilise standard occupancy grid mapping for each layer i and cell G m,n computing the posterior belief G m,n Si ∼ p(· | z 1:t , p 1:t ):
l(G m,n Si | z 1:t , p 1:t ) = l(G m,n Si | z t , p t )+ l(G m,n Si | z 1:t−1 , p 1:t−1 ) − l(G m,n Si ) ,(5)
where l(·) are the log odds of the binary random variable, p(G m,n Si | z t , p t ) is given by the projected semantic predictions, p(G m,n Si | z 1:t−1 , p 1:t−1 ) is the recursive map belief, and p(G m,n Si ) is the map prior. The model uncertainties and novelties are stored in the maps G U and G R with prior means µ U,0 and µ R,0 respectively. We fuse projected uncertainties u t given by Eq. (3) and novelty scores r(z t ) given by Eq. (4) using maximum likelihood estimation assuming normally distributed G U and G R . We maintain a hit map H : G → N W ×L counting the total number of times a grid cell was updated during a mission. Then, we update the means µ m,n U,t and µ m,n R,t for a grid cell G m,n by:
µ m,n U,t = µ m,n U,t−1 + 1 H(G m,n ) (u m,n t − µ m,n U,t−1 ) , µ m,n R,t = µ m,n R,t−1 + 1 H(G m,n ) (r(z t ) m,n − µ m,n R,t−1 ) .(6)
Last, we store a map T : G → N W ×L to count how often grid cells occur in the training data set to foster data diversity in our proposed planning objectives. Note that the maps H(·) and T (·) are different as the camera could provide a highfrequency image stream for mapping while images only at the planned measurement position are collected for training.
A key feature of our mapping approach is that we accumulate and update the information between missions by updating the map prior. After each UAV mission, the network is retrained on the collected training data. Re-training changes the semantic predictions, model uncertainty, and representation novelty estimates. Thus, we store all previously collected data and corresponding UAV positions. After re-training, we predict semantics, model uncertainties, and representation novelties of the stored data and sequentially fuse them according to Eq. (5) and Eq. (6). Our informed map prior strategy enhances map-based planning by avoiding exploring from scratch or replanning with outdated terrain knowledge.
C. Informative Path Planning
We develop IPP algorithms to guide a UAV to adaptively collect useful training data for our FCN. Our key idea is to link acquisition functions introduced in Sec. III-A to planning objective functions. Our planning strategies use the probabilistic terrain maps presented in Sec. III-B to guide the UAV online towards informative training data in an unknown terrain.
In general, IPP algorithms optimise an information criterion I : Ψ → R ≥0 over paths ψ = (p 1 , . . . , p P ) ∈ Ψ defined by P measurement positions p i ∈ R 3 :
ψ * = argmax ψ∈Ψ I(ψ), s.t. C(ψ) ≤ B ,(7)
where B ≥ 0 is the mission budget, e.g. flight time, and Ψ is the set of all possible paths of length P . The function C : Ψ → R ≥0 defines the cost of executing a path ψ:
C(ψ) = P −1 i=1 c(p i , p i+1 ) ,(8)
where c : R 3 × R 3 → R ≥0 computes the flight time between two measurement positions assuming constant acceleration and deceleration ±a, and maximum velocity v. The key insight of our work is to couple the AL acquisition functions with IPP information criteria I(·). This allows us to maximise model performance and minimise the labelling effort resulting from collecting images along a planned path ψ.
We propose four different replanning strategies in our framework, one local image-based and three global frontier, sampling and optimisation schemes with information criteria I(·) optimising Eq. (7) given the current terrain map states. The planners are illustrated in Fig. 5 evaluation, we compare each planner's performance in terms of segmentation performance over the total labelling cost.
In the following, we exemplarily present our planning objectives with respect to the globally mapped model uncertainties G U,t at a time step t (Sec. III-A1). In case of the representationbased objective (Sec. III-A2), we substitute the uncertainties G U,t with novelties G R,t , see Eq. (6). This variable can also be changed to capture other AL acquisition functions. Local planner. Our local image-based planner follows the direction of the highest estimated training data information in the image recorded at the current UAV position. Specifically, we choose the direction of the image edge e * zt with the highest average AL value normalised by the current training data counts T t (p t ) in the grid's subset spanned by the camera field of view from position p t projected to the flat terrain. This way, we select neighboring informative images while locally fostering training data diversity. Then, p * t+1 is reached by taking a predefined step size towards the direction of edge e * zt at a fixed altitude. This resembles the planner proposed by Blum et al. [31] and generalises it to any AL objective.
Frontier-based planner. Our global geometric planner guides the UAV towards frontiers of the explored terrain with the highest AL objective in the terrain map. We use the hit map H(·) to identify exploration frontiers. Particularly, we greedily choose the next-best measurement position p * t+1 from a set of candidate positions p c t+1 equidistantly sampled along the frontiers at a fixed altitude. As the planner acts greedily, optimising Eq. (7) reduces to selecting the path ψ * = (p * t+1 ):
p * t+1 = argmax p c t+1 I((p c t+1 )) = argmax p c t+1 ∥G U,t (p c t+1 )∥ 1 ∥T t (p c t+1 )∥ 1 ,(9)
where G U,t (p c t+1 ) and T t (p c t+1 ) are the globally mapped model uncertainties and training data counts within the camera field of view from position p c t+1 , and ∥·∥ 1 is the norm summing all elements in these subsets. This way, our frontier planner trades off both exploration of unknown space for data diversity and focusing on regions potentially valuable for AL. Optimisation-based planner. Our optimisation-based planner selects a path ψ * t+1 over a fixed horizon of multiple time steps. We utilise a two-step approach for efficient online replanning inspired by Popović et al. [3]. First, we greedily select a path ψ g t+1 of length P over a grid above the terrain. Second, we use an optimisation procedure to fine-tune ψ g t+1 in the continuous UAV workspace and return the next-best path ψ * t+1 .
First, we iteratively select a path ψ g t+1 = (p g t+1 , . . . , p g t+P ), where each measurement position p g t+i , i ∈ {1, . . . , P }, is greedily chosen over a sparse lattice F of discrete candidate positions p c at a fixed altitude:
p g t+i = argmax p c ∈F ∥G U,t (p c )∥ 1 c(p t+i−1 , p c )∥T t+i−1 (p c )∥ 1 ,(10)
where T t+i−1 (p c ) is the subset of the forward-simulated training data count map given by the camera field of view at position p c . The forward simulation of the current map T t based on the previously selected positions (p g t+1 , . . . , p g t+i−1 ) is crucial as one cannot forward-simulate model uncertainties. Forward-simulating T t linearly decreases uncertainty with the number of a grid cell's training set occurrences. This fosters data diversity and terrain exploration.
Second, we refine the greedy positions of ψ g t+1 in parallel in the continuous UAV workspace. To this end, we initialise an optimisation procedure with the greedy solution ψ g t+1 and extend Eq. (10) to an information criterion I(·) evaluating candidate paths ψ o t+1 = (p o t+1 , . . . , p o t+P ):
I(ψ o t+1 ) = P i=1 ∥G U,t (p o t+i )∥ 1 c(p o t+i−1 , p o t+i )∥T t+i−1 (p o t+i )∥ 1 .(11)
The candidate path ψ * t+1 = (p * t+1 , . . . , p * t+P ) maximising Eq. (11) is chosen and measurement position p * t+1 is executed. We found that normalising AL information of a path by its execution costs leads to more efficient budget allocation. This planning strategy supports any optimisation algorithm, which can optimise objective function Eq. (11). Sampling-based planner. Our sampling-based planner utilises Monte-Carlo tree search (MCTS) [65] to optimise a nextbest measurement position p * t+1 in a non-myopic fashion. We simulate a number of future paths ψ t+1 = (p n1 t+1 , . . . , p n P t+P ) of length P at a fixed altitude. Each tree node n i at depth i ∈ {0, . . . , P } is uniquely defined by its state S ni = {p ni t+i , T ni t+i ,B ni } consisting of a measurement position p ni t+i , forward-simulated training data count map T ni t+i along the tree's traversed path to node n i , and remaining mission budget B ni . The tree's root node n 0 is defined by S n0 = {p t , T t , B}, where p t , T t and B are the current UAV position, training data count map, and mission budget. At each node, the planner selects the next position from a discrete set of actions with different step sizes and orientations. While traversing the search tree, we use the upper confidence bound bandit algorithm [65] to choose a child node. When reaching a leaf node, we roll out the path by sampling actions uniformly at random in each subsequent node until the remaining budget is exceeded or path length P is reached. A simulated path's information I(ψ t+1 ) is computed by summing rewards along subsequent parent and child nodes n i , n i+1 given by:
R(n i , n i+1 ) = ∥G U,t (p ni+1 t+i+1 )∥ 1 c(p ni t+i , p ni+1 t+i+1 )∥T ni t+i (p ni+1 t+i+1 )∥ 1 .(12)
Note that T ni t+i (p ni+1 t+i+1 ) are the training data occurrences at the child node's position assuming the training data count after collecting a measurement at the parent's node position. This way, the reward estimates the next position's information value given the map state at replanning time t+i. After simulating a number of paths ψ t+1 , we select the root's child node n 1 with the highest average information value and the UAV moves to its associated measurement position p n1 t+1 . To show that our approach supports various planning algorithms, we proposed the four diverse planners above and showcase their integration into our modular framework. Further, we highlight that our planning strategies are agnostic to the acquisition functions introduced in Sec. III-A.
IV. EXPERIMENTAL RESULTS
Our experiments evaluate our proposed method and show the benefits of our mapping module (Sec. IV-B), Bayesian ensemble (Sec. IV-C), and AL planning objective functions (Sec. IV-D). We verify our planning framework's generality and analyse its AL performance on vastly differing realworld aerial datasets and in a photo-realistic simulator against classical coverage and random walk exploration data collection (Sec. IV-E). We conduct a sensitivity analysis validating the framework's robustness to the choice of model architecture, pre-training schemes and UAV starting positions (Sec. IV-F). Our framework consistently maximises semantic segmentation performance while minimising human labelling effort. Notably, our experiments show that our map-based planners outperform the local planning strategy for AL proposed by Blum et al. [31], which, to the best of our knowledge, is the only directly comparable approach to date.
A. Experimental Setup
Baselines. We compare our planning framework against three baselines: a traditionally-used coverage-based collection strategy [45], and two random walk-based exploration planners. The coverage strategy precomputes a static path maximising the area covered by the UAV to foster spatial coverage of training data. We precompute lawnmower-like patterns before each mission starts, alternate the pattern's orientations, and vary the step size between measurement positions.
We consider two random walk exploration planners, local and global planning. Similar to the local planner, the local exploration planner chooses for a given UAV position one of the four image edges at random and follows the edge direction with predefined step size. The global exploration planner randomly selects a UAV position in the continuous [7], and (c) in the photo-realistic Flightmare UAV simulator [12]. Steeper curves indicate better AL performance. We compare our planning strategies to the best-performing baseline in each setting: coverage on ISPRS Potsdam and the global random walk on RIT-18 and in Flightmare.
space above the terrain, similar to our map-based planners. For better budget management, we sample a step size uniformly at random between a minimum and maximum radius around the UAV, then also set its heading uniformly at random. This way, both exploration planners aim to foster data diversity while handling the budget properly. As they resemble the action spaces of the planners introduced in Sec. III-C, we can study the influence of our action space design and verify that our active planners maximise AL performance beyond random effects. Fig. 6 exemplifies the paths planned by all three baselines on the ISPRS Potsdam dataset [5]. Datasets. We evaluate our planning framework on two realworld orthomosaic datasets and in a photorealistic physicsbased UAV simulator resembling real-world deployment conditions. Detailed environment, sensor, and UAV mission settings are shown in Table I. Below, we highlight the key differences between the three scenarios.
First, we use the large 7-class urban aerial ISPRS Potsdam orthomosaic dataset [5]. This dataset is characterised by a dense spatial distribution of classes, such that the coverage and exploration baselines can collect visually and semantically different features easily. We sample 4000 train, 1000 validation, and 3500 test images uniformly at random from nonoverlapping regions in the orthomosaic. We use the ISPRS Potsdam dataset for the main experiments evaluating our mapping module (Sec. IV-B), Bayesian ensemble (Sec. IV-C), and planning objectives (Sec. IV-D).
Second, we evaluate our approach on the land cover RIT-18 orthomosaic dataset [7] consisting of semantics covering large connected areas, e.g. asphalt, vegetation, and lake, and local regions, e.g. building, with six classes in total. As the RIT-18 dataset does not provide different orthomosaics for training and testing, we evaluate the UAV's vision capabilities by sampling the test set from the same area. In contrast to the ISPRS Potsdam dataset, this does not allow us to draw conclusions about the model's generalisability, but about its performance in the deployed environment only. This is still a crucial skill for autonomous robot deployment. Our evaluation protocol on RIT-18 resembles that of Blum et al. [31].
Last, we test our framework in Flightmare, a photorealistic simulator with a physics engine for emulating UAV dynamics simulation [12]. We deploy a UAV in the provided 'Industrial' environment introducing 10 semantic classes of different spatial distributions, e.g. hangar, container, road, fence, and pipe. The scene covers a dense area leading to compactly distributed semantics easily explorable by the baseline approaches. As the 'Industrial' terrain is small, we evaluate the UAV's semantic segmentation performance in the deployed environment only.
We perform a study comparing the AL performance of the baseline strategies in Fig. 7. On the ISPRS Potsdam dataset, the coverage pattern is the superior baseline, while the global random walk exploration performs best on the RIT-18 dataset and in the Flightmare simulator. Note that, while MC dropout is used in Fig. 7 to predict semantic segmentation, we found that similar results hold true for deterministic network and ensemble inference. For visual clarity, we only compare our framework to the baselines with the strongest AL performance. Evaluation Metrics. Our AL planning pipeline aims to maximise semantic segmentation performance with minimal human labelling effort, i.e. minimal training data. In line with the standard in AL literature [8, 16, 19-21, 24, 31, 33, 34, 36, 61], our key evaluation metrics assess semantic segmentation performance (dependent variable) over the number of collected training images (independent variable). Higher semantic segmentation performance thanks to newly added images indicate better AL, and thus, planning performance. We choose mean Intersection-over-Union (mIoU), per-pixel accuracy, and per-pixel F1-score to access semantic segmentation performance. mIoU is used in popular semantic segmentation benchmarks [66,67]. It is defined as mIoU = T P T P +F P +F N , where T P , F P , T N , and F N are the true and false positives, and true and false negatives. Per-pixel accuracy acc and F1score f 1 are typically used in classification benchmarks [68]. They are defined as acc = T P T P +F P +T N +F N and f 1 = 2T P 2T P +F P +F N . RIT-18 and Flightmare have strongly imbalanced class distributions. Thus, we use the F1-score instead of accuracy for these scenarios. To make model performance trends easier to follow, we additionally fit trend lines for the experiments conducted on the ISPRS Potsdam and Flightmare datasets. As performance trends are less regular on the RIT-18 dataset due to the more challenging exploration of semantics, we show piecewise linear line plots for these experiments. Training Procedure. We utilise a lightweight Bayesian ERFNet for semantic segmentation as described in Sec. III-A. The model is pre-trained on the Cityscapes dataset [67] to start experiments and training after each of the 10 subsequent data collection missions from the same checkpoint. This also avoids catastrophic forgetting and accumulating train time. We re-train the model until convergence with batch size 8 and weight decay λ = (1 − p)/2N in Eq. (1), where p = 0.5 is the dropout probability, and N is the number of training images [20]. All other model hyperparameters follow the standard ERFNet [29], not tuned for maximal performance in our setting, and kept fixed with changing datasets and planners. Planning Hyperparameters. Our optimisation-based planner leverages the CMA-ES procedure as it has been shown to yield competitive performance in terrain monitoring tasks [3,41]. We fix a set of hyperparameters for all planners with reasonable length scales on the ISPRS Potsdam dataset, i.e. UAV step sizes, minimum and maximum action space radii, grid discretisation, and initial CMA-ES covariance. Only these hyperparameters dependent on the aerial dimensions are scaled accordingly with changing environment sizes. The scale-independent hyperparameters, e.g. number of MCTS simulations, are set in line with prior works [3,65]. We fix the UAV's starting position to the top-left corners of each terrain. Planning Strategies. We outline our planning strategies in detail in Sec. III-C. In our experiments, we refer to the planners in the legends as follows: the local planner is named Local, the frontier-based planner is named Frontier, the optimisationbased planner is named Optimisation, and the sampling-based planner is named Sampling. The baseline approaches are referred to as follows: the coverage pattern is called Coverage, and the local and global random walk exploration strategies are abbreviated with Rand-Glo and Rand-Loc.
B. Informative Mapping
The first set of experiments analyses the performance of our approach. It (i) verifies the superior AL performance of our planning framework over the baselines; (ii) shows that our global map-based planners outperform state-of-the-art local planning; and (iii) quantifies the benefit of our mapping module for the global map-based planners. The experiments are evaluated on the ISPRS Potsdam dataset [5]. To focus on evaluating the effect of our mapping module (Sec. III-B) on the map-based planners, we fix Bayesian model uncertainty as our planning objective estimated with MC dropout (Eq. (3)). Map priors are recomputed before each mission starts to allow for maximally informed global planning. Fig. 8 summarises the AL performance with the informed mapping strategy for each planner. All planners reach higher prediction performance than the coverage baseline (yellow). This supports the claim that our framework is generally applicable to different planning algorithms. Further, it suggests that active replanning is key to efficiently improving robot vision. Notably, our global map-based planners (orange, blue, green) exceed the coverage baseline's maximum prediction performance after ≤ 250 labelled images, while the baseline requires > 500 labelled images. Particularly, for the uncertaintybased objective, our map-based planners show stronger AL performance than the local planner (purple) proposed by Blum et al. [31]. In contrast to local planning, map-based planners drastically reduce training data requirements and tend to achieve higher final prediction performance. 10. Comparison of effects of our mapping module using our framework. The planners consistently benefit from recomputing informative map priors before a mission starts (purple). The performance gain of mapping a continuous RGB sensor stream (blue) instead of only mapping images at planned measurement positions (orange) is less significant. Combining both continuous sensor streams and informative map priors (yellow) also leads to consistent performance improvements. In particular, our informative mapping approach drastically improves the greedy frontier-based strategy. Fig. 11. Comparison of per-class AL performance of map-based frontier vs. coverage planning with a Bayesian model uncertainty-based planning objective estimated by MC dropout and computing informative prior maps before each mission starts. The frontier planner outperforms the coverage baseline (yellow) in almost all classes as our framework can capture complex task-dependent inter-class and intra-class model uncertainties.
To better understand the benefits of our active planning framework, Fig. 11 exemplarily compares the per-class AL performance of the map-based frontier planner (dashed lines) to the coverage baseline (solid lines) in the ISPRS Potsdam scenario. Our active frontier-based planning strategy shows higher AL performance in almost all classes, irrespective of their training data support. Interestingly, the 'car' class (blue) has lower training data support than the 'tree' (green) and 'vegetation' (red) classes but shows stronger IoU performance, even with non-targeted coverage planning. However, active planning improves the 'car' prediction performance even faster than the non-targeted baseline showing the benefit of our framework for classes with little training data support. Further, although the 'tree', 'background' (orange), and 'vegetation' classes have high training data support, they are difficult to distinguish as their visual appearance from a top-down view depends on the image resolution, altitude, and season. This leads to challenging predictions, which may be partially attributed to data instead of model uncertainty, which cannot be explained away with more training data [60]. Thus, not all classes with high training data support benefit to the same extent from active planning. At the same time, although the 'building' class (yellow) has high training data support and is reliably detected by both planners, the frontier-based planner still shows faster performance improvement as our framework can account for the differing visual appearance and geometry of office buildings, historical buildings, and townhouses. Overall, the results suggest that our framework can capture complex task-dependent inter-class and intra-class model uncertainties, which are too complex to capture with a single training data support heuristic, leading to superior AL performance over non-targeted baselines.
To support the claim that our new mapping module is important for the planning framework's performance, we perform an ablation study to measure its effect on our map-based planners. We consider two mapping setups where the UAV either maps training images at planned measurement positions only (pointwise sensor) or maps the images continuously as it moves (continuous sensor stream). Fig. 10 displays the AL performance of our map-based planners (i) recomputing informative prior maps before each mission starts based on previously collected data and the re-trained network (purple), (ii) mapping a continuous RGB image stream (blue) instead of mapping training images at planned measurement positions only (orange), and (iii) combining both informative prior maps and mapping continuous sensor streams (yellow).
All map-based planners show better AL performance with recomputed map priors as they exploit already mapped heterogeneous terrain information. This suggests that mapping and updating knowledge collected across missions with retrained networks, i.e. changing vision capabilities, is key to strong planning performance. In contrast, mapping more information during a single mission with a fixed network is less crucial. Mapping a continuous image stream instead of mapping training data information at planned measurement positions only leads to performance improvements for the greedy frontier-based planner, while both non-greedy planners do not benefit from mapping more information during a mission. Accordingly, combining both mapping continuous sensor streams and recomputing map priors leads to higher AL performance of the frontier-based and optimisation-based planner. The sampling-based planner does not show a performance gain when combining mapping of continuous sensor streams and recomputing map priors. Particularly, our greedy frontier-based planning strategy shows significant improvements by leveraging the informative mapping procedure. Our non-greedy optimisation-based and sampling-based planners benefit moderately from informative map priors while being more robust to less informed terrain maps. Qualitatively, Fig. 9 verifies that leveraging informative prior maps leads to more efficient terrain exploration across missions and targeted data collection within missions resulting in higher model performance with fewer training images.
C. Bayesian ERFNet Ensemble Study
The second experiment shows that our Bayesian ensemble provides reliable uncertainty estimates for AL planning objectives. Moreover, the ensemble achieves higher prediction performance than non-Bayesian and Bayesian ERFNet with MC dropout, presented in our prior work [30].
Input
Ground truth Prediction Error Uncertainty To confirm that our Bayesian ensemble of ERFNets delivers informative model uncertainties for planning and yields superior prediction performance, we train an ensemble on the ISPRS Potsdam dataset [5] with 4000 training images and compare it to the Bayesian ERFNet with MC dropout developed in our previous work [30] and a deterministic ERFNet. Qualitatively, Fig. 12 verifies high model uncertainty of our ensemble in misclassified or hard-to-predict regions. Thus, the ensemble's model uncertainties provide reliable information for planning objectives.
To assess our Bayesian ensemble's prediction capabilities and computational efficiency for online inference on UAVs, we study its performance with varying numbers of ERFNet models T = {2, . . . , 8} in Fig. 13. We compare the ensemble's performance to the deterministic ERFNet [29] and to our Bayesian ERFNet utilising T = 50 MC dropout samples for maximal performance [30]. To quantify the reliability of estimated uncertainties, we measure model calibration using the expected calibration error (ECE) metric [69]. Intuitively, model calibration is high, i.e. ECE low, when the model's probabilistic predictions match its accuracy on a test set.
For T = 8 models, our ensemble (blue) improves segmentation performance by 4.96% mIoU and ECE by 7.09% over the deterministic ERFNet (orange). Additionally, for T = 8 models, our ensemble improves segmentation performance by 0.90% mIoU and ECE by 1.68% compared to the Bayesian ERFNet with MC dropout. Overall, as the number of models increases, segmentation performance and calibration both improve. Favourably for online inference, with T ≈ 6 models, performance gains already converge. Further, the Bayesian ensemble performs on par with the Bayesian ERFNet (T = 50 MC dropout samples) already with T = 3 ERFNet models. Thus, our ensemble requires substantially fewer forward passes (≈ 16×), i.e. compute resources, at deployment. At train time, the ensemble's compute requirements scale linearly with the number of models T , while the MC dropout Bayesian ERFNet has constant compute requirements. However, training is performed offline, hence it is not time-critical. For details about efficient ensemble training, we refer to Huan et al. [39].
D. Comparison of Planning Objectives
Our third set of experiments shows that Bayesian model uncertainty-based objectives guarantee strong AL performance irrespective of the uncertainty estimation technique. Further, it verifies that our general framework supports various AL acquisition function paradigms, including representation-based and uncertainty-based objectives.
As we show in this experiment, Bayesian model uncertaintybased planning objectives outperform baselines with different uncertainty estimation techniques. We investigate our Bayesian ensemble's AL performance on the ISPRS Potsdam dataset. For a fair assessment, we evaluate the coverage baseline with ensemble inference. Fig. 14 summarises the results using our Bayesian ensemble of T = 4 ERFNets for all planning approaches. All planners show better performance than the coverage baseline (yellow), which confirms the intuition that active planning for AL benefits from Bayesian model uncertainty-based objective functions. Similar to our MC dropout-based uncertainty estimation in Fig. 8, mapbased planners (orange, blue, green) achieve higher prediction performance with fewer training images compared to the local planner (purple), further illustrating the advantage of mapbased planning in our framework. To further support our framework's generality under various uncertainty-based objective functions, we investigate its performance using a classical non-Bayesian entropy-based acquisition function [20,28]. Given an image z and a model with deterministic parameters W , p(y | z, W ) is the maximum likelihood estimate over labels y. Then, the prediction entropy is highest when the prediction is uniform, i.e. most uncertain. Qualitatively, Fig. 15 shows that non-Bayesian entropy is weakly correlated with prediction errors as it fails to estimate globally calibrated uncertainties.
H(y) = −p(y | z, W ) ⊤ log p(y | z, W )(13)
We replace the Bayesian model uncertainty, see Eq.
(3), with the entropy of a deterministic forward pass. For a fair comparison, the coverage baseline uses a deterministic forward pass as well. As shown in Fig. 16, the optimisation-based, frontier-based and local planners outperform the baseline, while the sampling-based planner performs similarly to the baseline. In line with results for Bayesian model uncertaintybased objectives, the optimisation-based and frontier-based planners show high prediction performance with substantially fewer training images compared to the local planner. Fig. 17 shows the effect of non-Bayesian entropy-based (orange) and Bayesian model uncertainty-based planning objectives estimated by either MC dropout (blue) or an ensemble (yellow) on the planners' performances. Particularly, the mapbased planners achieve higher AL performance using Bayesian model uncertainty-based objectives irrespective of the uncertainty estimation technique. Although the non-Bayesian objective yields competitive performance with multiple planners Input
Ground truth Prediction Error Novelty Fig. 18. Qualitative results of a deterministic ERFNet trained on the ISPRS Potsdam dataset [5]. Columns from left to right: RGB input, ground truth, prediction, error image (negative prediction log-likelihood), representation novelty (Eq. (4)). High novelty scores (yellower) in case of rare visual cues, such as the helipad (bottom row), suggest that our representation-based objective provides useful information for AL planning scenarios.
in early missions, generally, the Bayesian ensemble method leads to the best AL results. This could be due to two reasons. First, the ensemble shows higher prediction power (Fig. 13). Second, as suggested by our qualitative results (Fig. 15), non-Bayesian uncertainty is weakly calibrated, which results in a less informative planning objective.
To confirm that our framework is applicable to representation-based acquisition functions, we utilise the novelty score shown in Eq. (4) computed over the latent space of a deterministic ERFNet in our planning objective. For a fair assessment, we also utilise a deterministic ERFNet for the coverage baseline. Qualitatively, Fig. 18 visualises the representation novelties of a network trained and tested on disjoint areas of ISPRS Potsdam. Although the novelties do not correlate strongly with prediction errors (whiter), high novelty (yellower) is assigned to rare visual cues, such as the helipad (bottom row), which could be an informative objective to collect diverse training images. Fig. 19 depicts the AL results using representation novelties in the planning objective. All adaptive planners achieve higher segmentation performance than the coverage baseline (yellow). Further, our map-based optimisation (orange) and frontier (blue) planners require fewer training images than the local planner (purple) to reach high prediction performance. This validates that our framework generally supports various acquisition function paradigms and ensures higher AL performance than the baseline approaches, irrespective of the planning objective. Our experiments suggest that the mapbased planners outperform the local planner more significantly using Bayesian uncertainty objectives. This could be due to the better-calibrated Bayesian uncertainty estimates (Fig. 13) leading to more informative planning objectives.
E. Other Scenarios
The fourth set of experiments suggests that (i) our planning framework reduces the number of labelled images required Fig. 19. Comparison of AL performance with representation-based novelty objective and computing informative prior maps before missions start. All planners outperform the baseline (yellow) with fewer training images.
to maximise segmentation performance across substantially different environments, and (ii) our global map-based planning strategies outperform state-of-the-art local planning in most cases, irrespective of the chosen planning objective.
We support these claims with an evaluation of our framework on the RIT-18 dataset [7] and in the Flightmare simulator [12]. The RIT-18 semantics cover large areas leading to challenging exploration. The Flightmare simulator resembles real-world UAV control over an easy-to-explore photorealistic industrial terrain with strong random walk baseline performance. We access the framework's performance using the Bayesian model uncertainty estimated with MC dropout, see Eq. (3), and the representation novelty score given by Eq. (4). Fig. 20 summarises our planning results on the RIT-18 dataset [7]. Note that non-monotonic model performance improvements on RIT-18 are expected as semantics cover large areas leading to challenging exploration influencing the training class distribution. All map-based planning strategies show significantly higher final segmentation performance than the random walk baseline (yellow), irrespective of the chosen planning objective. This confirms that our framework reduces human labelling effort while maximising segmentation performance over vastly differing terrains. Particularly, in most cases, our map-based planners require fewer training images to achieve segmentation performance on par or higher than the local planner (purple). Notably, the local planner performs worse than the baseline using Bayesian model uncertainty showing that our map-based planners are more generally applicable than the local planner. Fig. 21 illustrates our planning results in the Flightmare simulator [12]. All planners using the Bayesian model uncertainty objective show higher AL performance than the random walk baseline (yellow). Using the representation-based objective, only our two map-based optimisation (orange) and sampling (green) planners result in higher final prediction performance than the baseline. Combined with the RIT-18 results (Fig. 20), this suggests that our Bayesian model uncertaintybased objectives are more robustly applicable across varying terrains compared to the representation novelty score proposed 4)). All map-based planners significantly outperform the random walk baseline (yellow). In most cases, our map-based planners lead to substantially higher AL performance than the local planning strategy (purple). by Blum et al. [31]. One possible explanation could be that Bayesian model uncertainty is more strongly correlated with the prediction errors, as indicated by our qualitative results in Fig. 12 and Fig. 18. In most cases, our map-based planners show higher AL performance in both terrains than local planning. This verifies that our map-based planners are crucial for informative data collection, while local planning is not robustly applicable to varying terrains and planning objectives.
F. Sensitivity Analysis
The fifth set of experiments analyses our framework under various task-dependent design choices. It (i) verifies our framework's AL performance with varying UAV starting positions; (ii) validates our framework's robustness to different pretraining schemes; and (iii) showcases our framework's applicability and superior performance over baselines with different model architectures. The experiments are evaluated on the ISPRS Potsdam [5] and RIT-18 [7] datasets using the Bayesian model uncertainty-based planning objective estimated by MC 4)). All planners outperform the random walk baseline (yellow) using the Bayesian model uncertainty objective. Using the representation novelty objective, only our map-based optimisation and sampling planners show higher final prediction performance than the baseline. dropout. If not stated otherwise, we utilise the Bayesian ERFNet (Sec. III-A) pre-trained on Cityscapes [67]. Fig. 22 summarises the AL performance for each planner averaged over three different starting positions at the top-left, top-right, and bottom-right corners of the ISPRS Potsdam and RIT-18 datasets. All our map-based planners, on average, reach higher AL performance than the coverage baseline (yellow) and local planner (purple) on both datasets. In contrast, the local planner, on average, does not perform better than the coverage baseline on the ISPRS Potsdam dataset, as indicated by their largely overlapping means and standard deviations. Further, as indicated by the large standard deviations of the local planner and random walk baseline (yellow) on the RIT-18 dataset, the local planning and random walk AL performances heavily depend on the UAV starting position in challenging to explore terrains. This verifies that our map-based planners are robust to varying UAV starting positions, while local planning and the baselines are sensitive to the UAV starting position. Fig. 23 summarises the AL performance for each planner averaged over three differently pre-trained Bayesian ERFNets. Each mission starts from the top-left corner of the ISPRS Potsdam and RIT-18 datasets with Bayesian ERFNet being randomly initialised, pre-trained on the Cityscapes dataset [67], or pre-trained on the Flightmare dataset [12]. Note that the standard deviations are mainly a result of the randomly initilised models having, as expected, weaker prediction performance than the pre-trained models irrespective of the planning approach. All our map-based planners, on average, show stronger AL performance than the baseline approaches (yellow) and the local planner (purple) on both datasets. Particularly, on the RIT-18 dataset, the local planner fails to outperform the random walk (yellow) irrespective of the pre-training scheme. These findings validate our map-based planners' robustness to varying model pre-training schemes. Fig. 24 summarises the AL performance of our planning framework utilising a Bayesian variant of U-Net [10]. We extend the U-Net architecture by adding dropout layers after each convolutional block with a dropout probability of 10% to perform MC dropout for computing the Bayesian uncertainty- [5] and (b) RIT-18 dataset [7] with the Bayesian model uncertainty-based planning objective estimated by MC dropout and computing informative prior maps before each mission starts. Results are averaged over three differently pretrained Bayesian ERFNets. Shaded regions indicate one standard deviation. Our map-based planners, on average, outperform the baseline approaches (yellow) and local planning (purple) on both datasets with less training data, irrespective of the pre-training scheme. based planning objective. We conduct experiments with the Bayesian U-Net pre-trained on the Flightmare dataset [12] using the ISPRS Potsdam dataset starting each mission from the top-left corner. All active planners exceed the maximum semantic segmentation performance of the coverage baseline (yellow) with less than half of the training images. This confirms the effectiveness of active planning for AL irrespective of the chosen model architecture. Further, our map-based frontier (blue) and optimisation (orange) planners outperform local planning (purple), while the sampling planner (green) performs on par with local planning. This showcases strong AL performance of our map-based planners and the applicability of our framework to different model architectures.
V. CONCLUSION AND FUTURE WORK
This paper proposed a novel and unified planning framework for active learning in aerial semantic mapping to improve a robot's semantic perception with minimal expert guidance.
A key aspect of our work is to link our planning objectives to Fig. 24. Comparison of AL performance on the ISPRS Potsdam dataset [5] using a Bayesian version of U-Net [10] pre-trained on the Flightmare dataset [12]. The Bayesian model uncertainty-based planning objective is estimated by MC dropout, and informative prior maps are computed before each mission starts. Our map-based optimisation (orange) and frontier (blue) planners outperform coverage (yellow) and local planning (purple), while the sampling planner (green) performs on par with local planning. active learning acquisition functions, enabling us to adaptively replan the robot's paths towards regions of informative training data. To ensure maximally informed online decision-making, our global planning algorithms leverage a sequentially updated probabilistic terrain map capturing semantics and acquisition function information. The framework is generally applicable to aerial robotic missions as it provides diverse acquisition functions, proposes various planning algorithms, is agnostic to the model architecture, and can be easily extended to other acquisition functions and planners.
Our experimental results show that our framework reduces the human labelling effort and maximises segmentation performance across varying terrains compared to traditionally used coverage and random walk data collection. Further, our mapbased planners outperform state-of-the-art local planners used in active learning. The results also verify the benefit of our mapping module for the active learning performance. Overall, our findings demonstrate how active learning combined with online planning enables efficient training data collection to improve robotic perception in initially unknown environments.
Future work concerns integrating varying altitudes into the planning algorithms and estimating the resulting data uncertainty to select multi-view consistent informative training data. To further reduce human labelling effort, combining the supervised AL paradigm with self-supervised training and continual learning across different terrains could be a promising avenue for future research.
Fig. 6 .
6Examples of paths planned by the baseline strategies on ISPRS Potsdam[5]. Orange lines show paths planned in one mission, black crosses indicate collected training images, and gray areas depict unexplored terrain.
Fig. 8 .Fig. 9 .
89Comparison of AL performance with a Bayesian model uncertaintybased planning objective estimated by MC dropout and computing informative prior maps before each mission starts. All active planners exceed the coverage baseline's performance (yellow) with less training data. Our global map-based planners outperform the local planning scheme (purple). Examples of paths planned on ISPRS Potsdam using the frontierbased planning strategy with (top) and without (bottom) our approach for precomputing informative prior maps. The priors are computed before each of four subsequent missions, with the UAV starting in the top-left corner. As shown by the planned paths (orange lines) and measurement positions (black crosses), using informative priors facilitates spatial exploration across missions and leads to more targeted training data collection within missions.
Fig. 10. Comparison of effects of our mapping module using our framework. The planners consistently benefit from recomputing informative map priors before a mission starts (purple). The performance gain of mapping a continuous RGB sensor stream (blue) instead of only mapping images at planned measurement positions (orange) is less significant. Combining both continuous sensor streams and informative map priors (yellow) also leads to consistent performance improvements. In particular, our informative mapping approach drastically improves the greedy frontier-based strategy.
Fig. 12 .
12Qualitative results with our ensemble of T = 8 ERFNets trained on ISPRS Potsdam. Columns from left to right: RGB input, ground truth, prediction, error image (negative prediction log-likelihood), model uncertainty (Eq. (3)). High model uncertainty (yellower) in misclassified regions (whiter) validates that our ensemble provides consistent uncertainty estimates as a basis for an AL planning objective in our framework.
Fig. 13 .
13Performance of our Bayesian ensemble with varying number T of ERFNets (blue), deterministic ERFNet (orange), and Bayesian ERFNet with T = 50 MC dropout samples [30] (black, dashed) on ISPRS Potsdam. For T = 8, the ensemble improves mIoU by 4.96% (middle) and reduces ECE by 7.09% (right) over a deterministic ERFNet, and improves mIoU by 0.90% and reduces ECE by 1.68% over the Bayesian ERFNet with MC Dropout.
Fig. 14 .Fig. 15 .
1415Comparison of AL performance with a Bayesian model uncertaintybased planning objective estimated by an ensemble of T = 4 ERFNets and computing informative prior maps before each mission starts. All active planners exceed the coverage baseline's performance with less training data. Our global map-based planners outperform the local planning scheme. Qualitative results of a deterministic ERFNet trained on ISPRS Potsdam. Columns from left to right: RGB input, ground truth, prediction, error image (negative prediction log-likelihood), prediction entropy. High entropy (yellower) is only weakly correlated with high errors (whiter).
Fig. 16 .Fig. 17 .
1617Comparison of AL performance with a non-Bayesian entropy-based planning objective over a deterministic forward pass and computing informative prior maps before each mission starts. The frontier-based, optimisationbased, and local planner outperform the coverage baseline's AL performance. Comparison of AL performance of uncertainty-based planning objectives on ISPRS Potsdam. Our Bayesian uncertainty-based objectives (blue, yellow) tend to perform better than the non-Bayesian entropy-based objective (orange). The Bayesian ensemble (yellow) achieves the highest AL performance across the planning strategies.
Fig. 20 .
20AL results on the RIT-18 dataset [7] using informative prior maps with (a) the Bayesian model uncertainty objective estimated by MC dropout (Eq. (3)), and (b) the representation novelty objective (Eq. (
Fig. 21 .
21AL results in the Flightmare simulator[12] using informative prior maps with (a) the Bayesian model uncertainty objective (Eq. (3)), and (b) the representation novelty objective (Eq. (
Fig. 22 .
22Comparison of AL performance on the (a) ISPRS Potsdam dataset[5] and (b) RIT-18 dataset[7] with the Bayesian model uncertainty-based planning objective estimated by MC dropout and computing informative prior maps before each mission starts. Results are averaged over three different UAV starting positions. Shaded regions indicate one standard deviation. Our mapbased planners consistently outperform the baseline approaches (yellow) and local planning (purple) on both datasets with less training data while showing less sensitivity to the UAV starting position.
Fig. 23 .
23Comparison of AL performance on the (a) ISPRS Potsdam dataset
. In our experimentalFig. 5. Our planning strategies for training data collection. Dark gray dots and lines indicate candidate measurements and paths evaluated based on their estimated information adding these images to the training set. Orange dots and lines indicate the most informative chosen measurements and paths. Light gray depicts unexplored terrain. In (c), black indicates the greedy initialisation, and ellipses indicate the optimisation of candidate paths in continuous space.(a) Image
(b) Frontier
(c) Sampling
(d) Optimisation
TABLE I
IENVIRONMENT, SENSOR, AND UAV MISSION PARAMETERS FOR THE THREE DATASETS IN OUR EXPERIMENTSDataset
Type
Classes
Task
Area [m×m]
FoV [px×px]
GSD [cm/px] Altitude [m]
Budget [s] Test Images
Potsdam [5]
Orthomosaic
7
Urban
900×900
400×400
15.0
30
1800
3500
RIT-18 [7]
Orthomosaic
6
Land Cover
261×568
400×400
8.0
15
400
3500
Flightmare [12] Unity/Gazebo
10
Industrial
150×130
480×720
8.3
20
150
1000
(a) ISPRS Potsdam
(b) RIT-18
(c) Flightmare
Fig. 7. Comparison of AL performance of our three baseline planning approaches with MC dropout inference (a) on ISPRS Potsdam dataset [5], (b) on the
RIT-18 dataset
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On Calibration of Modern Neural Networks. C Guo, G Pleiss, Y Sun, K Q Weinberger, Proc. of the Int. Conf. on Machine Learning (ICML). of the Int. Conf. on Machine Learning (ICML)C. Guo, G. Pleiss, Y. Sun, and K. Q. Weinberger, "On Calibration of Modern Neural Networks," in Proc. of the Int. Conf. on Machine Learning (ICML), 2017, pp. 1321-1330.
| {'fraction_non_alphanumeric': 0.05295622375597278, 'fraction_numerical': 0.01844683623727014, 'mean_word_length': 4.709026782762042, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 5, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 13, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Unmanned aerial vehicles (UAVs) are frequently used for aerial mapping and general monitoring tasks. Recent progress in deep learning enabled automated semantic segmentation of imagery to facilitate the interpretation of large-scale complex environments. Commonly used supervised deep learning for segmentation relies on large amounts of pixel-wise labelled data, which is tedious and costly to annotate. The domain-specific visual appearance of aerial environments often prevents the usage of models pre-trained on publicly available datasets. To address this, we propose a novel general planning framework for UAVs to autonomously acquire informative training images for model re-training. We leverage multiple acquisition functions and fuse them into probabilistic terrain maps. Our framework combines the mapped acquisition function information into the UAV's planning objectives. In this way, the UAV adaptively acquires informative aerial images to be manually labelled for model re-training. Experimental results on real-world data and in a photorealistic simulation show that our framework maximises model performance and drastically reduces labelling efforts. Our map-based planners outperform state-of-the-art local planning.", 'arxivid': '2302.03347', 'author': ['Julius Rückin ', 'Federico Magistri ', 'Cyrill Stachniss ', 'Marija Popović '], 'authoraffiliation': [], 'corpusid': 256627395, 'doi': '10.48550/arxiv.2302.03347', 'github_urls': ['https://github.com/dmar-bonn/ipp-al-framework.'], 'n_tokens_mistral': 28117, 'n_tokens_neox': 24976, 'n_words': 15392, 'pdfsha': 'cd8ec99afd1be81fe32332318ccbde52cc9b2e2b', 'pdfurls': ['https://export.arxiv.org/pdf/2302.03347v2.pdf'], 'title': ['An Informative Path Planning Framework for Active Learning in UAV-based Semantic Mapping', 'An Informative Path Planning Framework for Active Learning in UAV-based Semantic Mapping'], 'venue': []} |
arxiv |
INFN and University
TorinoItaly
10.1393/ncc/i2015-15003-8received 7 January 2015Colloquia: IFAE 2014 IL NUOVO CIMENTO 38 C (2015) 3 J/ψ and ψ(2S) production in p-Pb collisions at √ s NN = 5.02 TeV with ALICE at the LHC M. Leoncino for the ALICE Collaboration
The ALICE Collaboration has studied the inclusive J/ψ and ψ(2S) production in p-Pb collisions at √ sNN = 5.02 TeV, at the CERN LHC. The J/ψ measurement is performed in the µ + µ − and in the e + e − decay channels, down to zero pT. The results are in fair agreement with theoretical predictions. The ψ(2S) measurement has also been performed. In particular, a smaller ψ(2S) nuclear modification factor, with respect to the J/ψ one, has been observed. PACS 25.75.-q -Relativistic heavy-ion collisions. PACS 12.38.Mh -Quark-gluon plasma. PACS 14.40.Pq -Heavy quarkonia. c � CERN on behalf of the ALICE Collaboration Creative Commons Attribution 4.0 License
-Introduction
The suppression of charmonia, bound states of c andc quarks, is considered a clean signature of Quark-Gluon Plasma (QGP) formation in heavy-ion collisions [1]. In addition to the color screening mechanism, other effects may contribute to the charmonium production in Pb-Pb collisions. In particular, one can expect a recombination of c and c pairs from the medium (favoured by the large cc multiplicity typical at the LHC energies [2]). Furthermore, cold nuclear matter (CNM) effects, like shadowing [3,4] and initial state parton energy loss [5], are expected to play a role. It is thus important to study the production of charmonia in proton-nucleus collisions in order to disentangle the suppression contribution related to CNM from the one associated to the formation of a QGP.
-The ALICE detector and the p-Pb run
The ALICE detector consists of a central barrel dedicated to particle tracking and identification (in the range |η| < 0.9) and a forward spectrometer used for the detection of muons (in the interval −4 < η < −2.5). More details about the experimental setup can be found in [6]. The J/ψ resonance is detected both in the dielectron decay channel (using the central barrel detectors) and in the dimuon decay channel (using the forward -Inclusive J/ψ nuclear modification factor as a function of rapidity. Open boxes are uncorrelated systematic uncertainties, filled area indicate partially correlated uncertainties and the gray box at one the global uncertainty due to the T pPb . Theoretical models based on shadowing/energy loss [9][10][11] and CGC [12] are also shown. muon spectrometer), while, with the present statistics, the ψ(2S) can be studied only in the dimuon decay channel. Due to the energy asymmetry of the LHC beams in p-Pb collisions the nucleon-nucleon center-of-mass system is shifted by Δy = 0.465 in the direction of the proton beam. Data have been collected in two beam configurations with inverted beam directions, resulting in the following rapidity coverages: −4.46 < y cms < −2.96 at backward rapidity, −1.37 < y cms < 0.46 at midrapidity and 2.03 < y cms < 3.53 at forward rapidity.
-Results
The nuclear effects on J/ψ production in p-Pb collisions are quantified by means of the nuclear modification factor, which is defined by: R pPb = Y J/ψ /�T pPb � · σ J/ψ pp , where Y J/ψ is the efficiency corrected J/ψ yield, σ J/ψ pp is the production cross section in pp collisions in the same kinematical range at the same energy and T pPb is the nuclear thickness function estimated through the Glauber model [7]. Since pp data at √ s = 5.02 TeV are not available, the reference σ J/ψ pp is obtained with an interpolation procedure [8]. The results are reported in fig. 1: at mid and forward rapidity, the inclusive J/ψ production is suppressed with respect to that in pp collisions, whereas it is unchanged at backward rapidity. Models containing shadowing and/or energy loss [9][10][11] are in agreement with ALICE data (within uncertainties), while the CGC-based model [12] overestimates the suppression.
Since (fig. 2). At low transverse momentum data suggest a contribution from regeneration while, at higher transverse momenta, the suppression contribution starts to be dominant. The ψ(2S) analysis has been performed analogously to the J/ψ one. In fig. 3 (left) the double ratio [ψ(2S)/J/ψ] pPb / [ψ(2S)/J/ψ] pp is shown as a function of rapidity and is compared with the PHENIX result in d-Au collisions at √ s N N = 0.2 TeV. ALICE results show a similar trend compared with PHENIX data at midrapidity [13] indicating a suppression in the ψ(2S) production with respect to p-p collisions. This suppression is further investigated in fig. 3 (right) where the R ψ(2S) pPb is presented as a function of rapidity and is compared to R J/ψ pPb . The ψ(2S) is more suppressed with respect to the J/ψ. These results are compared to theoretical calculations used for the J/ψ [9][10][11], which should hold for the ψ(2S). The available theoretical predictions do not describe the ψ(2S) suppression, indicating that other mechanisms are required to explain it.
Fig. 1 .
1Fig. 1. -Inclusive J/ψ nuclear modification factor as a function of rapidity. Open boxes are uncorrelated systematic uncertainties, filled area indicate partially correlated uncertainties and the gray box at one the global uncertainty due to the T pPb . Theoretical models based on shadowing/energy loss [9-11] and CGC [12] are also shown.
the Bjorken x-values in the Pb nucleus in p-Pb collisions at √ s N N = 5.02 TeV are similar to the ones in Pb-Pb collisions at √ s N N = 2.76 TeV and assuming a factorization of shadowing effects, an expectation for the R AA based on the R pA can be derived by comparing R forward pPb × R backward pPb with R PbPb
Fig. 2 .Fig. 3 .
23-In the left plot the J/ψ R backward pPb × R forward pPb is compared to the R forward PbPb . In the right plot the J/ψ is compared " -Left: the double ratio [ψ(2S)/J/ψ] pPb / [ψ(2S)/J/ψ] pp . PHENIX data at √ sNN = 0.2 TeV at midrapidity is also shown. Right: the R
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| {'fraction_non_alphanumeric': 0.07013418176787937, 'fraction_numerical': 0.04841610181214553, 'mean_word_length': 3.7912524850894633, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The ALICE Collaboration has studied the inclusive J/ψ and ψ(2S) production in p-Pb collisions at √ sNN = 5.02 TeV, at the CERN LHC. The J/ψ measurement is performed in the µ + µ − and in the e + e − decay channels, down to zero pT. The results are in fair agreement with theoretical predictions. The ψ(2S) measurement has also been performed. In particular, a smaller ψ(2S) nuclear modification factor, with respect to the J/ψ one, has been observed. PACS 25.75.-q -Relativistic heavy-ion collisions. PACS 12.38.Mh -Quark-gluon plasma. PACS 14.40.Pq -Heavy quarkonia. c � CERN on behalf of the ALICE Collaboration Creative Commons Attribution 4.0 License', 'arxivid': '1511.06140', 'author': ['\nINFN and University\nTorinoItaly\n'], 'authoraffiliation': ['INFN and University\nTorinoItaly'], 'corpusid': 119294866, 'doi': '10.1393/ncc/i2015-15003-8', 'github_urls': [], 'n_tokens_mistral': 2495, 'n_tokens_neox': 2087, 'n_words': 1182, 'pdfsha': 'd9b3247de614a15fd1754ff095068c07468b6628', 'pdfurls': ['https://export.arxiv.org/pdf/1511.06140v1.pdf'], 'title': [], 'venue': []} |
arxiv |
Probabilistic exact cloning and probabilistic no-signalling
arXiv:quant-ph/9908017v2 17 Aug 1999
Arun Kumar
Dean Street
Quantum Optics and Information Group
SEECS
University of Wales
Bangor LL 57 IUTUK
Pati
Dean Street
Quantum Optics and Information Group
SEECS
University of Wales
Bangor LL 57 IUTUK
Probabilistic exact cloning and probabilistic no-signalling
arXiv:quant-ph/9908017v2 17 Aug 1999(August 7, 2018)
We show that non-local resources cannot be used for probabilistic signalling even if one can produce exact clones with the help of a probabilistic quantum cloning machine (PQCM). We show that PQCM cannot help to distinguish two statistical mixtures at a remote location. Thus quantum theory rules out the possibility of sending superluminal signals not only deterministically but also probabilistically. We give a bound on the success probability of producing multiple clones in an entangled system.
All the useful information that can be transmitted has a universal speed limit, namely, the speed of light. In quantum mechanics the situation seems to be changed when Einstein, Podolsky and Rosen put forward their famous thought experiment on two systems that had interacted in the past but are no longer in direct contact [1]. These kind of systems are described by entangled states which show non-local correlations that cannot be explained by any local hidden variable theory [2]. In [1] they proved that the measurement outcome of one system can instantaneously affect the result of the other. This suggested that one can exploit the non-local nature of the entangled states to send superluminal signals. However, it was shown that since the operators at space-like separated distance commute, the averages of the observable at distant site remain the same and do not depend on the operations carried out by the other (distant) party [3,4]. Subsequently, Herbert [5] put forward the idea of amplifying quantum states on one part of the system and performing a single measurement on many copies to get information about what the other party has done. It was then demonstrated by Wooters-Zurek [6] and Dieks [7] that such a multiplying device cannot exist due to linearity of quantum theory. This is now known as the "no-cloning theorem". Exact cloning of an unknown quantum state allows one to copy one part of an entangled system many times, which would give the possibility to find out what observable the other party has measured. This provides us with the means to communicate instantaneously. Since deterministic and exact cloning is ruled out by linear process, the superluminal signalling using non-local nature of entangled states was doomed. Further, it was found that it is the unitarity of quantum evolutions [8] which prevents us from cloning two non-orthogonal states exactly and this has been used against measuring a single quantum state [9]. In recent years various cloning machines which produce inaccurate copies [10,11] by unitary processes are conceived, but they cannot be used for superluminal signalling. In fact, Gisin [12] has shown that the fidelity criterion based on "no-signalling" and inaccurate copying are consistent with each other. The no signalling constraint has been used to generate optimal asymmetric clones [13].
A recent proposal by Duan and Guo [14] shows that two non-orthogonal states can be cloned exactly using unitary and measurement operations with a postselection. Furthermore, they [15] have shown that a set of states chosen secretly can be exactly cloned by a probabilistic cloning machine if and only if the states are linearly independent. Hardy and Song [16] have used no-signalling condition for the probabilistic cloning machine to find a limit on the number of state that can be cloned in a given Hilbert space of a quantum system. The exact quantum cloning have been studied from a state discrimination view by Chefles and Barnett [17]. They have also studied the interpolation between the inaccurate and exact cloning and proposed a network to implement the cloning operation [18]. Recently, we [19] have shown that the unitarity of quantum theory allows us to have a linear superposition of multiple exact clones along with a failure term iff the non-orthogonal states chosen secretly are linearly independent. This "novel cloning" machine is quite general one and all the probabilistic and deterministic cloning machines can be regarded as a special case of the "novel cloning" machine. We have also proposed a protocol which produces exact clones of an arbitrary qubit (universal cloning) with the help of classical communication and the use of entanglement [20].
It is known that if the quantum states can be cloned exactly then superluminal signals can be sent definitely. The question we address in this letter is whether exact probabilistic cloning allows us to send superluminal signals probabilistically. If a message can be transmitted instantaneously with certain non-zero probability less than unit, we call it probabilistic superluminal signalling. We show that given a probabilistic cloning machine which produces exact clones one cannot send superluminal signals probabilistically. Whereas deterministic superluminal signalling is impossible, there was some doubt that quantum mechanics might admit probabilistic superluminal signalling with the help of a probabilistic cloning machine. We give a bound on the success rate of producing M exact clones of one part of system using PQCM in a composite system. This result shows that again the quantum theory is in agreement with the principles of special relativity.
Suppose we have a singlet state consisting of two particles shared by Alice and Bob. The state is given by
|ψ − = 1 √ 2 (|0 1 − |1 0 ) = 1 √ 2 (|ψ |ψ ⊥ − |ψ ⊥ |ψ ),(1)
where |ψ = α|0 + β|1 and |ψ ⊥ = β * |0 − α|1 are mutually orthogonal spin states (or polarisations in case of photon). We call this basis {|ψ , |ψ ⊥ } as the qubit basis. Alice is in possession of particle 1 and Bob is in possession of particle 2. Alice can chose to measure the spin along the x-or the z-axis. It is known that given a shared EPR state between Alice and Bob, then the measurement outcomes of Bob are invariant under arbitrary local unitary transformation done by Alice. This is the basis for no-superluminal communication. The measurement outcomes of Bob is
p(a) = tr B (ρ B P a ),(2)
where the set of operator {P a } are projectors or could be some generalised measurements such as POVMs satisfying P † a P a = I. The density matrix ρ B = trρ AB = tr[(U A ⊗ I B )ρ AB (U A ⊗ I B ) † ] is invariant under a unitary operation by Alice. Hence Bob cannot distinguish two statistical mixtures resulting at his location due to the unitary operation done at a remote place. However, if Bob can produce exact clones (of any arbitrary input state) of his particle then he can distinguish two statistical mixtures. Is the same true with a probabilistic cloning machine?
For a single quantum system the probabilistic quantum cloning machine (PQCM) takes an input state |ψ i from a set S = {|ψ i }(i = 1, 2, .., K). This state is going to be cloned. Let |A = |0 ⊗M an ancilla (a collection of M blank states) and |P 0 , |P 1 be the 'check' states which after a measurement tells us whether cloning has been successful or not. Following [15] it can be proved that there is a unitary transformation U such that the following evolution holds
|ψ i |A |P 0 → U(|ψ i |A |P 0 ) = √ p i |ψ i ⊗M |P 0 + 1 − p i |Φ i |P 1(3)
if and only if the states {|ψ i } are linearly independent. Here, p i is the probability of successful cloning M states and |Φ i is the composite state of the input and blank states and these states need not be orthonormal. The unitary evolution together with a projection measurement yields 1 → M exact clones with a probability of success p i . To investigate the question of the possibility of the probabilistic superluminal signalling we carry out the action of PQCM on one part of a composite system (say) on Bob's particle 2. Bob attaches ancillas C and D and the evolution of the combined state is given by
|ψ − AB |A C |P 0 D → U BCD (|ψ − AB |A C |P 0 D ) = 1 √ 2 ( √ p ⊥i |ψ i A |ψ ⊥i ⊗M BC − √ p i |ψ ⊥i A |ψ i ⊗M BC )|P 0 D + 1 √ 2 ( 1 − p ⊥i |ψ i A |Φ i BC − 1 − p i |ψ ⊥i A |Φ i BC )|P 1 D ,(4)1 √ 2 (|ψ i |ψ ⊥i − |ψ ⊥i |ψ i ) → 1 √ 2 ( √ p ⊥i |ψ i |ψ ⊥i ⊗M − √ p i |ψ ⊥i |ψ i ⊗M ).(5)
If Bob can follow the above procedure, then the after Alice finds her particle in the basis {|0 , |1 } the reduced density matrix of the particle 2 (with ancillas) will be
ρ BC = tr DA (ρ ABCD ) = 1 2 (p 0 |0 0| ⊗M + p 1 |1 1| ⊗M ).(6)
On the other hand if Alice finds her particle in the basis |ψ 2 = cos θ|0 + sin θ|1 , |ψ ⊥2 = sin θ|0 − cos θ|1 }, then the reduced density matrix of the joint system (BC) would be
ρ BC = tr DA (ρ ABCD ) = 1 2 (p 2 |ψ 2 ψ 2 | ⊗M + p ⊥2 |ψ ⊥2 ψ ⊥2 | ⊗M ).(7)
Since the two statistical mixtures in (6) and (7) are different this would have allowed Bob to distinguish two preparation stages by Alice, thus allowing for superluminal signalling probabilistically. However, this is not possible. Since there are four states involved in the cloning process and Bob's particle belong to a two-dimensional Hilbert space only two of them can be linearly independent. It can be seen from the form of equation (5) that {|ψ 2 } and {|ψ ⊥2 } must be linearly dependent on the states {|ψ 1 } and {|ψ ⊥1 } . This means Bob's cloning machine should be able to clone the states from the set {|0 , |1 , cos θ|0 + sin θ|1 , sin θ|0 − cos θ|1 }. But this set is linearly dependent, hence Bob cannot clone all these four states exactly with a non-zero probability. Thus there is no way for Bob to make an inference based on the measurement results of his M clones and to know what Alice has done. Thus there cannot be any superluminal signalling even with a non-zero probability less than unit. The no-signalling constraint leads to the observation by Hardy and Song [16] that one cannot clone more than certain number of linear independent states via a PQCM. Here, we derive a bound on the success rate of producing M clones for N linearly independent states for a composite system. Let us consider an arbitrary composite system whose state is described by
|ψ AB = N A N B ij=1 a ij |x i |y j ,(8)
where {|x i } ∈ H A = C N A and {|y j } ∈ H B = C N B . Using the Schimdt decomposition we can write the bipartite state as
|ψ AB = N k=1 c k |α k |β k ,(9)
where we have put N A = N, where N is the dimension of the smallest Hilbert space. Though Hilbert space H B has N B number of linearly independent states in principle, only N of them can be cloned exactly [16]. We can write the bipartite state as [21]
|ψ AB = 1 √ N N k=1 |u k |v k ,(10)
where |u i are orthogonal basis for H A and |v i are non-orthogonal and linearly independent basis states for H B . Suppose the subsystem B is passed through a PQCM, then we have
|Ψ ABCD = U BCD (|ψ AB |A |P 0 ) = 1 √ N N k=1 √ p k |u k |v k ⊗M |P 0 + 1 − p k |u k |Φ k |P 1(11)
After a postselction of measurement result the machine will yield the state
|Ψ ABC = 1 √ N N k=1 √ p k |u k |v k ⊗M .(12)
We would like to know what is the success rate of producing M clones of two distinct states at Bob's site chosen from N linearly independent states. Let us define a basis in the Hilbert space H BCD as |X i , given by (using eq.(11) )
|X i = u i |Ψ ABCD = 1 N √ p i |v i ⊗M |P 0 + 1 − p i |Φ i |P 1 .(13)
Now taking the inner product of the two distinct basis we can get
N| X i |X j | ≤ √ p i p j | v i |v i | M + (1 − p i )(1 − p j )| Φ i |Φ j |.(14)
Simplifying the above inequality we have
1 2 (p i + p j ) ≤ (1 − N| X i |X j |) (1 − | v i |v j | M )(15)
The success probability in the case of composite system depends on the overlap of the actual states being cloned and also on the overlap of two non-orthogonal states belonging to a larger Hilbert space. In the limit of infinite number of cloning (when M → ∞), this bound approaches some kind of state discrimination bound for a composite system, given by
P = 1 2 (p i + p j ) ≤ (1 − N| X i |X j |).(16)
The maximum total success probability depends on the number of linearly independent states that can be cloned.
In conclusion, we have shown that even though a probabilistic quantum cloning machine can produce exact clones with certain non-zero probability still it is impossible to send superluminal signals with certain non-zero probability. Thus in the quantum theory the nosignalling condition is more stringent than that of the classical theory. In classical relativity we do not have a deterministic (with unit probability) superluminal signalling. But the quantum theory says that we cannot even have a probabilistic (less than unit) superluminal signalling. However, Kent [22] has shown that in a different quantum theory (in the context of time neutral cosmology) there is a superluminal signalling with non-zero probability. I thank P. Kok, S. L. Braunstein and L. M. Duan for useful discussions. I thank L. Hardy for useful correspondence. The financial support from ESPRC is gratefully acknowledged.
where the set {|ψ i } and {|ψ ⊥i }, (i = 1, 2) are linearly independent. The index i refers to two possible choices of basis onto which Alice might do a measurement. For example, {|ψ i } = {|0 , cos θ|0 + sin θ|1 } and {|ψ ⊥i } = {|1 , sin θ|0 − cos θ|1 }. Bob performs a measurement on the probing device P of the cloning machine. He keeps the states if the outcome is |P 0 and discards the result if the outcome is |P 1 . From Eq.(4) it is clear that after postselection of measurement result the antisymmetric Bell state becomes
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. A K Pati, Phys. Rev. A. in pressA. K. Pati, Phys. Rev. A (1999) (in press).
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| {'fraction_non_alphanumeric': 0.06430608954483442, 'fraction_numerical': 0.032680143867047005, 'mean_word_length': 3.754422169811321, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We show that non-local resources cannot be used for probabilistic signalling even if one can produce exact clones with the help of a probabilistic quantum cloning machine (PQCM). We show that PQCM cannot help to distinguish two statistical mixtures at a remote location. Thus quantum theory rules out the possibility of sending superluminal signals not only deterministically but also probabilistically. We give a bound on the success probability of producing multiple clones in an entangled system.', 'arxivid': 'quant-ph/9908017', 'author': ['Arun Kumar \nDean Street\nQuantum Optics and Information Group\nSEECS\nUniversity of Wales\nBangor LL 57 IUTUK\n', 'Pati \nDean Street\nQuantum Optics and Information Group\nSEECS\nUniversity of Wales\nBangor LL 57 IUTUK\n'], 'authoraffiliation': ['Dean Street\nQuantum Optics and Information Group\nSEECS\nUniversity of Wales\nBangor LL 57 IUTUK', 'Dean Street\nQuantum Optics and Information Group\nSEECS\nUniversity of Wales\nBangor LL 57 IUTUK'], 'corpusid': 14877683, 'doi': '10.1016/s0375-9601(00)00281-4', 'github_urls': [], 'n_tokens_mistral': 5171, 'n_tokens_neox': 4485, 'n_words': 2916, 'pdfsha': 'c99e692b24c0258af326bf225e93650a00055123', 'pdfurls': ['https://arxiv.org/pdf/quant-ph/9908017v2.pdf'], 'title': ['Probabilistic exact cloning and probabilistic no-signalling', 'Probabilistic exact cloning and probabilistic no-signalling'], 'venue': []} |
arxiv |
Deep Audio-Visual Singing Voice Transcription based on Self-Supervised Learning Models
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Deep Audio-Visual Singing Voice Transcription based on Self-Supervised Learning Models
Index Terms-Multimodel learningsinging voice transcrip- tionself-supervised-learningfeature fusion
Singing voice transcription converts recorded singing audio to musical notation. Sound contamination (such as accompaniment) and lack of annotated data make singing voice transcription an extremely difficult task. We take two approaches to tackle the above challenges: 1) introducing multimodal learning for singing voice transcription together with a new multimodal singing dataset, N20EMv2, enhancing noise robustness by utilizing video information (lip movements to predict the onset/offset of notes), and 2) adapting self-supervised learning models from the speech domain to the singing voice transcription task, significantly reducing annotated data requirements while preserving pretrained features. We build a self-supervised learning based audio-only singing voice transcription system, which not only outperforms current state-of-the-art technologies as a strong baseline, but also generalizes well to out-of-domain singing data. We then develop a self-supervised learning based video-only singing voice transcription system that detects note onsets and offsets with an accuracy of about 80%. Finally, based on the powerful acoustic and visual representations extracted by the above two systems as well as the feature fusion design, we create an audio-visual singing voice transcription system that improves the noise robustness significantly under different acoustic environments compared to the audio-only systems.
Deep Audio-Visual Singing Voice Transcription based on Self-Supervised Learning Models Xiangming Gu, Wei Zeng, Jianan Zhang, Longshen Ou, Ye Wang
Abstract-Singing voice transcription converts recorded singing audio to musical notation. Sound contamination (such as accompaniment) and lack of annotated data make singing voice transcription an extremely difficult task. We take two approaches to tackle the above challenges: 1) introducing multimodal learning for singing voice transcription together with a new multimodal singing dataset, N20EMv2, enhancing noise robustness by utilizing video information (lip movements to predict the onset/offset of notes), and 2) adapting self-supervised learning models from the speech domain to the singing voice transcription task, significantly reducing annotated data requirements while preserving pretrained features. We build a self-supervised learning based audio-only singing voice transcription system, which not only outperforms current state-of-the-art technologies as a strong baseline, but also generalizes well to out-of-domain singing data. We then develop a self-supervised learning based video-only singing voice transcription system that detects note onsets and offsets with an accuracy of about 80%. Finally, based on the powerful acoustic and visual representations extracted by the above two systems as well as the feature fusion design, we create an audio-visual singing voice transcription system that improves the noise robustness significantly under different acoustic environments compared to the audio-only systems.
Index Terms-Multimodel learning, singing voice transcription, self-supervised-learning, feature fusion.
I. INTRODUCTION
S INGING voice transcription (SVT), a task that automatically transcribes note events from singing audio signals, converts our singing voice to music notation. In addition to frame-level pitch estimation [1], [2], [3], [4], which focuses on predicting pitch contour / fundamental frequency (F0) at each time step, SVT pays attention to note events. Each note event includes onset time, offset time, and pitch (in hertz or semitone). This setup makes SVT more challenging but fits many music-related downstream applications. For example, the note events can be applied to music education [5], music therapy [6], human-computer interaction [7], and singing voice synthesis (SVS) [8], [9].
In this work, we aim to tackle two main challenges in the SVT that hamper its development:
Noise Robustness. In real-world applications, recorded singing data is often contaminated by musical accompaniment, applause, cheers, and other noises. Previously SVT models, e.g. [3], [10], [11], [12], [13], are all based only on audio signals, which result in their inability to deal with low signalto-noise ratio (SNR) scenarios. Among them, only [13] pro-Xiangming Gu, Wei Zeng, Jianan Zhang, Longshen Ou, Ye Wang are with School of Computing, National University of Singapore, Singapore. Ye Wang is the corresponding author.
Manuscript received xxx; revised xxx.
posed training an SVT system directly on polyphonic music with both singing and accompaniment. However, this method can not generalize to different SNR levels or noise types (e.g. babble/white/natural noise). Therefore, the solutions to building robust SVT systems still need to be explored. Label Insufficiency. Existing annotated SVT datasets are all small-scale, primarily because manual annotation of SVT is a highly time-consuming and demanding task. For instance, the largest manually annotated SVT dataset, MIR-ST500 [12] has about 30 hours, much less than 960 hours of speech recordings in LibriSpeech [14] (a benchmark dataset for automatic speech recognition). Several attempts have been made to mitigate the problem of insufficient labeled data [11], [13], [15]. Specifically, [11] separated SVT into onset/offset detection and pitch estimation. For onset/offset detection, they adopted virtual adversarial training (VAT) [16] to take advantage of unlabeled singing data; for pitch estimation, they directly used an existing model named PatchCNN [17]. The separated design made [11] not an end-to-end framework. Another work [13] exploited the benefits from frame-level pseudo labels. Using an existing F0 predictor, [13] firstly estimated framelevel pitch from singing data without note-level annotations. The estimated continuous pitch is quantized into semitone and then smoothed by median filters. After obtaining the pseudo labels, [13] followed the noisy student framework [18] to train the SVT system. However, the evaluation of [13] was conducted on Chinese singing only. Its generalization to other genres is unexplored. Besides, the unlabeled singing data in [11] and singing data with frame-level annotations in [13] are tiny compared to rich speech data. From another perspective, [15] formalized onset/offset detection as an object detection task and used YOLOX [19], which is pretrained on image data. However, the knowledge learned from images is difficult to transfer to singing data. For example, pitch estimation cannot be benefited from the pretrained YOLOX. To summarize, the problem of label insufficiency is still not well resolved.
To improve the noise robustness, we propose introducing multimodal learning by building an audio-visual system for the SVT task. The idea is motivated by both theoretical guarantees [20] as well as widespread empirical success of multimodal learning, such as audio-visual speech recognition [21], [22], [23], [24], audio-visual active speaker detection [25], etc. Additionally, in the singing domain, [26], [27] built audiovisual singing source separation systems. [28] built the first multimodal automatic lyric transcription system, MM-ALT, demonstrating that it is robust to the sound contamination, like musical accompaniment. Through our experiments, we observe that video modality can improve overall performance arXiv:2304.12082v1 [cs.SD] 24 Apr 2023 in onset/offset detection and pitch estimation. For example, videos of lip movements can successfully discriminate the transitions between contiguous notes, thus providing extra cues for onset/offset detection. Furthermore, compared to audioonly systems, our audio-visual SVT system demonstrates higher noise robustness in different acoustic environments, including the musical accompaniment, white noise, babble noise, and natural noise.
To deal with the label insufficiency, we propose adopting self-supervised learning (SSL) models in the SVT task. Previous research has demonstrated that SSL models perform well in downstream tasks after finetuning, even in low-resource scenarios [29], [30], [23], [31]. However, there are two obstacles in following this line of work. Firstly, there are no SSL models in the singing domain. To solve this, we choose SSL models trained on rich unlabeled speech data, considering the similarities between speech and singing data. [32] has shown that wav2vec 2.0 can be adapted from the speech domain to the ALT task. The adaptation of wav2vec 2.0 benefits from the fact that ALT and ASR are counterpart tasks, which means that the input-output pairs are essentially the same. However, in the scenario of SVT, the labels are the onset/offset/pitch scores instead of the texts in the ASR/ALT task. We must consider the task difference and domain shift between speech and singing, which is our second obstacle. To deal with this, we propose a new strategy for adapting SSL models from the speech domain to the SVT task inspired by [33]. In detail, we assume that the finetuning may distort the pretrained features. Therefore, we propose skipping the finetuning stage of SSL models on downstream tasks in the speech domain and finetune the SSL models directly on the SVT task using linear-probing and fullfinetuning (LP-FT). The resulting SVT systems demonstrate high performance on both in-domain (ID) distribution and out-of-domain (OOD) singing data. We summarize the main contributions of this work here:
• We propose a new pipeline to adapt the SSL models from the speech domain to the SVT tasks without distortion of pretrained features. Our audio-only SVT system outperforms state-of-the-art technologies on multiple benchmark SVT datasets significantly. It also generalizes to out-of-domain singing data with different languages and styles. • We initialize the task of video-only SVT and demonstrate that videos of lip movements can detect the onsets/offsets of note events through both quantitative results and qualitative analysis. • We curate the first multimodal SVT dataset: N20EMv2.
Our audio-visual SVT system shows higher robustness to different types of noise perturbations compared to audioonly SVT systems.
II. METHODOLOGY
A. Problem Setting of SVT
We first consider the conventional problem setting of audioonly SVT (A-SVT). Suppose the input waveform is x A whose duration is L, and the output note events can be represented as y. Concretely, y contains N note events:
≤ o 1 < f 1 ≤ o 2 < f 2 ≤ ... ≤ o N < f N ≤ L,
and p n is the pitch value. A-SVT task aims to train a neural network to map from x A to y. We model the training of the A-SVT system as a frame-level classification problem. Firstly, the raw waveform x A , which is a 1-D tensor, is fed into the audio-specific feature encoder φ A to extract the deep acoustic representations c A ∈ R T ×D , where T is the number of frames (i.e. time steps), and D refers to the number of dimensions. Since the duration of x is L, the frame length or frame resolution of acoustic representations is L T . Afterward, each frame of features is sent to a classifier θ to obtain the frame-level predictions for note events. To supervise the training of φ A and θ, we transform the notelevel annotations into frame-level annotations.
Following [12], the annotation for each frame has four targets, including onset, silence, pitch name, and octave. Since it is difficult to predict offset directly, our model predicts silence instead. Then the offset times f 1 , f 2 , ..., f N are located at the beginnings of silence. Onset labels O and silence labels S are 1-D tensors: O, S ∈ R T . The frames covering the onset times o 1 , o 2 , ..., o N are marked as onset frames and labeled as 1 (otherwise labeled as 0). Similarly, the frames covering the silent times (no notes) are marked as the silence frames and labeled as 1 (otherwise labeled as 0). Conventionally, the pitch values are labeled as MIDI note numbers from C2 (MIDI number 36, 65.41 Hz) to B5 (MIDI number 83, 987.77 Hz) 1 . We further split each pitch into a pair of values: octave and pitch name. The octave range is from 2 to 5, and the pitch name range is from C to B, representing 12 notes in each octave. For example, the octave of C3 (MIDI number 48) is three, while its pitch name is C. We also add an octave class and a pitch name class to represent the silence. Therefore, the labels for pitch name and octave are P ∈ R T ×13 , V ∈ R T ×5 . The frame-level predictions are concatenated and transformed back into the note events through post-processing, which will be elaborated on later.
Furthermore, we extend the type of input modality and propose the problem settings of video-only SVT (V-SVT) and audio-visual SVT (AV-SVT). For V-SVT, the difference is replacing the waveform x A with the videos of lip movements x V , and the audio-specific feature encoder φ A with the videospecific feature encoder φ V . While for AV-SVT, both audio and video modality are enabled (shown in Fig. 1). We first adopt modality-specific feature encoder φ A , φ V to obtain the features for each modality c A , c V . Subsequently, another module ψ is used to fuse the features: c = ψ(c A , c V ).
B. Single-Modal SVT System 1) A-SVT System: The model architecture of our A-SVT system is visualized in Fig. 2(c) and (d). The audio-specific feature encoder φ A is parameterized by wav2vec 2.0 Large [29], and the classifier θ is a linear layer. In detail, wav2vec 2.0 Large contains a CNN model to extract latent representations z A and a large Transformer to extract contextualized representations c A . The CNN has seven temporal convolution blocks with the kernel sizes of {10, 3, 3, 3, 3, 2, 2}, strides of {5, 2, 2, 2, 2, 2, 2} and 512 channels. The design ensures that the frame length of z A is about 20 ms. Transformer has 24 blocks with model dimension 1,024, inner dimension 4,096 for Feed-forward-network (FFN), and 16 heads for multihead self-attention (MHSA). The output of wav2vec 2.0 is c A ∈ R T ×1024 . Finally, the classifier θ is a linear layer. The output dimension is 20, including one dimension for onset prediction, one for offset prediction, five for octave prediction, and 13 for pitch name prediction.
2) V-SVT System: The video-specific feature encoder φ V is parameterized by AV-HuBERT Large [23]. AV-HuBERT includes two branches for audio and video input. In our implementation, we disable the audio branch as we only use AV-HuBERT to extract visual representations c V . AV-HuBERT has a hybrid ResNet-Transformer architecture. The videos are first handled by a modified ResNet-18 [23]. We set the input to the audio branch as zeros, so the fused features only contain the visual information. Afterward, Transformer, whose architecture is the same as that in wav2vec 2.0 [29], is adopted to extract contextualized representations c V . The V-SVT system has the same classifier design as our A-SVT system.
3) Model Training: Suppose the predicted logits for onset/silence/octave/pitch name areÔ,Ŝ,V ,P , respectively. To train the whole SVT system, we adopt the following loss:
L SVT = 1 T T t=1 [BCE(σ(Ô t ), O t , w o ) + BCE(σ(Ŝ t ), S t , w s ) +CE(V t , V t ) + CE(P t , P t )],(1)
where σ refers to the sigmoid activation function.
b t = 1 if t ≤ T b . Otherwise, M b t = 0.
Then the masked SVT loss in batch mode can be written as:
L SVT = 1 B b=1 Tmax t=1 M b t B b=1 Tmax t=1 M b
Algorithm 1 Adaptation of the SSL models from the speech domain to the SVT task Require: SSL model φ (0) which has been pretrained under objective L SSL , randomly initialized classifier θ (0) , learning rates γ 1 , γ 2 for θ and φ, iterations K 1 , K 2 for linear probing and full finetuning. Skip the stage of finetuning on the ASR task for k = 1 to
K 1 + K 2 do θ (k) = θ (k−1) − γ 1 ∂LSVT ∂θ (k−1) if k ≤ K 1 then φ (k) = φ (k−1) Linear Probing else φ (k) = φ (k−1) − γ 2 ∂LSVT ∂φ (k−1)
Full Finetuning end if end for
We propose a new strategy to adapt self-supervised learning (SSL) models from the speech domain to the SVT task. Before delving into our algorithm, we first recap the training of SSL models. Typically, SSL models are firstly pretrained under unsupervised objectives. Afterward, they are finetuned using labeled data pairs on downstream tasks. For wav2vec 2.0, the unsupervised objective is a combination of contrastive loss and diversity loss: L SSL = L m + αL d , as shown in Fig. 2 (a). Specifically, the latent representations z A are also sent to a quantization module to learn discrete units q A . The diversity loss L d ensures the equal usage of codebook entries of quantization modules. The contrastive loss can be written as:
L m = − log exp(sim(c A t , q A t )/κ) q A ∈Q A t exp(sim(c A t , q A t )/κ) ,(3)
where q A t ∈ Q A t refers to the candidate quantized representations, including one positive and K negatives, sim refers to cosine similarity, and κ is a temperature hyper-parameter. During the pretraining, the input to the Transformer will be randomly masked. In [29], wav2vec 2.0 was then finetuned on the ASR task using Connectionist Temporal Classification (CTC) loss [34]. For AV-HuBERT, the pretraining stage requires the participation of both audio and video modality. AV-HuBERT alternates feature clustering and masked prediction to perform SSL. The clusters, which are regarded as the target labels for masked prediction, are assigned by clustering audio-visual features. The masked prediction loss is a crossentropy loss. After pretraining, AV-HuBERT is finetuned on the speech recognition task using CTC loss [34] and sequenceto-sequence (S2S) [35] loss. We refer the readers to [29], [23] for more details.
Similar to [33], we find that finetuning on speech recognition tasks distorts the pretrained features of SSL models, thus affecting their performance on SVT tasks in both in-domain (ID) and out-of-domain (OOD) scenarios. Therefore, we skip finetuning the SSL models on speech recognition tasks. We then conduct linear probing on the classifier θ on SVT tasks before fully finetuning the feature encoder and classifier. We also assume that linear probing mitigates the catastrophic forgetting of SSL models. Furthermore, we adopt smaller learning rates for SSL models than the classifier, similar to [32]. The detailed algorithm is described in Alg. 1. Taking wav2vec 2.0 as an example, we visualize this procedure in Fig. 2, following the order of (a) → (c) → (d).
4) Post Processing:
Our post-processing procedure follows [12]. Given the predictionsV ,P of the SVT system, we can determine the predicted MIDI number (or silence) of each frame. Afterward, we traverse all frames to search for all note events. For each note event, we first determine its onset time. If onset predictionÔ t is larger than 0.4 (onset threshold) andÔ t is a local maximum, (t − 1) L T is the onset time. Then (t − 1) L T is the offset time under the condition of t = arg min(Ŝ t > 0.5) and t > t (Ŝ t is the silence prediction). The MIDI number of this note is the mode of predicted MIDI numbers between t-th and t -the frame. It is noticed that the frame resolution L T makes significant contributions to the accuracy of SVT tasks. C. Audio-Visual SVT System 1) Modality Fusion: Compared to the A/V-SVT systems, our AV-SVT system has a feature fusion module ψ to fuse the representations from both audio and video modality. The acoustic representations c A are extracted by wav2vec 2.0 φ A while the visual representations c V are extracted by AV-HuBERT φ V . To align the frame resolution between c A and c V , the frame-rate of video input x V is set as 50 Hz instead of 25 Hz in [23]. We include an experiment in the supplement to validate that video input with 50 Hz is empirically better than that with 25 Hz. Due to its performance superiority, we parameterize ψ using residual cross attention (RCA) proposed in [28]. We visualize the structure of RCA in Fig. 3, where MHSA stands for multi-head self-attention while MHCA refers to multi-head cross-attention. The basic idea of RCA is to add cross attention between acoustic features and visual features as shortcuts to augment self-attention.
MHCA MHSA Add & norm Add & norm Feed-Forward Positional Encoding Q K V V K Q MHSA MHCA Add & norm Add & norm Feed-Forward Q K V V K Q
2) Model Training: Both wav2vec 2.0 and AV-HuBERT are large-scale. Therefore, to mitigate the demanding requirements for GPU memories, we propose to train our AV-SVT system in two stages, similar to [36]. In the first stage, we train the A-SVT system and V-SVT systems separately, following Alg. 1. Then we use trained audio-specific feature encoder φ A and video-specific feature encoder φ V in our AV-SVT system. In the second stage, we fix the weights of feature encoders and train the feature fusion module and classifier together.
III. DATASETS
A. Benchmark A-SVT Datasets
MIR-ST500 [12] is the largest A-SVT dataset with human annotations. It has 500 Chinese pop songs (about 30 hours), including 400 songs for training and 100 songs for evaluation. TONAS [37] and ISMIR2014 [38] are two small datasets only to evaluate the A-SVT systems in out-of-domain (OOD) scenarios, considering their different styles, languages, and annotation processes. TONAS has 72 Flamenco songs (36 minutes in total duration), while ISMIR2014 has 14 songs sung by children, 13 by male adults and 11 by female adults (38 pop songs, 19 minutes in total duration). What is worthy of attention is that pitch values of MIR-ST500 and TONAS are annotated as semitones while ISMIR2014 is annotated in cent resolution (1 semitone = 100 cents).
B. N20EMv2 Dataset
There are no datasets to support the training and evaluation of video-only / audio-visual SVT systems, so we curate our own dataset N20EMv2 2 . It is based on the N20EM dataset [28] since N20EM provides synchronized audio and video of singing data. We refer the readers to [28] for the details about how N20EM was collected. Here we only highlight the changes of N20EMv2 compared to N20EM.
Firstly, in N20EMv2, each sample is a whole song instead of an utterance in N20EM. Since train/valid/test sets of the N20EM dataset have no overlapping songs, we keep the same data division in N20EMv2. The statistics of the N20EMv2 dataset are shown in Table I. The total duration of N20EMv2 is longer than that of N20EM since the silent utterances are removed in N20EM. Secondly, we use videos of lip movements at a frame rate of 50 Hz in N20EMv2 instead of 25 Hz in N20EM. Most importantly, we provide songlevel annotations for the A-SVT / V-SVT / AV-SVT tasks in N20EMv2. The label format is the same as MIRS-ST500 [12].
To improve the quality of annotation, we follow a coarse-tofine manner. The annotation process is shown in Fig. 4. In the first stage, the professional digital signal processing software Melodyne 3 is adopted to obtain coarse annotations. Then in the second stage, two experts manually adjust onset/offset/pitch by playing and comparing label and audio tracks simultaneously from an interface comprising spectrogram, waveform, and MIDI notes. In this stage, we set several rules to ensure consistency between different annotators. We include these rules and detailed annotation procedures in the supplement. Our coarse-to-fine annotation procedures ensure that our curated N20EMv2 dataset has higher annotation quality than the MIR-ST500 dataset, which is validated through additional experiments in the supplement.
IV. EXPERIMENTS
In this section, we first conduct experiments on our audioonly SVT system and video-only SVT system to (1) evaluate our model design and adaptation strategy; (2) demonstrate that our models can extract powerful acoustic and visual representations. Then we fuse the above features from audio and video modality and build our audio-visual SVT system. We test its noise robustness in the environments with different noise types and SNR levels. 2 We will release the N20EMv2 dataset soon.
A. Experimental Setup
We run our experiments based on the SpeechBrain platform [39] 4 . For data pre-processing, we first perform source separation on audio signals to isolate the vocal part using spleeter [40]. To meet the input requirements of wav2vec 2.0 [29], we down-sample the vocal audio to a 16 kHz sampling rate and convert it to mono-channel if necessary. We simulate noisy environments by mixing the vocal audio with noise according to different SNR, which we will explain later. We follow [23], [28] to process the video signal and perform data augmentation.
Our experiments are conducted on an AMD EPYC 7302P 16-core CPU and two RTX A5000 GPUs. Unless specified otherwise, we choose the following training configurations. For single-modal SVT experiments, we train the model using the Adam optimizer [41] for ten epochs, including two epochs for linear probing and eight epochs for full finetuning. The learning rate for the classifier layer is 3 × 10 −4 while the learning rate for the feature encoder is 5 × 10 −5 . We adopt the Newbob technique to schedule the above learning rates with factors of 0.8 and 0.9, similar to [32]. The conventional performance metrics of SVT systems include F1-scores of the COnPOff (Correct onset, pitch, and offset), COnP (Correct onset, pitch), and COn (Correct onset). Their definitions and implementations can be found in [42], [38] 5 . For fair comparisons with previous approaches, the pitch tolerance is set as 50 cents, the onset tolerance is set as 50 ms, and the offset tolerance is set as the maximum of 50 ms and 0.2×note duration. In experiments related to the N20EMv2 dataset, we also adopt the F1-score of the COff (Correct offset) metric to evaluate the performance on offset detection.
The metrics we mentioned above are computed on song level. However, loading a whole song to GPU memory is a bottleneck since the duration of each song is about 3-5 minutes, and self-supervised learning (SSL) models are largescale. To mitigate the demands for GPU memory, we split each song into segments with 5 s (the last segment may last 2.5-7.5 s). In our experiments, the segments are set not to overlap each other. This procedure is directly conducted on the samples and their corresponding frame-level annotations. We set the batch size as 8 for training and 1 for evaluation.
B. A-SVT Experiments
1) Comparison with state-of-the-art technologies:
Our work is the first attempt to adapt self-supervised learning (SSL) models from the speech domain to the SVT tasks. Specifically, the input to our A-SVT system is the raw waveform of singing data instead of manually designed acoustic features, such as constant-Q transform (CQT) in [12], generalized cepstrum of spectrum (GCoS) in [10], [11], and spectrogram in [13]. To demonstrate the superiority of this design, we first train our A-SVT system on the MIR-ST500 training set, which is marked as "Ours variant 1". The results are shown in Table II. The evaluation on the MIR-ST500 test set can be considered an in-domain (ID) test, while the evaluations on TONAS / ISMIR2014 are out-of-domain (OOD) tests. For ID testing, our A-SVT system outperforms the Efficient-b0 [12] and JDC note [13] significantly in terms of COnPOff / COnP / COn. Especially for the metric of COn-POff, our A-SVT system exceeds the previous state-of-theart (SOTA) performance by a large margin (6.61% F1-score). For OOD testing, our A-SVT system still performs better than EfficientNet-b0, which indicates the effectiveness of our model architecture design and proposed training strategy. We note that the performances on TONAS are much worse than the MIR-ST500 test set and ISMIR2014. The reason is that TONAS consists of Flamenco songs while other datasets are mostly pop songs, resulting in a large distribution shift.
We also train another A-SVT system on the mixture of MIR-ST500 and N20EMv2 training sets (marked as "Ours variant 2" in Table II) to take advantage of our new curated N20EMv2 dataset. Apart from a high performance for ID testing, "Ours variant 2" demonstrates much better generalization abilities on singing data from unseen domains (OOD test). Specifically, "Ours variant 2" achieves state-of-the-art performances in terms of COnPOff / COnP / COn on the TONAS dataset and COn on the ISMIR2014 dataset. Also the performance of "Ours variant 2" is close to state-of-the-art [11] in terms of COnPOff / COnP on the ISMIR2014 dataset even with quantization errors. In contrast to the MIR-ST500 / TONAS / N20EMv2 datasets, which are annotated in semitones, the pitch values in ISMIR2014 are annotated in cents, which puts our A-SVT system at a disadvantage. Following [12] and modern musical notation, our current design uses a 12-tonal equal temperament system with semitonal resolution, which is more practical in real-world applications. To summarize, wav2vec 2.0 can learn great acoustic representations for the SVT tasks.
2) Baseline performance for N20EMv2 dataset: We use the same A-SVT system trained on the mixture of MIR-ST500 and N20EMv2 training sets ("Ours variant 2") to build the baseline for the N20EMv2 valid/test set. As shown in Table III ("Tolerance 1" refers to the default onset/offset/pitch tolerance), we observe that the accuracy of predictions on the test set is consistently higher than that on the validation set. Moreover, the performance of onset detection is slightly better than offset detection. One reason could be that the model cannot confidently capture the ending of a note, as notes usually decay rather than cut off suddenly.
C. V-SVT Experiments 1) Baseline performance for N20EMv2 dataset: We build our V-SVT system based on the adaptation of AV-HuBERT. The curated N20EMv2 is the first dataset for the V-SVT task, so to build the baseline, we train our V-SVT system on the N20EMv2 training set and evaluate its performance on the N20EMv2 valid/test set. The results are summarized in Table III. We find that using the video of lip movements, our V-SVT system can achieve almost 80% F1-score in terms of onset/offset detection under the default tolerance. This result is noteworthy as it can compete with previous A-SVT systems' performances on the two metrics. Furthermore, we relax the tolerance to investigate the potential of our V-SVT system. In detail, we set the onset tolerance as 100 ms, the offset tolerance as the maximum of 100 ms and 0.2×note duration, and the pitch tolerance as 100 cents (labeled as "Tolerance 2" in Table III). We notice that the COn F1-score reaches about 89%, suggesting that within the range of ±50 ms, our V-SVT system can accurately detect almost all onsets. For pitch estimation, even with only video, our V-SVT system can hint at the discrimination of different pitches. Therefore, we conclude that AV-HuBERT can learn powerful visual representations for the SVT tasks.
2) Analysis of our V-SVT results: To interpret the high performance of our V-SVT system on onset/offset detection, we assume the reason is that our V-SVT system can detect the transitions of consecutive note events by recognizing small changes in the mouth shape. The pitch of each note event reflects the acoustic information, which is difficult to be captured by video only. The performances of our V-SVT system in COnPOff / COnP demonstrate that it can roughly differentiate between mouth shapes. However, the mouth shapes are not sufficient to predict the pitches. As visualized in Fig. 5 (a), in some cases, different mouth shapes of the same singer correspond to various pitch labels. Our V-SVT system can identify these cases. As shown in Fig. 5 (b), the mouth shapes are the same, but the ground truth MIDI numbers differ. Our V-SVT system will likely fail in these cases.
D. AV-SVT Experiments
To build our AV-SVT system, we follow the training strategy explained in Section II-C. We use the feature encoders (wav2vec 2.0 and AV-HuBERT) in our best-performing A-SVT and V-SVT systems described above. We then train the feature fusion module and the classifier on the N20EMv2 training set using a larger learning rate with 3 × 10 −3 for ten epochs. The motivation behind AV-SVT is taking advantage of video modality to augment the noise robustness under acoustic environments. Therefore, we synthesize noisy audio signals Fig. 6. Quantitative comparison of our A-SVT system and AV-SVT system on the N20EMv2 test set. We compare the COnPOff, and COn two metrics of two SVT systems under the same noise perturbations for each row. From top to bottom, we use musical accompaniment, babble noise, white noise, and natural noise as the perturbations.
using four different types of noise, including the musical accompaniment, babble noise, white noise, and natural noise 6 . The babble and natural noise are created based on MUSAN dataset [43]. We set different noise levels (SNR), including -10, -5, 0, 5, 10 dB, and ∞ (clean, no noise). We train and evaluate our AV-SVT system under each scenario and report the results in Fig. 6. To achieve fair comparisons with our A-SVT system, we follow the same training procedure as our AV-SVT system to train our A-SVT system. To enable the feature fusion module, we set video inputs as zeros.
1) Quantitative analysis of AV-SVT system: In Fig. 6, we compare our A-SVT / AV-SVT system on the N20EMv2 test set in terms of the COnPOff and COn metrics. The curves of COnP are similar to that of COnPOff; the curves of COff are similar to that of COn. We visualize the complete comparisons in the supplement. Our results show that the AV-SVT system consistently outperforms the A-SVT system across different noise levels under different noise types. The improvements brought by the video modality are significant in low SNR scenarios. The performance gaps between two SVT systems are narrowed with the increase of SNR since the contributions of the video modality are diluted in less noisy environments. With the assistance of video modality, our AV-SVT system exceeds the A-SVT system by a large margin in COn, which coincides with our assumption. It is undeniable that with the video modality, the overall performance (COnPOff) of the AV-SVT system can also be improved. Compared to the other three noise types, the improvements under natural noise are limited (the last row). As natural noise is short in intervals, we presume that temporary perturbations cause less damage to the A-SVT system than continued perturbations such as the musical accompaniment, babble noise, and white noise.
2) Qualitative analysis of AV-SVT system: To further present the effectiveness of our multimodal design, we first visualize the predictions of our A-SVT and AV-SVT systems in an environment of 0 dB babble noise in Fig. 7(a). From 120 s to 123 s in the selected song, there are seven notes in the ground truth. Our AV-SVT system predicts seven notes, while our A-SVT system only predicts five. The only wrongly predicted note (2-nd note) of the AV-SVT system is also close to the ground truth since the MIDI note number difference is 1. To further explain the predictions of our AV-SVT system, we also visualize some frames of lip movements. Firstly, we display three frames x V t1−1 , x V t1 , x V t1+1 located at the onset t 1 of the 3-rd note. We notice that from t 1 − 1 to t 1 + 1, the subject slightly opened her mouth, which marks the transition from silence to 3-rd note. Moreover, we also display three frames x V t2−1 , x V t2 , x V t2+1 located at the offset t 2 of the last note. It is observed that from t 2 − 1 to t 2 + 1, the subject gradually closed her mouth. As displayed in Fig. 7(b), we also visualize the predictions of our A-SVT and AV-SVT systems in an environment with 0 dB musical accompaniment. Fig. 7(a) and (b) further validate that our AV-SVT system captures the transitions of consecutive note events and improves the noise robustness compared to audio-only systems.
E. Ablation Study 1) Effectiveness of our adaptation strategy: To adapt the self-supervised learning (SSL) models from the speech domain to the SVT tasks, we propose skipping the finetuning of SSL models on the ASR task and directly finetuning SSL models on the SVT task in a linear-probing and full-finetuning (LP-FT) fashion. To prove the effectiveness of this adaptation strategy, we conduct an ablation study on our A-SVT system by creating two variants. For "variant 1", we keep the finetuning of SSL models on the ASR task, followed by full-finetuning for ten epochs on the SVT task (the order of (a) → (b) → (d) in Fig. 2). For "variant 2", we skip the finetuning of SSL models on the ASR task, followed by full-finetuning on the SVT task for ten epochs (the order of (a) → (d) in Fig. 2.
As displayed in Table IV, our A-SVT system using the proposed adaptation strategy consistently outperforms the two variants on all datasets in terms of all metrics, including ID testing (MIR-ST500, N20EMv2 valid/test set) and OOD testing (TONAS, ISMIR2014). The results indicate the superiority of our adaptation strategy.
2) Ablation on model architecture choice: As described in Section II-B2, AV-HuBERT can accept both audio and video modality. Therefore, it is possible to build A-SVT / AV-SVT systems using AV-HuBERT as the audio-specific feature encoder. However, in our preliminary experiments, (-11.26) it is difficult for AV-HuBERT to learn powerful acoustic representations for the SVT tasks. To prove this, we build an A-SVT system using AV-HuBERT. We disable the video branch of AV-HuBERT. Unlike wav2vec 2.0, which accepts the raw waveform as input, AV-HuBERT requires audio signals' log filterbank energy feature as input. Specifically, the 26dimensional features are extracted from the raw waveform at a stride of 10 ms. The four neighboring acoustic frames are then stacked together, resulting in a frame rate of 25 Hz [23]. From Table V, the performance of the A-SVT system based on AV-HuBERT is much worse than that based on wav2vec 2.0, especially for pitch estimation. We attribute the reason to the low frame rate of input to AV-HuBERT. Each frame of the feature extracted by AV-HuBERT is 40 ms, while the frame length of the wav2vec 2.0 feature is about 20 ms. Frame resolution is a significant factor in the accuracy of the A-SVT system. Therefore, we assume low frame resolution is one of the reasons why the performance of the AV-HuBERT-based A-SVT system is inferior to that of the wav2vec 2.0-based A-SVT system. We attempt to change the input frame rate to 50 Hz by stacking only two neighboring acoustic frames. Since the input dimension is also altered, we add another linear layer to adjust the dimensions. However, the performances become much worse than before. The reason is that AV-HuBERT was pretrained on acoustic features with a frame rate of 25 Hz.
Modifying the input and model structure during the finetuning will drastically deteriorate the adaptation of SSL models to downstream tasks.
3) Discussion on adaptation of SSL models: SSL has emerged as a paradigm in the recent deep learning community [44], [29], [45], [46], [31]. One key aspect is the excellence of SSL models in downstream tasks, even in low-resource, fewshot, zero-shot setups. We agree with [33] that pretrained features play an essential role. Therefore, during the finetuning, the distortion of pretrained features will result in performance drops on downstream tasks. From this perspective, modifying the input or model structure can distort the pretrained features of SSL models. In contrast, skipping the finetuning on the speech recognition tasks, linear probing before full finetuning, and adopting smaller learning rates for SSL models will preserve the pretrained features.
V. CONCLUSION In this work, we proposed an audio-visual singing voice transcription (AV-SVT) system based on self-supervised learning (SSL) models. We curated the first multimodal SVT dataset, N20EMv2, to implement our systems. We then proposed a new approach to adapt SSL models from the speech domain to the SVT tasks to mitigate the challenge of label insufficiency. Based on this, our audio-only SVT system outperformed state-of-the-art technologies significantly and generalized to out-of-domain singing data of different languages and styles. We then initialized the video-only SVT task and our system successfully detected about 80% onset and offset of notes. Our ablation studies demonstrate the effectiveness of our adaptation strategy and model choices. Finally, our audio-visual SVT system showed excellence in clean and noisy scenarios through the experiments. Hence, our attempts validated that introducing additional modality can improve the noise robustness of the SVT system compared to the audioonly systems.
VI. ADDITIONAL DETAILS OF DATASETS In this section, we includes more details about how we curate the N20EMv2 dataset and more comparisons between the N20EMv2 dataset and benchmark singing voice transcription (SVT) datasets.
1) Curation of our N20EMv2 dataset: In the main paper, we mention that two music experts accomplished the annotation process of the N20EMv2 dataset. To ensure inter-rater reliability, we set several rules as guidelines before annotation, which are important for ensuring a higher quality of the N20EMv2 dataset. Firstly, notes are segmented according to both pitches and syllables. Different syllables are always considered as separate notes, while detailed criteria about the onset/offset/pitch labeling are as follows:
• Pitch: Pitches with a duration longer than a semiquaver are considered individual notes as perceived by the annotators, while ornaments such as pitch bending at the beginning of the note or vibratos are not considered independent notes. The pitch of each note is annotated in semitonal resolution. • Onset: The onset time of each note is marked as the start of the vowel in each syllable. If a syllable begins with a non-vowel sonorant, the annotators deliberately determine when the vowel is pronounced as onset. For instance, if the lyrics of a note is "last" [la:st], the onset is placed at the beginning of "a" [a:] instead of "l" [l]. • Offset: The offset time of each note is marked when there are no significant patterns in the audio spectrogram or the next note starts. After the initial annotation, the two experts scrutinize each other's labeling results to reach final agreements.
2) Comparison with the benchmark SVT datasets: In Table VI, we compare our curated N20EMv2 dataset with three benchmark SVT datasets including TONAS [37], ISMIR2014 [38], and MIR-ST500 [12]. Firstly, our N20EMv2 is the first multimodal SVT dataset. Secondly, N20EMv2 is much more large-scale than TONAS / ISMIR2014 and has more accurate annotations than MIR-ST500. It is noticed that except for ISMIR2014 which labeled the pitch in cents resolution, all the other datasets labeled the pitch as semitones. Because almost all modern music is based on a 12-tonal equal temperament scale, we follow the convention of labeling the pitch in semitones to balance the labeling precision and efficiency.
VII. ADDITIONAL EXPERIMENTS
Besides the experiments conducted in the main paper, we also run two additional experiments to (a) evaluate the annotation quality of our curated dataset N20EMv2; (b) explore the effects of video input frame rate on the performance of our video-only SVT system. 1) Evaluation of N20EMv2 Annotation Quality: We evaluate the annotation quality of our N20EMv2 by comparing it to the MIR-ST500 dataset. Specifically, we train our audio-only SVT systems separately using the MIR-ST500 training set and N20EMv2 training set. The former is marked as "Ours variant 1" in the main paper. Afterward, we conduct out-of-domain (OOD) testing, reflecting the generalization abilities of two trained SVT systems on singing data from unseen domains. We select TONAS and ISMIR2014, two datasets for OOD testing.
As shown in Table VII, it is noticed that the SVT system trained on the N20EMv2 training set demonstrates better OOD testing performance even though the number of songs in the MIR-ST500 training set (400 songs) is larger than that in our N20EMv2 training set (123 songs). Training on our N20EMv2 dataset improves the performance of our audio-only SVT system on TONAS by 5.44% in COnPOff, 3.47% in COnP, 10.51% in COn. While on the ISMIR2014 dataset, two SVT systems perform similarly for onset detection. However, the SVT system trained on N20EMv2 shows performance gains in COnPOff and COnP. Through this experiment, we conclude that the annotation quality of our N20EMv2 dataset is better than MIR-ST500.
2) Ablation on video frame rate choice: As mentioned in the main paper, AV-HuBERT was pretrained on video of lip movements at a frame rate of 25 Hz [23]. However, the frame rate of acoustic features extracted by wav2vec 2.0 is 50 Hz [29]. To facilitate the feature fusion of audio and video modality and increase the frame resolution, we decide to use videos at a frame rate of 50 Hz. Unlike changing the frame length of acoustic features extracted by AV-HuBERT, there is no need to modify its model structure when changing the input video frame rate. Therefore, we conduct this ablation study to validate our choice of video frame rate by comparing our video-only SVT systems accepting the videos in different frame rates. We keep other training configurations the same for fair comparisons. As shown in Table VIII, the V-SVT system with 50 Hz video input performs better than that with 25 Hz video input, especially for onset detection. The improvement is not as significant as we expect. We attribute this phenomenon to that changing the video input frame rate distorts the pretrained features, which neutralizes the improvements brought by higher frame resolution. Considering the feature fusion and the slight performance gains, we set the frame rate of video input to AV-HuBERT as 50 Hz.
VIII. ADDITIONAL VISUALIZATION RESULTS
We visualize the complete quantitative comparison of the audio-only and audio-visual SVT systems on the N20EMv2 test set, including COnPOff, COnP, COn, COff four metrics in Fig. 8. Fig. 8. Complete quantitative comparison of audio-only SVT system and audio-visual SVT system on N20EMv2 test set. For each row, we compare the COnPOff, COnP, COn, COff four metrics of two SVT systems under the same noise perturbations. From the top to bottom, we use the musical accompaniment, babble noise, white noise, natural noise as the perturbations, respectively.
Fig. 1 .
1Problem setting of the audio-visual SVT. y = [(o 1 , f 1 , p 1 ), ..., (o n , f n , p n ), ..., (o N , f N , p N )], where o n and f n are the onset/offset time of n-th note, 0
Fig. 2 .
2For onset classification, we set positive weight w o as 15.0 to amortize the effects of imbalanced distribution in O. For silence classification, we set positive weight w s as 1.0. We propose a masked version of SVT loss to enable batch mode training and handle the samples with uneven duration. Suppose the numbers of frames for all samples in a batch are T 1 , ..., T b , ..., T B , where B is the batch size. We pad each sample as well as its frame-level annotations to the duration of T max = Adaptation procedure of wav2vec 2.0 from the speech domain to the SVT task. (a) Pretraining stage of wav2vec 2.0 with unsupervised training objectives; (b) Finetuning stage of wav2vec 2.0 on downstream ASR task; (c) Linear Probing stage of wav2vec 2.0 on the SVT task; (d) Full Finetuning stage of wav2vec 2.0 on the SVT task.
max{T 1 , ..., T b , ..., T B } with zeros. Then we construct the mask M ∈ R B×Tmax for each batch. Each element M
Fig. 3 .
3Architecture of the Residual Cross Attention (RCA) module.
Fig. 4 .
4Two stages of our annotation process for the N20EMv2 dataset.
Fig. 5 .
5Examples of (a) different mouth shapes for the same pronunciation with different pitches and (b) the same mouth shape for the same pronunciation with different pitches.
Fig. 7 .
7Qualitative comparison of our A-SVT and AV-SVT systems on the N20EMv2 test set (a) with 0 dB babble noise perturbation; (b) with 0 dB perturbation of the musical accompaniment.
TABLE I STATISTICS
IOF N20EMV2 TRAIN / VALID / TEST SETSet
Duration
Number of songs
Total
8 h 22 min
157
Train
6 h 26 min
123
Valid
47 min
16
Test
69 min
18
TABLE II CONPOFF
II/ CONP / CON F1-SCORE (%) OF DIFFERENT A-SVT SYSTEMS ON MIR-ST500 TEST SET / TONAS / ISMIR2014. WE COMPARE OUR SVT SYSTEM TO STATE-OF-THE-ART APPROACHES. THE BEST RESULTS ARE MARKED AS BOLD FACE WHILE THE SECOND-BEST RESULTS ARE HIGHLIGHTED USING UNDERLINE.Dataset
Metric (%) ↑
Tony [3]
HCN [10]
VOCANO [11]
EfficientNet-b0 [12]
JDCnote [13]
Ours variant 1
Ours variant 2
MIR-ST500
COnPOff
-
-
-
45.78
42.23
52.39
52.84
COnP
-
-
-
66.63
69.74
70.73
70.00
COn
-
-
-
75.44
76.18
78.32
78.05
TONAS
COnPOff
-
-
-
9.57
-
12.71
24.08
COnP
-
-
-
19.65
-
25.24
36.87
COn
-
-
-
42.41
-
52.77
64.38
ISMIR2014
COnPOff
50
59.4
68.38
49.55
-
52.36
62.42
COnP
68
-
80.58
63.63
-
70.38
75.91
COn
73
79.0
84.04
79.16
-
92.77
93.02
TABLE III CONPOFF
III/ CONP / CON / COFF F1-SCORE (%) OF OUR A-SVT / V-SVT SYSTEMS ON N20EMV2 VALID / TEST SET.Dataset
Metric
Audio
Video
(%) ↑
Tolerance 1
Tolerance 1 Tolerance 2
N20EMv2
valid
COnPOff
61.83
4.45
9.27
COnP
68.42
6.16
11.79
COn
92.18
77.14
88.69
COff
89.80
74.68
83.01
N20EMv2
test
COnPOff
73.06
6.84
15.25
COnP
79.56
8.79
18.53
COn
93.66
78.62
88.64
COff
91.78
78.83
84.48
TABLE IV CONPOFF
IV/ CONP / CON / COFF F1-SCORE (%) OF OUR SVT SYSTEM ON DIFFERENT TEST SETS. WE COMPARE THE PERFORMANCES OF OUR SVT SYSTEM WHEN USING DIFFERENT ADAPTATION STRATEGIES. CONP / CON / COFF F1-SCORE (%) OF AV-HUBERT-BASED AND WAV2VEC 2.0-BASED SVT SYSTEMS ON DIFFERENT TEST SETS.Dataset
Metric
Ours
Baseline
(%) ↑
variant 1
variant 2
N20EMv2
valid
COnPOff
61.83
56.89 (-4.94)
59.24 (-2.59)
COnP
68.42
63.39 (-5.03)
65.99 (-2.43)
COn
92.18
91.50 (-0.68)
91.17 (-1.01)
COff
89.80
89.09 (-0.71)
89.62 (-0.18)
N20EMv2
test
COnPOff
73.06
70.16 (-2.90)
69.90 (-3.16)
COnP
79.56
77.25 (-2.31)
76.84 (-2.72)
COn
93.66
93.08 (-0.58)
92.71 (-0.95)
COff
91.78
91.22 (-0.56)
91.21 (-0.57)
MIR-
ST500
COnPOff
52.84
50.78 (-2.06)
51.43 (-1.41)
COnP
70.00
68.75 (-1.25)
68.89 (-1.11)
COn
78.05
77.23 (-0.82)
77.98 (-0.07)
TONAS
COnPOff
24.08
21.63 (-2.45)
22.55 (-1.53)
COnP
36.87
34.60 (-2.27)
36.72 (-0.15)
COn
64.38
63.01 (-1.37)
63.48 (-0.90)
ISMIR2014
COnPOff
62.42
61.03 (-1.39)
57.97 (-4.45)
COnP
75.91
74.25 (-1.66)
72.21 (-3.70)
COn
93.02
91.84 (-1.18)
92.16 (-0.86)
TABLE V
CONPOFF / Dataset
Metric
Ours
AV-HuBERT
(%) ↑
25 Hz
50 Hz
N20EMv2
valid
COnPOff
61.83
45.50 (-16.33)
22.25 (-39.58)
COnP
68.42
52.22 (-16.20)
29.37 (-39.05)
COn
92.18
86.67 (-5.51)
80.12 (-12.06)
COff
89.80
87.23 (-2.57)
74.57 (-15.23)
N20EMv2
test
COnPOff
73.06
62.96 (-10.10)
33.37 (-39.69)
COnP
79.56
70.32 (-9.24)
44.60 (-34.96)
COn
93.66
90.63 (-3.03)
81.25 (-12.41)
COff
91.78
90.77 (-1.01)
77.21 (-14.57)
MIR-
ST500
COnPOff
52.84
42.95 (-9.89)
21.13 (-31.71)
COnP
70.00
61.06 (-8.94)
35.36 (-34.64)
COn
78.05
74.03 (-4.02)
66.26 (-11.79)
TONAS
COnPOff
24.08
19.84 (-4.24)
5.22 (-18.86)
COnP
36.87
32.13 (-4.74)
15.17 (-21.70)
COn
64.38
62.62 (-1.76)
43.33 (-21.05)
ISMIR2014
COnPOff
62.42
46.92 (-15.50)
25.40 (-37.02)
COnP
75.91
56.35 (-19.56)
35.89 (-40.02)
COn
93.02
86.43 (-6.59)
81.76
TABLE VI COMPARISONS
VIAMONG VARIOUS DATASETS FOR THE SVT TASK.Dataset
Size
Ave length
Total
length
Year
Multimodal
Singing voice
Genre
Method
Pitch resolution
TONAS
72 songs,
2983 notes
30 s
36 min
2013
No
Flamenco expert
Flamenco songs
Labeled by
experts
Semitone
ISMIR2014
38 songs,
2153 notes
15 to 86 s
19 min
2014
No
14 children,
13 adult male,
11 adult female
Pop songs,
children songs
Labeled by
experts
Cent
MIR-ST500
500 songs,
>160k notes
3 to 5 min
≈ 30 h
2021
No
Online
published songs
Pop songs
in Chinese
Labeled by
non-experts
Semitone
N20EMv2
(Ours)
157 songs,
38857 notes
2 to 5 min
≈ 8.4 h
2022
Yes
Recorded songs
from amateurs
Pop songs
in English
Labeled by
experts
Semitone
TABLE VII
VIICONPOFF / CONP / CON F1-SCORE (%) OF OUR A-SVT SYSTEM ON THE TONAS AND ISMIR2014 DATASETS. WE COMPARE THE PERFORMANCE OF OUR SVT SYSTEM WHEN USING DIFFERENT TRAINING SETS.TABLE VIII CONPOFF / CONP / CON / COFF F1-SCORE (%) OF OUR V-SVT SYSTEM ON THE N20EMV2 VALID/TEST SET. WE COMPARE THE PERFORMANCES USING DIFFERENT VIDEO INPUT FRAME RATES (HZ).Dataset
Metric (%) ↑
MIR-ST500
N20EMv2
TONAS
COnPOff
12.71
18.15 (+ 5.44)
COnP
25.24
28.71 (+ 3.47)
COn
52.77
63.28 (+10.51)
ISMIR2014
COnPOff
52.36
59.46 (+ 7.10)
COnP
70.38
73.19 (+ 2.81)
COn
92.77
91.88 (-0.89)
Dataset
Metric (%) ↑
25 Hz
50 Hz
N20EMv2
valid
COnPOff
4.14
4.45 (+ 0.31)
COnP
5.38
6.16 (+ 0.78)
COn
74.89
77.14 (+ 2.25)
COff
76.01
74.68 (-1.33)
N20EMv2
test
COnPOff
5.38
6.84 (+ 1.46)
COnP
7.31
8.79 (+ 1.48)
COn
76.40
78.62 (+ 2.22)
COff
78.77
78.83 (+ 0.06)
Two adjacent MIDI numbers are differed by one semitone.
t [BCE(σ(Ô b t ), O b t , w o ) + BCE(σ(Ŝ b t ), S b t , w s ) + CE(V b t , V b t ) + CE(P b t , P b t )](2)
Our code repo: https://github.com/guxm2021/SVT SpeechBrain 5 https://github.com/craffel/mir eval
We include some noisy audio samples in the supplement.
Yin, a fundamental frequency estimator for speech and music. A De Cheveigné, H Kawahara, The Journal of the Acoustical Society of America. 1114A. De Cheveigné and H. Kawahara, "Yin, a fundamental frequency estimator for speech and music," The Journal of the Acoustical Society of America, vol. 111, no. 4, pp. 1917-1930, 2002.
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| {'fraction_non_alphanumeric': 0.05784828572214127, 'fraction_numerical': 0.041516592889939787, 'mean_word_length': 4.1730194851372495, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Singing voice transcription converts recorded singing audio to musical notation. Sound contamination (such as accompaniment) and lack of annotated data make singing voice transcription an extremely difficult task. We take two approaches to tackle the above challenges: 1) introducing multimodal learning for singing voice transcription together with a new multimodal singing dataset, N20EMv2, enhancing noise robustness by utilizing video information (lip movements to predict the onset/offset of notes), and 2) adapting self-supervised learning models from the speech domain to the singing voice transcription task, significantly reducing annotated data requirements while preserving pretrained features. We build a self-supervised learning based audio-only singing voice transcription system, which not only outperforms current state-of-the-art technologies as a strong baseline, but also generalizes well to out-of-domain singing data. We then develop a self-supervised learning based video-only singing voice transcription system that detects note onsets and offsets with an accuracy of about 80%. Finally, based on the powerful acoustic and visual representations extracted by the above two systems as well as the feature fusion design, we create an audio-visual singing voice transcription system that improves the noise robustness significantly under different acoustic environments compared to the audio-only systems.', 'arxivid': '2304.12082', 'author': ['Journal Of L A T E X Class ', 'Files ', 'Vol ', 'Xxx ', 'No ', 'Xxx '], 'authoraffiliation': [], 'corpusid': 258298334, 'doi': '10.48550/arxiv.2304.12082', 'github_urls': ['https://github.com/guxm2021/SVT', 'https://github.com/craffel/mir'], 'n_tokens_mistral': 23154, 'n_tokens_neox': 19853, 'n_words': 11259, 'pdfsha': 'f08a76076b6c8139aa493dcf9e4822871abb4c3b', 'pdfurls': ['https://export.arxiv.org/pdf/2304.12082v1.pdf'], 'title': ['Deep Audio-Visual Singing Voice Transcription based on Self-Supervised Learning Models', 'Deep Audio-Visual Singing Voice Transcription based on Self-Supervised Learning Models'], 'venue': []} |
arxiv |
AMIL: Adversarial Multi Instance Learning for Human Pose Estimation
Pourya Shamsolmoali
Masoumeh Zareapoor
Shanghai
Jie Yang [email protected]
Shanghai
HUIYU ZHOU
Jiao Tong University
China
University of Leicester
United Kingdom
Jiao Tong University
China
Department of Informatics
Shanghai Jiao Tong University
ShanghaiChina
University of Leicester
LE1 7RHLeicesterUnited Kingdom
AMIL: Adversarial Multi Instance Learning for Human Pose Estimation
Authors' addresses: P. Shamsolmoali, M. Zareapoor, J. Yang (corresponding author), Institute of Image Processing and Pattern Recognition,CCS Concepts: • Theory of computation → Machine learning theory• Computing methodologies →Artificial intelligenceImage representationsAdditional Key Words and Phrases: Pose estimations, adversarial network, multiple instance learning, neural networks
Human pose estimation has an important impact on a wide range of applications from human-computer interface to surveillance and content-based video retrieval. For human pose estimation, joint obstructions and overlapping upon human bodies result in departed pose estimation. To address these problems, by integrating priors of the structure of human bodies, we present a novel structure-aware network to discreetly consider such priors during the training of the network. Typically, learning such constraints is a challenging task. Instead, we propose generative adversarial networks as our learning model in which we design two residual multiple instance learning (MIL) models with the identical architecture, one is used as the generator and the other one is used as the discriminator. The discriminator task is to distinguish the actual poses from the fake ones. If the pose generator generates the results that the discriminator is not able to distinguish from the real ones, the model has successfully learnt the priors. In the proposed model, the discriminator differentiates the ground-truth heatmaps from the generated ones, and later the adversarial loss back-propagates to the generator. Such procedure assists the generator to learn reasonable body configurations and is proved to be advantageous to improve the pose estimation accuracy. Meanwhile, we propose a novel function for MIL. It is an adjustable structure for both instance selection and modeling to appropriately pass the information between instances in a single bag. In the proposed residual MIL neural network, the pooling action adequately updates the instance contribution to its bag. The proposed adversarial residual multi-instance neural network that is based on pooling has been validated on two datasets for the human pose estimation task and successfully outperforms the other state-of-arts models. The code will be made available on https://github.com/pshams55/AMIL.
INTRODUCTION
Estimation of human pose from an image is a challenging task because of the information limitation of images and the large distinctions in the form of different parts of body. Previously, most of the works used graphical models to handle these problems [1,2,3]. Regardless of the progresses made by mentioned fascinating models and algorithms, the bottleneck seems to be the absence of operative feature representations that have the ability to characterize several stages of visual signs and accounting for the changes in the appearance of people. Most of the recent studies [4][5][6][7][8][9][10] from the advanced illustration of the human body (e.g. skeleton) moved to the low-level feature collection (e.g. local features) as the full-body pose estimation is remained an effortful task. Recently, deep learning widely attracts computer vision researchers. Deep neural networks have the skills to appropriately learn better feature representations. For example, a recent proposed model reported in [11] achieved the state-of-the-art performance for human pose estimation. The distinct style, that uses repeated top-down and bottom-up inference going through different scales for different accessible fields, support the model to capture inherent relationships between human body parts. Though, such method may estimate human pose with improbable outlines because of severe occlusion or overlapping with the other neighboring people. This model predicts some similar features from the other person or the background. Nonetheless, it is challenging to integrate the priors of human body structures into Deep Convolution Neural Networks (DCNNs), as Tompson et al. [12] mentioned that the low-level DCNNs process is usually difficult to implement, whilst DCNNs have the ability of feature learning. As a result, an irrational human pose can be formed by an ordinary DCNN. As stated in [13], in case of dense occlusions, ordinary DCNNs achieve poor results. To cope with this issue, priors regarding the combination of the human body joints required to be considered. The best way to handle this problem is to learn the real body joints' structures from a huge amount of training data. Although, learning from such a distribution is a challenging task. To solve these problems, we intend to learn the distribution of human body structures. Suppose there is a discriminator which can determine the best form based on the reasonability of the predicted poses. If the model gets properly trained and generates the samples that are quite similar to the real samples and the discriminator could not distinguish the real samples from the fake ones, the model would have effectively learned the structure of the human body. In [58] the authors propose a biologically inspired appearance model for robust visual tracking. Motivated in part by the success of the hierarchical organization of the primary visual cortex, they build an architecture containing five layers: whitening, rectification, normalization, coding and polling. Zhang et al. [59] propose a novel visual model based matching framework for robust tracking based on basis matching rather than point matching. In [60] the authors present a machine learning model to learn a codebook of visual elements for representing the leaf shape and venation patterns. Zhang et al. [61] propose a real-time visual tracking method based on structurally random projection and weighted least squares techniques. Chen et al. [63] proposed a hybrid model for human pose estimation. On one hand, used feature pyramid network which can localize the "simple" keypoints like eyes and hands but may fail to precisely recognize the occluded or invisible keypoints. Then again, used RefineNet to explicitly handle the "hard" keypoints by integrating all levels of feature representations from the GlobalNet with the hard keypoint mining loss. Due to the recent success of Generative Adversarial Networks (GAN) on several applications [14][15][16][17], we propose a discriminator to take the responsibility of checking various structures of the human body. The generator is the main human pose estimator to capture important features of the image. In the proposed model, our discriminator and generator have the same architecture. In this paper, the adversarial training approach is used to empower the discriminator to differentiate improbable poses and guide the generator. Once the training is completed, the generator can be used as a human pose estimator and the discriminator can be ignored. At the present, convolution neural networks (CNNs) are the most effective deep learning algorithms in GAN for human pose estimation [13,17,18,64]. Chen et al. [65] proposed a statistical GAN based on the human biological structure. In [66] the authors presented a joint mining method based on GAN, which consists of two stacked hourglasses with a similar architecture. The typical CNNs architecture is a stack of convolutional, pooling, non-linear and fully connected layers, accompanied by a loss function [19,20]. It is built for taking the advantages of pooling, connections, shared weights and the use of different layers to learn high-level representations of natural images, and it has shown significant performance over numerous benchmark pose estimation datasets [20,21]. Despite that, DCNNs demand large proper labeled training data to reach superior results, however the labeling work is actually sluggish and costly by hand and if the amount of the training data is limited or with inferior quality, it will lead to suboptimal models. To minimize the influence of noisy training pairs, we model CNNs in a weakly supervised learning framework. In place of assigning labels to all the generated images, we serve the generated images as a bag and treat the main label as the bag label. This is called Multiple Instance Learning (MIL) [22,23]. For binary MIL, a bag is labeled positive if the bag holds as a minimum positive instance, and it is labeled negative if all its instances are negative. Consequently, integrating MIL into a deep learning algorithm would fully exploit the training set's potentiality and attain better performance. Figure1 shows the performance of the proposed model on some samples. The major contributions of this paper are three folds.
We design a residual MIL using neural networks based on pooling to learn the configuration and structure of different human body parts through adversarial training (AMIL). The training procedure of generative adversarial networks is used to train the proposed system to solve the complex human pose estimation problems. To our best knowledge, we are the first one to use MIL and GAN for improving human pose estimation.
We also proposed a multi-task network for predicting the pose heatmaps to achieve better performance. Being evaluated on two human pose estimation datasets (MPII and LSP) [24,25], the proposed model considerably outperforms the other state-of-the-art approaches, and is able to constantly generate better pose estimation compared to the other methods. This paper is organized as follows: Section 2 presents the related work. Section 3 illustrates the detailed description of the proposed model. In Section 4, we present the experiment results, performance evaluation of the proposed model, and we describe how AMIL improves the human pose estimation on two datasets. Section 5 concludes the paper.
RELATED WORK
This paper is related to the work using heatmaps for human pose estimation based on multiple instance neural networks and adversarial networks.
Human Pose Estimation
Most of the current human pose estimations methods use deep neural networks to predict the key-points of the human body in the images. DeepPose [27] is one of the first deep-learning based methods for human pose estimation, expressing the pose estimation obstacle as a regression problem by a regular convolutional model. Some modern approaches aim to predict human body structures, commonly named heatmaps or support maps that describe the probabilities to detect every keypoint at different positions. The precise place of a keypoint is predicted by calculating the maximum in combination of heatmaps. In comparison with direct regression models, heatmap-based approaches have the ability to leverage the allocated properties and are appropriate for training [27]. Several CNN architectures have been designed to capture the key evidence and cues of human body parts. Ukita and Uematsu [28] proposed a weakly-supervised learning model for human pose estimation. The authors used fully-annotated images which take a pose annotation and an action label to obtain initial pose models for every action. Papandreou et al. [29] used a CNN and geometric embedding descriptors that learn to perceive specific keypoints and estimate their relative movements that help to group keypoints into individual pose cases.
In [11], the authors used a multilevel structure to sufficiently enlarge the receptive field to learn the longrange spatial dealings. Furthermore, transitional supervision is adapted to generate intermediate assessment maps and to let them be processed over each stage. Recently, some techniques focus on dealing with the multi-person pose prediction problem. The method reported in [30] predicts multiple person poses in an image. The authors used the advantages of CNNs to produce keypoints and proposed integer linear programming to match the joints for each person in a group. In [31], the authors present a model that estimates the multiple person keypoint heatmaps and the affinity fields, and uses a greedy algorithm to assemble the joints that fit to the same person.
Multiple Instance learning
Multiple Instance Learning (MIL) is a different method for supervised learning. In MIL, in place of using positive or negative singletons, samples are collected to form a "bag", and each bag can have several instances [32]. Present works [23,33,34] indicate that, MIL delivers higher human action recognition accuracy. Ronchi and Perona [35] proposed a method to analyze the impact of errors in the algorithms for multi-instance pose estimation. The model calculates the sensitivity of a human pose with respect to the instance size, number and form of visible keypoints, mash of the instances, and the affiliate score of instances. However, the proposed model does not have significant performance in multiple human pose estimation in an image. Babenko et al. [36] showed the high performance and stability of MIL models on visual tracking and object detection. Yun et al. [37] proposed a geometric relational feature based on the distances between all the pairs of joints. They applied a method associated with MIL in which the sequence is represented by a bag of body-pose features. This model is accurate in detecting body joints but not efficient. Pathak et al. [38] proposed a new MIL model for semantic segmentation learning by using a fully convolution neural network and multi-class MIL loss. Hoffman et al. [39] provide a new formulation of a joint MIL method that contains samples from the object data and the labels, and executes field transfer learning to increase the underlying detector representation. The proposed model is efficient; however, in complex cases, the model captures a small portion of the image background as part of the objects.
Xu [40] proposed a MIL based decision neural network that attempts to bond the semantic gap in content-based image retrieval. In this model, the locally unsupervised learning in each subnet is to find the hidden structure in the "unlabeled" training data. While the negative or positive labels given to each image in the MIL-based application could be denoted as a kind of supervised information that considerably affects the training results. Zeng and Ji [41] proposed a deep CNN model, called as "multi-instance multi-task CNN", while the number of images, representing a multi-task problem, is considered as the inputs and a collective sub-CNN is linked to all the input images to build the instance representations. These models demonstrate the potential of CNN to capture the ambiguous multiple-to-multiple relationship in multiinstance multi-task learning on a data set with a limited number of the labeled samples. However, such ability for individual conformation cannot be differentiated in the experiment. Zhang et al. [42] proposed a different framework for object detection, by reformulating it as a MIL problem and additionally integrate it into a self-paced learning system. The proposed model is able to provide insightful metric measurements and learns patterns under co-salient areas in a self-learning way by using MIL. The proposed model shows that the bag level demonstration from the hierarchical compound neural network layers generate more dissimilar features than those formed by basically combing instance level representations.
Generative Adversarial Networks
Goodfellow et al. [43] introduced GANs for generating natural images such that it allows for unsupervised training while minimizing the blur consequence of using variational autoencoders. However, there are some concerns about hard training and instability of GANs. To efficiently train GANs, several models have been proposed to use deep convolutional architectures [18]. These models presented some elements in their networks to improve the stability, for instance, using batch normalization to avoid diversity loss and removing the fully connected layer (FC). The combination of deep convolution and GAN leads to an impressive configuration to handle the training of GAN. Gulrajani et al. [20] proposed to use gradient penalty instead of the weight clipping strategy. Gradient penalty is a loss function and an extra term that adapts the discriminator's gradient norm. The overall performance proves that this strategy is more stable and faster than the other methods. Another work by Berthelot et al. [44] presents a balance term based on the relational control theory, to make equilibrium between the generator and the discriminator. Meanwhile, if the model is collapses or reaches its last state, a conversion measure is used to control the process. In [67] the authors present a dual discriminator generative adversarial network which, unlike GAN, equipped with two discriminators; and together with a generator, it also has the analogy of a min-max game, wherein a discriminator rewards high scores for samples from data distribution whilst another discriminator, conversely, favoring data from the generator, and the generator produces data to fool both two discriminators. Hoang et al. [68] proposed to use multiple generators, instead of using a single one to overcome the collapsing problems. In [69] the authors proposed to use several generator and discriminator to increase the performance of GAN. Nevertheless, the model required high computational resources. GANs have a great success on generating images [18]; hence, it is highlighted for unsupervised learning. The idea of uncertain GAN [45] is presented for combining the class information. These methods merge the loss of uncertain GAN and the L 1 or L 2 gap between the generated and the ground-truth images. Another technique is to creating heatmaps of labels similar to semantic segmentation [46], or human pose recognition [13]. Chu et al. [47] proposed to integrate CNNs and a multi-context attention model into an end-to-end framework to recognize human poses. The model used hourglass networks to produce heatmaps from features at multiple resolutions with different semantics. Using the adversarial training approach can bring certain benefits. In this paper, we use adversarial training methods [44] to increase the performance of the pose estimator. Pose estimation can be considered as a conversion from an RGB image to a multichannel heatmap. The proposed network can well achieve this translation. Dissimilar to the other works, in the proposed network, the discrimination objective is not only to distinguish a fake image from a real one, but also to enforce the geometric constraint to the model. Table 1 summarizes some key achievements for pose estimation, where we listed each approach, experimental results, computational complexity, and efficiency.
The traditional pose estimation pipeline is based on three major points: 1) extraction of local features; 2) dictionary learning and usage of feature encoding, and 3) classification of the actions. Dense and improved trajectories [27], is popular since it highlights local changes in the spatio-temporal domain. On the other hand, local feature descriptors are also pooled to obtain image and video-level representations [18]. In addition to local interest points or feature descriptors, mid-level features of body parts [17] or deep features [54] are used frequently to discover abstract representations from videos. A detailed discussion on different video-based feature extraction techniques can be found in [56]. Nevertheless, instead of detecting local interest points, an alternative way is to extract all the local descriptors while filtering out the effects of the un-representative ones throughout the learning process. Specifically, the MIL paradigm assumes the images and videos as bags of instances (local descriptors) without concerning about their discriminative skills. However, in the MIL origination, it is required that a given bag contains at least one class precise descriptor. Fig. 2. The proposed model framework. We propose a combination of MI_RNet for pose estimation as the generator (on the left) with a discriminator (on the right) to differentiate the ground-truth heatmaps from the generated heatmaps by input heatmaps reconstruction.
THE PROPOSED ADVERSARIAL MULTI-INSTANCE LEARNING
As presented in Figure 2, the proposed AMIL model consists of two networks, the pose generator and the pose discriminator. The first network, generator, is a multiple instance residual neural network (MI-RNet) architecture. The inputs to the generator are the RGB images after the processing unit; it generates a set of heatmaps for each input image that specifies the confidence score for every keypoint on different locations of the image. The other network, the discriminator, has architecture similar to the generator; it encodes the heatmaps along with the RGB image and decodes them into a new set of heatmaps in order to distinguish real heatmaps from fake ones. Moreover, after training the pose generator with the guidance of the pose discriminator, the human body priors are extracted, which helps to increase the prediction accuracy.
Pose Generator
The generative network objective is to learn a mapping from an RGB image to keypoint heatmaps. Figure 3 illustrates the architecture of the generator. If the model extracts clear information of the body parts, it offers significant materials for describing the geometric information of a human pose. The goal of the generative network is to learn and project an image y onto the corresponding pose heatmaps x, ( ) = {̂} while ̂ is the predicted heatmap. MI-RNet has the ability to learn contextual features from the input images. Furthermore, in the pose discriminator, the adversarial loss is introduced and used for presenting the error between the ground-truth heatmaps and the generated ones. This method supports the generator to learn the f spatial dependencies from the input images and the human body patterns.
The basic block of the pose generator is expressed as follows:
{ { , } = ( −1 , ) ≥ 2 { , } = ( ) = 1
X n is the output initiation tensor of the n th weighted generative network for pose detection. Y is the image feature, captured after pre-training by using the proposed MI-RNet. In the proposed model, the final heatmap output ̂n is achieved from X n through the FC layers with the step size of 1 without padding. Specifically, the performance of the final FC layer performances as a linear classifier is gained as the final predicted heatmaps. Consequently, the specified training set { , } =1 and M represents the number of the images that are assigned for training. Moreover, the adversarial loss for the pose discrimination is proposed and considered jointly with the errors between the ground-truth heatmaps and the generated ones. This method supports the generator to learn the features and spatial dependencies out of the images and also detect all the human body configurations. • In case that bag X i is negative, therefore all the instances in X i will be negative, i.e., whenever Y i = 0, then all y ij = 0; • In case bag X i is positive, therefore at least one instance in X i will be positive, i.e., whenever Y i = 1, then ∑ =1 ≥ 1.
The aim of MIL is to train a bag classifier to estimate a new bag label. In the MI network, instance-tobag connections are various under different hypotheses. Therefore, a constant MI hypothesis on instance labels and bag labels are not signified. Accordingly, we endeavored to create a MI model to predict the bag labels. Unlike the pixels having the spatial relation, in MIL, the bag instances are a set of features that do not have a precise order. Hence, a significant asset of the MI data is the invariance to input permutation. MI networks [22] have three phases: (1) learning an instance embedded by the instance modifier; (2) executing a permutation-invariant MIL pooling to create an improvised bag; (3) bag classification depending on the bag embedding. Every phase has the permutation-invariant assets plus the essential theorem of symmetric functions [48]. Deep residual learning is proposed recently [49] and impressively performed in object detection by taking advantages of deep neural networks. In this work, as represented in Figure 3, three FC layers and three proposed MIL adjusting pooling layers are implemented. For each middle FC layer that can learn instance features, a FC layer for predicting instance scores and a proposed MIL adjust pooling layer follows it. During the training, the supervision is added to each level. During the testing, we compute the mean score for each level. We apply the residual connections after each MIL adjust pooling layer to concatenate the instance features. The task of the first FC layer is to produce a bag of feature vector. The next fully connected layers learn the residuals of bag representation. The sizes of all the FC layers are set to 128. MI-RNet is formulated as: [50]. The output of the l th layer of an instance x ij is represented as . In , , the k th index means multiple bag features from all different levels of instance features that have been learned by MIL adjust pooling. MI-RNet, by utilizing multiple hierarchies, can achieve better bag classification accuracy. We could expound it in two folds: first, for instance, better feature training can be achieved at bottom layers; and second, for testing, the average of multiple bag probabilities will be calculated to catch a better label. The weights of different levels are set equally in this work. The proposed MI-RNet is not similar to the standard residual learning [49] which learns representation residuals by convolution, ReLU and batch normalization; the model learns the residuals of the bag representation through the FC layers, ReLU and MIL adjust pooling. At the last stage of the network, the final bag representation is connected to the bag label via a FC layer with one neuron and softmax activation [50].
MIL Adjust Pooling
Other MIL pooling approaches find it is difficult to set the contextual information among the instances in a bag, as the pooling functions are of feed-forward procedures and the instance weights are calculated separately. Motivated by [51,52], we recommend using adjust pooling. To demonstrate the process, f(.) represents the instance transformer and f(X) = {f(x 1 ), f(x 2 ), ..., f(x K )} signifies the instance embedding corresponding to the bag X. The proposed adjust pooling can be stated as a weighted-sum pooling step:
( ) = ∑ ( ), =1(2)
where w i denotes the instance weight which defines the influence of the instance i th onto its bag embedding. According to these weights, to combine the instance embeddings into a single bag embedding in a weighted-sum pooling model, we use a non-linear squeezed function. The non-linear squeezed function [51] is formulated as follows:
( ) = || ( )|| 2 1 + || ( )|| 2 ( ) || ( )|| .(3)
The instance weight w i can be computed using an adjustable method. To illustrate the pooling process, a provisional instance weight is defined as b i . Afterwards, the instance weight w i is appointed via a straightforward function as follows. The superscript t signifies the t th iteration. First, t = 1 and 1 = 0, which shows that each instance equally contributes to embedding a single bag. Later, the instance weights are updated regularly while their similarities are considered in the last updated embedded bag. Hence, in the t th iteration, the embedded bag is s t (X); consequently the update function for temporary instance weight is as follows:
+1 = + ( ). ( ).(5)
After every feed forward pass, we can implement the bag embedding s T (X). L 2 norm is performed on s T (X) to demonstrate the probability of the positive bag indicated as ||s||. Therefore, the proposed MI-RNet can be optimized in the following form while Y denotes the bag label, m + = 0.9 and m − = 0.1.
( ) = (0, + − || ||) 2 + (1 − ) max (0, || || − − ) 2 .(6)
Subject to the major proposition of symmetric functions [48,57], the permutation-invariant symmetric functions W can be formulated as follows:
( ) = (∑ ∅( )). ∈(7)
where ρ and φ denote the transformations. To verify the permutation-invariant of the proposed adjust pooling, we demonstrate that the proposed pooling method satisfies the permutation-invariant symmetric requirements. The aim of the proposed pooling model is to employ weighted-sum pooling. In this model, the weight reflects other instances fitting to the same bag and its total value is calculated by t time's iteration. At the beginning (t = 1), adjust pooling starts with mean pooling:
While ∀ ∈ [1, ] 1 = . The mean pooling is a classic symmetric function. In the t th iteration, the pooling function is calculated as:
( ) = ∑ . ( ) = ∑ (∑ ( ) −1 ( )) ( ). >1(9)
At the t th iteration, the bag embedding is s t (X), which is the result of the symmetric function and retains the property of permutation-invariant. Based on the symmetric function shown in Eq. (7), we know the adjust procedure is part of ∅ and L 2 norm that signifies the position of and calculates the bag length.
Training the Generator
To use multiple instance learning for pose estimation, different regions in each image should be considered as a bag. If B = {x 1 , x 2 ...x m }, x m is the m-th region in the image. The loss function of the bag B is calculated as follows:
= − ∑ log( ( = 1| ) =1 ,(10)
Here p(S i = 1|B) denotes the possibility that the bag is correctly classified into the i th class and y i = {0, 1} 1×S is the label matrix, while S is the total sum of the classes. Based on the MIL theory, if all the instances in the bag are negative, B is a negative bag for the i th class:
( = 0| ) = ∏ (1 − ( = 1| )) , =1(11)
p(S i = 1|x j ) is the probability of the j th image region selected as the i th class, and
( = 1| ) = 1 − exp(− ℎ ).(12)
where ℎ is the i th output of the proposed MI-RNet model before the loss layer for j th region, and λ is a constant positive value and h i j ∈ [0, ∞). Eq. (12) not only significantly improves the classification results but also simplifies the gradient's calculation. For the MI-RNet, we propose a loss function to perform in parallel with the learning of the pooling function along with the instance-level classifier. In the typical MIL, the supervision of the bag-level should be handled in the loss function. As an example, the cross entropy can be used as the bag-level loss function, where C is the number of possible classes and ∈ {1, … , }. On the other hand, due to unavailability of the instance-level labels , we just rely on the bag-level label .
= − × log( ) − (1 − ) × log(1 − ).(13)
Furthermore, as the adjust pooling function is considered for gathering the instance-level predictions to attain the bag-level prediction, the ideal parameters of pooling should be dependent to the efficiency of the instance-level, which is used during the training procedure. Eq. (14) is derived from Eq. (13) to estimate the instance-level loss
= − × log( ) − (1 − ) × log(1 − ),(14)
while the relevant likelihoods to the class c, are denoted by ∈ [0,1] and = 1 ( ≥ 0.5) also 1(· ) is an pointer function. In fact, the instance-level loss is a weight of the uncertainty of , which furthermore signifies the discriminative skill of the instance-level classifier. Additionally, we propose to minimize the loss difference between the instance level and the bag-level as follows,
= || − 1 ∑ =1 ||.(15)
Adjust pooling is used as a bridge between instances and bags to minimize Eq. (15), and to fit it for the current status of the instance-level classifier. The task of the pooling layer is to efficiently transfer the instance-level discriminative skill into the bag-level classification skill. While training in the perspective of error back propagation, the loss at the bag-level is back propagated to the instance-level classifier over the pooling layer. Therefore, the learned pooling is considered to be ideal for training. To wrap up, the following loss function is proposed to mutually minimize the bag-level loss and the gap between the baglevel and the instance-level losses,
= + λ( ) 2 = + λ ( − 1 ∑ =1 ) 2 ,(16)
while λ=1 and is the Lagrange multiplier and for the mathematical simplicity, the square of difference Eq. (15) is used. In the proposed model the gradient descent approach is used to optimize the loss function for training the deep networks. In a mini-batch, the losses of different bags and instances are considered separately and then summarized. Meanwhile, losses of negative and positive bags are not computed similarly. For the positive bags where = 1, the loss function presented in Eq. (16) is used. However, for the negative bags while = 0, the instance-level loss is directly adopted,
= 1 ∑ , =1(17)
where Eq. (14) is used to calculate and = 0. Therefore, the gradients are formulated as,
ℎ = (1 − ).(18)
The methods of dynamic pooling [51], adaptive pooling [52], dynamic routing [56], and adjust pooling are adopted in the part-to-whole connection strategy. The softmax and sigmoid functions are also used to sort out the learned weights and then perform the weighted-sum pooling.
It is worthy to mention that the task of the softmax function is different. In dynamic pooling [51], softmax is applied into the individual bags to extract the relationships among them and in the adaptive pooling [52] to handle the same task sigmoid is used. In the dynamic routing [56], the softmax is used for weighting all the parent capsules with only a single child capsules. Hence, each particular weight means that the ratio of the consistent capsule in the overhead layer is sent to the child capsule. However, in the adjust pooling; the weight shows the instance involvement to the bag embedding. The softmax function is applied to all the instance contributions of the same bag and pushes them to interact with each other.
Pose Discriminator
To empower the training of the model in extracting the configurations of human body joints, the pose discriminator is designed. The discriminator's task is to recognize actual images from the generated ones. The discriminator input contains the heatmaps of the ground-truth images or the generated images, which are integrated with the corresponding actual images of persons. The discriminator should learn from the input pairs that the pose demonstrated by the heatmaps is accurate and matches the human pose in the input images. Meanwhile, the other task of the discriminator is to reconstruct a new set of heatmaps. The reconstructed heatmaps, similar to the real ones, help to determine the efficiency of the discriminator. The loss is calculated as the error between the real and the reconstructed heatmaps. For every single training image, the generated and the base heatmaps will be fed to the discriminator. The batches of the heatmaps will be reconstructed to compute l real and l fake . The discriminator is updated at each iteration by using the collected gradient, which is computed according to l real and l fake . When the input contains the ground-truth heatmaps, the discriminator is trained to identify it and create a similar one, whilst reducing the error between the reconstructed heatmaps and the groundtruth ones. The loss is formulated as, Pixel loss (l D ) is used to optimize the discriminator. The discriminator for each particular pose based on the bunch of heatmaps generates a value for each pixel. This value is the discriminator's error rate. The value shows the correctness level of a particular pixel from the view of the discriminator. For instance, if the left elbow is more accurate than the right elbow, a proper trained discriminator will create a heatmap of the left elbow that has a larger error at the position of the right elbow. This is dissimilar to a conventional GAN that only judges the properness of the whole input. Here, on the input heatmaps, the discriminator offers detailed comments and advises which parts of the heatmaps do not yield a real pose. In addition to the adversarial loss, L 2 loss is also applied on the predicted poses to measure the difference between the generated and the actual ground truth heatmaps. The final loss is the sum of the adversarial loss and an L 2 loss. A variable k t is used for balance controlling between the generator and the discriminator [44]. The variable k t is defined as follows and updated at every iteration t.
+1 = + ( − ),(20)
where k t is limited between 0 and 1, and is a hyperparameter. From Eq. (19), k t shows the amount of the emphasis put on l fake . If the generator performs better than the discriminator, l fake is smaller than ϒl real , and the generated heatmaps are very similar to the real ones. Therefore, k t will be increased to make l fake more dominant; and accordingly the discriminator will be trained to better recognize the generated heatmaps. Similarly, if the discriminator performs better than the generator, k t will be decreased to slow down the training on l fake thus the generator can keep up its performance with the discriminator.
Adversarial Networks Training
The training of the proposed adversarial network is based on supervised learning. The goal of the generator is to reduce the gap between ̂ and (̂, ) , however, the discriminator attempts to increase it. To differentiate different poses, the discriminator tries to detect the important aspect of the real pose distribution throughout the reconstruction process. Simultaneously, the generator tries to improve the quality of human pose heatmaps thus it may fool the discriminator and to allow the discriminator to reconstruct the same heatmaps. The discriminator can be eliminated after the completion of the training. The generated heatmaps ̂= ( ) will be used to conclude the last outcome. To conduct the estimations, the original image and its flipped form are evaluated, and their output heatmaps are averaged. During the training, the location with the largest confidence score in each joint's heatmap is extracted. Then, the model converts the location to the original coordinate space with respect to the input image size.
EXPERIMENTS
The proposed model is evaluated on the two benchmark pose estimation datasets, MPII Human Pose [1] and extended Leeds Sports Poses (LSP) [3]. The MPII dataset contains 25,000 images with 40,000 annotated samples, around 28,000 for training, and 11,000 for testing. The whole body images are annotated with 16 different landmarks and several directions to the camera. The images are taken from videos in YouTube and the contents include daily human activities. Compared to the other human pose datasets, MPII has affluent information, for example, fully unannotated image frames. During the training, only keypoint positions are used. The LSP dataset contains 11,000 training images and 1,000 testing images that show different sports activities. The performance of the proposed model is demonstrated using the UCF YouTube action dataset [55]. The proposed network is composed of three fully connected layers that all have a size of 128 and adjust pooling function. The weights of the fully connected layers are initialized by a normal distribution and biases are initialized to 0. T is assigned to 3 which denotes the iteration times of the adjust pooling. The Adam optimizer is used to optimize the network [53].
The details of the hyper-parameter optimization process such as learning rate and weight decay are illustrated in Table 2. For the MPII and LSP datasets, after every 20 iterations, the learning rates are decayed with the base 0.01 and 0.005 respectively. The provided hyper-parameters are specified by the model selection system based on the highest validation performance. Five times of 10-fold cross validation independently are run and we use the average results as the final results. The proposed model is implemented in TensorFlow 1.3.0 GPU as the backend deep learning engine. Python 3.6 is used for all the implementations. All the implementations of the network are conducted on a workstation equipped with an Intel i7-6850K CPU with 64 GB Ram and an NVIDIA GTX Geforce 1080 Ti GPU and the operating system is Ubuntu 16.04.
Evaluation Metrics
The experiments are based on two metrics. PCK is used to measure the performance on LSP. For MPII, PCKh is used.
Percentage of Correct Keypoints (PCK) [54]: PCK shows the percentage of the precise detection that is located within a tolerance range. The tolerance range is a portion of the torso size. It can be formulated as,
|| −̂|| 2 || ℎ − ℎ || 2 ≤ ,(21)
while y i is the ground-truth location of the i th keypoint and ̂ is the estimated location of the i th keypoint. The fraction r is limited between 0 and 1.
Percentage of Correct Keypoints with respect to head (PCKh) [1]: PCKh is very similar to the PCK, where the tolerance range is a fraction. Table 3 reports the performance of the proposed AMIL model and the other approaches on the LSP dataset based on PCK. The results of the proposed AMIL model is shown in Table 4 with the MPII training set, and the results are computed at r = 0.2. AMIL has the best detection rate through all the tolerance range. Moreover, at the tighter distance (0.05 < r < 0.1), the proposed model demonstrates much better outcomes. Figure 4 presents exemplar qualitative performance of the proposed model. We can see that our proposed model achieves better understanding which leads to correct human body joint detection and pose estimation. In Figure 4, there are a range of images in different poses (highly twisted, partly occluded, complex structures in the wild and invisible body limbs) that our proposed AMIL successfully predicts the joints and estimate the poses. This is due to the shape prior learning and correct extraction of the proper features in the training process. However, some state-of-art models locate some of the body parts to the wrong place due to the absence of the correct body configuration constraints. The proposed discriminator contains the body constraints; hence the network successfully pinpoints the precise body position even in some challenging situations. Tompson et al. [12] Insafutdinov et al. [30] Newell et al. [11] Chu et al. [47] Chou et al. [17] Chen et al. [13] AMIL (ours) Tompson et al. [12] Insafutdinov et al. [30] Newell et al. [11] Chu et al. [47] Chou et al. [17] Chen et al. [13] AMIL (ours)
Results and Evaluations
Normalized distance Performance of AMIL and previous methods at r = 0.5 on the MPII dataset is presented in Table 4 and Figure 5 (a). AMIL is trained with the LSP training set. Figure 5 (b) shows the performance of AMIL in comparison with other models on LSP dataset.
Analysis
Here, we illustrate the influence of different factors on AMIL. The experiments have been conducted on the MPII test dataset and the accuracy through training iterations has been recorded.
MIL and MI-RNet
To evaluate the performance of the proposed AMIL model, we have conducted experiments on several network configurations. In Figure 6 (a), we compare the prediction of the standard MIL [32] and Residual MIL on the MPII dataset. Figure 6 (b) presents the performance of MI_RNet with and without adjust pooling. As the result shows, by using residual MIL and adjust pooling, there are significant improvements in the pose estimation. The discriminator shows satisfactory performance even while the person image is not provided. The reason is that the pose even could be estimated by only the pre-trained pose information. The image of the person is additional information; however the discriminator does not require this information all the time. Meanwhile, we compare the result of the MI-RNet generator trained with that of the discriminator while applying adjusts pooling with the standard MIL. These networks are trained by using the heatmaps. The performance of the proposed body-structure-aware adopted GANs on the MPII validation set increases by 0.7% compared to the standard model. This result proves that the discriminator guides the generator to generate more reliable poses that look similar to the ground truth heatmaps. In fact, separately adding the MI-RNet with adjust pooling or discriminator increases the pose estimation accuracy. However, adopting them separately improved the results by 11.2% and 0.6% respectively, though by both design adoption makes the overall improvement of 11.8%. This high performance is due to the provision of sufficient and reliable features to the discriminator. Table 5 presents the performance of AMIL while having a varying number of FC layers. As the results show, the performance acquired with three FC layers and adjusted pooling is the best. In the proposed model, the adjust pooling has a key role in extracting deep features. Figure 7 shows the confusion matrix comparison of the standard MIL and the proposed model on the LSP dataset. To check the advantage of the adversarial training, we compare the performance of AMIL with and without adversarial training on the MPII dataset. Figure 8(a) shows the significant improvement of the proposed model while using adversarial training and Figure 8(b) presents the effect of adversarial loss. As the results show, the AMIL has faster convergence and more stable performance while taking adversarial training. We also figure out that the learning rate decay approach is supportive; which resulted in more stable performance. The performance of AMIL on LSP and MPII datasets with and without the learning rate decay presented in Figure 8 (c) and (d) respectively. Fig. 7. Confusion matrix comparison on standard MIL and proposed AMIL on LSP dataset Figure 9 presents the interchange concerning the computation cost and number of iterations of six pose estimation models on the MPII and LSP dataset. As the results show AMIL requires less computation costs as compared to the other state-of-the-art models. Between the iterations 200 and 250, Tompson et al. [12] performs similar to AMIL. However, after this period, AMIL outperforms the other models.
Adversarial Training Performance
We believe that the presented algorithm is run by epochs. In respective epoch, we randomly partition the vertex set of particular image V as I mini-batches V 1 , V 2 , … , V I , and in the i-th iteration, we run a forward pass to estimate the human pose for nodes in Vi , a back propagation is used to compute the gradients, and update the history. In every epoch, the scanning of all the nodes is executed, rather than just training nodes, to check that the history of each node is updated at least once per epoch.
From the results, we notice that AMIL requires approximately 3× fewer parameters and time to achieve comparable accuracy to the original MIL. Furthermore, AMIL did not use the depth-wise divisible FC, and just used the simple FC layers. It is possible to use AMIL as a meta-architecture to even obtain a more efficient network. Figure 10 shows the performance of proposed AMIL model in more complex cases. As the results show, the proposed model perfectly estimated the human poses in a single or group activities. Tompson et al. [12] Chu et al. [47] Chen et al. [13] Chou et al. [17] Newell et al. [11]
CONCLUSION
This work has presented an adversarial multi-instance neural network with adjust pooling to solve the human pose estimation problem. The proposed model has the combination of a generator and a discriminator with a similar architecture. The generator is operates based on the predicted feature heatmaps of the human body keypoints, and the discriminator is to distinguish implausible poses and advice useful hints to the generator for improving the heatmaps. After completion of the training, we can remove the discriminator, hence it does not affect the time of other tasks. We evaluated AMIL on two widely used human pose estimation benchmark datasets and the overall results proved that the proposed model outperformed several state-of-the-art approaches and generated better human pose prediction. We will explore hierarchical learning models in the future to incorporate the structural information into the deep models. Additionally, we plan to apply the proposed model to more extensive real applications, such as image segmentation and weakly supervised learning.
Fig. 1 .
1The pose estimation and joint detection of the proposed model on several poses. The head and neck are indicated by purple and red respectively. The blue and green lines are specified on the right side. The light green, yellow and orange have shown on the left side.
Fig. 3 .
3The architecture of the proposed residual multiple instance neural network. The first FC layer produces a bag feature vector. The next FC layers learn the residuals of bag representation. The sizes of all the FC layers are set to 128.3.1.1 Multi Instance Residual Neural Network (MI-RNet) ArchitectureMIL focuses on handling the intricate data in the form that all the bags X = {X 1 , X 2 , ..., X N } and instance features of i th bag X i = {x i1 , x i2 , ..., x imi }, x ij ∈ R d×1 , while N and m i represent the number of the bags and instances in bag X i correspondingly. Assume Y i ∈ {0, 1} and y ij ∈ {0, 1} distinctly are the label of bag X i and instance x ij , where 0 and 1 represent negative and positive respectively. In MIL, simply bag labels are provided throughout the training, and there are two limitations for MIL:
( 1 )
1For example, a single bag X i have various instances x ij , in the MI-RNet. It is composed of l layers which contain a non-linear transformation (. ), while l indicates the layer. (. ) is a compound of several actions such as rectified linear units (ReLU), or inner product (or FC)
Fig. 4 .
4Qualitative results of AMIL. The blue lines indicate the right arm and green lines indicate the body right side, the Caribbean green lines indicate the left arm and yellow and orange lines indicate the left side of body.
. 5. (a) PCKh on MPII dataset, and (b) PCK on the LSP dataset.
Fig. 6 .
6PCKh on the MPII dataset. (a) Performance comparison of standard MIL and residual MIL. (b) Shows the performance of MI_RNet with and without the proposed adjust pooling.
Fig. 8 (
8a) and (b). PCKh on the MPII dataset. Show the performance of AMIL with and without adversarial Training and adversarial Loss. (c) and (d) present the performance of AMIL with and without learning rate (LR) decay on LSP and MPII datasets.
Fig. 9 .
9Computation time vs number of iterations while training on MPII and LSP datasetFig. 10. Performance of AMIL in more complex cases. The blue lines indicate the right arm and green lines indicate the body right side, the Caribbean green lines indicate the left arm and yellow/orange lines indicate the left side of the body.
Table 1 .
1Summary of some key achievements for pose estimation.Ref.
Approach
Experimental
Results on MPII
dataset
Optimization
Efficiency
[6]
CNN and heatmap labeling
PCKh = 88.1
per-pixel Softmax loss
---
[11]
Hourglass CNN
PCKh = 90.9
Rmsprop
---
[12]
CNN and body joints relations
PCKh = 90.2
Coarse optimization
Input Convolution stages
[13]
GAN and structure-aware CNN
PCKh = 92.1
Confidence discriminator
---
[17]
GAN and stacked hourglass CNN
PCKh = 91.8
per-pixel loss
---
[26]
Deep CNN
PCKh = 90.8
Optimal Back-propagation
---
[27]
Dense correspondences between
image and a human body
PCKh = 91.7
---
Annotation pipeline
[35]
MIL algorithm error optimization
---
Optimal MIL
---
[37]
MIL geometric relation
PCKh = 91.1
---
---
[47]
CNN and multi context attention
PCKh = 91.5
RMSprop algorithm
Generate attention maps
Ours
Adversarial MIL and heatmap
labeling
PCKh = 92.3
Gradient descent and Pixel
loss
Adjust pooling
Table 2 :
2The optimization procedure of the hyper-parametersDataset
Learning rate
Weight decay
Iterations
Decay steps
MPII
0.001
0.01
350
20
LSP
0.001
0.005
350
20
Table 3 .
3Human pose detection on the LSP dataset based on PCK.Methods Head
Sho.
Elb.
Wri.
Hip
Knee
Ank.
Mean
[12]
94.8
88.7
81.3
76.8
83.6
86.7
81.9
84.7
[25]
94.9
88.7
81.5
76.9
83.5
86.9
82.3
84.9
[4]
95.2
89.0
81.5
77.0
83.7
87.0
82.8
85.2
[30]
96.8
95.2
89.3
84.4
88.4
83.4
78.0
88.6
[6]
97.1
92.1
88.1
85.1
92.2
91.5
88.7
90.7
[11]
97.0
92.3
88.2
85.2
92.2
91.6
88.9
90.8
[47]
98.1
93.7
89.3
86.9
93.4
94.0
92.5
92.6
[62]
98.4
93.8
89.7
87.4
93.9
94.0
92.8
92.9
[13]
98.5
94.0
89.8
87.5
93.9
94.1
93.0
93.1
[17]
98.2
94.9
92.2
89.5
94.2
95.0
94.1
94.0
Ours
98.4
95.1
92.2
89.7
94.0
95.3
94.2
94.2
Table 4 .
4Human pose detection on the MPII dataset based on PCKh.Methods Head
Sho.
Elb.
Wri.
Hip
Knee
Ank.
Mean
[12]
95.8
90.3
80.5
74.3
77.6
69.7
62.8
79.6
[25]
96.1
91.9
83.9
77.8
80.9
72.3
64.8
82.0
[26]
96.1
92.0
84.1
77.9
81.1
72.3
64.9
82.1
[4]
97.7
95.0
88.2
82.9
87.9
82.6
78.4
88.2
[30]
96.9
95.3
89.4
84.5
88.5
83.5
77.9
88.6
[6]
97.9
95.1
89.9
85.4
89.4
85.6
81.8
89.7
[11]
98.2
96.2
91.2
87.1
90.2
87.5
83.6
90.9
[47]
98.5
96.3
91.9
88.1
90.6
88.0
85.0
91.5
[62]
98.3
96.5
92.1
88.0
91.1
88.9
85
91.6
[17]
98.2
96.8
92.2
88.0
91.3
89.1
84.9
91.8
[13]
98.6
96.4
92.4
88.6
91.5
88.6
85.7
92.1
Ours
98.8
96.5
92.5
88.5
91.5
88.8
85.8
92.3
Table 5 .
5Human pose detection on the MPII dataset based on PCKh with different AMIL network settingMethods
Head Sho.
Elb.
Wri.
Hip
Knee
Ank.
Mean
AMIL with 1 FC
95.8
90.3
80.5
74.3
77.6
69.7
62.8
79.6
AMIL with 2 FC
96.1
91.9
83.9
77.8
80.9
72.3
64.8
82.0
AMIL with 3 FC
98.1
96.0
90.1
88.1
91.1
88.3
84.9
91.8
AMIL with 3 FC
and adjust pooling
98.8
96.5
92.5
88.5
91.5
88.8
85.8
92.3
( ) = ∑ 1 ( ),(8)
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A Biologically Inspired Appearance Model for Robust Visual Tracking. S Zhang, X Lan, H Yao, H Zhou, D Tao, X Li, IEEE Transactions on Neural Networks and Learning Systems. 2810S. Zhang, X. Lan, H. Yao, H. Zhou, D. Tao and X. Li, A Biologically Inspired Appearance Model for Robust Visual Tracking, IEEE Transactions on Neural Networks and Learning Systems, 28(10), 2357-2370, 2017.
S Zhang, X Lan, Y Qi, P C Yuen, Robust Visual Tracking via Basis Matching, IEEE Transactions on Circuits and Systems for Video Technology. 27S. Zhang, X. Lan, Y. Qi, P. C. Yuen, Robust Visual Tracking via Basis Matching, IEEE Transactions on Circuits and Systems for Video Technology, 27(3), 421-430, 2017.
Computer Vision Cracks the Leaf Code. P Wilf, S Zhang, S Chikkerur, S A Little, S L Wing, T Serre, Proceedings of the National Academy of Sciences of the United States of America. the National Academy of Sciences of the United States of America113P. Wilf, S. Zhang, S. Chikkerur, S. A. Little, S. L. Wing, T. Serre, Computer Vision Cracks the Leaf Code, Proceedings of the National Academy of Sciences of the United States of America, 113(12), 3305-3310, 2016.
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Fast and robust animal pose estimation, bioRxiv. J M Graving, D Chae, H Naik, L Li, B Koger, B R Costelloe, I D Couzin, J. M. Graving, D. Chae, H. Naik, L. Li, B. Koger, B. R. Costelloe, I. D. Couzin, Fast and robust animal pose estimation, bioRxiv, 2019.
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| {'fraction_non_alphanumeric': 0.05042027673606621, 'fraction_numerical': 0.02746670115091168, 'mean_word_length': 4.301727684080626, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 1, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Human pose estimation has an important impact on a wide range of applications from human-computer interface to surveillance and content-based video retrieval. For human pose estimation, joint obstructions and overlapping upon human bodies result in departed pose estimation. To address these problems, by integrating priors of the structure of human bodies, we present a novel structure-aware network to discreetly consider such priors during the training of the network. Typically, learning such constraints is a challenging task. Instead, we propose generative adversarial networks as our learning model in which we design two residual multiple instance learning (MIL) models with the identical architecture, one is used as the generator and the other one is used as the discriminator. The discriminator task is to distinguish the actual poses from the fake ones. If the pose generator generates the results that the discriminator is not able to distinguish from the real ones, the model has successfully learnt the priors. In the proposed model, the discriminator differentiates the ground-truth heatmaps from the generated ones, and later the adversarial loss back-propagates to the generator. Such procedure assists the generator to learn reasonable body configurations and is proved to be advantageous to improve the pose estimation accuracy. Meanwhile, we propose a novel function for MIL. It is an adjustable structure for both instance selection and modeling to appropriately pass the information between instances in a single bag. In the proposed residual MIL neural network, the pooling action adequately updates the instance contribution to its bag. The proposed adversarial residual multi-instance neural network that is based on pooling has been validated on two datasets for the human pose estimation task and successfully outperforms the other state-of-arts models. The code will be made available on https://github.com/pshams55/AMIL.', 'arxivid': '2003.08002', 'author': ['Pourya Shamsolmoali ', 'Masoumeh Zareapoor ', 'Shanghai ', 'Jie Yang [email protected] ', 'Shanghai ', '\nHUIYU ZHOU\nJiao Tong University\nChina\n', '\nUniversity of Leicester\nUnited Kingdom\n', '\nJiao Tong University\nChina\n', '\nDepartment of Informatics\nShanghai Jiao Tong University\nShanghaiChina\n', '\nUniversity of Leicester\nLE1 7RHLeicesterUnited Kingdom\n'], 'authoraffiliation': ['HUIYU ZHOU\nJiao Tong University\nChina', 'University of Leicester\nUnited Kingdom', 'Jiao Tong University\nChina', 'Department of Informatics\nShanghai Jiao Tong University\nShanghaiChina', 'University of Leicester\nLE1 7RHLeicesterUnited Kingdom'], 'corpusid': 212747939, 'doi': '10.1145/3355612', 'github_urls': ['https://github.com/pshams55/AMIL.'], 'n_tokens_mistral': 21909, 'n_tokens_neox': 19227, 'n_words': 12022, 'pdfsha': '68d3dd927eb51e954d696f12576019561a87c787', 'pdfurls': ['https://arxiv.org/pdf/2003.08002v1.pdf'], 'title': ['AMIL: Adversarial Multi Instance Learning for Human Pose Estimation', 'AMIL: Adversarial Multi Instance Learning for Human Pose Estimation'], 'venue': []} |
arxiv |
Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps Dyi-Shing Ou
February 6, 2018
Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps Dyi-Shing Ou
February 6, 2018
This article extends the theorem of the absence of wandering domains from unimodal maps to infinitely period-doubling renormalizable Hénon-like maps in the strongly dissipative (area contracting) regime. The theorem solves an open problem proposed by several authors[64,44], and covers a class of maps in the nonhyperbolic higher dimensional setting. The classical proof for unimodal maps breaks down in the Hénon settings, and two techniques, "the area argument" and "the good region and the bad region", are introduced to resolve the main difficulty.The theorem also helps to understand the topological structure of the heteroclinic web for such kind of maps: the union of the stable manifolds for all periodic points is dense.Mathematics Subject Classification (2010) 37E30 · 37C70 · 37E20 · 37D45
Introduction
This article studies the question of the existence of wandering domains for Hénonlike maps. A Hénon-like map is a real two-dimensional continuous map that has the form F(x, y) = ( f (x) − ε(x, y), x) (1.1) where f is a unimodal map (will be defined later) and ε is a small perturbation. For renormalization purposes, the Hénon-like maps in consideration are all real analytic and strongly dissipative (the Jacobian ∂ ε ∂ y is small 1 ). One can see from the definition, Hénon-like maps are a generalization of classical Hénon maps [30] (two-parameters polynomial maps) to the analytic settings and extension of unimodal maps to higher dimensions. Strongly dissipative Hénon-like maps are the maps that are close to unimodal maps. They share some dynamical properties with unimodal maps. For example, the tool of unimodal renormalization can be adopted to Hénon-like maps [12,44,28], the renormalization operator is hyperbolic [12,Theorem 4.1], and an infinitely renormalizable Hénon-like map has an attracting Cantor set [12,Section 5.2]. However, there are also some properties that make Hénon-like maps distinct from unimodal maps. For example, the Cantor set for infinitely renormalizable Hénon-like maps is not rigid [12,Theorem 10.1] and a universal model can not be presented by a finite dimensional family of Hénon-like maps [29]. In the degenerate case, unimodal maps do not have wandering intervals [24,50,51,45]. It is natural to ask whether this property can be promoted to Hénon-like maps.
The study of wandering domains has a broad interest in the field of dynamics. In one-dimension, the problem has been widely studied and there are many important consequences due to the absence of wandering intervals/domains. However, only a few systems in higher dimensions were known not having wandering domains.
In real one-dimension, showing the absence of wandering intervals in a system is important to solve the classification problem. For circle homeomorphisms, a sequence of works [25,67,32,57] follows after Denjoy [14] showed that a circle homeomorphism with irrational rotation number (the average rotation angle is irrational) does not have a wandering domain if the map is smooth enough. Those maps are conjugated to the rigid rotation with the same rotation number by a classical theorem from Poincare [62]. For multimodal maps, a full family is a family of multimodal maps that exhibits all relevant dynamical behavior. A multimodal map that does not have a wandering interval is conjugated to an element in a full family [24,50,51,43,8,45].
In complex dimension one, Sullivan's no-wandering-domain theorem [65] fully solves the problem for rational maps. The theorem says that a rational map on the Riemann sphere does not have wandering Fatou components. As a consequence, this theorem completes the last puzzle for the classification of Fatou components [19,35]. Thus, the main interest turns to transcendental maps. In general, there are transcendental maps that have wandering domains [3,4,31,65,16,7,18]. There are also some types of transcendental maps that do not have wandering domains [23,17,6,53].
In real higher dimensions, the problem for wandering domains is still wide open. There is no reason to expect the absence of wandering domains [66], especially when the regularity is not enough as in one-dimension [48,49,9]. The classification problem fails between any two different levels of differentiability for diffeomorphisms on d-manifold with d = 1, 4 [26,27]. Examples are found in polynomial skew-product maps having wandering domains [2]. Non-hyperbolic phenomena also play a role in building counterexamples [13,39,37]. A relevant work by Kiriki and Soma [39] found Hénon-like maps having wandering domains by using a homoclinic tangency of some saddle fixed point [36,38]. On the other hand, there are studies [58,41,40,55] suggests that some types of systems may not have wandering domains. However, only a few [56,9] were discovered not having wandering domains.
In complex higher dimension, counterexamples in transcendental maps can be constructed from one-dimensional examples [20] by taking direct products. For polynomial maps, very little was known about the existence of wandering Fatou components until recent developments on polynomial skew-products [42,2,60,59,61], which are the maps of the form F(z, w) = ( f (z, w), g(w)).
The first example was given by Astorg, Buff, Dujardin, Peters, and Raissy [2], who found a polynomial skew-product possessing a wandering Fatou component as the quasi-conformal methods break down. The reader can refer to the survey [63] for more details about other relevant work on polynomial skew-product [42,60,59,61]. For complex Hénon maps 2 , a recent paper by Leandro Arosio, Anna Miriam Benini, John Erik Fornaess, and Han Peters [1] found a transcendental Hénon map exhibiting a wandering domain. Nevertheless, the problem is still unsolved [5] for complex polynomial Hénon maps [33,34].
In this paper, a wandering domain is a nonempty open set that does not intersect the stable manifold of any saddle periodic points. This definition is weaker than the classical notion because it excludes the condition having a disjoint orbit. The reason to drop this condition is to allow the usage study the topological structure of attractors which will be discussed later. Dropping the condition also makes the conclusion of the theorem stronger compared to the classical definition. In fact, this condition is redundant in the unimodal setting (See Remark 6.2).
The main result of this article, Theorem 10.16, is stated as follow.
Theorem A strongly dissipative infinitely period-doubling renormalizable Hénonlike map does not have wandering domains.
The theorem covers a class of maps in the higher dimensional nonhyperbolic setting [44,Corollary 6.2] and solves an open problem proposed by van Strien [44], Lyubich, and Martens [44]. The result does not overlap with the previous work by Kiriki and Soma [39]. The Hénon-like maps in this article are real analytic and the fixed points are far away from having a homoclinic tangency, while the examples they found having wandering domains only have finite differentiability and their construction relies on the existence of a homoclinic tangency of a fixed point [36,38]. The condition of being infinitely renormalizable is imposed to the theorem to gain a self-similarity between different scale. A map is renormalizable means that a higher iterate of the map has a similar topological structure on a smaller scale. Several papers [9,46,47] in different contexts show that this condition will ensure the absence of wandering domains. In this paper, we center on infinitely renormalizable maps of period-doubling combinatorics type which is one of the most fundamental types of maps. The reader will see later in the proof that the condition infinitely renormalizable is essential because that the area where bad things (nonhyperbolic phenomena) happens, called the bad region, becomes smaller when a map gets renormalized more times.
The theorem is important because it helps us to understand the structure of attractors. An attracting set is a closed set such that many points evolve toward the set. Hénon maps are famous for its chaotic limiting behavior since Hénon first discovered the strange attractor in the classical Hénon family [30]. The ω-limit set of a point is an attractor which characterizes the long-time behavior of a single orbit. For an infinitely period-doubling renormalizable Hénon-like map, the map has only two types of ω-limit set [22,44]: a saddle periodic orbit and the renormalization Cantor attractor. From this dichotomy, a wandering domain is equivalently a non-empty open subset of the basin of the Cantor attractor. The theorem of the absence of wandering domains implies that the union of the stable manifolds is dense. In other words, the basin of the Cantor set has no interior even though it has full Lebesgue measure.
Two tools are introduced to prove the theorem: the bad region and the area argument (thickness). The bad region is a set in the domain where the length expansion argument from unimodal maps breaks down. This is where the main difficulty of extending the theorem occurs. The solution to this is the area argument which is also a dimension two feature. These two concepts make the Hénon-like maps different from the unimodal maps.
The tools may be used to prove the nonexistence of wandering domains in other contexts. One is infinitely renormalizable Hénon-like maps with arbitrary combinatorics [28]. The definition of the bad region carries over to the arbitrary combinatorics case directly. It is also possible to generalize the area argument because the tip of those maps also has a universal shape [28,Theorem 6.1]. However, the expansion argument breaks down for other combinatorics. This may be solved by studying the hyperbolic length instead of the Euclidean length.
Outline of the article In this article, chapters, sections, or statements marked with a star sign "*" means that the main theorem, Theorem 10.16, does not depend on them. Terminologies in the outline will be defined precisely in later chapters. Chapters 2, 3, 4, and 5 are the preliminaries of the theorem. The chapters include basic knowledge and conventions that will be used in the proof. Most of the theorems in Chapter 4 and Section 5.1 can be found in [12,44].
The proof for the nonexistence of wandering domains is motivated by the proof of the degenerate case. A Hénon-like map is degenerate means that ε = 0 in (1.1). In this case, the dynamics of the map degenerates to the unimodal dynamics. In Chapter 7, a short proof for the nonexistence of wandering intervals for infinitely renormalizable unimodal maps is presented by identifying a unimodal map as a degenerate Hénonlike map. The proof assumes the contrapositive, there exists a wandering interval J. Then we apply the Hénon renormalization instead of the standard unimodal renormalization to study the dynamics of the rescaled orbit of J that closest approaches the critical value. The rescaled orbit is called the J-closest approach (Definition 6.1). The proof argues that the length of the elements in the rescaled orbit approaches infinity by a length expansion argument which leads to a contradiction. The expansion argument motivates the proof for the Hénon case.
The proof of the main theorem is covered by Chapters 6,8,9,and 10. The structure is explained as follows.
Assume the contrapositive, a Hénon-like map has a wandering domain J. In Chapter 6, we study the rescaled orbit {J n } n≥0 of J that closest approaches to the tip, called the J-closest approach. Each element J n belongs to some appropriate renormalization scale (the domain of the r(n)-th renormalization R r(n) F for some nonnegative integer r(n)). The transition between two constitutive sequence elements J n → J n+1 is called one step. Motivated by the expansion argument from the degenerate case, we estimate the change rate of the horizontal size l n in each step. The horizontal size of a set is the size of its projection to the first coordinate (Definition 6.8). Our final goal is to show that the horizontal size of the sequence elements approaches infinity to obtain a contradiction.
In the degenerate case, the expansion argument says that the horizontal size expands at a uniform rate and hence the horizontal size of the sequence elements approaches infinity. Unfortunately, the argument breaks down in the non-degenerate case. There are two features that make the non-degenerate case special:
1. The good region and the bad region. 2. Thickness.
The good region and the bad region, introduced in Chapter 8, divide the phase space of a Hénon-like map into two regions by how similar the Hénon-like map behaves like unimodal maps. Each renormalization scale (domain of the n-th renormalization R n F for some n) has its own good region and bad region, and the size of the bad regions contract super-exponentially as the renormalization applies to the map more times ([12, Theorem 4.1] and Definition 8.1). When the elements in a closest approach stay in the good regions of some appropriate scale, we show that the expansion argument can be generalized to the Hénon-like maps. Thus, the horizontal size expands at a uniform rate (Proposition 9.2). However, when an element J n enters the bad region of the renormalization scale of the set, the expansion argument breaks down. At this moment, another quantity, called the thickness, offers a way to estimate the horizontal size of the next element J n+1 (Definition 10.2). The reader should imagine the thickness of a set is the same as its area. We will show that the thickness has a uniform contraction rate proportion to the Jacobian of the map (Proposition 10.6). For a strongly dissipative Hénon-like map, the Jacobian is small and hence the contraction is strong. This strong contraction yields the main obstruction toward our final goal.
The breakthrough is the discovery that the elements in a closest approach can at most enter the bad regions finitely many times (Proposition 10.15). When an element J n enters the bad region, the horizontal size contracts and the following element J n+1 belongs to a deeper renormalization scale. But the size of the bad region in the deeper scale (scale of J n+1 ) is much smaller than the bad region of the original scale (scale of J n ). Roughly speaking, we found that the contraction of the size of the bad region is faster than the contraction of the horizontal size so that the elements cannot enter the bad region infinitely times. The actual proof is more delicate because another quantity, the time span in the good regions (Definition 10.9), also involves in the competition. The two-row lemma (Lemma 10.13) is the key lemma that gives an estimate for the competitions between the contraction of the thickness, the expansion of the horizontal size in the good region, the time span in the good region, and the size of the bad region when the closest approach enters the bad region twice. The conclusion follows after applying the two-row lemma inductively (Lemma 10.14).
In summary, the horizontal size of the elements in a closest approach expands in the good regions, while contracts in the bad regions. However, the contraction happens only finitely many times. This shows that the horizontal size approaches infinity which is a contradiction. Therefore, wandering domains cannot exist.
Schwarzian derivative
In this section, we recall the definition and the properties of Schwarzian derivative. The proof for the properties stated in this section can be found in [52]. These properties will be used only in Chapter 3.
Definition 2.2 (Schwarzian Derivative) Assume that f is a C 3 real valued function on an interval. The Schwarzian derivative of f is defined by
(S f )(x) = f (x) f (x) − 1 2 f (x) f (x) 2 = f (x) f (x) − 3 2 f (x) f (x) 2 whenever f (x) = 0. The map f is said to have negative Schwarizan derivative if S f (x) < 0 for all x ∈ I with f (x) = 0.
Negative Schwarzian derivative is preserved under iteration.
Proposition 2.3
If f has negative Schwarzian derivative, then f n also has negative Schwarzian derivative for all n > 0.
Proposition 2.4 (Minimal Principle)
Assume that J is a bounded closed interval and f : J → R is a C 3 map with negative Schwarzian derivative. If f (x) = 0 for all x ∈ J, then | f (x)| does not attain a local minimum in the interior of J.
Unimodal Maps
In this chapter, we give a short review over the procedure for unimodal renormalization. The goal is to introduce the hyperbolic fixed point for the renormalization operator (Proposition 3.8) and establish the estimations for its derivative (Subsection 3.2.2). (1) , and p (2) are defined as in Definition 3.4.
(x) = f (x) andx = x. If x = c (0) , definê x = c (0) . A B C A p(0) p (1) p (2)
The renormalization of a unimodal map
To define the period-doubling renormalization operator for unimodal maps, we introduce a partition on I that allows us to define the first return map for a renormalizable unimodal map.
Definition 3.4 Assume that f ∈ U has a unique fixed point p(0) ∈ I with a negative multiplier. Let p (1) =p(0) and p (2) be the point such that f (p (2) ) = p (1) and p (2) > c (0) . Define A = (−1, p (1) ) ∪ (p (2) , 1), B = (p (1) , p(0)), and C = (p(0), p (2) ). The sets A = A( f ), B = B( f ), and C = C( f ) form a partition of the domain D ≡ I. See Figure 3.1 for an illustration.
The property "renormalizable" is defined by using the partition elements.
Definition 3.5 (Renormalizable)
A unimodal map f ∈ U is (period-doubling) renormalizable if it has a fixed point p(0) with a negative multiplier and f (B) ⊂ C. The class of renormalizable unimodal maps is denoted as U r . Remark 3.6 Most of the articles define the unimodal renormalization by using the critical orbit. However, here we choose to use an orbit that maps to the fixed point with a negative multiplier instead. The purpose of doing this is to make the partition consistent with the partition defined for Hénon-like maps (Definition 4.14) because Hénon-like maps do not have a critical point.
For a renormalizable unimodal map, an orbit that is not eventually periodic follows the paths in the following diagram.
A / / 9 9 B / / C o o
This allows us to define the first return map on B and the period-doubling renormalization. Definition 3.7 (Renormalization) Assume that f ∈ U r . The renormalization of f is the map R f = s • f 2 • s −1 where s is the orientation-reversing affine rescaling such that s(p(0)) = −1 and s(p (1) ) = 1.
The renormalization operator is a map R : U r → U . If the procedure of renormalization can be done recurrently infinitely many times, then the map is called infinitely (period-doubling) renormalizable. The class of infinitely renormalizable unimodal maps is denoted as I .
The fixed point of the renormalization operator
In this section, we study the fixed point g of the renormalization operator. The map g is also important for the Hénon case because it also defines the hyperbolic fixed point of the Hénon renormalization operator [12,Theorem 4.1].
The existence and uniqueness of the fixed point was proved in [15,11]. Here, the theorem is restated in the coordinate system used in this paper.
Proposition 3.8 There exists a unique constant λ = 2.5029... and a unique solution g ∈ I of the Cvitanović-Feigenbaum-Coullet-Tresser functional equation
g(x) = −λ g 2 − x λ (3.1)
for −1 ≤ x ≤ 1 with the following properties:
1. g is analytic in a complex neighborhood of [−1, 1]. 2. g is even.
3. g is concave on [−c (1) , c (1) ]. 4. g(c (1) ) = − 1 λ c (1) and g (c (1) ) = −λ . 5. g has negative Schwarzian derivative. Corollary 3.9 The map g satisfies the following property
g 2 n 1 (−λ ) n x = 1 (−λ ) n g (x) (3.2)
for all n ≥ 0 and all x ∈ I.
Proof The proof follows from the functional equation (3.1).
In the remaining part of the section, the notations for the unimodal maps will be applied to the map g. For example, c ( j) = c ( j) (g) j≥0 is the critical orbit and the sets A = A(g), B = B(g), and C = C(g) form a partition of the domain D = I.
A backward orbit of the critical point
In this section, we establish a backward orbit b (2)
→ b (1) → c (0) of the critical point c (0) . Let b (1) ∈ [0, c (1) ] be the point such that g(b (1) ) = 0. Set b (2) = 1 λ b (1) . Lemma 3.10 We have g b (2) = b (1) .
Proof Since g is even, the only two roots of g are −b (1) and b (1) . By the functional equation (3.2), we have
g 2 b (2) = − 1 λ g −b (1) = 0. Thus, g b (2) = −b (1) or b (1) . Also, g b (2) = −b (1) because b (2) ∈ (0, b (1) ) and g (x) > 0 on (0, b (1) ). Therefore, g b (2) = b (1) .
Estimations for the derivative
Apply the chain rule to the functional equation (3.1), we have
g (x) = g − x λ g • g − x λ (3.3)
for x ∈ I. We will use this formula to derive the values for the derivative of g at some particular values.
Lemma 3.11 The slope at b (2) is
g (b (2) ) = −1. (3.4)
Proof From (3.3) and g is even, we have
g (b (1) ) = g −b (2) g • g −b (2) = −g b (2) g b (1) .
We solve g (b (2) ) = −1.
Let q(−1) = −1 (the fixed point with a positive multiplier) and q(0) be the fixed point with a negative multiplier. From the functional equation (3.1), we get q(0) = 1 λ . Lemma 3.12 The slopes at the fixed points satisfy the relation
g (q(−1)) = g (q(0)) 2 . (3.5) Proof From (3.3), compute g (q(−1)) = g (−1) = g 1 λ g • g 1 λ = g (q(0)) g • g (q(0)) = g (q(0)) 2 .
Finally, we prove that the map g is expanding on A and C.
Proposition 3.13 The slope of g is bounded below by
g (x) ≥ g (q(0)) > 1 for all x ∈ [q(−1),q(0)] ∪ [q(0),q(−1)].
Proof It is enough to prove the case when x ∈ [q(0),q(−1)] since g is even.
First, we consider the interval [b (2) , c (1) ]. We have b (2) < q(0) < c (1) . By (3.4) and Proposition 3.8, the derivatives of the boundaries are g (b (2) ) = −1 and g (c (1) ) = −λ . We get |g (q(0))| > 1 by the minimal principle (Proposition 2.4).
Next, we consider the interval [q(0),q(−1)]. From (3.5), we also get |g (q(−1))| > 1. Therefore, the proposition follows from the minimal principle (Proposition 2.4).
Hénon-like Maps
In this chapter, we give an introduction to the theory of Hénon renormalization in the strongly dissipative regime developed by [12,44]. Their theorems are adopted to fit the notations and the coordinate system used in this article.
The class of Hénon-like maps
Definition 4.3 (Hénon-like map) Assume that I v ⊃ I h I are closed intervals. A Hénon-like map is a smooth map F :
I h × I v → R 2 of the form F(x, y) = ( f (x) − ε(x, y), x)
where f is a unimodal map and ε is a small perturbation. The function h will also be used to express the x-component, h y (x) = h(x, y) = π x F(x, y). A representation of F will be expressed in the form F = ( f − ε, x).
The function spaces of the Hénon-like map is defined as follows.
I h (δ ) × I v (δ ) → C. 2. Given ε > 0 and f ∈ U δ (I h ). Denote H δ (I h × I v , f , ε) to be the class of Hénon- like maps F ∈ H δ (I h × I v ) with the form F = ( f − ε, x) such that ε < ε. 3. Denote H δ (I h × I v , ε) = ∪H δ (I h × I v , f , ε)
where the union is taken over all f ∈ U δ (I h ).
Remark 4.5
The domain I h × I v used in this article is slightly larger than the domain studied in the two original papers [12,44] [12,44] also hold in the larger domain and rephrase them in the notations used in this article without reproving. See also Remarks 4.18,4.24,and 10.17. From the definition, it follows immediately that Lemma 4.6 Given I v ⊃ I h I, δ > 0, ε > 0, and f ∈ U δ (I h ).
1. If ε 1 < ε 2 then H δ ( f , ε 1 ) ⊂ H δ ( f , ε 2 ). 2. If I ⊂ I h 1 ⊂ I h 2 ⊂ I v and f ∈ U δ (I h 2 ), then U δ (I h 1 ) ⊃ U δ (I h 2 ) and H δ (I h 1 × I v , f , ε) ⊃ H δ (I h 2 × I v , f , ε).
An important property of a Hénon-like map is that it maps vertical lines to horizontal lines; it maps horizontal lines to parabola-like arcs.
Example 4.7 (Degenerate case) Assume that F(x, y) = ( f (x) − ε(x, y), x) is a Hénonlike map. The map is called a degenerate Hénon-like map if ∂ π x F ∂ y = ∂ ε ∂ y = 0; a nondegenerate Hénon-like map if ∂ π x F ∂ y = ∂ ε ∂ y = 0. If F is degenerate, then ε only depends on x. In this case, without lose of generality, we will assume the Hénon-like map has the representation F(x, y) = ( f (x), x) where f = π x F and ε = 0.
For the degenerate case, the dynamics of the Hénon-like map is completely determined by its unimodal component. So it will also be called as the unimodal case in this article.
The degenerate case is an important example in this article. A proof for the nonexistence of wandering intervals for unimodal maps will be presented in Chapter 7 by identifying a unimodal map as a degenerate Hénon-like map. The expansion argument in the proof motivates the proof for the non-degenerate case. The difference between the degenerate case and the non-degenerate case produces the main difficulty (explained in Chapter 8 and Chapter 10) of extending the proof to the non-degenerate case.
= (−1 + a(1 − x 2 ) − by, x) where a, b > 0. These are Hénon-like maps F a,b ∈ H δ (I h × I v , −1 + a(1 − x 2 ), b[|I v | + 2δ ]) for all δ > 0 and I v ⊃ I h .
Local stable manifolds and partition of a Hénon-like map
To study the dynamics of a Hénon-like map, we need to find a domain D ⊂ I h × I v that turns the Hénon-like map into a self-map. Also, to renormalize a Hénon-like map, we need to find a subdomain C ⊂ D that defines a first return map. Motivated from unimodal maps, one can construct a partition of the domain I h × I v to find the domains. In the unimodal case, an orbit that maps to the fixed point p(0) with an expanding multiplier splits the domain D into a partition {A, B,C} (Definition 3.4). For a strongly dissipative Hénon-like map, the orbit becomes components of the stable manifold of the saddle fixed point p(0). These components are vertical graphs that split the domain into multiple vertical strips. Definition 4.9 A set Γ is a vertical graph if there exists a continuous function γ :
I v → I h such that Γ = {(γ(t),t);t ∈ I v }. The vertical graph Γ is said to have Lipschitz constant L if the function γ is Lipschitz with constant L.
In this paper, a local stable manifold is a connected component of a stable manifold. Inspired by [12], the partition will be the vertical strips separated by the associated local stable manifolds.
First, we study the local stable manifolds of the saddle fixed point p(−1) which contains an expanding positive multiplier. 1. If the connected component that contains the fixed point p(−1) is a vertical graph, let W 0 (−1) be the component. 2. Assume that W 0 (−1) exists. If F −1 (W 0 (−1)) has two components, one is W 0 (−1) and the other is a vertical graph. Let W 2 (−1) be the one that is disjoint from W 0 (−1).
p(-1) W 0 (-1) F -1 W 2 (-1) p(0) W 0 (0) F -1 W 1 (0) F -1 W 2 (0) A B C A -1.0 -0.
If the the local stable manifolds W 0 (−1) and W 2 (−1) exists, define D = D(F) ⊂ I h × I v to be the open set bounded between the two local stable manifolds. See The second property follows from the definition of the local stable manifolds and ε > 0 is sufficiently small.
Next, we study the local stable manifolds of the other saddle fixed point p(0) with an expanding negative multiplier to define a partition of D.
Definition 4.12 (The local stable manifolds of p(0)) Given I v ⊃ I h I, δ > 0, and F ∈ H δ (I h × I v ). Assume that F has a saddle fixed point p(0) with an expanding negative multiplier. Consider the stable manifold of p(0).
1. If the connected component that contains p(0) is a vertical graph, let W 0 (0) be the component. 2. Assume that W 0 (0) exists. If F −1 (W 0 (0)) has two components, one is W 0 (0) and the other is a vertical graph. Let W 1 (0) be the one that is disjoint from W 0 (0). 3. Assume that W 0 (0) and W 1 (0) exist. If F −1 (W 1 (0)) has two components and one component is a vertical graph located to the right of W 0 (0). Let W 2 (0) be the component.
See Figure 4.1 for an illustration.
Remark 4.13 At this moment, the numbers 0 and −1 in the notation of the fixed points p(0) and p(−1) (and also the local stable manifolds) do not have a special meaning. After introducing infinitely renormalizable Hénon-like maps, the notation p(k) will be used to define a periodic point with period 2 k . See Definition 5.2. The numbers are introduced here for consistency.
The local stable manifolds split the domain D into vertical strips. These strips define a partition of the domain.
Remark 4.15
The local stable manifolds W 0 (−1), W 1 (0), W 0 (0) , W 2 (0), and W 2 (−1) are associated to the points p(−1) = −1, p (1) , p(0), p (2) , and 1 respectively (Definition 3.4).
For a strongly dissipative Hénon-like map, the local stable manifolds are vertical graphs and the dynamics on the partition is similar to the unimodal case.
Proposition 4.16 Given δ > 0 and intervals I h and I v with I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈ H δ (I h × I v , ε) the following properties hold:
1. The sets W 0 (0), W 1 (0), W 2 (0), A, B, and C exist. The local stable manifolds are vertical graphs with Lipschitz constant c ε .
F(A) ⊂
A ∪W 1 (0) ∪ B. 3. F(C) ⊂ B.
4. If z ∈ A then its orbit eventually escapes A, i.e. there exists n > 0 such that F n (z) / ∈ A.
Proof The first property is proved by graph transformation. See [44,Chapter 3].
The second and third properties follows from the definition of the local stable manifolds. See also [44,Lemma 4.2].
The last property holds because the only fixed points are p(−1) and p(0) so the local unstable manifold of p(−1) must extends across the whole set A. See also [44,Lemma 4.2].
By the definition of B, its iterate F(B) is contained in the right component of D\W 0 (0). With the third property of Proposition 4.16, we can define the condition "renormalizable" as follows.
(I h × I v , ε) ⊂ H δ (I h × I v , ε).
Remark 4.18
The notion of "renormalizable" here is similar to [12, Section 3.4] (which they called pre-renormalization) but not exactly the same. The "renormalizable" in their paper is called CLM-renormalizable here to compare the difference. In their article, the set "C" (they named the set D) where they define the first return map is a region bounded between W 0 (0) and a section of the unstable manifold of p(−1). In this article, the set C is defined to be the largest candidate (around the critical value) that is invariant under F 2 which only uses the local stable manifolds of p(0). Thus, the sets B and C in this article is slightly larger than theirs. As a result, the property "renormalizable" in this article is stronger than theirs. If a Hénon-like map is renormalizable then it is also CLM-renormalizable. Although the converse is not true in general, the hyperbolicity of the renormalization operator [12, Theorem 4.1] allows us to apply the notion of renormalizable to an infinitely CLM-renormalizable map. This makes the final result, Theorem 10.16, also works for CLM-renormalizable maps. See Remarks 4.24 and 10.17 for more details.
Their definition has some advantages and disadvantages. Their notion of renormalizable does not depend on the size of the vertical domain I v . However, their sets B and C are too small. It may requires more iterations for an orbit to enter their B and C. See the proof of [44,Lemma 4.2]. This is the reason for adjusting their definition.
For a renormalizable Hénon-like map, an orbit that is disjoint from the stable manifold of the fixed points follows the paths in the following diagram.
A / / finite iterations 9 9 B / / C o o
Therefore, a renormalizable map has a first return map on C.
Renormalization operator
When a Hénon-like map is renormalizable, the map has a first return map on C. However, the first return map is no longer a Hénon-like map by a direct computation
F 2 (x, y) = (h x (h y (x)), h y (x)).
The paper [12] introduced a nonlinear coordinate change H(x, y) ≡ (h y (x), y) that turns the first return map into a Hénon-like map. The next proposition defines the renormalization operator.
Proposition 4.19 (Renormalization operator) Given δ > 0 and intervals I h , I v with I v ⊃ I h I. There exists ε > 0 and c > 0 so that for all F ∈ H r δ (I h × I v , ε) there exists an R-symmetric orientation reversing affine map s = s(F) which depends continuously on F such that the following properties hold:
Let Λ (x, y) = (s(x), s(y)) and φ = Λ • H.
1. The map x → h y (x) is injective on a neighborhood of C(F) and hence φ is a diffeomorphism from a neighborhood of C(F) to its image.
The renormalization RF
≡ φ • F 2 • φ −1 is an Hénon-like map defined on I h R (δ R ) × I v R (δ R )f R − R c f I h R (δ R ) < c ε and ε R I h R (δ R )×I v R (δ R ) < c ε 2 . Proof See [12, Section 3.5].
Remark 4.20 The rescaling φ preserves the orientation along the x-coordinate and reverses the orientation along the y-coordinate.
A map is called infinitely renormalizable if the procedure of renormalization can be done infinitely many times. The class of infinitely renormalizable Hénon-like map is denoted as
I δ (I h × I v , ε) ⊂ H δ (I h × I v , ε).
Assume that F ∈ I δ (I h × I v , ε), we define F n = R n F. The subscript n is called the renormalization level. The subscript is also used to indicate the associated renormalization level of an object. For example, H n , s n , and Λ n are the functions in Proposition 4.19 that corresponds to F n . The vertical domain I v n satisfies I v 0 = I v and I v n+1 = s n (I v n ) for all n ≥ 0. The vertical graphs W t n ( j) are the local stable manifolds of F n . The sets A n , B n , and C n form a partition of the dynamical domain D n that associates to F n . The points p n (−1) and p n (0) are the two saddle fixed points of F n .
Also, define Φ j n = φ n+ j−1 • · · · • φ n and λ n = s n (x).
Recall g ∈ U is the fixed point of the renormalization operator R, and λ is the rescaling constant defined in 3.8. Let G(x, y) = (g(x), x) be the induced degenerate Hénon-like map.
The renormalization operator is hyperbolic. The next proposition lists the properties of infinitely renormalizable Hénon-like maps.
I v with I v ⊃ I h I. There exists ρ < 1 (universal), ε > 0, c > 0 such that for all F ∈ I δ (I h × I v , ε) there exists 0 < δ R < δ , an interval I h R with I h ⊃ I h R
I, and b ∈ R such that the following properties hold:
Let F n = R n F be the sequence of renormalizations of F. Remark 4.23 The Hénon-renormailzation is an operation that renormalizes around the critical value. However, the renormalization F n converges to the fixed point G of the unimodal-renormalization that renormalizes around the critical point. This is because of the nonlinear rescaling H maps the domain from C to B in the degenerate case. See Chapter 7 for a more detail explanation.
Then F n ∈ H δ R (I h R × I v n ) for all n ≥ 0. Also, the sequence has a representation F n = ( f n − ε n , x) with f n ∈ U δ R (I h R ) that satisfies 1. f n − g I h R (δ R ) < cρ n F − G I h R (δ R )×I v (δ R ) 2. ε n+1 I h R (δ R )×I v n+1 (δ R ) < c ε n 2 I h R (δ R )×I v n (δ R ) , 3. f n+1 − s n • f 2 n • s −1 n I h R (δ R ) < c ε n I h R (δ R )×I v n (δ R ) , 4. |λ n − λ | < cρ n F − G I h R (δ R )×I v (δ R ) , and 5. ε n (x, y) = b 2 n a(x)y(1 + O(ρ n )) (universality) for all n ≥ 0 where a(x
Remark 4.24
Although infinitely CLM-renormalizable in general does not imply infinitely renormalizable, the hyperbolicity provides a connection between the two notions of infinitely renormalizable. Assume that F is infinitely CLM-renormalizable. The hyperbolicity of the renormalizable operator [12, Theorem 4.1] says that R n F converges to the fixed point G. This means that R n F is also infinitely renormalizable for all n sufficiently large. This makes Theorem 10.16 also applies to infinitely CLM-renormalizable Hénon-like maps. See Remark 10.17 for more details.
From now on, for any infinitely renormalizable map F, we fix a representation F n = ( f n − ε n , x) such that the maps f n and ε n satisfy the properties given in Proposition 4.21. Also, we neglect the subscript of the supnorms f n − g = f n − g I h R (δ R ) and ε n = ε n I h R (δ R )×I v n (δ R ) whenever the context is clear. Corollary 4.25 There exists a constant c > 1 such that
F n − G < cρ n F − G and ε n+t < (c ε n ) 2 t for all t ≥ 1.
Lemma 4.26 Assume that ε > 0 small enough such that Proposition 4.21 holds. There exists a constant c 1 > 0 such that the inequalities hold
∂ ε n ∂ x (x, y) , ∂ ε n ∂ y (x, y) ≤ c 1 ε n (4.1) for all F ∈ I δ (I h × I v , ε) and (x, y) ∈ I h × I v n . In addition, if F is non-degenerate, there exists N = N(F) ≥ 0, δ R > 0, and c 2 > 0 such that ∂ ε n ∂ y (x, y) ≥ c 1 |I v n | ε n (4.2)
for all (x, y) ∈ I h × I v n and n ≥ N.
Proof The first inequality (4.1) follows from Lemma 2.1. By the universality (and the proof of [12, Theorem 7.9]) of the infinitely renormalizable Hénon-like maps, the perturbation ε and its derivative has the asymptotic form
ε n (x, y) = b 2 n a(x)y(1 + O(ρ n )) and ∂ ε n ∂ y (x, y) = b 2 n a(x)(1 + O(ρ n )).
Since a is a positive map on a compact set that covers the whole domain, the second inequality follows.
To study the wandering domains, it is enough to consider Hénon-like maps that are close to the hyperbolic fixed point G. By Corollary 6.4 later, for any integer n ≥ 0, we show an infinitely renormalizable Hénon-like map F has a wandering domain in D(F) if and only if F n has a wandering domain in D(F n ). Also, the maps F n converge to the hyperbolic fixed point G as n approaches to infinity by Proposition 4.21. Thus, we focus on a small neighborhood of the fixed point G.
Definition 4.27 Given δ > 0 and I I h ⊂ I v . If ε is small enough such that Propo- sition 4.21 holds, defineÎ δ (I h × I v , ε) to be the class of non-degenerate Hénon-like maps F ∈ I δ (I h × I v , ε) such that F n ∈ H δ (I h × I v n , ε), F n − G < ε, |λ n − λ | < ε, s n (x) − (−λ )x I h < ε, and (4.2) holds for all n ≥ 0.
In the remaining part of the article, we will study the dynamics and the topology of Hénon-like maps in this smaller class of maps.
Structure and Dynamics of Infinitely Renormalizable Hénon-Like Maps
In this chapter, we study the topology of the local stable manifolds and the dynamics on the partition for a infinitely renormalizable Hénon-like map.
Rescaling levels
This section introduces a finer partition of C, called the rescaling levels, based on the maximum possible rescalings of a point in C.
For each two consecutive levels of renormalization n and n + 1, the maps F 2 n and F n+1 are conjugated by the nonlinear rescaling φ n . The rescaling φ n relates the two renormalization levels as follow.
Lemma 5.1 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all F ∈ I δ (I h × I v , ε) the following properties hold for all n ≥ 0:
1. φ n (p n (0)) = p n+1 (−1), 2. φ n (W k n (0)) = W k n+1 (−1) for k = 0, 2, and 3. φ n : C n → D n is a diffeomorphism.
The itinerary of a point follows the arrows in the diagram.
A n+1 F n+1 F n+1 o o A n+2 F n+2 F n+2 o o B n+1 F n+1 B n+2 F n+2 C n φ n / / φ n = = φ n F F C n+1 φ n+1 / / φ n+1 7 7 φ n+1
? ?
C n+2 φ n+2 / / · · ·
The diagram says, if z 0 ∈ C n , then we can rescale the point. The rescaled point z 1 = φ n (z 0 ) enters the domain D n+1 of the next renormalization level n + 1 by Lemma 5.1. On the renormalization level n + 1, the rescaled point z 1 belongs to one of the sets A n+1 , B n+1 , or C n+1 if it is disjoint from the stable manifolds. The process of rescaling stops if z 1 belongs to A n+1 or B n+1 and z 0 can be rescaled at most one time. If z 1 belongs to C n+1 , we can continue to rescale the point. The rescaled point z 2 = φ n+1 (z 1 ) enters the domain D n+2 of the next renormalization level n + 2. Similarly, the process of rescaling stops if z 2 belongs to A n+2 or B n+2 and z 0 can be rescaled at most two times. If z 2 belongs to C n+2 , we can rescale again and repeat the procedure until the rescaled point enters the sets A or B of some deeper renormalization level. Motivated from the diagram, we define the finer partition C n ( j) on C n by the maximal possible rescalings as follows.
Definition 5.2 For consistency, set C n (0) = A n ∪ W 1 n (0) ∪ B n .
Given a positive integer j. The j-th rescaling level in C is defined as C n ( j) = Φ j n −1 (C n+ j (0)) and the j-th rescaling level in B is defined as B n ( j) = F −1 n (C n ( j)). Also, set p n ( j) = Φ j n −1
(p n+ j (0)) and W t n ( j) = Φ j n −1 (W t n+ j (0)) for t = 0, 2.
The diagram explains the definition of a rescaling level.
C n ( j)
F 2 j n / / Φ j n C n ( j) Φ j n D n+ j F n+ j / / D n+ j
From the definition, the relations of the rescaling levels between two different renormalization level are listed as follow.
Proposition 5.3 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
1. p n ( j) is a periodic point of F n with period 2 j for j ≥ 0. 2. W t n ( j) is a local stable manifold of p n ( j) for j ≥ 0 and t = 0, 2. 3. Φ k n (W t n ( j)) = W t n+k ( j −k) and Φ k n (p n ( j)) = p n+k ( j −k) for j ≥ k −1 and t = 0, 2. 4. The map Φ k n : C n ( j) → C n+k ( j − k)
is a diffeomorphism for j ≥ k, and 5. For each j ≥ 0, the set C n ( j) contains two components. The left component C l n ( j) is the set bounded between W 0 n ( j − 1) and W 0 n ( j) and the right component C r n ( j) is the set bounded between W 2 n ( j) and W 2 n ( j − 1). The partition and the local stable manifolds W t n ( j) are illustrated in Figure 5.1. The sets {C n ( j)} j≥1 form a partition of C n and the sets {B n ( j)} j≥1 form a partition of B n .
Next, we introduce the tip to study the geometric structure of the rescaling levels in C. Recall from [12, Section 7.2] that Definition 5.4 (Tip) Assume that ε > 0 is sufficiently small. The tip τ of an infinitely renormalizable Hénon-like map F ∈Î δ (I h × I v , ε) is the unique point such that
{τ} = ∩ ∞ j=N Φ j 0 −1 (D j ∩ I h × I h ) for all N ≥ 0.
The tip is an analog of the critical value in the non-degenerate case. Roughly speaking, the tip generates the attracting Cantor set of a Hénon-like map. See [12, Chapter 5] for more information. From Proposition 5.3, a rescaling level C n ( j) contains two components which are both bounded by two local stable manifolds. The following proposition lists the geometric properties of the local stable manifolds.
Proposition 5.5 Given δ > 0 and I v ⊃ I h I. There exists ε > 0, c > 0 and c > 1 such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
1. W t n ( j) is a vertical graph with Lipschitz constant c ε n for all j ≥ −1 and t = 0, 2. The partition and the local stable manifolds of two renormalization levels F 0 and F 1 from the left to the right. The rescaling levels 1, 2, 3, and below 4 are shaded from light to dark as shown in the legend. Finally, we study the geometric structure of the rescaling levels in B. In the degenerate case, each rescaling level B n ( j) contains two components which are bounded by local stable manifolds. In the non-degenerate case, fix an integer j, the rescaling level B n ( j) also contains two components when ε is small enough. The geometric properties of the boundary local stable manifolds are listed as follow.
2. 1 c 1 λ 2 j < z (t) n ( j) − τ n < c 1W 0 (−1) W 1 (0) W 0 (0) W 0 (1) W 0 ( j) τ W 2 ( j) W 2 (1) W 2 (0) W 2 (−1) A B C C(1) C(2) λ −2 j λ −2 j C(2) C(1) A
Proposition 5.6 Given δ > 0 and I v ⊃ I h I. For all j ≥ 0 and d > 0, there exists ε = ε( j, d) > 0 and c = c( j) > 0 such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
1. F −1 n (W 0 n ( j)) has exactly two components W l n ( j) ⊂ [q l ( j) − d, q l ( j) + d] × I v n and W r n ( j) ⊂ [q r ( j) − d, q r ( j) + d] × I v n . 2.
Both components W l n ( j) and W r n ( j) are vertical graphs with Lipschitz constant c ε n .
Proof The proof is similar to Proposition 5.5.
Remark 5.7 Unlike Proposition 5.5, here the constant ε is not uniform on j ≥ 0. For a non-degenerate Hénon-like map, the structure of the local stable manifolds is similar to degenerate case when j is large. The local stable manifold W 0 n ( j) is far away from the tip and hence the pullback F −1 n (W 0 n ( j)) is the union of two vertical graphs in B n . However, the structure turns to be different when j is large. The local stable manifold is close to the tip and the vertical line argument in Chapter 8 shows that the pullback
F −1 n (W 0 n ( j)) is a concave curve in B n .
Asymptotic behavior near G
In this section, we estimate the derivatives of a Hénon-like map that is close to the hyperbolic fixed point G. Define v n ∈ I h to be the critical point of f n and w n = f n (v n ) be the critical value. The first lemma proves that a Hénon-like map acts like a quadratic map on B.
Lemma 5.8 Given δ > 0 and I v ⊃ I h I. There exists a > 0 (universal), ε > 0, and an interval I B ⊂ I h (universal) such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
The interior of I B containsq(0) and q(0),
I B × I v n ⊃ B n , 1 a |x − v n | ≤ f n (x) ≤ a |x − v n | , and 1 2a (x − v n ) 2 ≤ | f n (x) − f n (v n )| ≤ a 2 (x − v n ) 2 for all x ∈ I B .
Proof The lemma is true because the map F is close to the hyperbolic fixed point G and the map g is concave on the compact set [−c (1) , c (1) ] by Proposition 3.8.
The next lemma shows that a Hénon-like map is expanding on A and C in the x-coordinate when it is close enough to the fixed point G.
Lemma 5.9 Given δ > 0 and I v ⊃ I h I. There exists E > 1 (universal), ε > 0, and a union of two intervals I AC ⊂ I h such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
The interior of I AC contains q(−1),q(0) , q(0), andq(−1),
I AC × I v n ⊃ A n ∪ W 2 n (0) ∪C n , and ∂ h n ∂ x (x, y) ≥ E for all (x, y) ∈ I AC × I v n .
Proof The lemma is true because the map g is expanding on A and C by Proposition 3.13 and the Hénon-like map F is close to the hyperbolic fixed point G of the renormalization operator.
Relation between the tip and the critical value
In Lemma 5.8, we proved that a Hénon-like map behaves like a quadratic map when a point is close to the critical point v n of f n for the representation F n = ( f n − ε n , x). However, the critical point v n and the critical value w n in the estimates depend on the representation.
In this section, we show that the critical value w n (for any representation) is ε nclose to the tip τ n in Proposition 5.13. This allows us to replace v n and w n by the representation independent quantity τ n . This makes the quadratic estimations in Lemma 5.8 useful when a point is ε n -away from the tip.
To estimate the distance from the tip to the critical value, we write τ n = (a n , b n ). Since the rescaling φ n maps a horizontal line to a horizontal line, we focus on the horizontal slice that intersects the tip in each renormalization level. Define the restriction of the rescaling map φ to the slice as
η n (x) = π x • φ n (x, b n ) = s n • h n (x, b n ).
By the definition of the tip, the quantities satisfy the recurrence relations φ n (τ n ) = τ n+1 , η n (a n ) = a n+1 , and s n (b n ) = b n+1 .
First, we prove a lemma that allows us to compare the critical value between two renormaliztion levels.
Lemma 5.10 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that
for all F ∈Î δ (I h × I v , ε) we have |w n+1 − η n (w n )| < c ε n for all n ≥ 0.
Proof First, we compare the critical points v n and v n+1 . By Proposition 4.21, we have
f n+1 − (s n • f 2 n • s −1 n ) I h < c f n+1 − s n • f 2 n • s −1 n I h (δ ) < c ε n
for some constant c > 0 when ε > 0 is sufficiently small. Since the critical point of the map f n+1 is nondegenerate and the map (
s n • f 2 n • s −1 n ) is a small perturbation of f n+1 , the root s n (v n ) of (s n • f 2 n • s −1 n )
is also a small perturbation of the root v n+1 of f n+1 . That is, there exists c > 0 such that
|v n+1 − s n (v n )| ≤ c ε n .
The constant c can be chosen to be independent of F because F is close to G.
Moreover, by the quadratic estimates in Lemma 5.8, we get
f n+1 (v n+1 ) − s n • f 2 n (v n ) ≤ | f n+1 (v n+1 ) − f n+1 (s n (v n ))| + f n+1 (s n (v n )) − s n • f 2 n (v n ) ≤ a 2 |v n+1 − s n (v n )| 2 + f n+1 (s n (v n )) − s n • f 2 n • s −1 n (s n (v n )) ≤ ac 2 2 ε n 2 + c ε n ≤ c ε n for some constant c > 0.
Finally, we compare the critical values w n and w n+1 . Compute
|w n+1 − η n (w n )| = f n+1 (v n+1 ) − s n ( f 2 n (v n ) − ε n ( f n (v n ), b n )) ≤ f n+1 (v n+1 ) − s n • f 2 n (v n ) + λ n |ε n ( f n (v n ), b n )| ≤ c ε n + 2λ ε n = (c + 2λ ) ε n for all n ≥ 0 whenever ε is small enough such that λ n ≤ 2λ .
The rescaling maps {η n } n≥0 can be viewed as a non-autonomous dynamical system (system that depends on time). An orbit is defined as follows.
Definition 5.11 (Orbit of Non-Autonomous Systems) Let Y n be a complete metric space, X n ⊂ Y n be a closed subset, and f n : X n → Y n+1 be a continuous map for all n ≥ 1. A sequence {x n } ∞ n=1 is an orbit of the non-autonomous system { f n } ∞ n=1 if x n ∈ X n and x n+1 = f n (x n ) for all n ≥ 1. A sequence {x n } ∞ n=1 is an ε-orbit of the non-autonomous system { f n } ∞ n=1 if x n ∈ X n and |x n+1 − f n (x n )| < ε for all n ≥ 1. Next, we state an analog of the shadowing theorem for non-autonomous systems.
Lemma 5.12 (Shadowing Theorem for Non-Autonomous Systems) For each n ≥ 1, let Y n be a complete metric space equipped with a metric d (the metric depends on n), X n ⊂ Y n be a closed subset, and f n : X n → Y n+1 be a homeomorphism. Also assume that the non-autonomous system { f n } ∞ n=1 has a uniform expansion. That is, there exists a constant L > 1 such that | f n (a) − f n (b)| ≥ L |a − b| for all a, b ∈ X n and n ≥ 1. The result from Lemma 5.10 shows that the sequence of critical values w n is an εorbit of the expanding non-autonomous system η n . With the help from the Shadowing Theorem, we are able to obtain the goal for this section.
If {x n } ∞ n=1 is an ε-orbit of { f n } ∞ n=1 , there exists a unique orbit {u n } ∞ n=1 of { f n } ∞ n=1 such that d(x n , u n ) ≤ ε L − 1 for all n ≥ 1. In addition, if {X n } ∞(I h × I v , ε) we have | f n (v n ) − π x (τ n )| < c ε n for all n ≥ 0.
Proof Fix n ≥ 0. The critical values {w j } j≥n form an ε n -orbit and the tips {τ j } j≥n form an orbit of the perturbed maps {η j } j≥n . Also, the perturbed maps are uniform expanding by Lemma 5.9 and λ n > 1. Therefore, the proposition follows by Lemma 5.12.
In addition, we can also estimate the distance from the critical point to the preimage of the tip.
Corollary 5.14 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that
for all F ∈Îδ (Î h × I v , ε) we have v n − π y (τ n ) < c ε n for all n ≥ 0.
Proof The corollary is true because the map f n behaves like a quadratic map near the critical point by Lemma 5.8.
Closest Approach
The proof for the nonexistence of wandering domains begins from this chapter. We assume the contrapositive: there exists a wandering domain J.
In this chapter, we construct a rescaled orbit {J n } ∞ n=0 of an wandering domain J which is called the J-closest approach. Then we define the horizontal size l n , the vertical size h n , and the rescaling level k n of an element J n .
Recall the definition of wandering domains.
Definition 6.1 (Wandering Domain) Assume that F ∈ H δ (I h × I v ), D(F) exists, and F is an open map (diffeomorphism from D(F) to the image). A nonempty con- nected open set J ⊂ D(F) is a wandering domain of F if the orbit{F n (J)} n≥0
does not intersect the stable manifold of a periodic point.
Remark 6.2
The classical definition of wandering intervals contains one additional condition: the elements of the orbit do not intersect. This condition is redundant for case of the unimodal maps. Assume that J is an nonempty open interval that does not contain points from the basin of a periodic orbit. If the elements in the orbit of J intersect, then take a connected component A of the union of the orbit that contains at least two elements from the orbit. Then, there exists a positive integer n such that f n (U) ⊂ U. It is easy to show that f n has a fixed point in the interior of U by applying the Brouwer fixed-point theorem several times which leads to a contradiction. Therefore, the orbit elements of J are disjoint.
The following proposition allow us to generate wandering domains by iteration and rescaling.
Proposition 6.3 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all open maps F ∈ H r δ (I h × I v ), the following properties hold:
1. A set J ⊂ D(F) is a wandering domain of F if and only if F(J) is a wandering domain of F. 2. A set J ⊂ C(F) is a wandering domain of F if and only if φ (J) ⊂ D(RF) is a wandering domain of RF.
Proof The proposition is true because the stable manifold of a periodic orbit is invariant under iteration and the rescaling of a stable manifold is also a stable manifold. The converse follows from the second property of Proposition 6.3.
Also, we define the rescaling level of a wandering domain in B.
Definition 6.5 (Rescaling level) Assume that U ⊂ A n ∪ B n is a connected set that does not intersect any of the stable manifolds. Define the rescaling level k(U) as follows. If U ⊂ B n , set k(U) to be the integer such that U ⊂ B n (k(U)); otherwise if U ⊂ A n , set k(U) = 0.
To study the dynamics of a wandering domain, we apply the procedure of renormalization. If a wandering domain is contained in A 0 or B 0 , then its orbit will eventually leave A 0 and B 0 and enter C 0 . If the orbit of the wandering domain enters C 0 , we rescale the orbit element by φ 0 , φ 1 , · · · as many times as possible until it lands on one of the sets A n or B n of some renormalization level n, then study the dynamics of the rescaled orbit by the renormalized map F n . If the rescaled orbit enters C n again, then we rescale it and repeat the same procedure again. By this process, we construct a rescaled orbit as follows.
Definition 6.6 (Closest approach) Assume that ε > 0 is sufficiently small and F ∈ I δ (I h × I v , ε). Given a set J ⊂ A ∪ B such that it does not intersect any of the stable manifolds. Define a sequence of sets {J n } ∞ n=0 and the associate renormalization levels {r(n)} ∞ n=0 by induction such that J n ⊂ A r(n) ∪ B r(n) for all n ≥ 0.
1. Set J 0 = J and r(0) = 0. 2. Write the rescaling level of J n as k n = k(J n ) whenever J n is defined.
3. If J n ⊂ A r(n) , set J n+1 = F r(n) (J n ) and r(n + 1) = r(n). 4. If J n ⊂ B r(n) , set J n+1 = Φ k n r(n) • F r(n) (J n ) and r(n + 1) = r(n) + k n . The transition between two constitutive sequence elements, one iteration together with rescaling (if possible), is called one step. The sequence {J n } ∞ n=0 is called the rescaled iterations of J that closest approaches the tip, or J-closest approach for short.
The itinerary of a closest approach is summarized by the following diagram.
A r(n) F r(n) F r(n) o o A r(n+1) B r(n) Φ kn r(n) •F r(n) / / A r(n+1) ∪ B r(n+1) / / 4 4 B r(n+1)
Example 6.7 In this example, we explain the construction of a closest approach and demonstrate the idea of proving the nonexistence of wandering domains.
Let F = ( f − ε, x) be a Hénon-like map such that f (x) = 1.7996565(1 + x)(1 − x) − 1 and ε(x, y) = 0.025y.
The map F is numerically checked to be seven times renormalizable. Given a set J = (−0.950, −0.947) × (0.042, 0.045) ⊂ A. We show that the set is not a wandering domain by contradiction.
If J is a wandering domain, we construct a J-closest approach as shown in Figure 6.1. Set J 0 = J and r(0) = 0. The set J 0 is contained in A r(0) . The next element is defined to be J 1 = F r(0) (J 0 ) and r(1) = r(0) = 0. The set J 1 is also contained in A r (1) .
Set J 2 = F r(1) (J 1 ) and r(2) = r(1) = 0. The set J 2 is contained in B r(2) (1). Set k 2 = 1, r(3) = r(2) + k 2 = 1, and J 3 = Φ k 2 r(2) • F r(2) (J 2 ) = φ 0 • F 0 (J 2 )
. The set J 3 is contained in A r (3) . Set J 4 = F r(3) (J 3 ) and r(4) = r(3) = 1.
From the graph, we see that the sizes of the elements {J n } grow as the procedure continues and the element J 4 ⊂ B 1 becomes so large that it intersects some local stable manifolds. This leads to a contradiction. Therefore, J is not a wandering domain.
Motivated from the example, we study the growth of the horizontal size and prove the sizes of the elements approach to infinity to obtain a contradiction. The size is defined as follow.
Definition 6.8 (Horizontal and Vertical size) Assume that J ⊂ R 2 . Define the horizontal size as y 1 ), (x 2 , y 2 ) ∈ J} = |π x J| and the vertical size as
l(J) = sup {|x 1 − x 2 | ; (x 1 ,h(J) = sup {|y 1 − y 2 | ; (x 1 , y 1 ), (x 2 , y 2 ) ∈ U} = π y J .
If J is compact, the horizontal endpoints of J are two points in the set that determines l(J).
for all J ⊂ I h × I v .
For simplicity, we start from a closed subset J of a wandering domain such that int(J) = J. Then consider the J-closest approach {J n } n≥0 instead to ensure the horizontal endpoints exist. Note that the sequence element J n is also a subset of a wandering domain of F r(n) . For elements in a closest approach, set l n = l(J n ) and h n = h(J n ). Our final goal is to show that the horizontal size l n approaches to infinity and hence wandering domains cannot exist.
*The Degenerate Case
In this chapter, we study the relationship between the unimodal renormalization and the Hénon renormalization by identifying a unimodal map as a degenerate Hénonlike map. The main goal is to present a short proof for the nonexistence of wandering intervals for an infinitely renormalizable unimodal map at the end of this chapter. It is well known that a unimodal map (under some regularity condition) does not have wandering interval [50,51,43,8,45]. Here, we give a different proof by using the Hénon renormalization instead of the unimodal renormalization. The expansion argument introduced in the proof motivates the proof for the non-degenerate case.
Local stable manifolds and partition
First, we adopt the notations from unimodal maps and Hénon-like maps. Let F be a degenerate Hénon-like map F(x, y) = ( f (x), x).
We use the super-scripts "u" and "h" to distinguish the difference between the notations for unimodal maps and Hénon-like maps to avoid confusion. For example, p u (−1) = −1 and p u (0) are the fixed points of f ; p h (−1) = (−1, −1) and p h (0) are the saddle fixed points of F; A u , B u ,C u ⊂ I is the partition defined for f ; A h , B h ,C h ⊂ I h × I v is the partition defined for F. The next lemma gives the relations between the local stable manifolds for the degenerate Hénon-like map with the fixed points and their preimages for the unimodal maps. Recall that p (1) and p (2) are the points such that f (p (2) ) = p (1) , f (p (1) ) = p u (0), and p (1) < p u (0) < p (2) (Definition 3.4); W 0 (−1) and W 2 (−1) are the local stable manifolds of p h (−1) (Definition 4.10); W 0 (0),W 1 (0),W 2 (0) are the local stable manifolds of p h (0) (Definition 4.12). It follows from the definition that the partition for unimodal maps and degenerate Hénon-like maps coincide.
Corollary 7.2 (Partition) Assume that F ∈ H δ (I h × I v ) is a degenerate Hénon-like map. Then A h = A u × I v , B h = B u × I v , C h = C u × I v , and D h = I × I v .
Renormalization operator
Next we compare the renormaliztion operator for Hénon-like maps with the renormalization operator for unimodal maps. Recall the definitions of the rescaling maps. For a degenerate renormalizable Hénon-like map F, the rescaling map has the form φ = Λ • H where Λ (x, y) = (s h (x), s h (y)), s h is the affine rescaling map, and H(x, y) = ( f (x), y) is the nonlinear rescaling term. The renormalized map is
RF = φ • F 2 • φ −1 .
For a renormalizable unimodal map f , s u is the affine rescaling and R f = s u • f 2 • (s u ) −1 is the renormalization about the critical point.
Although the Hénon renormaliation rescales the first return map around the "critical value", the operation acts like the unimodal renormalization which rescales the first return map around the "critical point". This is because of the nonlinear rescaling term H for the Hénon-renormalization. Let
H • F 2 • H −1 (x, y) = ( f 2 | B u 0 (x), x)
is the first return map on B h 0 . Therefore, the two renormalizations coincide
RF(x, y) = (s u • f 2 • (s u ) −1 (x), x) = (R f (x), x).
This also explains why R n F converges to the fixed point g of the unimodal renormalization operator. The observation is summarized as follows. In fact, if F is infinitely renormalizable, then the affine term Λ n :
B n ( j) → B n+1 ( j − 1)
is a bijection for all n ≥ 0 and j ≥ 1 where B n (0) ≡ A n ∪W 2 n (0) ∪C n .
From now on, we remove the super-script from s because the maps are the same.
For an infinitely renormalizable Hénon-like map, we also adopt the subscript used for the renormalization levels to the degenerate case. Assume that a degenerate Hénon-like map F(x, y) = ( f (x), x) is infinitely renormalizable. Let F n = R n F and f n = R n f . Then F n (x, y) = ( f n (x), x) by the second property of Lemma 7.3.
Next proposition proves an important equality which will be used to prove the nonexistence of wandering intervals for infinitely renormalizable unimodal maps. The expansion argument comes from this proposition. Proof Prove by induction on j. It is clear that the equality holds when j = 0.
Assume that the equality holds for some j. Then
(s n+ j • f n+ j ) • (s n+ j−1 • f n+ j−1 ) • · · · • (s n • f n ) • f n =(s n+ j • f n+ j ) • f n+ j • s n+ j−1 • · · · • s n =(s n+ j • f n+ j • f n+ j • s −1 n+ j ) • s n+ j • s n+ j−1 • · · · • s n = f n+ j+1 • s n+ j • s n+ j−1 • · · · • s n .
Therefore, the lemma is proved by induction. By Lemma 7.3 and Proposition 7.3, we get Corollary 7.5 Assume that F ∈ I δ (I h × I v ) is a degenerate Hénon-like map. Then Φ j n • F n = F n+ j • Λ n+ j−1 • · · · • Λ n for all integers n ≥ 0 and j ≥ 0.
Nonexistence of wandering intervals
In this section, we present a proof for the nonexistence of wandering intervals for infinitely renormalizable unimodal maps by identifying a unimodal map as a degenerate Hénon-like map and using the Hénon renormalization. A wandering interval is a nonempty interval such that its orbit does not intersect itself and the omega limit set does not contain a periodic point.
Proposition 7.6 A infinitely renormalizable unimodal map does not have a wandering interval.
Proof (Sketch of the proof) Prove by contradiction. Assume that f is an infinitely renormalizable unimodal map that has a wandering interval J u . Without lose of generality, we may assume that the map is close to the fixed point g of the renormalization operator because the sequence of renormalizations R n f converges to g as n approaches to infinity. Let F = ( f , x). Then F is a degenerate infinitely renormalizable Hénon-like map. Assume that J u ⊂ I is a wandering interval of f 0 . Let J h = J u × {0} and J n ⊂ A r(n) ∪ B r(n) be the J h -closest approach. The projection π x J n is a wandering interval of f r(n) and the horizontal size l n is the length of the projection. If J n ⊂ A r(n) , then l n+1 > El n for some constant E > 1 because g is expanding on A(g) by Proposition 3.13 and the map f r(n) is close to g. If J n ⊂ B r(n) (k n ), then J n+1 = F r(n+1) • Λ r(n)+k n −1 • · · · • Λ r(n) (J n ) by Corollary 7.5. The rescaling maps Λ r(n) , · · · ,Λ r(n)+k n −1 expands the horizontal size. The map F r(n+1) also expands the horizontal size because Λ r(n)+k n −1 •· · ·•Λ r(n) (J n ) ⊂ A r(n+1) ∪ C r(n+1) , g is expanding on A(g) ∪ C(g) by Proposition 3.13, and the map f r(n+1) is close to g. Thus, l n+1 > E l n for some constant E > 1. This shows that the horizontal size l n approaches to infinity which yields a contradiction. Therefore, wandering intervals cannot exist.
In the proof, we showed that the horizontal size expands at a definite size in each size. This motivates the proof for the non-degenerate case. In the remain part of the article, we will study the growth rate or contraction rate of the horizontal size. In Chapter 9, we will show this is also true for the non-degenerate case under some conditions.
The Good Region and the Bad Region
In this chapter, we group the sub-partitions of {B n ( j)} ∞ j=1 and {C n ( j)} ∞ j=1 into two regions by the following phenomena. Assume that {J n } ∞ n=0 is the J-closest approach and J n ∈ B r(n) (k n ) for some n.
When J n is far from the center, i.e. k n is small, the topology of B r(n) (k n ) and the dynamics of F r(n) behave like the unimodal case. It can be proved that the boundaries of B n (k n ) are vertical graphs of small Lipschitz constant. Also, studying the iteration of horizontal endpoints of a wandering domain provides a good approximation to the expansion rate of the horizontal size. Chapter 9 will show the expansion argument works in this case. This group is called "the good region".
However, when J n is close to the center, i.e. k n is large, the topology of B r(n) (k n ) and dynamics of F r(n) is different from the unimodal case. In this group, the two boundary local stable manifolds of C r(n) (k n ) are so close to the tip τ r(n) that they only intersect the image F r(n) (D r(n) ) once. Thus, the preimage of a local stable manifold becomes concave and hence B r(n) ( j) becomes an arch-like domain. See the left graph of Figure 6.1 and the next paragraph. Also, the expansion argument fails. The iteration of horizontal endpoints of a wandering domain fails to provide an approximation for the change rate of the horizontal size. In fact, we show that the x-coordinate of the two iterated horizontal endpoints can be as close as possible in the next paragraph. This group is called "the bad region".
The vertical line argument in Figure 8.1 explains why the expansion argument fails in the bad region. The construction is as follows. Draw a vertical line (dashed vertical line in the figure) so close to the tip that its intersection with the image of F r(n) only has one component. Take the preimage of the intersection. Unlike the case in the good region, the preimage is not a vertical graph. Instead, it is a concave curve that has a y-extremal point close to the center of the domain. When a sequence element J n is in the bad region, it is close to the center. The size of J n is small because the size of bad region is small and hence F r(n) acts like a linear map. If the line ← → UV connecting horizontal endpoints U and V of J n is also parallel to the concave curve as in Figure 8.1a, the image of the horizontal endpoints will also be parallel to the vertical line as Figure 8.1b shows. In this case, the iterated horizontal endpoints forms a vertical line that has no x-displacement. Therefore, the horizontal size shrinks and the horizontal endpoints fail to estimate the change of horizontal size when a sequence element enters the bad region.
From the vertical line argument, it becomes crucial to group the sets {C n ( j)} j≥1 by how close the set to the tip is. The size of the image is ε n . To avoid a vertical line intersecting the image only once, the line has to be ε n away from the tip. This motivates the definition of the boundary sequence {K n } n≥0 , the good region, and the bad region.
Definition 8.1 (Good and Bad Regions)
Fixed b > 0. Assume that ε > 0 is sufficiently small so that Proposition 5.5 holds and F ∈Î δ (I h × I v , ε). For each n ≥ 0, define K n = K n (b) to be the largest positive integer such that |π x z − π x τ n | > b ε n for all z ∈ W 0 n (K n ) ∩ I h × I h . The set C n ( j) (resp. B n ( j)) is in the good region if j ≤ K n ; in the bad region if j > K n . The sequence K n is called the boundary for the good region and the bad region. See
Remark 8.3
Here the boundary sequence {K n } n≥0 depends on the constant b and we make b flexible. In the theorems of this chapter, we will prove that each property holds for all b that satisfies certain constraints. At the end, we will fix a constant b sufficiently large that makes all theorems work. So the sequence {K n } n≥0 will be fixed in the remaining article.
Remark 8.4
One can see that the bad region is a special feature for the Hénon case. For the degenerate case, ε n = 0 and hence K n = ∞. This means that there are no bad region for the degenerate case. Hence, the boundary is K = 2 and the good region contains the lightest part and the bad region contains the two darker parts.
Our goal in this chapter is to study the geometric properties for the good region and the bad region. The main theorem is stated as follows.
Proposition 8.5 (Geometric properties for the good region and the bad region)
Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 1 such that for all F ∈Î δ (I h × I v , ε) and b > b the following properties hold for all n ≥ 0:
The boundary K n is bounded by
1 c 1 b ε n ≤ λ K n ≤ c 1 b ε n . (8.1)
For the good region 1 ≤ j ≤ K n , we have 1. C r n ( j) ∩ F n (D n ) = φ , 2. |π x z − π x τ n | > b ε n for all z ∈ C n ( j) ∩ F n (D n ), 3. |π x z − v n | > 1 c b ε n for all z ∈ B n ( j), and 4. 1 c 1 λ 2 j < |π x z − π x τ n | < c 1 λ 2 j for all z ∈ C n ( j) ∩ F n (D n ).
For the bad region j > K n , we have 1. |π x z − π x τ n | < cb ε n for all z ∈ C n ( j) ∩ F n (D n ) and 2. |π x z − v n | < c b ε n for all z ∈ B n ( j).
This proposition will be proved by the lemmas in this chapter.
First, we estimate the bounds for the boundary K n .
Lemma 8.6 Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 1 such that for all F ∈Î δ (I h × I v , ε) and b > b we have
1 c 1 b ε n ≤ λ K n ≤ c 1 b ε n for all n ≥ 0.
Proof In the proof, we apply Proposition 5.5 to relate the rescaling level K n with the x-coordinate of the local stable manifold. Assume that ε > 0 is small enough.
For the upper bound, by the definition of K n and Proposition 5.5, we have
c 1 λ 2K n ≥ z (0) n (K n ) − τ n ≥ b ε n .
for some constant c > 1. Thus,
λ K n ≤ c b 1 ε n .
For the lower bound, by the definition of K n , there exists z ∈ W 0 n (K n + 1) ∩ I h × I h such that |π x z − π x τ n | ≤ b ε n . Apply Proposition 5.5 , we get
1 c 1 λ 2(K n +1) ≤ z (0) n (K n + 1) − τ n ≤ |π x z − π x τ n | + π x z − π x z (0) n (K n + 1) ≤ b ε n + c I h ε n .
for some constant c > 0. We solved
λ K n ≥ 1 λ 1 c (b + c |I h |) 1 ε n ≥ 1 λ 1 2c b 1 ε n when b ≥ c I h .
Properties for the good region
To prove the properties, the strategy is to first estimate the x-location of the local stable manifolds W t n ( j). Since the local stable manifolds bounds C n ( j), the properties for C n ( j) follows. Properties for B n ( j) follows directly by the quadratic estimations from Lemma 5.8 and Proposition 5.13. Lemma 8.7 Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 1 such that for all F ∈Î δ (I h × I v , ε) and b > b we have
1 c 1 λ 2 j ≤ |π x z − π x τ n | ≤ c 1 λ 2 j
for all z ∈ W t n ( j) ∩ I h × I h with t ∈ {0, 2}, 0 ≤ j ≤ K n , and n ≥ 0.
Proof In the proof, we apply Proposition 5.5 to relate the rescaling level j with the x-coordinate of the local stable manifold W t n ( j). Assume that ε > 0 is sufficiently small. We prove the case for t = 0 and the other case is similar.
Let z ∈ W 0 n ( j) ∩ I h × I h and b > b where b is given by Lemma 8.6. To prove the lower bound, apply Proposition 5.5, we get
|π x z − π x τ n | ≥ z (0) n ( j) − τ n − π x z − π x z (0) n ( j) ≥ 1 c 1 λ 2 j − c ε n I h ≥ 1 c − c I h ε n λ 2K n 1 λ 2 j
for some constant c > 1. By Lemma 8.6, there exists c > 1 such that
|π x z − π x τ n | ≥ 1 c − cc 2 I h b 1 λ 2 j ≥ 1 2c 1 λ 2 j .
Here we assume that b ≥ 2c 2 c 2 I h . Similarly, to prove the upper bound, apply Proposition 5.5, we get
|π x z − π x τ n | ≤ z (0) j ( j) − τ n + π x z − π x z (0) n ( j) ≤ c 1 λ 2 j + c ε n I h ≤ c + c I h ε n λ 2K n 1 λ 2 j .
By Lemma 8.6, we get
|π x z − π x τ n | ≤ 1 c + cc 2 I h b 1 λ 2 j ≤ 3 2c 1 λ 2 j .
We prove the first property for the good region.
Lemma 8.8 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and b > 0 such that for all F ∈Î δ (I h × I v , ε) and b > b we have
C r n ( j) ∩ F n (D n ) = φ
for all 1 ≤ j ≤ K n and n ≥ 0.
Proof Since ∪ K n j=1 C r n ( j) is bounded by the local manifolds W 2 n (K n ) and W 2 n (0), it suffices to prove the local stable manifold W 2 n (K n ) is far away form the image. We have
f n (v n ) − ε n ≤ sup z ∈D n h n (z ) = sup z ∈D n f n (π x z ) + ε n (z ) ≤ f n (v n ) + ε n .
Apply Proposition 5.13, Lemma 8.6, and Lemma 8.7, there exists constants c > 0 and a > 1 such that
π x z − sup z ∈D n h n (z ) ≥ (π x z − π x τ n ) − |π x τ n − f n (v n )| − f n (v n ) − sup z ∈D n h n (z ) ≥ 1 a 1 λ 2K n − c ε n − ε n ≥ b a 3 − c − 1 ε n for all z ∈ W 2 n (K n ) ∩ I h × I h . The coefficient is positive when b > 0 is large enough. Consequently, C r n ( j) ∩ F n (D n ) = φ for all 1 ≤ j ≤ K n .
By the previous lemma, it is enough to prove the rest of the properties for the left component C l n ( j). The second property follows directly from the definition of K n . Lemma 8.9 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and b > 0 such that for all F ∈Î δ (I h × I v , ε) and b > b we have |π x z − π x τ n | > b ε n for all z ∈ C n ( j) ∩ F n (D n ) with 1 ≤ j ≤ K n and n ≥ 0.
Proof By Lemma 8.8, only the left component C l n ( j) intersects the image. Also the set C l n ( j) is bounded by the local stable manifolds W 0 n ( j − 1) and W 0 n ( j). Thus, the lemma follows from the definition of K n .
The third property of the good regions follows by the previous proposition and the quadratic estimates from Lemma 5.8. Corollary 8.10 Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 0 such that for all F ∈Î δ (I h × I v , ε) and b > b so that the following property hold for all n ≥ 0:
If z ∈ I B × I v n satisfies |h n (z) − π x τ n | ≥ b ε n , then
|π x z − v n | ≥ c b ε n . (8.2)
In particular, (8.2) holds for all z ∈ B n ( j) with 1 ≤ j ≤ K n .
Proof Assume that b > 0 is large enough such that Lemma 8.9 holds. Let z ∈ I B × I v n be such that |h n (z) − π x τ n | > b ε n . Apply Proposition 5.13, we get
| f n (π x z) − f n (v n )| ≥ |h n (z) − π x τ n | − | f n (π x z) − h n (z)| − |π x τ n − f n (v n )| ≥ (b − 1 − c) ε n > b 2 ε n for some c > 0 when b > 2(1 + c).
Moreover, by the quadratic estimates from Lemma 5.8, there exists a > 1 such that
| f n (π x z) − f n (v n )| ≤ a 2 (π x z − v n ) 2
for all n ≥ 0 when ε > 0 is small enough. Therefore,
|π x z − v n | ≥ 1 a b ε n .
The fourth property for the good region gives an estimate for the x-location of C l n ( j) in terms of the rescaling level j. To prove the property, we use the boundary stable manifolds W 0 n ( j − 1) and W 0 n ( j) to estimate the x-location of C l n ( j). < |π x z − π x τ n | < c 1 λ 2 j for all z ∈ C n ( j) ∩ F n (D n ) with 1 ≤ j ≤ K n and n ≥ 0.
Proof We use the property that C l n ( j) is bounded by W 0 n ( j − 1) and W 0 n ( j) then apply the estimations from the local stable manifolds Lemma 8.7. Assume that b > 0 is large enough such that Lemma 8.7 and Lemma 8.8 hold.
For all z ∈ C l n ( j) ∩ F n (D n ) with 1 ≤ j ≤ K n , there exists z 1 ∈ W 0 n ( j − 1) ∩ I h × I h and z 2 ∈ W 0 n ( j) ∩ I h × I h such that π y z = π y z 1 = π y z 2 since the local stable manifolds are vertical graphs. From Lemma 8.7, we obtain
1 c 1 λ 2 j ≤ |π x z 2 − π x τ n | ≤ |π x z − π x τ n | ≤ |π x z 1 − π x τ n | ≤ cλ 2 1 λ 2 j .
Properties for the bad region
We prove the first property for the bad region by applying Lemma 8.7 to the boundary local stable manifolds W 0 n (K n ) and W 2 n (K n ).
Lemma 8.12
Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 0 such that for all F ∈Î δ (I h × I v , ε) and b > b we have |π x z − π x τ n | < cb ε n for all z ∈ C n ( j) ∩ F n (D n ) with j > K n and n ≥ 0.
Proof Since the bad region ∪ j>K n C n ( j) is bounded by the local stable manifolds W 0 n (K n ) and W 2 n (K n ), it is sufficient to estimate the location of W 0 n (K n ) and W 2 n (K n ). Assume that z ∈ W t n (K n ) ∩ I h × I h with t ∈ {0, 2}. By Lemma 8.6 and Lemma 8.7, there exists c > 1 such that
|π x z − π x τ n | ≤ c 1 λ 2K n ≤ c 3 b ε n for all b > 0 sufficiently large.
The second property for the bad region follows from the quadratic estimates Lemma 5.8.
Corollary 8.13
Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 0 such that for all F ∈Î δ (I h × I v , ε) and b > b we have |π x z − v n | < c b ε n for all z ∈ B n ( j) with j > K n and n ≥ 0.
Proof Assume that z ∈ B n ( j). Then F n (z) ∈ C n ( j) ∩ F n (D n ). By Proposition 5.13 and Lemma 8.12, there exists c > 0 such that
| f n (π x z) − f n (v n )| ≤ |h n (z) − π x τ n | + | f n (π x z) − h n (z)| + |π x τ n − f n (v n )| ≤ (cb + 1 + c) ε n < 2cb ε n
for all b > 0 sufficiently large. Also, by Lemma 5.8, we have
| f n (π x z) − f n (v n )| ≥ 1 2a (π x z − v n ) 2
for some constant a > 0. Combine the two inequalities, we obtain
|π x z − v n | ≤ √ 4ac b ε n .
The Good Region and the Expansion Argument
Our goal in this chapter is to prove the expansion argument in the good region, Proposition 9.2: the horizontal size of the closest approach expands when the wandering domains stay in the good region. This shows that Hénon-like maps behaves like unimodal maps in the good region. From now on, fix b > 0 sufficiently large so that Proposition 8.5 holds and the sequence {K n } n≥0 depends only on F.
To prove the horizontal size expands, we iterate the horizontal endpoints to estimate the expansion of the horizontal size. The vertical line argument in Chapter 8 showed that the iteration of the horizontal endpoints fails to approximate the expansion rate when the line connecting the two horizontal endpoints is parallel to the preimage of a vertical line. The following condition, R-regular, provides a criteria to ensure "parallel" does not happen in the good region.
Definition 9.1 (Regular) Let R > 0. A set U ⊂ D(F) is R-regular if h(U) l(U) ≤ R 1 ε 1/4 . (9.1)
To see R-regular implies not parallel, we estimate the slope of the preimage of a vertical line. Assume that γ : I v → I h is the vertical graph of the preimage of some vertical line x = x 0 by the Hénon-like map F n and the vertical graph is in the good region. Then h n (γ(y), y) = x 0 .
Apply the derivative in terms of y to the both sides, we solved γ (y) = ∂ ε n ∂ y (γ(y), y) f n (γ(y)) − ∂ ε n ∂ x (γ(y), y)
. By Lemma 5.8 and Proposition 8.5, we get
f n (y) − ∂ ε n ∂ x (γ(y), y) ≥ 1 a |γ(y) − v n | − 1 δ ε n ≥ c a ε n − 1 δ ε n ≥ c 2a ε n
when ε is small enough. This yields
γ (y) ≤ c ε n (9.2)
for some constant c > 0. The condition R-regular says that the vertical slope of the line determined by the horizontal endpoints (x 1 , y 1 ) and (x 2 , y 2 ) of J is bounded by
|x 2 − x 1 | |y 2 − y 1 | ≥ l(J) h(J) ≥ 1 R ε n 1/4 . (9.3)
From (9.2) and (9.3), we get
|x 2 − x 1 | |y 2 − y 1 | γ (y) .
This concludes that the line connecting the horizontal endpoints is not parallel to the preimage of a vertical line if the wandering domain is R-regular. Now we state the main proposition of this chapter.
Proposition 9.2 (Expansion argument) Given δ > 0 and I v ⊃ I h I. There exists ε > 0, E > 1, and R > 0 such that for all F ∈Î δ (I h × I v , ε) the following property hold: Assume that J ⊂ A ∪ B is a R-regular closed subset of a wandering domain for F and {J n } ∞ n=0 is the J-closest approach. If k n ≤ K r(n) for all n ≤ m, then J n is R-regular for all n ≤ m + 1 and l n+1 ≥ El n (9.4) for all n ≤ m.
Proof The proof is based on the estimations for the expansion rate developed later in this chapter. The expansion rate will be computed in three different cases:
1. J n ⊂ A r(n) , proved in Lemma 9.3. 2. J n ⊂ B r(n) (k n ) for the intermediate region 1 ≤ k n < K where K is some constant, proved in Lemma 9.14. 3. J n ⊂ B r(n) (k n ) for the region close to the center K ≤ k n ≤ K r(n) , proved in Lemma 9.4.
Here we assume the three lemmas to prove this proposition. Fixed R > 0 to be the constant given by Lemma 9.4. Also, fixed K ≥ 1 to be an integer large enough such that cE K 2 > 1 from (9.7) where c > 0 and E 2 > 1 are the constants in Lemma 9.4. Let E 1 > 1 be the expansion constant in Lemma 9.3 and E 3 > 1 be the expansion constant in Lemma 9.14. Set E = min(E 1 , cE K 2 , E 3 ) > 1. Let ε > 0 be small enough such that Proposition 8.5, Lemma 9.3, Lemma 9.4, and Lemma 9.14 hold.
Assume that F ∈Î δ (I h × I v , ε) and J ⊂ A ∪ B is a closed R-regular subset that is a wandering domain of F. We prove that J n is R-regular by induction then (9.4) follows by the three lemmas.
For the base case, J 0 is R-regular by assumption. If J n is R-regular and k n ≤ K r(n) for some n ≥ 0. If J n ⊂ A r(n) , then J n+1 is Rregular and l n+1 ≥ E 1 l n ≥ El n by Lemma 9.3. If J n ⊂ B r(n) (k n ) with 1 ≤ k n ≤ K, then J n+1 is R-regular and l n+1 ≥ E 3 λ k n l n ≥ El n by Lemma 9.14. If J n ⊂ B r(n) (k n ) with K ≤ k n ≤ K n , then J n+1 is R-regular and l n+1 ≥ cE k n 2 l n ≥ cE K 2 l n ≥ El n by Lemma 9.4. Therefore, the theorem is proved by induction.
Case J n ⊂ A r(n)
In this section, we compute the expansion rate when a wandering domain J lies in A.
We use the property that F n is close to the fixed point G then apply the properties for g in Section 3.2 to estimate the expansion. where J = F n (J).
Proof Let E > 1 be the constant defined in Lemma 5.9 and ∆ E > 0 be small enough such that E ≡ E − ∆ E > 1. Assume that ε > 0 is small enough such that Lemma 5.9 holds and R δ ε n 3/4 < ∆ E (9.5) for all n ≥ 0.
To prove the inequality, let (x 1 , y 1 ), (x 2 , y 2 ) ∈ J such that x 2 − x 1 = l(J). Then h(J) ≥ |y 2 − y 1 |.
Compute
l(J ) ≥ |π x [F n (x 2 , y 2 ) − F n (x 1 , y 1 )]| ≥ |π x [F n (x 2 , y 2 ) − F n (x 1 , y 2 )]| − |π x [F n (x 1 , y 2 ) − F n (x 1 , y 1 )]| .
By the mean value theorem, there exists ξ ∈ (x 1 , x 2 ) and η ∈ (y 1 , y 2 ) such that
π x [F n (x 2 , y 2 ) − F n (x 1 , y 2 )] = ∂ h n ∂ x (ξ , y 2 )(x 2 − x 1 )
and
π x [F n (x 1 , y 2 ) − F n (x 1 , y 1 )] = ∂ ε n ∂ y (x 1 , η)(y 2 − y 1 ).
Since (x 1 , y 1 ), (x 2 , y 2 ) ∈ A n ⊂ I AC × I v n , we have (ξ , y 2 ) ∈ I AC × I v n . By Lemma 4.26 and Lemma 5.9, we get
l(J ) ≥ El(J) − 1 δ ε n h(J) = E − 1 δ ε n h(J) l(J) l(J).
Also, by J is R-regular and (9.5), this yields
l(J ) ≥ E − R δ ε n 3/4 l(J) ≥ E l(J). (9.6)
To prove that J is R-regular, we apply (9.6) and h(J ) = l(J). We get
h(J ) l(J ) ≤ 1 E .
Also assume that ε is small enough such that 1 E ≤ R ε n −1/4 for all n ≥ 0. This proves that J is R-regular.
9.2 Case J n ⊂ B r(n) (k n ), K ≤ k n ≤ K r(n)
In this section, we prove the horizontal size expands when a wandering domains is in the center of good region. Unfortunately, the rescaling trick, Proposition 7.4, does not work in the nondegenerate case. In the degenerate case, the affine rescaling coincide with the rescaling for the unimodal renormalization about the critical point. Thus, the affine rescaling Λ n maps rescaling levels in B n to renormalization levels in B n+1 (Proposition 7.3).
However, in the non-degenerate case, the affine rescaling has no geometrical and dynamical meaning. So the proofs for Proposition 7.4 and Corollary 7.5 do not apply to the non-degenerate case. We have to find another strategy to estimate the expansion rate for the horizontal size.
The idea of the proof is as follows. When the sequence element J n enters B r(n) , the step from J n to J n+1 contains an iteration F r(n) and a composition of rescalings Φ k n r(n) . On the one hand, the horizontal size contracts by the iteration F r(n) because the Hénon-like map acts like a quadratic map. Lemma 5.8 says that the contraction becomes strong as J n approach to the center. On the other hand, the horizontal size expands by the rescaling Φ k n r(n) (Lemma 5.9). Proposition 8.5 says the number of rescaling k n becomes large as J n approach to the center. We will show the expansion compensates with the contraction. This yields the following lemma.
Lemma 9.4 Given δ > 0 and I v ⊃ I h I. There exists ε > 0, E > 1, R > 0, and c > 0 such that for all F ∈Î δ (I h × I v , ε) the following property holds for all n ≥ 0:
Assume that J ⊂ B n (k) is an R-regular closed set and 1 ≤ k ≤ K n , then J ⊂ C n+k (0) = A n+k ∪W 1 n+k (0) ∪ B n+k is R-regular and
l(J ) ≥ cE k l(J) (9.7) where J = Φ k n • F n (J).
In the remaining part of this chapter, we will fixed K > 0 to be sufficiently large so that (9.7) provides a strict expansion to the horizontal size for all k ≥ K.
To prove this lemma, we set up the notations. Given a closed set J ⊂ B n (k). Let (x 1 , y 1 ), (x 2 , y 2 ) ∈ J be such that l(J) = |x 2 − x 1 |. Then h(J) ≥ |y 2 − y 1 |. We define (x ( j)
1 , y ( j) 1 ) = Φ j n • F n (x 1 , y 1 ) and (x ( j) 2 , y ( j) 2 ) = Φ j n • F n (x 2 , y 2 ) for j = 0, · · · , k. Also, let x ∈ {x 1 , x 2 } be such that |x − v n | = min i=1,2 |x i − v n |.
We first prove the following estimate.
Lemma 9.5 Given δ > 0 and I v ⊃ I h I. For all R > 0, there exists ε = ε(R) > 0, E > 1, a > 1, and R > 0 such that for all F ∈Î δ (I h × I v , ε) the following property holds for all n ≥ 0: Assume that J ⊂ B n (k) is an R-regular closed set and k ≤ K n then
x ( j) 2 − x ( j) 1 ≥ 1 2a |x − v n | (λ E) j l(J) and y ( j) 2 − y ( j) 1 x ( j) 2 − x ( j) 1 ≤ R 1 ε n for j = 0, · · · , k.
The constants E, a, and R does not depend on R.
Proof Let ε > 0 be small enough so that Lemma 4.26, Lemma 5.8, and Proposition 8.5 hold. Let E > 1 be the constant defined in Lemma 5.9 and E be a constant such that E > E > 1. We prove the lemma by induction on j.
For the case j = 0, we have
x (0) 2 − x (0) 1 = |π x (F n (x 2 , y 2 ) − F n (x 1 , y 1 ))| ≥ |π x (F n (x 2 , y 2 ) − F n (x 1 , y 2 ))| − |π x (F n (x 1 , y 2 ) − F n (x 1 , y 1 ))| (9.8)
Apply the mean value theorem, there exists ξ ∈ (x 1 , x 2 ) and η ∈ (y 1 , y 2 ) such that
π x (F n (x 2 , y 2 ) − F n (x 1 , y 2 )) = f n (ξ ) − ∂ ε n ∂ x (ξ , y 2 ) (x 2 − x 1 ) (9.9)
and
π x (F n (x 1 , y 2 ) − F n (x 1 , y 1 )) = − ∂ ε n ∂ y (x 1 , η)(y 2 − y 1 ). (9.10)
Then ξ ∈ I B since (x 1 , y 1 ), (x 2 , y 2 ) ∈ B n ⊂ I B × I v n . By Lemma 5.8, (9.9) yields
|π x (F n (x 2 , y 2 ) − F n (x 1 , y 2 ))| ≥ f n (ξ ) − ∂ ε n ∂ x (ξ , y 2 ) l(J) ≥ 1 a |x − v n | − 1 δ ε n l(J). (9.11)
Also, since J is R-regular, (9.10) yields
|π x (F n (x 1 , y 2 ) − F n (x 1 , y 1 ))| ≤ 1 δ ε n h(J) ≤ R δ ε n 3/4 l(J). (9.12)
Combine (9.8), (9.11), and (9.12), we get
x (0) 2 − x (0) 1 ≥ 1 a |x − v n | − 1 δ ε n 1/2 + R ε n 1/4 ε n l(J)
By Proposition 8.5, c |x − v n | > ε n for some constant c > 1. Also, assume that ε = ε(R) is small enough such that c δ ε n 1/2 + R ε n 1/4 < 1 2a for all n ≥ 0. We obtain
x (0) 2 − x (0) 1 ≥ 1 2a |x − v n | l(J).
Moreover, by applying y
y (0) 2 − y (0) 1 x (0) 2 − x (0) 1 ≤ 2a |x − v n | .
By Proposition 8.5 again, we get
y (0) 2 − y (0) 1 x (0) 2 − x (0) 1 ≤ R 1 ε n where R = 2ac.
Assume that the two inequalities are true for j ≤ k. We prove the inequalities for
j + 1 ≤ k. We have (x ( j) 1 , y ( j) 1 ), (x ( j) 2 , y ( j)
2 ) ∈ C n+ j (k − j). By the mean value theorem, there exists ξ j ∈ (x ( j)
1 , x ( j) 2 ) ⊂ I AC and η j ∈ (y ( j) 1 , y ( j) 2 ) such that π x φ n+ j (x ( j) 2 , y ( j) 2 ) − φ n+ j (x ( j) 1 , y ( j) 2 ) = −λ n+ j ∂ h n ∂ x (ξ j , y ( j) 2 ) x ( j) 2 − x ( j) 1 (9.13)
and
π x φ n+ j (x ( j) 1 , y ( j) 2 ) − φ n+ j (x ( j) 1 , y ( j) 1 ) = λ n+ j ∂ ε n+ j ∂ y (x ( j) 1 , η j ) y ( j) 2 − y ( j) 1
. (9.14)
Apply Lemma 5.9 to (9.13), we have
π x φ n+ j (x ( j) 2 , y ( j) 2 ) − φ n+ j (x ( j) 1 , y ( j) 1 ) ≥ λ n+ j E x ( j) 2 − x ( j) 1 (9.15)
for some constant E > 1. Also apply the induction hypothesis to (9.14), we have
π x φ n+ j (x ( j) 1 , y ( j) 2 ) − φ n+ j (x ( j) 1 , y ( j) 1 ) ≤ λ n+ j δ ε n+ j y ( j) 2 − y ( j) 1 ≤ λ n+ j R δ ε n x ( j) 2 − x ( j) 1 . (9.16)
By the triangular inequality, (9.15), and (9.16),we get
x ( j+1) 2 − x ( j+1) 1 ≤ π x φ n+ j (x ( j) 2 , y ( j) 2 ) − φ n+ j (x ( j) 1 , y ( j) 2 ) − π x φ n+ j (x ( j) 1 , y ( j) 2 ) − φ n+ j (x ( j) 1 , y ( j) 1 ) ≤λ n+ j E − R δ ε n x ( j) 2 − x ( j) 1 (9.17)
Assume that ε is sufficiently small such that λ n+ j E − R δ ε n > λ E for all n ≥ 0 and j ≥ 0 since E > E > 1 and |λ n − λ | < ε. Apply the induction hypothesis to (9.17), we get
x ( j+1) 2 − x ( j+1) 1 ≥ λ E x ( j) 2 − x ( j) 1 ≥ 1 2a |x − v n | (λ E) j+1 l(J).
Moreover, by applying y
( j+1) 2 − y ( j+1) 1 = λ n+ j y ( j) 2 − y ( j) 1
, (9.17), and the induction hypothesis, we get
y ( j+1) 2 − y ( j+1) 1 x ( j+1) 2 − x ( j+1) 1 ≤ 1 E − 1 δ R ε n y ( j) 2 − y ( j) 1 x ( j) 2 − x ( j) 1 < R 1 ε n since E − 1 δ R ε n > 1
when ε is small enough. Therefore, the two inequalities are proved by induction.
We also need a lemma to relate the rescaling level k n with the distance from the wandering domain J n to the center. Lemma 9.6 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈Î δ (I h × I v , ε) the following property holds for all n ≥ 0:
If (x, y) ∈ B n (k) with k ≤ K n then k is bounded below by
λ k > c |x − v n | .
Proof Let ε be sufficiently small. Apply Proposition 8.5 and triangular inequality, we have
1 c 1 λ 2k < |h n (x, y) − π x τ n | ≤ | f n (x) − f n (v n )| + | f n (v n ) − π x τ n | + |h n (x, y) − f n (x)|
for some constant c > 0 since F n (x, y) ∈ C n (k). By Lemma 5.8, Lemma 5.13, and Proposition 8.5, we get
1 c 1 λ 2k ≤ a 2 (x − v n ) 2 + (c + 1) ε n ≤ a 2 + (c + 1)c (x − v n ) 2
for some constants a > 0 and c > 0. Therefore, the lemma is proved. Now we are ready to prove the main lemma of this section.
Proof (Proof of Lemma 9.4) By Lemma 9.5, we have
l(J ) ≥ x (k) 2 − x (k) 1 ≥ 1 2a |x − v n | (λ E) k l(J).
for some constant a > 0. Apply Lemma 9.6, we get
l(J ) ≥ c 2a E k l(J)
for some constant c > 0. This proves (9.7). It remains to show that J is R-regular. By Proposition 4.21, we have
ε n+k ≤ ε n+1 ≤ c ε n 2
for some constant c > 0 since k ≥ 1. Also, the Hénon-like map F n maps the x-th coordinate to y-th coordinate and Φ k n rescales the y-th coordinate affinely, we have h(J ) = y Set R = R c. Then J is R-regular. Here, we fixed R so ε is a constant.
9.3 Case J n ⊂ B r(n) (k n ), 1 ≤ k n < K In Lemma 9.4, the expansion of horizontal size only works for levels k ≥ K which are close to the center. In this section, we prove the expansion argument also holds in the intermediate region 1 ≤ k < K.
Although the rescaling trick does not apply to the non-degenerate case, we still can apply it to the limiting degenerate Hénon-like map G. In the limiting case, the horizontal size expands by the rescaling trick. Because K is a fixed number, we will show the horizontal size also expands in the intermediate region of a non-degenerate Hénon-like map when it is close enough to the limiting function G.
Observe in the limiting case, we have where [(−λ )g] j means the function x → (−λ )g(x) is composed j times.
The following is the rescaling trick, Lemma 7.4, for the limiting case.
Lemma 9.7 (Rescaling trick) Assume that j ≥ 0 is an integer. Then
[(−λ )g] j • g(x) = g((−λ ) j x) (9.19) for all − 1 λ j ≤ x ≤ 1 λ j .
Proof The lemma follows either from the functional equation 3.1 or Proposition 7.4.
By the rescaling trick, we are able to estimate the derivative for the limiting case as follows.
Lemma 9.8 There exists universal constants E, E > 1 such that for all integer j ≥ 0 we have
Eλ j ≤ d [(−λ )g] j • g dx (x) ≤ E λ j (9.20)
for all and 1 λ j+1 ≤ |x| ≤ 1 λ j .
Proof By the rescaling trick and chain rule, we get
d [(−λ )g] j • g dx (x) = (−λ ) j g ((−λ ) j x)
for all |x| ≤ 1 λ j . By Proposition 3.13, there exists E > 1 such that
g (x) ≥ E
for all 1 λ ≤ |x| ≤ 1. Also, by compactness, there exists E > 0 such that
g (x) ≤ E
for all x ∈ I. This yields (9.20) since 1
λ ≤ (−λ ) j x ≤ 1 for all 1 λ j+1 ≤ |x| ≤ 1 λ j .
Next, we need to do some hard work to make these expansion estimates also work on Hénon-like maps that are close enough to the fixed point G.
One of the difficulty is the subpartitions on B and C are not rectangular in the non-degenerate case. The following lemmas, Lemma 9.9, Lemma 9.10, and Corollary 9.11, allow us to apply the function Φ j n • F n on a rectangular neighborhood of B n ( j).
Lemma 9.9 Given δ > 0 and
I v ⊃ I h I. For all d > 0, there exists ε = ε(d) > 0 and d = d (d) such that for all F ∈Î δ (I h × I v , ε) we have s n • h n ([a − d , b + d ], y) ⊂ [(−λ g)(a) − d, (−λ g)(b) + d]
for all [a − d , b + d ] ⊂ I AC,r , y ∈ I v n , and n ≥ 0 where I AC,r is the right component of I AC defined in Lemma 5.9.
Proof By the compactness of I h , there exists E ≥ 1 such that |g (x)| ≤ E for all x ∈ I h . Then
s n • h n (a − d ) ≥(−λ )g(a − d ) − (−λ )g(a − d ) − (−λ )h n (a − d ) − (−λ )h n (a − d ) − s n • h n (a − d ) ≥(−λ )g(a) − λ Ed − λ F n − G − s n (x) − (−λ )x I h ≥(−λ )g(a) − d when d is small enough such that λ Ed < d 2 and ε is small enough such that λ F n − G + s n (x) − (−λ )x I h < d 2 for all n ≥ 0. Similarly, s n • h n (b + d ) ≤(−λ )g(b + d ) + (−λ )g(b + d ) − (−λ )h n (b + d ) + (−λ )h n (b + d ) − s n • h n (b + d ) ≤(−λ )g(b) + λ Ed + λ F n − G + s n (x) − (−λ )x I h ≤(−λ )g(b) + d.
Therefore, the lemma is proved.
Recall that q( j) = g(q c ( j)) from Definition ??. Lemma 9.10 Given δ > 0 and I v ⊃ I h I. For all integer j ≥ 1, there exists ε( j) > 0 and d C ( j) > 0 such that for all F ∈Î δ (I h × I v , ε) the rescaling Φ j n is defined on
[q( j − 1) − d C ( j), q( j) + d C ( j)] × I v n for all n ≥ 0 and [q(0) − d C (1), q(1) + d C (1)] ⊂ I AC .
In addition,
φ n ([q(k −1)−d C (k), q(k)+d C (k)]×I v n ) ⊂ [q(k −2)−d C (k −1), q(k −1)+d C (k −1)]×I v n+1 (9.21) for all F ∈Î δ (I h × I v , ε( j)), n ≥ 0, and 2 ≤ k ≤ j.
Proof We prove by induction on j ≥ 1.
For the base case j = 1, Φ 1 n = φ n is defined on I AC,r × I v n by Lemma 5.9 where I AC,r is the right component of I AC .
Assume that there exists d > 0 such that Φ j n is defined on [q( j − 1) − d, q( j) + d] × I v n for all n ≥ 0. For the case j + 1, we know that (−λ g)(q( j)) = q( j − 1) and (−λ g)(q( j + 1)) = q( j). By Lemma 9.9, there exists d > 0 and ε( j + 1) ≥ ε( j) such that
s n • h n ([q( j) − d , q( j + 1) + d ], y) ⊂ [q( j − 1) − d, q( j) + d] for all F ∈Î δ (I h × I v , ε( j + 1)), y ∈ I v n , and n ≥ 0. Then π x • φ n ([q( j) − d , q( j + 1) + d ] × I v n ) ⊂ [q( j − 1) − d, q( j) + d] × I v n+1 . Therefore, Φ j+1 n = Φ j n+1 • φ n is defined on [q( j) − d , q( j + 1) + d ] × I v
n by the induction hypothesis. The relation (9.21) follows from the definition of d( j).
Recall q l ( j) = − |q c ( j)| and q r ( j) = |q c ( j)| from Definition ??. Corollary 9.11 Given δ > 0 and I v ⊃ I h I. For all integer j ≥ 1, there exists ε( j) > 0, d B ( j) > 0, E > 1, and E > 1 such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
1. F n ([q l ( j − 1) − d B ( j), q l ( j) + d B ( j)] × I v n ) ⊂ [q( j − 1) − d C ( j), q( j) + d C ( j)] × I v n and F n ([q r ( j − 1) − d B ( j), q r ( j) + d B ( j)] × I v n ) ⊂ [q( j − 1) − d C ( j), q( j) + d C ( j)] × I v n . That is, Φ j n •F n is defined on ([q l ( j −1)−d B ( j), q l ( j)+d B ( j)]∪[q r ( j)−d B ( j), q r ( j − 1) + d B ( j)]) × I v n . 2. B l n ( j) ⊂ [q l ( j −1)−d B ( j), q l ( j)+d B ( j)]×I v n and B r n ( j) ⊂ [q r ( j −1)−d B ( j), q r ( j)+ d B ( j)] × I v n .
Here B l n ( j) and B r n ( j) are the left and right components of B n ( j) respectively.
Proof Fixed j ≥ 1. There exists d > 0 and ε > 0 small enough such that Φ j n is defined on [q( j − 1) − d , q( j) + d ] × I v n for all n ≥ 0. By the continuity of g, there exists d > 0 such that
g([q l ( j − 1) − d, q l ( j) + d]) ⊂ [q( j − 1) − d 2 , q( j) + d 2 ].
Then
h n (q l ( j − 1) − d, y) ≥ g(q l ( j − 1) − d) − h n − g I h (δ )×I v n (δ ) ≥ q( j − 1) − d for all n ≥ 0.
Here, we assume that ε is small enough such that h n − g I h (δ )×I v n (δ ) < d 2 . Similarly,
h n (q l ( j) + d, y) ≥ g(q l ( j) + d) − h n − g I h (δ )×I v n (δ ) ≥ q( j) − d . Thus, F n ([q l ( j − 1) − d, q l ( j) + d] × I v n ) ⊂ [q( j − 1) − d , q( j) + d ] × I v n . This proves that Φ j n • F n is defined on [q l ( j − 1) − d, q l ( j) + d] × I v n . Similarly, we can choose d > 0 to be small enough such that F n ([q r ( j − 1) − d, q r ( j) + d] × I v n ) ⊂ [q( j − 1) − d , q( j) + d ] × I v n .
Consequently, the first property is proved.
By Proposition 5.6, we may also assume that ε = ε( j) is small enough such that
W l n ( j − 1) ⊂ [q l ( j − 1) − d, q l ( j − 1) + d] × I v n , W l n ( j) ⊂ [q l ( j) − d, q l ( j) + d] × I v n , W r n ( j) ⊂ [q r ( j)−d, q r ( j)+d]×I v n , and W r n ( j −1) ⊂ [q r ( j −1)−d, q r ( j −1)+d]×I v n .
This proves the second property.
The next lemma will show that the expansion of a Hénon-like map is close to the limiting case when ε is small. Lemma 9.12 Given δ > 0 and I v ⊃ I h I. For allε > 0 and integer j ≥ 1, there exists ε = ε(ε, j) > 0 such that
∂ π x • Φ j n • F n ∂ x (x, y) − d [(−λ )g] j • g dx (x) <ε (9.22) for all F ∈Î δ (I h × I v , ε), n ≥ 0, and (x, y) ∈ ([q l ( j − 1) − d B ( j), q l ( j) + d B ( j)] ∪ [q r ( j) − d B ( j), q r ( j − 1) + d B ( j)]) × I v n .
Proof The Lemma is true because j is fixed and the Hénon-like maps F n are close to the fixed point G when ε is small. The proof is left as an exercise to the reader.
Corollary 9.13 Given δ > 0 and I v ⊃ I h I. For all integer j ≥ 1, there exists ε( j) > 0,d B ( j), E > 1, and E > 1 such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
Eλ j ≤ ∂ π x • Φ j n • F n ∂ x (x, y) ≤ E λ j for all x ∈ [q l ( j − 1) −d B ( j), q l ( j) +d B ( j)] ∪ [q r ( j) −d B ( j), q r ( j − 1) +d B ( j)]
and y ∈ I v n .
Proof By the continuity of g and (9.20), we may assume thatd B ( j) < d B ( j) is small enough and E > E > 1 such that
Eλ j ≤ d [(−λ )g] j • g dx (x) ≤ E λ j for all 1 λ j+1 −d B ( j) ≤ |x| ≤ 1 λ j +d B ( j).
By Lemma 9.12, we get
√ Eλ j ≤ ∂ π x • Φ j n • F n ∂ x (x, y) ≤ √ E λ j for all 1 λ j+1 −d B ( j) ≤ |x| ≤ 1 λ j +d B ( j)
, y ∈ I v n , and n ≥ 0 when ε is small enough.
By applying the estimates from the limiting case, we are able to estimate the expanding rate for the intermediate case as follows.
Lemma 9.14 Given δ > 0 and I v ⊃ I h I. For all K > 0 and R > 0, there exists ε = ε(K, R) > 0 and E > 1 such that for all F ∈Î δ (I h × I v , ε) the following properties hold for all n ≥ 0:
Assume that J ⊂ B n (k) is a connected closed R-regular set and k ≤ min K, K n . Then J ⊂ C n+k (0) = A n+k ∪W 1 n+k (0) ∪ B n+k is R-regular and
l(J ) ≥ Eλ k l(J) (9.23) where J = Φ k n • F n (J).
Proof Since K is fixed, we may assume that ε = ε(K) > 0 is sufficiently small such that the properties in Corollary 9.13 hold for all j ≤ K. Given a connected closed R-regular set J ⊂ B n (k) with k ≤ min K, K n and n ≥ 0. For convenience, let G = Φ k n • F n and denote G x = π x • G. We have J = G(J). To prove (9.23), assume the case that J ⊂ B l n (k). The other case J ⊂ B r n (k) is similar. Let (x 1 , y 1 ), (x 2 , y 2 ) ∈ J such that l(J) = x 2 − x 1 . From Corollary 9.13, G is defined on [q l (k − 1) − d(k), q l (k) + d(k)] × I v n and x 1 , x 2 ∈ [q l (k − 1) − d(k), q l (k) + d(k)]. We can apply the mean value theorem. There exists ξ ∈ (x 1 , x 2 ) and η ∈ (y 1 , y 2 ) such that
G x (x 2 , y 2 ) − G x (x 1 , y 2 ) = ∂ G x ∂ x (ξ , y 2 )(x 2 − x 1 )
and
G x (x 1 , y 2 ) − G x (x 1 , y 1 ) = ∂ G x ∂ y (x 1 , η)(y 2 − y 1 ).
By triangular inequality and J is R-regular, we get
l(J ) ≥ |G x (x 2 , y 2 ) − G x (x 1 , y 2 )| − |G x (x 1 , y 2 ) − G x (x 1 , y 1 )| (9.24) ≥ ∂ G x ∂ x (ξ , y 2 ) l(J) − ∂ G x ∂ y (x 1 , η) h(J) ≥ ∂ G x ∂ x (ξ , y 2 ) − ∂ G x ∂ y (x 1 , η) R ε n −1/4 l(J).
The first term ∂ G x ∂ x (ξ , y 2 ) can be bounded by Corollary 9.13. That is
Eλ k < ∂ G x ∂ x (ξ , y 2 ) < E λ k (9.25)
for some constants E > E > 1.
To bound the second term ∂ G x ∂ y (x 1 , η), compute by the chain rule and Corollary 9.13. We get
E λ k > ∂ G x ∂ x (x 1 , η) = ∂ π x • Φ k n ∂ x • F n (x 1 , η) ∂ h n ∂ x (x 1 , η) + ∂ π x • Φ k n ∂ y • F n (x 1 , η) ≥ ∂ π x • Φ k n ∂ x • F n (x 1 , η) ∂ h n ∂ x (x 1 , η) − ∂ π x • Φ k n ∂ y • F n (x 1 , η)
and
∂ G x ∂ y (x 1 , η) = ∂ π x • Φ k n ∂ x • F n (x 1 , η) ∂ h n ∂ y (x 1 , η) = ∂ π x • Φ k n ∂ x • F n (x 1 , η) ∂ h n ∂ x (x 1 , η) ∂ h n ∂ y (x 1 , η) ∂ h n ∂ x (x 1 , η) ≤ E λ k + ∂ π x • Φ k n ∂ y • F n (x 1 , η) f n (x 1 ) − 1 δ ε n −1 1 δ ε n .
Assume that ε = ε(K) is small enough such that ∂ π x •Φ k n ∂ y (x, y) < E for all k ≤ K and n ≥ 0. This is possible because of (9.18), the definition ofÎ , and K is a fixed bounded number. Apply Lemma 5.8, we get
∂ G x ∂ y (x 1 , η) ≤ 2E λ k 1 a |x 1 − v n | − 1 δ ε n −1 1 δ ε n .
for some constant a > 1. Also, by Proposition 8.5, we obtain
∂ G x ∂ y (x 1 , η) ≤ 2E λ k c a ε n 1/2 − 1 δ ε n −1 1 δ ε n ≤ 4E a δ c λ k ε n 1/2 (9.26)
for some constant c > 0 when ε is small enough. Combine (9.24), (9.25), and (9.26), we obtain
l(J ) ≥ Eλ k − 4E ac δ λ k ε n 1/2 R ε n −1/4 l(J) ≥ √ Eλ k l(J) (9.27)
when ε is small enough.
To prove that J is R-regular, we apply (9.27) and h(J ) = ∏ k(J)−1 j=0 λ j+n l(J). Assume that ε = ε(K) is small enough such that ∏ i−1 j=0 λ j+n ≤ 2λ i for all 1 ≤ i ≤ K and n ≥ 0. Thus,
h(J ) l(J ) ≤ 2λ k l(J) √ Eλ k l(J) = 2 √ E ≤ R ε n+k −1/4
when ε = ε(R) is small enough.
The Bad Region and the Thickness
In the good region, we showed the expansion argument holds by studying the iteration of the horizontal endpoints. However, in the bad region, the iteration of horizontal endpoints fails to estimate the change rate of the horizontal size. In fact, the x-displacement of the endpoints can shrink as small as possible by the vertical line argument in Chapter 8. Even worse, the case of entering the bad region is unavoidable. The next lemma shows that an infinitely renormalizable Hénon-like map must have a wandering domain in the bad region if it has any wandering domain. If F has a wandering domain in D, then F j also has a wandering domain J in D j by Corollary 6.4. By iterating the wandering domain, we can assume without lose
of generality that J ⊂ F j (D j ). Set J C = Φ j 0 −1 (J ). Then J C ⊂ C is a wandering domain of F. Moreover, since J ⊂ F j (D j ), we have J C ⊂ Φ j 0 −1 (F j (D j )) ⊂ F(D). Let J B = F −1 (J C )
. Then J B ⊂ B is a wandering domain of F in the bad region.
The case of entering the bad region becomes the main difficulty for proving the nonexistence of wandering domain.
In this chapter, we will first introduce a new quantity "thickness" that approximates the horizontal or vertical cross-section of a wandering domain. When a sequence element J n enters the bad region, we showed by the vertical line argument in Chapter 8 that the next sequence element J n+1 can turn so vertical that the iteration of horizontal endpoints fails to estimate the horizontal size of J n+1 . However, the horizontal size l n+1 is not zero because J n+1 has area as shown in Figure 8.1c. This is because the Hénon-like map is non-degenerated. The Jacobian is not zero. Thus, thickness (horizontal cross-section) provides an approximation for the horizontal size l n+1 of J n+1 .
Before giving a precise definition for the thickness and rigorous computation for its change rate, here we present a lax estimation on the thickness by using an area argument to explain the relations between the thickness, horizontal size, and vertical size in a closest approach.
Assume that we start from a square subset J 0 of a wandering domain. Let {J n } ∞ n=0 be the J-closest approach, a n be the area of J n , and w n be the thickness of J n . Assume that J 0 , J 1 , · · · , J n−1 stays in the good region and J n enters the bad region. Since J 0 is a square, we have l 0 = h 0 . Some assumptions are made here to simplify the argument. The contribution from the rescaling is neglected. When the wandering domain is R-regular, assume that the horizontal size is comparable to the vertical size, i.e. l n ∼ h n . Also, assume that the thickness w n is determined by the horizontal cross-section which can be approximated by w n ∼ a n h n . In the good region, the estimations in Chapter 9 determines the relation of the horizontal size. Proposition 9.2 says that l m+1 ∼ El m for all m ≤ n − 1 where E > 1 is a constant.
However, for the wandering domain J n+1 , the horizontal size l n fails to estimate l n+1 because J n enters the bad region. We need to use the thickness to approximate the horizontal size. That is, l n+1 ∼ w n+1 . The only known relation between the horizontal size and the thickness prior entering the bad region is w 0 = l 0 . This is because J 0 is a square. To relate l n+1 with l 0 , we need to go back to study the change rate of the thickness in each step.
We use the area to study the change rate of thickness. In each step, the change rate of the area is determined by the Jacobian JacF 0 ∼ ε 0 of the Hénon-like map, i.e a m+1 ∼ ε r(m) a n . We get
w m+1 ∼ a m+1 h m+1 ∼ ε r(m) a m l m ∼ ε r(m) a m h m ∼ ε r(m) w m .
This allows us to relate l n+1 with l 0 by
l n+1 ∼ w n+1 ∼ n ∏ m=0 ε r(m) w 0 ∼ n ∏ m=0 ε r(m) l 0 .
Consequently, the horizontal size becomes extremely small when the J-closest approach leaves the bad region.
One can see two problems from the estimations. One problem is the wandering domain J n+1 fail to be R-regular after the J-closest approach leaves the bad region. This is because
h n+1 l n+1 ∼ l n l n+1 ∼ E n l 0 ∏ n m=0 ε r(m) l 0 = E n n ∏ m=0 ε r(m) −1 .
Thus, the expansion argument does not work for the later sequence element even if the J-closest approach does not enter the bad region again. This problem will be resolved by introducing the largest square subset. Another problem is the strong contraction of horizontal size when the wandering domain enters the bad region. We will show that this strong contraction happens every time when the J-closest approach enters the bad region. If the the J-closest approach enters the bad region infinitely many time, then the horizontal size may fails to tend to infinity because this strong contraction happens infinity many times. This problem will be resolved in Section 10.3 by proving the closest approach J n can only enter the bad region at most finitely many times.
Finally, after combining all of the ingredients in this article together, we will show wandering domains do not exist in Section 10.4.
Thickness and largest square subset
When the wandering domain J n enters the bad region, there are two issues that stop us to proceed. First, the horizontal size of J n+1 cannot be estimated by the expansion argument in Chapter 9. Instead, it is determined by its horizontal cross-section that is not comparable to the horizontal size of J n . Second, J n+1 fail to be R-regular. The estimations for the expansion rate of the horizontal size in Proposition 9.2 does not apply to the later steps J n+1 → J n+2 → · · · in the sequence.
To resolve the two issues, we need the following:
1. A quantity to approximate the horizontal cross-section of a wandering domain, called the thickness. 2. Keep track of the thickness in each step of the closest approach. This will provides the information for the horizontal size when the wandering domain enters the bad region. 3. A method to select a subset from the wandering domain J n+1 that makes the subset to be R-regular and has approximately the same horizontal size as J n+1 . The subset will be defined to be a largest square subset of J n+1 .
In this section, we define thickness and largest square subset then study the properties of these two objects in a closest approach. First, define Proof The lemma follows from compactness.
To keep track of the thickness in each step, the following two lemmas estimate the change rate of a square under iteration and rescaling.
Lemma 10.4 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈Î δ (I h × I v , ε) the following property holds for all n ≥ 0:
If I ⊂ D n is a square, there exists a square I ⊂ F n (I) such that
l(I ) ≥ c ε n |I v n | l(I).
Proof The lemma is trivial when F is degenerate. We assume that F is non-degenerate. By the definition ofÎ and (4.2) we have ∂ ε n ∂ y > 0 for all n ≥ 0. Write I = [x, a] × [y 1 , y 2 ]. Fixed b > 0 to be sufficiently small. Let (x 1 , x) = F n (x, y 2 ), (x 2 , x) = and
I = x − 1 2 W, x + 1 2 W × [x,
x +W ]. To prove that I ⊂ F n (I) for some b > 0 sufficiently small, it suffice to prove the inequality
h n (t, y 2 ) < x − 1 2 W < x + 1 2 W < h n (t, y 1 ) (10.1)
that corresponds to the four points on a horizontal cross section at y = t for x ≤ t ≤ x + W . See Figure 10.2. If this is true, then by the mean value theorem, there exists η ∈ (y 1 , y 2 ) such that
l(I ) = W = b ∂ ε n ∂ y (x, η)l(I).
Also, by (4.2), we obtain
l(I ) ≥ bc |I v n | ε n l(I)
since F ∈Î δ (I h × I v , ε) which proves the lemma.
First, we prove the left inequality of (10.1)
h n (t, y 2 ) < x − 1 2 W.
By the mean value theorem and the compactness of the domain, there exists ξ ∈ (x,t) and E > 1 such that
h n (t, y 2 ) − x 1 = |h n (t, y 2 ) − h n (x, y 2 )| = ∂ h n ∂ x (ξ , y 2 ) |t − x| ≤ EW.
We get
x − 1 2 W − h n (t, y 2 ) = x − 1 2 W − x 1 − h n (t, y 2 ) − x 1 ≥ x 2 − x 1 2 − 1 2 W − EW = 1 2 − 1 2 + E b x 2 − x 1 > 0 when b < 1 1+2E .
Note that b can chosen to be universal. Thus, the left inequality is proved.
Similarly, we prove the right inequality of (10.1)
x + 1 2 W < h n (t, y 1 ).
By the mean value theorem, there exists ξ ∈ (x,t) such that
h n (t, y 1 ) − x 2 = |h n (t, y 1 ) − h n (x, y 1 )| = ∂ h n ∂ x (ξ , y 1 ) |t − x| ≤ EW.
Similarly, we get
h n (t, y 1 ) − x + 1 2 W = x 2 − x + 1 2 W − x 2 − h n (t, y 1 ) ≥ x 2 − x 1 2 − 1 2 W − EW = 1 2 − 1 2 + E b x 2 − x 1 > 0.
Thus, the right inequality is proved.
Lemma 10.5 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all F ∈ I δ (I h × I v , ε) the following property holds for all n ≥ 0:
If I ⊂ C n is a square, there exists a square I ⊂ φ n (I) such that l(I ) = λ n l(I). Proof
Let I = [x 1 , x 2 ] × [y 1 , y 2 ], W = l(I), x = 1 2 [h n (x 2 , y 1 ) + h n (x 1 , y 1 )], and I = [x − 1 2 W, x + 1 2 W ] × [y 1 , y 2 ]
. Then I is a square with l(I ) = l(I). First we prove that I ⊂ H n (I). It suffice to prove the inequality
h n (x 2 ,t) < x − 1 2 W < x + 1 2 W < h n (x 1 ,t)
that corresponds to the four points on a horizontal cross section at y = t for y 1 ≤ t ≤ y 2 . See Figure 10.3.
To prove the left inequality, by the mean value theorem, there exists ξ ∈ (x 1 , x 2 ) and η ∈ (y 1 ,t) such that
h n (x 1 , y 1 ) − h n (x 2 , y 1 ) = ∂ h n ∂ x (ξ , y 1 ) (x 2 − x 1 ) and ε n (x 2 ,t) − ε n (x 2 , y 1 ) = ∂ ε n ∂ y (x 2 , η)(t − y 1 ).
By Lemma 5.9, there exists E > 1 such that
x − 1 2 W − h n (x 2 ,t) = [x − h n (x 2 , y 1 )] − [ε n (x 2 , y 1 ) − ε n (x 2 ,t)] − 1 2 W ≥ 1 2 ∂ h n ∂ x (ξ , y 1 ) (x 2 − x 1 ) − ∂ ε n ∂ y (x 2 , η) (t − y 1 ) − 1 2 W ≥ E 2 − 1 δ ε n − 1 2 W > 0
when ε > 0 is sufficiently small. Thus, the left inequality is proved. Similarly, to prove the right inequality, by the mean value theorem, there exists η ∈ (y 1 ,t) such that
ε n (x 1 ,t) − ε n (x 1 , y 1 ) = ∂ ε n ∂ y (x 1 , η ) (t − y 1 ) . Compute h n (x 1 ,t) − x + 1 2 W = [h n (x 1 , y 1 ) − x] − [ε n (x 1 ,t) − ε n (x 1 , y 1 )] − 1 2 W ≥ 1 2 ∂ h n ∂ x (ξ , y 1 ) (x 2 − x 1 ) − ∂ ε n ∂ y (x 1 , η ) (t − y 1 ) − 1 2 W ≥ E 2 − 1 δ ε n − 1 2 W > 0.
Thus, the right inequality is proved. Finally, let I = Λ n (I ). Then I ⊂ φ n (I) and
l(I ) = λ n l(I ) = λ n l(I).
As before we abbreviate w n = w(J n ) for a closest approach {J n } ∞ n=0 . The next proposition allows us to estimate the contraction rate of the thickness for a closest approach. For the case that J n ⊂ B r(n) , let I be a largest square of J n . By Proposition 10.4, there exists a square I 0 ⊂ F r(n) (I) ⊂ F r(n) (J n ) such that
l(I 0 ) ≥ c ε r(n) I v r(n) l(I).
Also by Proposition 10.5, there exists a square I j+1 ⊂ φ r(n)+ j (I j ) ⊂ Φ j r(n) • F r(n) (J n ) such that l(I j+1 ) = λ r(n)+ j l(I j )
for all 0 ≤ j < k n . We get
w n+1 ≥ l(I k n ) = k n −1 ∏ j=0 λ r(n)+ j l(I 0 ) ≥ c ε r(n) I v r(n) l(I) = c ε r(n) I v r(n) w n .
Remark 10.7 The original proof was based on the area and horizontal cross-section estimates briefly mentioned in the beginning of this chapter instead of tracking the size of largest square subset. However, the area argument is discarded by two reasons. First, to estimate the horizontal cross-section of a set, we need to find the lower bound of a/l. This means that we need to repeat the arguments in Chapter 9 to find the upper bound for l and the lower bound for a. This makes the argument several times longer than the current one. Second, to select a subset from the wandering domain after it enters the bad region, the area approach makes it hard to find the upper bound of l for the subset.
Since ε n decreases super-exponentially and |I v n | increases exponentially, we can simplify Corollary 10.8 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈Î δ (I h × I v , ε) the following property holds:
Assume that J ⊂ A ∪ B is a compact subset of a wandering domain of F and {J n } ∞ n=0 is the J-closest approach. Then w n+1 ≥ c ε r(n) 3/2 w n for all n ≥ 0.
Double sequence
Next, we study the number of times that a closest approach enters the bad region by defining a double sequence (two-dimensional sequence/sequence of two indices) of sets. The double sequence consists of rows. Each row is a closest approach in the sense of Definition 6.6. When the sequence first enters the bad region in a row, the horizontal size of the next step is dominated by its thickness. Add a new row by selecting a largest square subset then generate the closest approach starting from the subset. Thus, each row in the double sequence corresponds to enter the bad region once.
J (0) 0 / / Square · · · / / Good J (0) n (0) −1 / / J (0) n (0) Φ k (0) r(n (0) ) •F (0) r(n (0) ) / / Bad J (0) n (0) +1 largest square subset . . . . . . . . . largest square subset J ( j−1) 0 / / · · · · · · / / J ( j−1) n ( j−1) −1 / / J ( j−1) n ( j−1) Φ k ( j−1) r(n ( j−1) ) •F ( j−1) r(n ( j−1) ) / / J ( j−1) n ( j−1) +1 J ( j) 0 / / J ( j) 1 / / · · · Good
The precise definition of the double sequence is as follows. Figure 10.4 illustrates the construction. r ( j) (n ( j) +1) . 6. If the procedure never stop, i.e. enters the bad region infinitely many times, set j = ∞.
The two dimensional sequence J ( j) n n≥0,0≤ j≤ j is called a double sequence generated by J or a J-double sequence. The integer j is the number of rows for the double sequence (enters the bad region j times).
To be consistent and avoid confusion, the superscript is assigned for the row and the subscript is assigned for the renormalization level or the index of sequence element in the closest approach. For example, abbreviate A
( j) n = A(F ( j) n ), B ( j) n = B(F ( j) n ), C ( j) n = C(F ( j) n ), D ( j) n = D(F ( j) n ), l ( j) n = l(J ( j) n ), h ( j) n = h(J ( j) n ), w ( j) n = w(J ( j) n ), and k ( j) n = k(J ( j) n ) as before.
In the following, we abbreviate r ( j) (n) = r(n) when the context is clear, for example F
( j) r(n ( j) +1) = F ( j) r ( j) (n ( j) +1) . Also, write ε ( j) = ε ( j) r(n ( j) ) , K ( j) = K ( j) r(n ( j) ) , and k ( j) = k ( j) n ( j) .
For convenience, let m ( j) = n ( j) + 1. In this example, we choose the same Hénon-like map as in Example 6.7. Select an initial square set J (1). In this example, ε is chosen to be so large that C r 0 (1) intersects the image F 0 (D 0 ). Thus K
n (0) +1 = Φ k (0) r(n (0) ) • F (0) r(n (0) ) (J (0) n (0) ) = φ (0) 0 • F (0) 0 (J (0) 1 ).(0)
The double sequence in this example is chosen in purpose to demonstrate the set J (0) n (0) +1 turns so vertical that the thickness dominates the horizontal size as in Figure 10.5d. Select a largest square subset J
0 = F (0) n (0) +1 .
The procedure is repeated until the sequence does not enter the bad region again.
Next, we study the relation between horizontal size and thickness in a double sequence. For each row j, the first set J ( j) 0 is a square so l
( j) 0 = w ( j)
0 . When the row stays in the good region (n < n ( j) ), the next horizontal size can be estimated by expansion argument l From the discussion, the horizontal size of any set in the double sequence can be estimated as follows.
Proof By definition and Proposition 4.21, we have
ε ( j+1) = ε ( j+1) r(n ( j+1) ) ≤ ε ( j+1) 0 = ε ( j) r(n ( j) +1) ≤ c ε ( j) r(n ( j) ) 2 k ( j) = c ε ( j) 2 k ( j)
for some constant c > 0. Apply logarithm to the both side, we get
ln ε ( j+1) ≤ 2 k ( j) ln ε ( j) + ln c ≤ 2 k ( j) −1 ln ε ( j) . (10.3)
Here we assume that ε > 0 is small enough such that
− 1 2 ln ε ( j) > ln c for all j ≥ 0. Since J ( j)
n ( j) enters the bad region, we have k ( j) > K ( j) . By Proposition 8.5 and the change base formula, we get
2 k ( j) > 2 K ( j) = λ K ( j) ln 2 ln λ ≥ c 1 ε ( j)ln ε ( j+1) ≤ c 2 1 ε ( j) 3α ln ε ( j) < 1 ε ( j) 2α ln ε ( j) .
Note that ln ε ( j) < 0. Here we also assume that ε is small enough such that
c 2 1 ε ( j) α ≥ c 2 1 ε α > 1
for all j ≥ 0. This proves the proposition.
Closest approach cannot enter the bad region infinitely many times
A strong contraction on the horizontal size occurs each time when the double sequence (or closest approach) enters the bad region as proved in Proposition 10.11. The contraction produces an obstruction to the expansion argument. This section will resolve the problem by proving the double sequence can have at most finitely many rows.
Although entering the bad region produces an obstruction to the expansion argument, it also provides a restriction to the sequence element J ( j) n ( j) : its horizontal size l ( j) n ( j) cannot exceed the size of bad region (Proposition 8.5). The Two Row Lemma, which is the final key toward the proof, studies the interaction between the obstruction and restriction between two consecutive rows as illustrated in Figure 10.6. Assume the two rows j and j + 1 both enters the bad region. On row j + 1, the sequence element J n ( j+1) on the same row, the horizontal size expand. This means that the initial sequence element J ( j+1) 0 is restricted by both the size of bad region and the amount of expansion on row j + 1.
/ / · · · E / / expansion E n ( j+1) l ( j+1) n ( j+1) < ε ( j+1) ∼ ε ( j) ε ( j) −α
On row j, the thickness determines the horizontal size of the next row j + 1. The contraction of thickness produces the contraction of horizontal size from l ( j)
0 to l ( j+1) 0 .
This cause the obstruction toward the expansion argument.
The following lemma summarize the discussion.
m ( j) > ln E −2 ln ε ( j) m ( j+1) + 1 ε ( j) α + 1 −2 ln ε ( j) ln l ( j) 0 (10.5) for all 0 ≤ j ≤ j − 2.
Proof The idea of the proof comes from Figure 10.6.
On row j + 1, J
n ( j+1) is in the bad region since j + 1 ≤ j − 1. The size of J ( j+1) n ( j+1) cannot exceed the size of bad region. Let z 1 , z 2 ∈ J ( j+1) n ( j+1) be such that |π x z 2 − π x z 1 | = l ( j+1) n ( j+1) . Apply Proposition 8.5 to bound the horizontal size. We get l ( j+1)
n ( j+1) ≤ π x z 2 − v ( j+1) n ( j+1) + π x z 1 − v ( j+1) n ( j+1) ≤ 2c ε ( j+1) n ( j+1) = 2c ε ( j+1)
for some constant c > 0. Also, the horizontal size expands on row j + 1. Proposition 10.11 yields
E n ( j+1) l ( j+1) 0 ≤ l ( j+1) n ( j+1) ≤ 2c ε ( j+1) .
Apply natural logarithm to the both sides, we get ln l ( j+1) 0 < −n ( j+1) ln E + 1 2 ln ε ( j+1) + ln 2c
= −m ( j+1) ln E + 1 2 ln ε ( j+1) + (ln E + ln 2c) .
On row j, the thickness contracts. Proposition 10.11 provides the contraction as
2m ( j) ln ε ( j) ≤ ln l ( j+1) 0 − ln l ( j) 0 < −m ( j+1) ln E + 1 2 ln ε ( j+1) + (ln E + ln 2c) − ln l ( j) 0 .
Since ln ε ( j) < 0, we solved m ( j) > ln E −2 ln ε ( j) m ( j+1) + To simplify the second term, apply Proposition 10.12. We obtain
m ( j) > ln E −2 ln ε ( j) m ( j+1) + 1 4 1 ε ( j) 2α + ln E + ln 2c 2 ln ε ( j) + 1 −2 ln ε ( j) ln l ( j) 0 = ln E −2 ln ε ( j) m ( j+1) + 1 ε ( j) α 1 4 1 ε ( j) α + ln E + ln 2c 2 ln ε ( j) ε ( j) α + 1 −2 ln ε ( j) ln l ( j) 0 > ln E −2 ln ε ( j) m ( j+1) + 1 ε ( j) α + 1 −2 ln ε ( j) ln l ( j) 0 .
Here we assume that ε is sufficiently small such that ln E + ln 2c 2 ln ε ( j) ε ( j) α > 1 for all j ≥ 0 to assimilate the constants.
If the double sequence has infinite rows, then the obstruction and restriction both happens infinitely many times. For this to happen, the obstruction must beats (or balance with) the restriction. However, it is not possible to compare the contraction of the horizontal size with the size of bad region directly because the time span in the good regions also interacts with the obstruction and restriction as (10.5) shows. So we turn to analyze the relation between the time span in the good regions versus the the number of rows in a double sequence.
If the double sequence enters the bad region twice, we apply the Two Row Lemma to row 0 and 1. The restriction from the bad region says the the horizontal size is bounded by the size of bad region ε (1) . At this moment, there are no information for the expansion on row 1. So the restriction comes only from the size of bad region.
To balance the obstruction with the restriction, the total contraction ε (0) m (0) on row 0 must have at least the same order as the size of bad region. Thus, the time span in the good regions m (0) must be large (≈ ε (0) −1 ) by Proposition 10.12 because the contraction and the size of the bad region come from the perturbation on two different rows.
If a double sequence enters the bad region three times, we apply the Two Row Lemma twice. First, we apply the lemma to row 1 and 2. The previous paragraph says that m (1) is large (≈ ε (1)
−1
). Then, apply the Two Row Lemma again to row 0 and 1. Unlike the previous paragraph, now the expansion on row 1 is determined when the Two Row Lemma is applied to row 1 and 2. So the restriction comes from both the size of bad region ε (1) and the expansion of horizontal size E m (1) ∼ E ε (1) −1 .
To balance the obstruction with the restriction, the total contraction ε (0) m (0) on row 0 must have at least the same order as E − ε (1) −1 ε (1) . This yields a larger estimate (compare to the previous paragraph) for the time span in the good regions m (0) because of the expansion.
If the double sequence enters the bad region infinite times, we start from any arbitrary row j + k + 1 then apply the Two Row Lemma recurrently to the rows j, j + 1, · · · , and j +k +1 in reverse order. It is important that the contribution of obstruction and restriction comes from the perturbation in two different rows as illustrated in Figure 10.6. With the help from Proposition 10.12, the contribution from different rows makes the time span in the good regions increases each time when the Two Row Lemma is applied. This gives the following lemma Lemma 10.14 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all non-degenerate open maps F ∈Î δ (I h × I v , ε) the following property holds:
Let J ⊂ A∪B be a square subset of a wandering domain of F and J ( j) n n≥0,0≤ j≤ j be a J-double sequence. Then the time span in the good regions n ( j) for row j is bounded below by m ( j) = n ( j) + 1 > 2 k ε ( j) α + 1 −2 ln ε ( j) ln l for all j and k with 0 ≤ j ≤ j − 2 and 0 ≤ k ≤ ( j − 2) − j where α > 0 is a universal constant.
In particular for the case j = 0 m (0) = n (0) + 1 > 2 k ε (0) α + 1 −2 ln ε (0) ln l for all 0 ≤ k ≤ j − 2.
Proof We prove (10.6) holds for all 0 ≤ j ≤ j − k − 2 by induction on k ≤ j − 2. Let ε be small enough such that Proposition 10.11, Proposition 10.12, and Lemma 10.13 hold.
For the base case k = 0. Apply (10.5), we have
m ( j) > ln E −2 ln ε ( j) m ( j+1) + 1 ε ( j) α + 1 −2 ln ε ( j) ln l ( j) 0 > 1 ε ( j) α + 1 −2 ln ε ( j) ln l ( j) 0 for all j with 0 ≤ j ≤ j − 2.
Assume that there exists k with 1 ≤ k ≤ j − 2 such that (10.6) holds for all j with 0 ≤ j ≤ j − k − 2. If k + 1 ≤ j − 2 and 0 ≤ j < j − (k + 1) − 2, then k ≤ j − 2 and 1 ≤ j + 1 ≤ j − k − 2. The induction hypothesis yields m ( j+1) > 2 k ε ( j+1) α + 1 −2 ln ε ( j+1) ln l For the first term of (10.9), we have ln 1
ε ( j) < 1 ε ( j) .
Together with (10.2), we get
ln E −2 ln ε ( j) 2 k ε ( j+1) α > 2 k ln E 2 1 ε ( j) α ε ( j) −2α −1 > 2 k+2 1 ε ( j) α .
Here, we assume that ε is small enough such that ln E 8 > ε ( j) and α ε ( j) −2α − 2 > α for all j ≥ 0. For the second term of (10.9), apply Proposition 10.11. We get ln E −2 ln ε ( j)
1 −2 ln ε ( j+1) ln l ( j+1) 0 > ln E 2 ln ε ( j+1) m ( j) + ln E −2 ln ε ( j) 1 −2 ln ε ( j+1) ln l ( j) 0 .
Combine the results to (10.9), we obtain m ( j) > 2 k+2 1
ε ( j) α + ln E 2 ln ε ( j+1) m ( j) + 1 −2 ln ε ( j) 1 + ln E −2 ln ε ( j+1) ln l ( j) 0 . Then 1 + ln E −2 ln ε ( j+1) m ( j) >2 k+2 1 ε ( j) α + 1 −2 ln ε ( j) 1 + ln E −2 ln ε ( j+1) ln l ( j) 0 .
Solve for m ( j) , we get m ( j) > 2 k+2 1 + ln E −2 ln ε ( j+1) To simplify the inequality, we assume that ε is small enough such that ln E −2 ln ε ( j+1) ≤ ln E −2 ln ε < 1 for all j ≥ 0. Therefore,
m ( j) > 2 k+1 ε ( j) α + 1 −2 ln ε ( j) ln l ( j) 0
and the lemma is proved by induction.
The lemma shows that the restriction beats the obstruction because (10.7) approaches infinity as the total number of rows in a double sequence increases. This proves Proposition 10.15 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all non-degenerate open maps F ∈Î δ (I h × I v , ε) the following property holds:
Let J ⊂ A∪B be a square subset of a wandering domain of F and J ( j) n n≥0,0≤ j≤ j be a J-double sequence. Then the number of rows j for the double sequence is finite.
Nonexistence of wandering domain
Finally, the main theorem is concluded as follows.
Theorem 10.16 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that every non-degenerate open Hénon-like map F ∈ I δ (I h × I v , ε) does not have wandering domains.
Proof Assume that ε > 0 is small enough such that Proposition 4.21 holds and F ∈ I δ (I h × I v , ε). There exists 0 < δ R < δ and I I h R ⊂ I h such that F n ∈ H δ R (I h R × I v n , ε) for all n ≥ 0.
Prove by contradiction. Assume that F has a wandering domain. Let ε > 0 be small enough such that Proposition 10.11 and Proposition 10.15 holds for δ R and I h R × I h R . By Proposition 4.21, there exists N ≥ 0 such that F N ∈Î δ R (I h R × I h R , ε ). Set Assume that F is a strongly dissipative infinitely CLM-renormalizable Hénonlike map. By the hyperbolicity of the renormalization operator [12, Theorem 4.1], there exists N ≥ 0 such that F n is sufficiently close to the fixed point G for all n ≥ N. This means that F n is renormalizable for all n ≥ N and hence F N is infinitely renormalizable in the sense of this article. Thus, we can apply the theorem to F N to conclude F does not have wandering domains.
As a consequence, the absence of wandering domains provides the information of the topology as follows.
Corollary 10.18 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for any non-degenerate open map F ∈ I δ (I h × I v , ε), the union of the stable manifolds for the period doubling periodic points is dense in the domain. s c Affine rescaling about the critical point, page 9 τ
The tip of an infinite renormalizable Hénon-like map, page 21 U Class of unimodal maps, page 7 U r Class of renormalizable unimodal maps, page 8, 9 U δ Class of unimodal maps with holomorphic extension on a δ -neighborhood, page 11 w Thickness, page 57 W t (−1) Local stable manifolds of p(−1), page 13 W t (0) Local stable manifolds of p(0), page 15 W t n ( j) Local stable manifold of p n ( j), page 20
Definition 3. 1 (
1Unimodal Map) Let I = [−1, 1]. A unimodal map in this paper is a smooth map f : I → I such that 1. the point −1 is the unique fixed point with a positive multiplier, 2. f (1) = −1, and 3. the map f has a unique maximum at c ∈ int(I) and the point c is a non-degenerate critical point, i.e. f (c) = 0 and f (c) = 0.The class of analytic unimodal maps f : I → I is denoted as U .
Definition 3. 2 (
2Critical Orbit) For a unimodal map f ∈ U , let c (0) = c (0) ( f ) ∈ I be the critical point of f . The critical orbit is denoted as c (n) = f n (c (0) ) for all n > 0.
Definition 3. 3 (
3Reflection) Assume that f ∈ U and x ∈ I. If x = c (0) , define the reflection of x to be the pointx ∈ I such that f
Fig. 3 . 1 :
31The partition {A, B,C} of a unimodal map. The parabola is the graph of a unimodal map. The points p(0), p
Definition 4 . 4 (
44Class of Hénon-like maps) Assume that I v ⊃ I h I and δ > 0. 1. Denote H δ (I h × I v ) to be the class of real analytic Hénon-like maps F : I h × I v → R 2 that have the following properties: (a) It has a representation F = ( f − ε, x) such that f ∈ U δ (I h ). (b) It has a saddle fixed point p(−1) near the point (−1, −1). The fixed point has an expanding positive multiplier. (c) The x-component h(x, y) has a holomorphic extension to
Example 4 . 8 (
48Classical Hénon maps) The classical Hénon family is a two-parameter family of the form F a,b (x, y)
Definition 4.10 (The local stable manifolds of p(−1) and the iteration domain D) Given I v ⊃ I h I, δ > 0, and F ∈ H δ (I h × I v ). Consider the stable manifold of the saddle fixed point p(−1).
Fig. 4 . 1 :
41Local stable manifolds and partition A, B, C for a Hénon-like map F. The shaded area is the image of the Hénon-like map. The vertical graphs are the local stable manifolds W 0 (−1), W 1 (0), W 0 (0), W 2 (0), and W 2 (−1) from left to right. The arrows illustrates the construction of each local stable manifold.
Figure 4 .
41 for an illustration.The domain D turns the Hénon-like map into a self-map.
Proposition 4. 11
11Given δ > 0 and intervals I h and I v with I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈ H δ (I h × I v , ε) the following properties hold:1. The sets W 0 (−1), W 2 (−1), and D exist. The two local stable manifolds are vertical graphs with Lipschitz constant c ε .2. F(D) ⊂ D. Proof The first property follows from the graph transformation. The techniques were developed in [44, Chapter 3]. See [44, Lemma 3.1, 3.2].
Definition 4 .
414 (A, B, and C) Given I v ⊃ I h I, δ > 0, and F ∈ H δ (I h ×I v ). Assume that F has a saddle fixed point p(0) with an expanding negative multiplier, the local stable manifolds in Definition 4.12 exist, and D exists.1. Define A = A(F) ⊂ I h ×I vto be the union of two sets. One is the open set bounded between W 0 (−1) and W 1 (0); the other is the open set bounded between W 2 (0) and W 2 (−1). 2. Define B = B(F) ⊂ I h × I v to be the open set bounded between W 0 (0) and W 1 (0). 3. Define C = C(F) ⊂ I h × I v to be the open set bounded between W 0 (0) and W 2 (0).
Definition 4 . 17 (
417Renormalizable) Assume that ε > 0 is sufficiently small. A Hénonlike map F ∈ H δ (I h × I v , ε) is (period-doubling) renormalizable if ithas a saddle fixed point p(0) with an expanding negative multiplier and F(B) ⊂ C. The class of renormalizable Hénon-like maps is denoted by H r δ
for some δ R > 0 and intervals I s(I v ). 3. The domain I h R ×I v R contains D(RF), and the rescaling φ maps φ (C(F)) = D(RF). 4. The fixed points satisfy the relation φ (p(0)) = p RF (−1) where p RF (−1) is the saddle fixed point of RF with an expanding positive multiplier. 5. The renormalization has a representation RF = ( f R − ε R , x) where f R ∈ U . The representation satisfies the relations
Proposition 4 . 21 (
421Hyperbolicity of the Renormalization operator) Given δ > 0 and intervals I h ,
) is a universal analytic positive function. The value δ R in the estimates can be replaced by any positive number that is smaller than δ R . Proof See [12, Theorem 3.5, 4.1, 7.9, and Lemma 7.4].
Remark 4.22The constant b is called the average Jacobian of F. See[12, Section 6].
λ 2 j for all j ≥ −1 and t = 0, 2 where z(t) n ( j) is the intersection point of W t n ( j) with the horizontal line through τ n . See Figure 5.2. Proof See [44, Lemma 3.4 and Proposition 3.5].
Fig. 5 . 1 :
51Fig. 5.1: The partition and the local stable manifolds of two renormalization levels F 0 and F 1 from the left to the right. The rescaling levels 1, 2, 3, and below 4 are shaded from light to dark as shown in the legend.
Fig. 5. 2 :
2The structure of the partition of the domain. The figure shows the partition and the local stable manifolds on the horizontal cross section that intersects the tip.
Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈Î δ
Corollary 6. 4
4Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all open maps F ∈ H r δ (I h × I v , ε), F has a wandering domain in D(F) if and only if RF has a wandering domain in D(RF). Proof Assume that J ⊂ D(F) is a wandering domain. If J ⊂ C, then RF has a wandering domain by Proposition 6.3. If J ⊂ A, there exists n ≥ 1 such that F n (J) ⊂ B by Proposition 4.16. If J ⊂ B, then F(J) ⊂ C by Proposition 4.16. Thus, RF has a wandering domain by Proposition 6.3.
Fig. 6 . 1 :
61The construction of a closest approach J n . The graphs are the domains and the partitions of F 0 and F 1 from the left to the right.
Figure 10 .
101 illustrates the horizontal size and the vertical size of a set J. For a Hénonlike map F ∈ H δ (I h × I v ), it follows from the definition that h(F(J)) = l(J)
Lemma 7. 1 (
1Fixed points and local stable manifolds) Assume that F ∈ H δ (I h × I v ) is a degenerate Hénon-like map. Then1. p h ( j) = (p u ( j), p u ( j)) for j = −1, 0, 2. the local stable manifold W 0 ( j) is the vertical line x = p u ( j) for j = −1, 0, 3. the local stable manifold W 2 (−1) is the vertical line x =p u (−1), 4. the local stable manifold W 1 (0) is the vertical line x = p(1) , and 5. the local stable manifold W 2 (0) is the vertical line x = p(2) .
the partition for F and D h 1 be the domain for RF. The rescaling mapφ (x, y) = (s h • f (x), s h (y)) maps C h 0 to D h 1 .This means the operation f in the x-component maps C u 0 to B u 0 and the affine map s h maps C u 0 back to the unit size I. Thus, the two affine maps s u and s h are the same and
Lemma 7. 3 (
3Renormalization operator) Assume that F ∈ H δ (I h × I v ) is a degenerate Hénon-like map. Then F is Hénon renormalizable if and only if f is unimodal renormalizable. When the map is renormalizable, we have 1. s h = s u and 2. RF(x, y) = (R f (x), x).
Proposition 7. 4 (
4Rescaling trick) Assume that f ∈ I . Then
(s n+ j−1 • f n+ j−1 ) • · · · • (s n • f n ) • f n = f n+ j • s n+ j−1 • · · · • s n for all integers n ≥ 0 and j ≥ 0.
Figure 8 . 2 .
82Remark 8.2 It is enough to consider the subdomain I h × I h ⊂ I h × I v n in the definition because F n (D n ) ⊂ I h × I h .
Fig. 8. 1 :
1Vertical line argument. The scales in (a) and (b) are chosen to be the same for the reader to compare the change of horizontal size.
Fig. 8 . 2 :
82Good and bad region. The shaded areas represent the rescaling levels 1, 2, and below from light to dark. In this example, one can see a tiny light area on the center bottom part of the graph. This is because C r (2) intersects the image F(D).
Corollary 8. 11
11Given δ > 0 and I v ⊃ I h I. There exists ε > 0, b > 0, and c > 1 such that for all F ∈Î δ (I h × I v , ε) and b >
Lemma 9. 3
3Given δ > 0 and I v ⊃ I h I. For all R > 0, there exists ε = ε(R) > 0 and E > 1 such that for all F ∈Î δ (I h × I v , ε) the following property hold for all n ≥ 0:Assume that J ⊂ A n is an R-regular closed set. Then J ⊂ C n (0) = A n ∪W 1 n (0)∪B n is R-regular and l(J ) ≥ El(J)
1
= l(J) and the previous inequality, we have
F n (x, y) = ([(−λ )g] j • g(x), (−λ ) j x)
Lemma 10. 1 *
1Given δ > 0 and I v ⊃ I h I. There exists ε > 0 such that for all nondegenerate Hénon-like maps F ∈Î δ (I h × I v , ε) the following property holds. If F has a wandering domain in D then F has a wandering domain in the bad region of B and a wandering domain in the bad region of C.Proof Recall K 0 is the boundary of the good and bad region for F 0 = F. See Definition 8.1. Let j > K 0 .
Definition 10. 2 (
2Square, Largest square subset, and Thickness) A set I ⊂ R 2 is a square if I = [x 1 , x 2 ] × [y 1 , y 2 ] with x 2 − x 1 = y 2 − y 1 . This means that I is a closed square with horizontal and vertical sides. Assume that J ⊂ R 2 . Define the thickness of J to be the quantity w(J) = sup {l(I)} where the supremum is taken over all square subsets I ⊂ J. A subset I ⊂ J is a largest square subset of J if I is a square such that l(I) = w(J). The definition is illustrated as in Figure 10.1. Lemma 10.3 A largest square subset of a compact set exists.
Fig. 10 . 1 :
101Comparison of the horizontal size l, the vertical size h, and the thickness w for J. In this picture, I is a largest square subset of J.
Fig. 10 . 2 :
102Four points on the cross section y = t.F n (x, y 1 ), and W = b(x 2 − x 1 ) = b [ε n (x, y 2 ) − ε n (x, y 1 )] > 0. Define x = x 1 +x 2 2
Fig. 10 . 3 :
103Four points on the cross section y = t.
Proposition 10. 6 Proof
6Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and c > 0 such that for all F ∈Î δ (I h × I v , ε) the following property holds:Assume that J ⊂ A ∪ B is a compact subset of a wandering domain of F and {J n } ∞ n=0 is the J-closest approach. Let ε > 0 be small enough such that Lemma 10.4 and Lemma 10.5 holds. The sets {J n } ∞ n=0 are compact by the continuity of Hénon-like maps and rescaling. For the case that J n ⊂ A r(n) , let I be a largest square of J n . By Proposition 10.4, there exists a square I ⊂ F r(n) (I) ⊂ J n+1 such that l(I ) ≥ c ε w n+1 ≥ l(I ) ≥ c ε r(n)
Fig. 10 . 4 :
104Construction of a double sequence.
Definition 10 . 9 (
109Double sequence, Row, and Time span in the good regions) Given δ > 0 and I v ⊃ I h I. Assume that ε > 0 be sufficiently small so that Proposition 8.5 holds and F ∈Î δ (I h × I v , ε) is a non-degenerate open map. Given a square subset J ⊂ A∪B of a wandering domain for F. Define J j ∈ N ∪ {0, ∞} 3 by induction on j such that the following properties hold.1. For j = 0, set J The super-script j is called row. The initial set J For a row j, if there exists some n ≥ 0 such that k
n) , set n ( j) to be the smallest integer with this property. The set J
j) is the first set in row j that enters the bad regions. The nonnegative integer n ( j) is called the time span in the good regions for row j. Otherwise, if the row never enters the bad region, set n ( j) = ∞ and j = j and the construction stops.
5 .
5If n ( j) < ∞, construct a new row j + 1 by defining J
Example 10.10 Figure 10.5 gives an example of constructing a double sequence.
the bad region. Set n (0) = 1. By the construction, J
9.2). When the row first enters the bad region n = n ( j) , the expansion argument fails. The vertical line argument in Chapter 8 shows that the only way to estimate the horizontal size l
j) +1 is to use the thickness w
j) +1 . That is, l
j) +1 ≥ w
j) +1 . Proposition 10.6 provides the relation between l
j) +1 and l
using the thickness. Finally, the horizontal size l ( j+1) 0 and thickness w ( j+1) 0 of the first set J ( j+1) 0 in the next row j + 1 is obtained by the thickness w
Fig. 10. 5 :
5Construction of a double sequence. The left and right are the graphs for F . The arrows indicate the iteration and rescaling in the construction of the double sequence. The sub-figures (a), (b), (c), and (d) are the zoomed double sequence elements. The scale of (a), (b), (c), and (d) are chosen to be the same for the reader to compare the change of the horizontal size.
constant c > 0. Let α = ln 2 6 ln λ > 0. Combine (10.3) and (10.4), we obtain
Fig. 10 . 6 :
106Relations of horizontal size and thickness in two rows j and j + 1.
j+1) enters the bad region. The size of bad region provides the restriction to the horizontal size l
Lemma 10 . 13 (
1013Two Row Lemma) Given δ > 0 and I v ⊃ I h I. There exists ε > 0, E > 1, α > 0 (universal) such that for all non-degenerate open maps F ∈Î δ (I h × I v , ε) the following property holds:Let J ⊂ A∪B be a square subset of a wandering domain of F and J ( j) n n≥0,0≤ j≤ j be a J-double sequence. Then the time span in the good regions n ( j) = m ( j) − 1 for row j is bounded below by
ln ε ( j) .
6.4, F N has a wandering domain J in D(F N ) ⊂ I h (F N ) × I v N . If J ⊂ B(F N ), then J ⊂ I h R × I v N and so F 2 (J) ⊂ B(F N ) ∩ I h R × I h R . If J ⊂ A(F N ), there exists n > 0 such that F n (J) ⊂ B(F N ) by Proposition 4.16. If J ⊂ C(F N ), then F(J) ⊂ B(F N ). Without lose of generality, we may assume that J ⊂ B(F N ) ∩ I h R × I h R . Hence, J ⊂ B(F) is a wandering domain of the restrictionF.LetĴ be a nonempty square subset of J and J( j) n n≥0,0≤ j≤ j be aĴ-double sequence. By Proposition 10.15, j is finite. Then the second property of Proposition 10.a contraction. Therefore, F does not have wandering domains. Remark 10.17 The result for Theorem 10.16 also applies to infinitely CLM-renormalizable maps if all levels of renormalization are defined on a sufficiently large domain. This is because of the hyperbolicity of the Hénon renormalization operator.
. Their domain is equivalent to the dynamical interval [ f 2 (c), f (c)] for unimodal maps which does not include the fixed point with positive multiplier. The larger domain is necessary in this article to study the rescaled orbit of a point. See Proposition 4.11, Proposition 4.16, and Proposition 5.3. Their work also holds on the larger domain I h × I v . See for examples [12, Footnote 7, Section 3.4] and [44, Lemma 3.3, Proposition 3.5, Theorem 4.1]. However, reproving their theorem on the larger domain is not the aim here. This article will assume the results from
n=1 is uniformly bounded, then the non-autonomous system { f n } ∞ n=1 has exactly one orbit {u n } ∞ n=1 . This Lemma is an analog of the Anosov's Shadowing Theorem. See [10, Exercise 5.1.3, Corollary 5.3.2] for the version of autonomous systems. The proof is left to the reader.
Feigenbaum-Cvitanović functional equation, 9 good region, 5, 32, 33, 35, 37-41, 42 Hénon-like map, 1, 12 degenerate, 13, 29 horizontal endpoints, 29 horizontal size, 5, 29, 42, 66, 68 Lipschitz continuous, 6 local stable manifold, 13, 15, 13-16, 20, 21, 22 degenerate, 30 thickness, 4, 5, 55, 57, 57, 61 time span in the good regions, 5, 63, 66, 68, 70 tip, 5, 21, 21, 24, 26, 28 two row lemma, 68 unimodal case, see degenerate Hénon-like map unimodal map, 1, 4, 7, 11 unimodal maps, 29 vertical graph, 13 vertical line argument, 33 vertical size, 29 wandering domain, 26, 27 double sequence, see double sequence nonexistence, 73 sequence, see closest approach wandering interval, 2, 4, 32, 32Index
bad region, 4, 5, 32, 33, 35, 40-41, 55
Cantor set, 2, 21
closest approach, 5, 28
step, 28
critical point, 7, 26
orbit, 7
orbit of renormalization fixed point, 10
critical value, 24, 26
double sequence, 64
number of rows, 64, 72
expansion argument, 41, 42
degenerate, 32
infinitely renormalizable, 17
Jacobian, 6
length, 6
partition
degenerate, 30
Hénon-like map, see also local stable man-
ifold, 14, 15, 15, 21
unimodal map, 8
projection, 6
regular, 41
renormalizable
Hénon-like, 16
unimodal map, 8
renormalization
degenerate, 31
fixed point, 9
renormalization level, 17
renormalization operator
Hénon-like map, 17
hyperbolicity, 18
unimodal map, 9
rescaling level, 20, 20
Hénon-like, 27
rescaling trick, 31, 49
row, 63, 66, 68, 70, 72
Schwarzian derivative, 7
negative, 7
self-map, 13, 14
shadowing theorem, 25
square, 57, 57, 59
largest square subset, 57, 64
sup norm, 6
This article assumes ε is small. This implies that ∂ ε ∂ y is also small in the analytic setting.
The study of complex Hénon maps covers a broader class of functions motivated by the classification of polynomial automorphisms[21]. It allows the map f in (1.1) to be any polynomial or analytic map.
For the case j = ∞, this means that the sequence is defined for all finite positive integers j.
Proposition 10.11 Given δ > 0 and I v ⊃ I h I. There exists ε > 0 and E > 1 such that for all non-degenerate open maps F ∈Î δ (I h × I v , ε) the following property holds:Let J ⊂ A∪B be a square subset of a wandering domain of F and J ( j) n n≥0,0≤ j≤ j be a J-double sequence. Thenn for all n < n ( j) and all 0 ≤ j ≤ j.Proof Let ε > 0 be small enough such that Proposition 9.2 and Corollary 10.8 hold.With the help Corollary 10.8, we are able to compare l ( j+1) 0 with l ( j) 0 by using the thickness. That iswhere c > 0 is a constant. Apply nature logarithm to both sides, we getHere we assume that ε is small enough so that 2 3 ln c ≥ 1 3 ln ε ( j) for all 0 ≤ j ≤ j − 1 to assimilate the constants.The second inequality follows directly from Proposition 9.2, the definition of n ( j) , and a square is R-regular when ε is small enough.The next proposition provides the relation of the perturbation ε between two rows.for all 0 ≤ j ≤ j − 1.
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| {'fraction_non_alphanumeric': 0.08777841550795208, 'fraction_numerical': 0.034918812834422615, 'mean_word_length': 3.022791032547978, 'pattern_counts': {'":': 0, '<': 78, '<?xml version=': 0, '>': 332, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 263, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This article extends the theorem of the absence of wandering domains from unimodal maps to infinitely period-doubling renormalizable Hénon-like maps in the strongly dissipative (area contracting) regime. The theorem solves an open problem proposed by several authors[64,44], and covers a class of maps in the nonhyperbolic higher dimensional setting. The classical proof for unimodal maps breaks down in the Hénon settings, and two techniques, "the area argument" and "the good region and the bad region", are introduced to resolve the main difficulty.The theorem also helps to understand the topological structure of the heteroclinic web for such kind of maps: the union of the stable manifolds for all periodic points is dense.Mathematics Subject Classification (2010) 37E30 · 37C70 · 37E20 · 37D45', 'arxivid': '1705.05036', 'author': [], 'authoraffiliation': [], 'corpusid': 119167027, 'doi': '10.1007/s00222-019-00902-4', 'github_urls': [], 'n_tokens_mistral': 59837, 'n_tokens_neox': 52180, 'n_words': 33765, 'pdfsha': '28f726f5447f1d8633161c543cb78b5dfa968da3', 'pdfurls': ['https://arxiv.org/pdf/1705.05036v6.pdf'], 'title': ['Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps Dyi-Shing Ou', 'Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps Dyi-Shing Ou'], 'venue': []} |
arxiv |
XTab: Cross-table Pretraining for Tabular Transformers
Bingzhao Zhu
Xingjian Shi
Nick Erickson
Mu Li
George Karypis
Mahsa Shoaran
XTab: Cross-table Pretraining for Tabular Transformers
The success of self-supervised learning in computer vision and natural language processing has motivated pretraining methods on tabular data. However, most existing tabular self-supervised learning models fail to leverage information across multiple data tables and cannot generalize to new tables. In this work, we introduce XTab, a framework for cross-table pretraining of tabular transformers on datasets from various domains. We address the challenge of inconsistent column types and quantities among tables by utilizing independent featurizers and using federated learning to pretrain the shared component. Tested on 84 tabular prediction tasks from the OpenML-AutoML Benchmark (AMLB), we show that (1) XTab consistently boosts the generalizability, learning speed, and performance of multiple tabular transformers, (2) by pretraining FT-Transformer via XTab, we achieve superior performance than other state-of-the-art tabular deep learning models on various tasks such as regression, binary, and multiclass classification.
Introduction
With the increasing number of datasets represented as tables with rows and columns, tabular machine learning makes the foundation of many real-world applications. While deep learning has achieved tremendous success in the fields of computer vision (CV) (He et al., 2022;Liu et al., 2021) and natural language processing (NLP) (Devlin et al., 2018;Vaswani et al., 2017), tabular deep learning models are not used as commonly as tree-based models (Grinsztajn et al., 2022;Gijsbers et al., 2022). The primary challenge of tabular deep learning is the diversity of tabular tasks. Unlike text, which can be standardized as a sequence of tokens, tables are highly data-specific. Tabular data can vary in the number and types of columns. This makes it difficult for tabular deep learning models to transfer the knowledge learned from one table to another, leading to poor generalization abilities. Therefore, self-supervised learning for tabular data (He et al., 2022;Devlin et al., 2018), particularly one that is able to bootstrap the learning on new tables, is still an open problem.
There is an ongoing effort in migrating self-supervised pretraining techniques from CV (Chen et al., 2020) and NLP (Devlin et al., 2018) to tabular tasks. With selfsupervised pretraining, tabular deep models have demonstrated improved performance (Ucar et al., 2021;Bahri et al., 2021;Majmundar et al., 2022). However, existing methods generally pretrain the tabular model on data from the same domain as the downstream task. As a result, the data-specific models cannot generalize to new tables.
Another direction of deep tabular learning aims to leverage Transformers, which drives the recent progress in NLP (Vaswani et al., 2017) and CV (Dosovitskiy et al., 2020) for tabular tasks. Inspired by the success of the attention mechanism, Transformers were adapted to tabular data (Gorishniy et al., 2021;Somepalli et al., 2021;Wu et al., 2021;Wang & Sun, 2022) and demonstrated strong performance (Grinsztajn et al., 2022). The core idea of tabular transformers is to consider the table columns as tokens, similar to words in a sentence. Therefore, tabular transformers can process tables with variable numbers of columns, thus making transferable learning (Wang & Sun, 2022) feasible.
In this paper, we present XTab, a general framework for cross-table pretraining of tabular transformers. To resolve the issue that tables may vary in the number and types of columns, XTab decomposed the tabular transformers to two components: data-specific featurization and projection layers that capture the characteristics of each table, and a cross-table-shared block that stores the common knowledge. On a diverse collection of data tables, XTab trains these dataspecific blocks and the shared block jointly via federated learning (Collins et al., 2022). Once pretrained, XTab can bootstrap the learning process on a new table by initializing the shared block with pretrained weights. To verify our design, we conducted extensive experiments on AutoML Benchmark (AMLB) (Gijsbers et al., 2022). Our results show that transformers pretrained and initialized with XTab consistently outperform transformers with random initialization. By pretraining FT-Transformer (Gorishniy et al., 2021) with XTab, we outperform the state-of-the-art tabular deep learning models.
The contributions of the paper are summarized as follows:
• XTab offers a framework to account for cross -table variations and enable cross-table knowledge transfer. • Given the large diversity of tabular datasets, we propose to pretrain on tabular datasets with federated learning. This allows us to perform distributed pretraining across a large collection of tables. • To the best of our knowledge, we are the first to show that cross-table pretraining can boost the learning speed and performance on new tables. This is different from table understanding tasks (Yin et al., 2020), the focus of which is to extract the semantical information from tables.
Related work
Tabular self-supervised learning. Inspired by the success of pretraining in CV and NLP, previous papers studied tabular self-supervised learning (Yoon et al., 2020;Ucar et al., 2021;Somepalli et al., 2021;Bahri et al., 2021;Majmundar et al., 2022;Rubachev et al., 2022;Wang & Sun, 2022). Among those works, Yoon et al. (2020); Ucar et al. (2021) proposed an auto-encoder framework with a pretext task to reconstruct the missing part of a (2022) further incorporated the label columns of tabular tasks in pretraining and proposed "target-aware" objectives leading to higher performance. As existing approaches only pretrain on one (Bahri et al., 2021;Ucar et al., 2021) or a few relevant tables (Wang & Sun, 2022), the pretrained tabular model lacks generalizability. XTab alleviates this issue by pretraining on a large number of tables. Tabular transformers. Transformer models are gaining popularity in the realm of deep learning for tabular data. For example, FT-Transformer has demonstrated superior performance on tabular classification/regression tasks (Gorishniy et al., 2021). Saint introduces the row-wise attention and captures the inter-sample interactions using transformer (Somepalli et al., 2021). Fastformer proposes to use additive attention on tabular tasks, which is a lightweight attention mechanism with linear complexity to the length of input sequences (Wu et al., 2021). TransTab features transfer learning in tabular tasks using transformers (Wang & Sun, 2022) and also supports the cross-table transfer. Our approach is different from TransTab in that TransTab has limited ability in generalizing to tables from new domains, while XTab is able to generalize to new domains.
Cross-table transfer learning. Pretrained vision and text models can be adapted to a wide range of tasks (Bommasani et al., 2021). One reason is that the sentences and images share general representations across various tasks. As for tabular learning, one may question if there is shared knowledge across tables as two different tables can have totally different numbers of columns and the associated semantic meanings. We argue that different tables share a similar prior given the recent success of zero-shot hyperparameter optimization (HPO) in AutoML (Winkelmolen et al., 2020), which learns a general hyperparameter configuration applicable to a wide range of tabular tasks. Unlike pretrained models in NLP (Devlin et al., 2018), XTab does not attempt to learn a universal tokenizer for all tables, as the meaning and context of each
Methods
Previous works have proposed various pretraining methods for tabular learning (Bahri et al., 2021;Ucar et al., 2021;Rubachev et al., 2022;Somepalli et al., 2021). However, existing pretrained models are still domain-specific since they were pretrained on the training set of each individual tabular prediction task. As a result, existing pretrained models lack generalizability and fail to cover downstream tasks on other types of tables. Here, we propose XTab to pretrain transformer models using the information from multiple tables. With cross-table pretraining, XTab aims to learn the shareable knowledge that can boost the performance for various downstream regression and classification tasks.
Model structure
The model structure of XTab is described in Figure 1. During the pretraining phase, we sample mini-batches of rows from different tables (one batch per since they are specific to each dataset and the pretraining objectives. Among all pretraining losses, reconstruction loss and contrastive loss do not require information from the label column, whereas supervised losses use the groundtruth data in the label columns of each table. Using groundtruth information during the pretraining phase is referred to as "target-aware pretraining" (Rubachev et al., 2022;Wang & Sun, 2022) or "pre-finetuning" (Aghajanyan et al., 2021) in previous works.
A key challenge in cross-table pretraining lies in the variations of input tables. Previous works on transferable tabular learning either require tables to come from similar domains (Levin et al., 2022) or use additional information (e.g., column names) to identify the shared knowledge across tables. XTab is designed to be applicable to previously unseen tables with no assumption on the domain or column name format. To this end, XTab contains model blocks that carry the data-specific information (green blocks in Figure 1), as well as the shared backbone that stores the common knowledge (grey blocks in Figure 1). Once pretrained, only a shared backbone is kept for all downstream tasks. For each downstream task, featurizers and projection heads are randomly initialized and the entire model is finetuned on the downstream training data until a stopping criterion is met.
PROJECTION HEADS AND OBJECTIVES
There exist various pretraining objectives for tabular prediction tasks (Rubachev et al., 2022;Majmundar et al., 2022;Bahri et al., 2021;Ucar et al., 2021;Wang & Sun, 2022;Yoon et al., 2020). Among them, table reconstruction and contrastive learning are the most popular and effective objectives for tabular tasks. In addition to the self-supervised pretraining objectives, we also tested the pre-finetuning setting using supervised loss.
Reconstruction loss: Reconstruction loss is a selfsupervised training objective shown to be effective on various tabular tasks (Rubachev et al., 2022;Majmundar et al., 2022). The reconstruction objective aims to recover the original sample x from a corrupted view of the samplex. The reconstruction projection head takes the representation ofx as input, and generates an estimate of the original input x. The reconstruction loss is calculated by comparing x andx. Specifically, we use Cross-Entropy loss to measure the reconstruction error of categorical columns and Mean Squared Error (MSE) for numerical columns.
Contrastive loss: Similar to the reconstruction objective, we also generatex as a corrupted sample. x and its corresponding corruptionx are considered as a positive pair of samples, whereas x and other samples in the batch form negative sample pairs. In general, contrastive loss aims to minimize the distance between positive pairs of samples and maximize the distance for negative pairs. Following Supervised loss: In addition to reconstruction and contrastive losses that do not require labels in pretraining, one can directly pretrain a model using the supervised objective. With supervised losses, the projection head aims to predict the values under a certain field (or column), as predefined by each dataset. The supervised prediction tasks included regression and classification.
In XTab, the projection heads are data-specific. Different pretraining datasets do not need to share common objectives. For example, we can simultaneously pretrain XTab on both regression and classification tasks, or a mixture of reconstruction and contrastive losses. The diversity of pretraining objectives ensures that the shared backbone is widely adaptable to various downstream tables.
Federated pretraining
XTab introduces data-specific featurizers and projection heads (green blocks in Figure 1) to account for the variations across table columns and pretraining objectives. During pretraining, both the time and space complexity increase linearly as we include more tabular datasets. As a result, it is challenging to quickly pretrain XTab using a single machine on a large collection of tabular tasks. To alleviate this issue, we fit XTab into the federated learning framework (McMahan et al., 2017). With the federated setting, XTab involves only marginal overhead in wall-clock time with more pretraining tasks. Federated learning makes it feasible to pretrain XTab on a cluster of commercially available GPUs (NVIDIA T4 GPUs, 16GB memory).
We use the Federated Averaging (FedAvg) algorithm to pretrain XTab (McMahan et al., 2017;Li et al., 2019). We have a central server and multiple clients. Each client only hosts one dataset. Therefore, we can distribute the data-specific components of XTab across clients such that each client stores one featurizer, one projection head, and the shared transformer. During pretraining, each client calculates the gradient using the local dataset:
w k,i+1 ← w k,i − α∇ k ,(1)
where k denotes the client (or table) index and i shows the current iteration. α is the learning rate and (k) is the loss function. w represents the trainable parameters which contains two components: w (S) for the shareable modules across all pretraining tasks, and w (NS) for the non-shareable parts (w = stack[w (NS) , w (S) ]). All clients operate synchronously during pretraining with the same learning rate and batch size.
The central server is responsible for aggregating the local gradients from clients. FedAvg allows clients to make multiple local updates before an aggregation step is made on the central server. Let N denote the number of local updates per aggregation. The central server performs:
w (S) i+N ← w (S) i + K k=1 (w (S) k,i+N − w (S) i ).(2)
The aggregation is only performed on the shared weights. The term w
(S) k,i+N − w (S) i
is the gradient learned by client k since the last weight aggregation. The central server simply accumulates the gradients from all clients. Such unitary scalarization was recently shown to perform well in multitask learning (Kurin et al., 2022). i+N ]. Therefore, we force all clients to train on a shared backbone with data-specific featurizers and projection heads.
The number of local steps N is a key parameter to control communication efficiency. With N = 1, FedAvg corresponds to the distributed version of stochastic gradient descent (SGD). With N > 1, multiple local updates are performed between model aggregation steps at the server, thereby reducing the communication cost between the central server and clients. Unless otherwise specified, we choose N = 5 throughout the paper. The ablation study on N is shown in Figure 9 of the Appendix.
Federated learning was originally proposed as a privacypreserving approach to learning from distributed data. The collaboration of multiple clients to train a single shared model makes a good fit with our goal of cross-table pretraining. In this work, XTab leverages the distributed nature of federated learning to scale with a large number of pretraining tasks.
Experiments
We evaluate the performance of XTab on supervised tabular learning tasks, including binary and multiclass classification and regression. We tested on the following pretraining settings:
• XTab with various pretraining objectives, including reconstruction loss, contrastive loss, and supervised loss. • XTab with various transformer backbones, including FT-Transformer, Fastformer, and Saint-v. • XTab with the transformer backbone partially-or fullypretrained from other tasks. • XTab with different numbers of pretraining tasks.
During finetuning, we randomly initialize a new featurizer and projection head for each downstream task. All downstream tasks use the pretrained transformer backbone. We finetune all the model components using the training set of each downstream task. We included two different finetuning settings:
• Light finetuning: finetune XTab for a fixed number of epochs (3 epochs). • Heavy finetuning: finetune XTab with an early stopping patience of 3 epochs. The maximum number of epochs is set to infinity in this case.
For all finetuning settings, we retrieve the best model checkpoint based on validation scores, and use it to report the performance on the test data. The baseline models share the same model architecture and finetuning configurations as XTab, but with randomly initialized parameters instead of using the pretrained backbones. We find that XTab generally outperforms the baseline models in all scenarios and beats other deep learning models on tabular tasks. Ablation study on the number of pretraining datasets is in Appendix D.
Datasets
We use the public OpenML-AutoML Benchmark (AMLB: openml.github.io/automlbenchmark/) (Gijsbers et al., 2022) for pretraining and evaluation. AMLB is a recently proposed benchmark for automated machine learning, consisting of 104 tabular tasks (71 classification and 33 regression). We included the details of each dataset in Table 13 in the Appendix. Out of the 104 tabular datasets, we used 52 datasets for pretraining and the remaining 52 tasks for finetuning and evaluation. We split the pretraining and finetuning datasets by the alphabetical order of the task names (Table 13 in the Appendix).
Data split: For all downstream (or finetuning) tasks, AMLB reserves 10% of the tabular data for testing. Over the remaining data, we randomly partition 87.5% (7/8) into the training set and use 12.5% (1/8) for validation. We repeated 5 trials with different test folds for all tabular datasets. All methods use the same split within the same trial.
Data pre-processing: Following Bahri et al. (2021); Somepalli et al. (2021); Wang & Sun (2022), we limit the discussion to tables with numerical and categorical columns. Each Category is represented by a distinct integer to index the embedding in the lookup table of the categorical featurizer (see Section 3.1.1 for details). We normalized the numerical features by subtracting the mean and dividing them by the standard deviation. For regression tasks, we also apply the Standardization to the labels. The normalization parameters are calculated using the training set only to avoid information leakage. Missing entries are filled with the mean values of numerical columns, or treated as an additional category for categorical columns.
Experimental setup
We used a federated pretraining setting as detailed in Section 3.2. Both pretraining and finetuning were performed on a cloud cluster of NVIDIA T4 GPUs (16 GB memory). We used about 30 thousand GPU hours for all experiments.
Model configuration and training: Our default model configuration of transformer variants is the same as Gorishniy et al. (2021), with 3 transformer blocks, a feature embedding size of 192 and 8 attention heads. The feed forward networks ( Figure 1) have two layers with the same size as the embedding. We apply a dropout ratio of 20% to attention layers and 10% for feed forward networks. We Evaluation metrics: We choose the evaluation metrics as suggested by AMLB (Gijsbers et al., 2022). We use root mean-squared error (RMSE) for regression tasks, area under the receiver operating characteristic curve (AUC) for binary classification, and log loss for multi-class classification. The same evaluation metrics are applied to validation sets for early stopping. The efficacy of the pretrained transformer backbones is estimated by the downstream performance.
Comparison with baseline transformers
Cross-table pretraining improves downstream task performance. As shown in Figure 2, we compare the downstream prediction performance of FT-Transformer before (baseline) and after cross-table pretraining. Reconstruction objective is used for pretraining and all downstream tasks are finetuned for 3 epochs (light finetuning). We checkpoint the pretrained backbone after a certain number of pretraining heavy finetuning settings. We also compared the performance of pretraining objectives in terms of the model rank with (c) light and (d) heavy finetuning. We observe a consistent improvement of XTab compared to baseline models with all objectives. The reconstruction pretraining objective achieves the best performance, with 71.0% win rate under light finetuning and 56.1% for heavy finetuning at 2000 pretraining steps.
steps and finetune downstream tasks from various checkpoints (250/500/1000/1500/2000). In Figure 2(a), we show the win rate of the pretrained transformer on all downstream tasks with respect to baseline. Both classification and regression tasks benefit from our proposed cross-table pretraining. As the backbone is pretrained for more steps, we observe an increase in the win rate. We also calculate the rank of the model for each downstream task ( Figure 2(b)). Model rank is an integer from 1 to 6, with a lower number indicating better performance. Equal values are assigned a rank that is the average of the ranks of those values. The rank of the model improves with XTab pretraining. To further validate the advantage of XTab over transformers without cross-table pretraining, we further look into the normalized prediction performance and error reduction rate (Figure 2(c, d)). We min-max normalize the prediction performance of all models, such that the worst model receives a score of 0 and the best model receives 1. Similarly, errors are also normalized to the best and worst models. Negative numbers indicate a model with lower error (1 − AUC scores for binary classification) or loss (log loss for multiclass classification and RMSE for regression) than baseline. The mean error (or loss) is indicated by the stars. FT-Transformers pretrained with XTab on average obtain higher normalized performance and reduced error compared to traditional random initialization.
XTab with different pretraining objectives and finetun- ing settings. We extensively test XTab with various pretraining objectives and finetuning settings. Figure 3 summarizes the downstream performance using reconstruction, contrastive and supervised objectives as described in Section 3.1.3. We use FT-Transformer as the backbone. Figure 3(a, b) plot the win rate of XTab under the light and heavy finetuning settings, respectively. We finetune on all downstream tasks for 3 epochs with light finetuning, and use an early stopping patience of 3 for heavy finetuning. We observe a consistent improvement of XTab over the baseline with no cross-table pretraining. The advantage of XTab is more significant in the light finetuning setting compared to heavy finetuning. For example, XTab with the reconstruction objective achieves a 71.0% win rate with light finetuning, but only 56.1% with heavy finetuning. The difference is caused by catastrophic forgetting of deep models (Ramasesh et al., 2021;Kaushik et al., 2021). As tabular transformers are relatively small (<1M parameters for the FT-Transformer backbone), they are more vulnerable to catastrophic forgetting during the finetuning phase. It is possible to alleviate this issue with additional techniques (Ramasesh et al., 2021;Kaushik et al., 2021), but this is outside the scope of the paper. Figure 3(c, d) compare different objectives by ranking the models with light and heavy finetuning. All approaches are pretrained for 2000 steps. Each dot in Figure 3(c, d) represents a trial of downstream experiments (5 trials per dataset) and error bars indicate the standard deviations across trials. The advantage of crosstable pretraining is shown by a win rate >50% and a model rank value lower than the baseline. A more detailed comparison involving the normalized performance and error reduction rate is presented in Appendix A. We conclude that XTab consistently enhances the downstream performance of tabular transformers across multiple pretraining objectives and finetuning settings. Among all pretraining objectives tested, reconstruction loss performs better than contrastive or supervised losses.
XTab is applicable to various types of transformers.
XTab offers a framework to pretrain the shared model components across tabular tasks. Therefore, the choice of transformer backbone is flexible, as long as the model can process tables with variable columns. In Figure 4, we plug three transformer variants into XTab including FT-Transformer, Fastformer, and Saint-v. The explanation of transformer backbones can be found in Section 3.1.2. We pretrain all transformers using reconstruction objective, and finetune on the downstream tasks with the light and heavy settings, Figure 4(a, b). We show that XTab is applicable to various types of transformers and all models benefit from the proposed cross-table pretraining, achieving a higher win rate compared to the baseline.
Additional experimental results are presented in the Appendix. In Appendix B, we pretrain on different components of transformers to identify the shareable components in XTab. In Appendix C, we look into the downstream performance with only a portion of the training set used for finetuning. In Appendix D, we compare XTab backbone pretrained on different numbers of tasks and find that more pretraining tasks lead to improved performance. In Appendix E, we study the federated pretraining setting by changing the number of local updates per global aggregation (i.e., N ), and find that larger N leads to reduced downstream performance.
Performance compared to traditional baselines
To compare the performance of XTab and various tabular models, we run experiments on the full AutoML Benchmark (Gijsbers et al., 2022). We split the benchmark into 2 folds, each consisting of 52 tabular datasets. We pretrain on fold #1 and evaluate the downstream performance on fold #2 and vice versa. We pretrain XTab with the FT-Transformer backbone using reconstruction loss. 20 datasets are excluded since they could not fit into the GPU memory (16 GB, see Table 13 in the Appendix for details). We report the performance on the remaining 84 tasks. In addition to XTab, we include the following methods:
Tree-based models: performance. TransTab is included for comparison on classification tasks (regression not enabled yet with TransTab) under the supervised learning (TransTab-sl) and contrastive learning (TransTab-cl) settings (Wang & Sun, 2022). Please refer to Appendix I.3 for how the TransTab ranks are calculated, and Table 12 for results on classification tasks only. Table 1 shows the performance of models with the default hyperparameters and hyperparameter optimization (HPO). With the default hyperparameter, we pretrain XTab for 2000 rounds, whereas the number of pretraining rounds is tuned under the HPO setting. We use the AutoGluon default hyperparameters for tree-based models as they outperform the official defaults to give a strong baseline (Erickson et al., 2020). CatBoost is the state-of-the-art model on tabular tasks, which agrees with the recent finding in Grinsztajn et al. (2022). With cross-table pretraining, XTab improves the performance over FTT under light (FTT-l/XTab-l) and heavy (FTT-h/XTab-h) finetuning. Using more finetuning time, XTab-best achieves second place in the benchmark and beats other deep learning models. The success of XTab using the default configuration ensures that the pretrained backbone is widely applicable to tabular tasks, without the need for case-by-case tuning.
With HPO, we randomly search for data-specific hyperparameters on the validation performance. The detailed search space of each model is in Appendix I. We allow a maximum number of 100 HPO trials within a 1-hour time budget. Table 1 shows that gradient-boosted trees (i.e., XGBoost, LightGBM, CatBoost) achieve higher ranking with HPO, since they are generally faster to train. The search space is also smaller for tree models as they have fewer meaningful hyperparameters and well-known highly performant search spaces. The ranks are calculated separately for default hyperparameters and HPO and are not comparable across the two settings. The advantage of XTab over FTT increases as we allocate less training time for downstream tasks (XTab-l ← XTab-h ← XTab-best ← XTab with HPO). Therefore, one should use pretrained foundation models instead of randomly initialized weights for tabular transformers, especially with a tight training budget.
Conclusion
In this paper, we present XTab to improve the performance of deep tabular models. XTab pretrains tabular transformers with a diverse collection of data tables, and can improve the tabular prediction performance of an unseen table from arbitrary domains. XTab handles the cross-table variations by separating the models into data-specific and shared components, and encourages the shared components to learn general knowledge for tabular prediction. We also propose to combine self-supervised pretraining with federated learning to improve pretraining efficiency, where client-side nodes perform table reconstruction tasks followed by backbone averaging updates at the server. Our results suggest that finetuning from the pretrained transformer is superior to training tabular transformers from scratch. One limitation of XTab is that it still falls behind CatBoost. This motivates future works on bridging the gap between pretrained tabular deep learning models and tree models. Another interesting direction is to combine XTab with language/vision foundation models for improving multimodal learning.
Software and Data
The AutoML Benchmark (AMLB) is publicly available at openml.github.io/automlbenchmark. The code and sample pretrained checkpoints are attached to https: //github.com/BingzhaoZhu/XTab. Bahri, D., Jiang, H., Tay, Y., and Metzler, D. Scarf: Selfsupervised contrastive learning using random feature corruption. arXiv preprint arXiv:2106.15147, 2021.
A. XTab performance with various pretraining/finetuning settings
Here, we extensively present the performance of XTab with reconstruction, contrastive, and supervised pretraining objectives, under light and heavy finetuning. Downstream performance is compared in terms of win rate, model rank, normalized performance, and error reduction rate in Figure 5.
B. Identifying the shareable components in XTab
In XTab, we separate a model into data-specific components (e.g., featurizers and projection heads) and shareable components (Transformer blocks). Only the shareable components are pretrained and contain general knowledge of tabular learning. Therefore, identifying the shareable (or pretrainable) components is critical to the success of cross-table pretraining. In Figure 6, we run an experiment to pretrain on different FT-Transformer components with the supervised objective. For example, pretraining tasks may share only the first Transformer block and the later two blocks are marked as data-specific. We also let the pretraining tasks share all Transformer blocks, [CLS] token, and all blocks with [CLS] token. As expected, pretraining on the [CLS] token does not lead to improved downstream performance, since [CLS] token is directly related to downstream prediction and thereby highly data-specific. From Figure 6, we find that it is most beneficial to pretrain on all Transformer blocks without the [CLS] token. Featurizers and projection heads are not shareable since the input/output spaces can be different across tasks.
C. Finetuning on subsampled datasets
In addition to light and heavy finetuning, we further tune the pretrained backbone using datasets of different sizes. The backbone is a FT-Transformer model pretrained with the reconstruction objective. We subsampled the training sets of downstream tasks (i.e., finetuning set) by 25%, 50%, and 75%. The finetuning is performed on the reduced datasets to simulate the cases where training data is insufficient. Figure 7 shows the downstream performance with (a) light and (b) heavy finetuning.
All settings in Figure 7 show a clear improvement over the baseline. However, the advantage of XTab does not become more significant with reduced finetuning data. This is partially due to the fact that sufficient finetuning data is still needed to train featurizers and projection heads from scratch. For the same reason, XTab is not compatible with zero-shot learning.
D. Tuning the size of pretraining set
The pretrained backbone is expected to host general knowledge that is shared across multiple pretraining tasks. We use different numbers of tabular tasks to pretrain the FT-Transformer using the reconstruction objective. Figure 8 compares the backbone pretrained on 1 task (Adult income, OpenML task id 359983), 18 tasks, and 52 tasks (selected by the alphabetical order of the task names) with light finetuning. Figure 8(a) shows the win rate and Figure 8(b) compares the model rank. Figure 8 indicates that XTab benefits from more pretraining tasks. With many tables involved in cross-table pretraining, XTab can better learn the general knowledge which benefits the downstream performance. We pretrain the FT-Transformer backbone using 1 task, 18 tasks and 52 tasks. We compare the downstream prediction performance using (a) win rate and (b) model rank of different approaches. As we use more tasks for pretraining, we observe an improvement in downstream performance.
E. Tuning parameters of federated pretraining
XTab uses federated learning to account for a large number of pretraining tasks. We have several clients which perform optimization locally for one task, and a central server that aggregates the gradients from all client nodes. We tune the hyperparameter N in FedAvg (see Section 3.2), which indicates the number of local optimization steps between the aggregation steps at the server. We pretrained FT-Transformers with the reconstruction objective and various choices of N . Figure 9 compares the downstream performance with N =1, 5, and 10. We notice that the downstream performance decreases as N takes larger numbers. As N increases, there is less communication overhead between the central server and clients. Therefore, we can use N to control the trade-off between the communication cost of federated pretraining and the downstream performance.
F. Comparison to pretraining without external tasks
Without external tasks, models are simply pretrained on the downstream training set. Indeed, this is a key difference between XTab and existing tabular pretraining models. SubTab (Ucar et al., 2021), SCARF (Bahri et al., 2021) and SAINT (Somepalli et al., 2021) all use the downstream data for both pretraining and finetuning. Here, we run the experiments to compare XTab against models pretrained without external tasks. We used the "heavy" setting and reconstruction loss. The model details are described as follows:
• w/o external task: random initialization → pretrain on downstream task → finetune on downstream task.
• baseline: random initialization → finetune on downstream task • w/ external tasks (XTab): XTab initialization (using external tables) → pretrain on downstream task → finetune on downstream task Figure 9. Comparison of federated pretraining settings in XTab. We test FedAvg with different values of N , which represents the number of local optimization steps per global aggregation. We compare the downstream prediction performance in terms of (a) win rate and (b) model rank. Both figures suggest that the downstream performance decreases with more local steps in FedAvg.
Here, "w/o external task" is pretrained using the downstream training set. Comparing "w/o external task" and "w/ external task", the only difference lies in whether we use the XTab-pretrained transformer as initialization, which can indicate the importance of leveraging cross-table information. "Baseline" model does not use pretraining. From Table 2, we learn that "w/ external task" has a win rate of 55.7% over "w/ external task". Pretraining methods generally outperform baseline. This comparison helps illustrate the benefits of XTab in leveraging information across tasks.
G. Implementation of Saint-v
In Figure 10, we show the difference between the original Saint implementation (Somepalli et al., 2021) and our proposed variation, Saint-v, to fit into cross-table pretraining. Saint and Saint-v both have a row attention layer to account for the cross-sample interaction. The main difference between Saint and Saint-v lies in the reshaping operation. Saint increases the size of token embeddings by a factor equal to the sequence length. The number of trainable parameters in Saint is dependent on the token count (Somepalli et al., 2021), making it infeasible for cross-table training. Saint-v transposes the first (batch) and second (number of tokens) dimensions of the input, without altering the dimension of token embeddings. Therefore, Saint-v can be used to process tables with variable columns.
H. Visualization of pretrained weights
To understand the impact of cross-table pretraining on Transformer parameters, we visualize the weight distribution before and after pretraining ( Figure 11). Here, we ignore the layer normalization and bias terms. Before pretraining, Transformer weights are initialized with Kaiming uniform distribution (He et al., 2015). The weight distribution converges to a normal distribution with increased pretraining steps.
I. Benchmark configurations I.1. Tree-based models
As tree-based models are known to achieve state-of-the-art performance on tabular tasks ( For gradient-boosted trees (i.e., XGBoost, LightGBM, CatBoost), we apply early stopping to determine the optimal number of boosting rounds (early stopping rounds = adaptive). Specifically, we use the an early stopping patience of 300 if the training table has less than 10k rows. The patience is reduced by a factor of num rows/10k if the row count goes beyond 10k. A minimal early stopping patience of 20 is set to all tables regardless of the table size.
For Random Forest, we use max features to indicate the number of features to consider when making a split. Here max features = auto means max features=sqrt(n features) where n features denotes the column count of the training table.
I.2. Neural network and FastAI
We use the tabular neural network from AutoGluon which is implemented on top of PyTorch (Erickson et al., 2020). We use ReLU activation between layers. The default hyperparameters and search space of HPO are listed in Table 7.
We also include the FastAI tabular model in this benchmark, which is essentially a neural network that automatically configures the embedding sizes of input features (Howard & Gugger, 2020). We use the AutoGluon implementation and default hyperparameters/HPO search spaces suggested by AutoGluon. Detailed configurations of FastAI tabular model is listed in Table 8.
I.3. TransTab
We use the official implementation of TransTab v0.0.3 (Wang & Sun, 2022). Since regression tasks are not yet supported by this version, the model rank and training time in Table 1 are reported only on classification tasks. Specifically, we report the rank of TransTab models to all other methods. For example, if we have the AUC scores of model 1 > TransTab > model 2, then model 1 ranks #1, model 2 ranks #2, and TransTab gets a ranking of #1.5. TransTab rank is #0.5 with TransTab > model 1 > model 2, and #2.5 with model 1 > model 2 > TransTab. The inclusion of TransTab in the comparison will not alter the rank of other models, but the rank shows the relative standing of TransTab with respect to other models. Therefore, we can compare the ranking of all methods in Table 1 even without TransTab regression performance. In Table 12, we show the regular ranking of TransTab on classification tasks.
The hyperparameters of TransTab is listed in Table 9. We test both the conventional supervised learning setting (TransTab-sl) and the contrastive learning setting which follows the pretraining-finetuning process (TransTab-cl). We use the target-aware contrastive learning objective as it is shown to perform better than its unsupervised counterpart in Wang & Sun (2022). Hyperparameters are kept as default whenever possible. We use the column type information from AutoML Benchmark to identify numerical and categorical columns. TransTab-cl performs better than TransTab-sl in our benchmark, as shown in Table 1.
I.4. FT-Transformer
Table 10 summarize the general hyperparameters of FT-Transformer. We include three configurations of FT-Transformer in the benchmark:
FTT-l: FT-Transformer with light training. FT-Transformer is trained for maximum 3 epochs. We save the model after each epoch and retrieve the best checkpoint based on the validation performance.
FTT-h: FT-Transformer with heavy training. FT-Transformer is trained with an early stopping patience of 3. We save the model after each epoch and retrieve the best checkpoint based on the validation performance.
FTT-best: FT-Transformer for the best performance. FT-Transformer is trained with an early stopping patience of 20. We save the model after each 0.5 epoch (i.e., val check interval = 0.5 in Table 10). At the end of training, we retrieve the best 3 checkpoints based on the validation performance (i.e., top k = 3 in Table 10). The checkpoints are averaged using model soup for improved prediction performance (Wortsman et al., 2022).
From FTT-l → FTT-h → FTT-best, we achieve better tabular prediction performance with increased training time.
I.5. XTab
XTab uses exactly the same structure as FT-Transformer, but with pretrained parameters to initialize the model. Similar to FTT-l/FTT-h/FTT-best, we have XTab-l/XTab-h/XTab-best that follow the same finetuning configurations. We pretrain XTab with the reconstruction loss and FT-Transformer as the backbone. N = 1 is used for federated pretraining since it achieves the best performance in Figure 9. With default hyperparameters, we pretrain the backbone for 2000 rounds, and the number of pretraining iterations is considered as a hyperparameter in HPO. Table 11 summarizes the details of XTab. Table 13 shows the statistics of all datasets from the AutoML Benchmark (Gijsbers et al., 2022), including the task name, type, and table dimensions. We equally split the benchmark into 2 folds for pretraining and downstream evaluation. Therefore, there is minimal overlap between pretraining tasks and downstream tasks. The success of XTab in this setting demonstrates the ability of learning general knowledge across all downstream tasks.
J. Dataset statistics
K. Raw prediction performance
Here, we present the raw prediction performance on AutoML Benchmark in Table K, 15 and 16. Please refer to Table 1 for the aggregated comparison. 20 datasets are excluded from the benchmark since they fail to fit into the 16 GB GPU memory. We report the performance on the remaining 84 downstream tasks. All experiments are repeated for 5 trials and we report the average performance.
Submission and Formatting Instructions for ICML 2023 Table 13. Dataset statistics of AutoML Benchmark. We split the benchmark into 2 folds. We use fold 1 to pretrain XTab and fold 2 to evaluate downstream performance, and vice versa. 20 out of the 104 datasets failed during our experiments. They are marked with symbols and excluded from the comparison. Table 14. Raw prediction performance on AutoML Benchmark of the following models: Random Forest (RF), XGBoost, LightGBM, CatBoost, tabular neural network from AutoGluon (NN), FastAI tabular model, and TransTab with contrastive pretraining (TransTab-cl). All models use the default hyperparameters as specified in Appendix I. We use AUC scores as the evaluation metric for binary classification (↑), log loss for multicloass classification (↓) and RMSE for regression tasks (↓ Table 15. Raw prediction performance on AutoML Benchmark of the following models: FT-Transformer with light finetuning (FTT-l), XTab with light finetuning (XTab-l), FT-Transformer with heavy finetuning (FTT-h), XTab with heavy finetuning (XTab-h), FT-Transformer with model soup (FTT-best), and XTab with model soup (XTab-best). All models use the default hyperparameters as specified in Appendix I. We use AUC scores as the evaluation metric for binary classification (↑), log loss for multicloass classification (↓) and RMSE for regression tasks (↓). Zoom in for better view.
Bahri et al. (2021); Chen et al. (2020), we used InfoNCE loss for contrastive cross-table pretraining. The contrastive projection heads are similar to those used in SimCLR (Chen et al., 2020), mapping the representations to the space where we apply the contrastive loss.
from the central server, and apply the weights to the transformer backbone w k,i+N =
Figure 2 .
2Tabular prediction performance of XTab using various evaluation criteria under the light finetuning setting. (a) The win rate of the pretrained transformer with respect to baseline. (b) The average rank of the models. (c) The normalized prediction performance. (d) The average error reduction rate compared to baseline. Each dot indicates a trial of the downstream task (5 trials per dataset). The error bars show standard deviations in (b) and (c). As the backbone is pretrained for more steps, we observe an increase in all evaluation criteria. use ReGLU (Shazeer, 2020) as the activation function and layer normalization (Ba et al., 2016) in the feed forward layers. The projection heads are ReLU networks with 2 layers and a hidden dimension of 192. All model components use Kaiming initialization (He et al., 2015) with the bias terms fixed at zeros. The batch size is fixed at 128 for both pretraining and finetuning. Both stages use AdamW as the optimizer, with a learning rate of 1e-4. Following Gorishniy et al. (2021); Rubachev et al. (2022), we also apply a weight decay of 1e-5 to all components excluding featurizers, [CLS] tokens, layer normalization and bias terms.
Figure 3 .
3Comparison of different pretraining objectives under the light (a, c) and heavy (b, d) finetuning settings. We show the win rate of XTab with different objectives with (a) light and (b)
Figure 4 .
4XTab with transformer variants including FT-Transformer, Fastformer, and Saint-v. We use different transformer models as the shared backbone in XTab. We calculate the win rate of the pretrained backbone over randomly initialized transformers. (a) shows the results for light finetuning and (b) represents heavy finetuning. FT-Transformer, Fastformer, and Saint-v all benefit from our proposed cross-table pretraining, achieving >50% win rate in all experiments.
Figure 5 .
5The figure is similar toFigure 2in the main paper, but contains more pretraining/finetuning configurations. See the caption and explanation there for more details.
Figure 6 .
6Comparison of XTab with various pretrained components in FT-Transformer. We run this study to understand which component carries general knowledge of tabular tasks and benefits from cross-table pretraining. Several settings are tested, sharing the first block of Transformer, all blocks, [CLS] token, all blocks with [CLS] token, or no component (baseline). Performance is compared in terms of (a) win rate and (b) model rank with light finetuning. Pretraining on the Transformer blocks leads to improved performance, whereas sharing the data-specific [CLS] token is hardly beneficial.
Figure 7 .
7Downstream prediction performance with different sizes of finetuning set. We subsample the rows of tables (i.e., samples) used for finetuning to a fraction of 25%, 50%, 75%, and 100% (no subsampling). The comparison is performed with (a) light and (b) heavy finetuning.
Figure 8 .
8Comparison of XTab pretrained on different numbers of tabular tasks.
Grinsztajn et al., 2022), we include popular tree ensemble methods in the benchmark such as XGBoost (Chen & Guestrin, 2016), LightGBM (Ke et al., 2017), CatBoost (Dorogush et al., 2018), and Random Forest.Tables 3, 4, 5, and 6 include the default hyperparameters used for tree-based models and the search space of HPO. We use the default hyperparameters, early stopping strategy, and feature preprocessing logic implemented in AutoGluon 0.5.3 release for each of these models(Erickson et al., 2020), which achieves state-of-the-art performance on AutoML Benchmark(Gijsbers et al., 2022). The HPO search space is kept the same as
Figure 10 .
10Model structure of Saint and Saint-v. The difference lies in the reshaping operation. Here, b refers to batch size, n is the length of the sequence, and d is the dimension of embedding. The parameter count of Saint is dependent on the number of table columns (i.e., n), whereas Saint-v is applicable to all tables with the same structure.
Figure 11 .
11Parameters of FT-Transformer before cross-table pretraining (left), 50 steps after cross-table pretraining (middle), and 500 steps after pretraining (right). The model weights are initialized using a Kaiming uniform distribution. With XTab pretraining, the weights converge to a normal distribution.Hollmann et al. (2022).
200, 100), (1000, 500, 200)] * This indicates both the layer count and hidden dimension at each layer.
XTab is irrelevant to the number of training samples. Thus, XTab also works for large tables.table varies. Instead, we aim to learn a
weight initialization that is generalizable to various down-
stream tasks. Concurrent to our work, tabular prior-data
fitted networks (TabPFN) (Hollmann et al., 2022) learns
a prior model on synthetic tabular data and demonstrated
promising results on small numerical tabular classification
tasks with ≤ 1000 samples. Different from TabPFN, the
inference complexity of
The featurizers convert a sample to feature embeddings E ∈ R c×d . Here, c denotes the number of columns and d is the embedding dimension. Each row of a table is considered as an input sample, and each column is a token. The embedding of[CLS] token is appended to the feature embedding for prediction stack[E,[CLS]] ∈ R c+1×d . In this work, we limit our discussion to tables with numerical and categorical columns. Text cells are treated as categorical attributes. Our tokenizer is similar toGorishniy et al. (2021). For numerical features, we multiply the numerical value x k at the k-th column with a trainable vector W k ∈ R d and add a bias term b k . For categorical columns, XTab learns an embedding matrix ∈ R Ncat×d as a lookup table, where N cat is the total number of categories of the dataset. During the forward pass, we retrieve the categorical feature embeddings from the embedding matrix. Wang & Sun, 2022). As long as the backbone can process input sequences of variable lengths, XTab is flexible on the exact implementation. In this work, we present three backbone variants:3.1.1. FEATURIZERS
XTab allows tables to have different numbers of columns
and arbitrary column types. Featurizers are data-specific to
handle various types and numbers of columns in the input.
3.1.2. BACKBONES
As the shared component across multiple pretraining
datasets, transformers can handle input sequences with vari-
able lengths. Therefore, it is possible to pretrain a tabular
transformer that can be applied to all tabular datasets. Com-
pared with other deep learning architectures like multi-layer
perceptron (MLP), transformers are favorable for cross-table
knowledge transfer since they can handle variable input se-
quences (FT-Transformer: Feature Tokenizer Transformer (FT-
Transformer) is a simple yet well-performing transformer
model for tabular prediction tasks (Gorishniy et al., 2021).
The transformer module in FT-Transformer consists of a
Multi-Head Self-Attention (MHSA) block and a Feed For-
ward block (Vaswani et al., 2017). Recent work has found
FT-Transformers to beat other deep learning methods on
tabular data (Grinsztajn et al., 2022).
Fastfromer: Conventional Transformer-like architectures
have a quadratic complexity to the length of input sequence
(Vaswani et al., 2017), making them inefficient for tables
with large numbers of columns. Fastfromer is an efficient
transformer architecture which uses additive attention in
place of MHSA (Wu et al., 2021). With additive attention,
Fastformer only considers the interaction between each to-
ken and the global representation, achieving a linear com-
plexity.
Saint-v: Saint has introduced the row-wise attention in
addition to the column-wise attention of FT-Transformer
and Fastformer (Somepalli et al., 2021). The original im-
plementation of Saint is sensitive to the sequence length
and can not handle variable-column tables (Somepalli et al.,
2021). We present a variation of Saint (Saint-v) to fit into
our cross-table pretraining setting. Saint-v consists of both
column-and row-wise attention blocks, and the detailed
model structure is depicted in Appendix G.
Table corruption :
corruptionSelf-supervised learning objectives, in-
cluding both contrastive and reconstruction losses, require
a corrupted view of the input sample. In this work, we fol-
low Bahri et al. (2021); Rubachev et al. (2022) to randomly
resample features and construct a corrupted sample. Specifi-
cally, we randomly select a fraction of features at each row
of the table. Those features are corrupted by resampling
from the empirical marginal distribution of the column. For
all datasets, the corruption ratio was set to 60% as suggested
in Bahri et al. (2021). In other words, for each sample x and
its corrupted viewx, 60% of entries are resampled whereas
40% of features remain unchanged.
Tree-based models provide strong performance on tabular tasks(Grinsztajn et al., 2022). We include Random Forest (RF) and gradient-boosted tree variants: XGBoost (Chen & Guestrin, 2016),LightGBM (Ke et al., 2017) andCatBoost (Dorogush et al., 2018). Neural networks: We include the AutoGluon neural networks implemented on top of PyTorch(Erickson et al., 2020) and the FastAI tabular model(Howard & Gugger, 2020). Transformers: We include the FT-Transformer which is a direct counterpart of XTab without pretraining. The finetuning settings of FTT/XTab include light (FTT-l/XTab-l) and heavy (FTT-h/XTab-h) finetuning as described above. We further introduce FTT-best/XTab-best, which incorporates an early-stopping patience of 20 and model soup of the top 3 checkpoints (Wortsman et al., 2022) to achieve betterTable 1. Comparison of tabular prediction performance with default model configuration and hyperparameter optimization (HPO). Mean training time and model rank (± standard deviation) are calculated across 84 datasets from AutoML Benchmark. We perform 5 independent trials for each task. XTab outperforms its counterpart FTT in all scenarios thanks to cross-table pretraining, whereas CatBoost is the overall best model. The best overall method (CatBoost) and the best deep learning approach (XTabbest) are highlighted in bold. 51 ± 2.00 † CPU training time.* Only evaluated on classification tasks.Methods
Time (s)
Rank
Default hyperparameter
RF
66.8 †
7.14 ± 3.81
XGBoost
43.1 †
5.06 ± 3.08
LightGBM
23.9 †
5.23 ± 3.25
CatBoost
322.8 †
2.98 ± 2.66
FastAI
89.6
7.24 ± 3.44
NN
188.8
7.40 ± 3.43
TransTab-sl *
539.7
11.04 ± 2.75
TransTab-cl *
312.0
10.79 ± 3.00
FTT-l
189.2
10.19 ± 2.43
XTab-l
189.8
9.21 ± 2.57
FTT-h
532.5
7.29 ± 2.20
XTab-h
506.3
6.93 ± 2.09
FTT-best
810.9
4.94 ± 2.25
XTab-best
755.9
4.39 ± 2.36
HPO
RF
1084.4 †
5.00 ± 2.40
XGBoost
862.3 †
3.69 ± 2.45
LightGBM
285.0 †
4.40 ± 1.93
CatBoost
1529.3 †
3.25 ± 2.10
FastAI
549.7
5.24 ± 2.38
NN
1163.5
5.32 ± 2.20
FTT
2221.1
4.58 ± 2.08
XTab
2335.3
4.
Table 2 .
2Comparison to pretraining without external tasks.w/o external task baseline w/ external tasks
win rate (against w/o external task)
50%
35.2%
55.7%
Table 3 .
3XGBoost hyperparameter space.Parameter
Default
HPO search space
learning rate
0.1
UniformLog[exp(-7), 1]
max depth
6
UniformInt[1, 10]
subsample
1
Uniform[0.2, 1]
colsample bytree
1
Uniform[0.2, 1]
colsample bylevel
1
Uniform[0.2, 1]
min child weight
1
UniformLog[exp(-16), exp(5)]
reg alpha
0
UniformLog[exp(-16), exp(2)]
reg lambda
1
UniformLog[exp(-16), exp(2)]
gamma
0
UniformLog[exp(-16), exp(2)]
n estimators
10000
UniformInt[100, 4000]
booster
gbtree
gbtree
early stopping rounds adaptive *
adaptive
* The early stopping rounds depends on the size of data with a minimal patience
of 20 and maximal patience of 300 rounds.
Table 4 .
4LighGBM hyperparameter space. The early stopping rounds depends on the size of data with a minimal patience of 20 and maximal patience of 300 rounds.Parameter
Default
HPO search space
num leaves
31
UniformInt[5, 50]
max depth
inf
UniformInt[3, 20]
learning rate
0.05
UniformLog[exp(-3), 1]
n estimators
10000
UniformInt[50, 2000]
min child weight
1e-3
UniformLog[exp(-5), exp(4)]
reg alpha
0
Categorical[0, 0.1, 1, 2, 5, 7, 10, 50 , 100]
reg lambda
0
Categorical[0, 0.1, 1, 5, 10, 20, 50, 100]
subsample
1
Uniform[0.2, 0.8]
early stopping rounds adaptive *
adaptive
*
Table 5 .
5CatBoost hyperparameter space.Parameter
Default
HPO search space
learning rate
0.05
UniformLog[exp(-5), 1]
random strength
1
UniformInt[1, 20]
l2 leaf reg
3
UniformLog[exp(-3), 1]
bagging temperature
1
Uniform[0, 1]
leaf estimation iterations
1
UniformInt[1, 20]
iterations
10000
UniformInt[100, 4000]
early stopping rounds
adaptive *
adaptive
* The early stopping rounds depends on the size of data with a minimal patience
of 20 and maximal patience of 300 rounds.
Table 6 .
6Random forest hyperparameter space.Parameter
Default
HPO search space
n estimators
300
UniformInt[10, 1000]
max features
auto
Categorical[auto, 0.5, 0.25]
max leaf nodes
inf
UniformInt[100, 4000]
Table 7 .
7Neural network hyperparameter space.Parameter
Default
HPO search space
num epochs
300
300
early stop patience
20
20
learning rate
3e-4
UniformLog[1e-4, 0.1]
weight decay
1e-6
UniformLog[1e-12, 0.1]
num layers
4
Categorical[2, 3, 4]
hidden size
128
Categorical[128, 256, 512]
Table 8 .
8FastAI hyperparameter space.Parameter
Default
HPO search space
num epochs
30
Uniform[5, 30]
early stop patience
20
20
learning rate
1e-2
UniformLog[5e-5, 0.1]
weight decay
1e-6
UniformLog[1e-12, 0.1]
layers *
none
Table 9 .
9TransTab hyperparameter for the base and pretraining settings.Parameter
supervised learning contrastive pretraining
num partition
4
overlap ratio
0.5
max pretrain epochs
50
pretrain batch size
128
pretrain learning rate
1e-4
max epochs
50
50
batch size
238
128
learning rate
1e-4
1e-4
num layers
2
2
hidden dim
128
128
patience
5
5
num attention heads
8
8
Table 10 .
10FT-Transformer hyperparameter space.Parameter
Default
HPO search space
num epochs
inf
inf
early stop patience
20
20
num blocks
3
3
hidden size
192
192
num attention heads
8
8
batch size
128
Categorical[128, 32, 8, 1]
val check interval
1 or 0.5
Categorical[0.5, 1]
top k
1 or 3
Categorical[1, 3, 5]
Table 11 .
11XTab hyperparameter space.Parameter
Default
HPO search space
All default parameters and search spaces from FT-Transformer
N FedAvg
1
1
pretrain objective
reconstruction
reconstruction
num pretrain rounds
2000
Categorical[0, 250, 1000, 2000]
Table 12 .
12Thistable is similar to Table 1, but compares the tabular models on 48 classification tasks. Since TransTab v0.0.3 does not
supports regression tasks, we include this table for classification tasks only.
Methods
Time (s)
Rank
Default hyperparameter
RF
11.39
7.58 ± 4.19
XGBoost
11.90
5.10 ± 3.41
LightGBM
8.62
5.58 ± 3.54
CatBoost
229.36
3.02 ± 2.87
FastAI
27.01
7.27 ± 3.79
NN
73.64
6.96 ± 3.66
TransTab-sl
342.49
12.33 ± 2.68
TransTab-cl
331.98
11.60 ± 3.13
FTT-l
74.91
10.94 ± 2.54
XTab-l
74.48
10.06 ± 2.88
FTT-h
309.64
7.23 ± 2.17
XTab-h
291.19
7.35 ± 1.92
FTT-best
544.77
5.33 ± 2.43
XTab-best
472.35
4.63 ± 2.28
Table 16 .
16Raw prediction performance on AutoML Benchmark under the HPO setting. All models use the HPO search spaces as specified in Appendix I.name
task type
metrics RF
XGB
LGBM
CAT
FastAI
NN
FTT
XTab
APSFailure
binary
AUC
0.9891
0.9929
0.9905
0.9923
0.9825
0.9896
0.9859
0.9875
Amazon employee access
binary
AUC
0.8629
0.8526
0.8555
0.8995
0.8535
0.8329
0.7945
0.7929
Australian
binary
AUC
0.9331
0.9382
0.9399
0.9362
0.9272
0.9211
0.9184
0.9132
Click prediction small
binary
AUC
0.6976
0.7017
0.6953
0.7105
0.681
0.6964
0.675
0.6757
Higgs
binary
AUC
0.8126
0.8365
0.8345
0.8367
0.8485
0.8435
0.8458
0.8329
KDDCup09 appetency
binary
AUC
0.8186
0.8307
0.8041
0.8367
0.762
0.8168
0.8159
0.8127
MiniBooNE
binary
AUC
0.9813
0.9866
0.9863
0.9865
0.9845
0.9878
0.9823
0.9799
PhishingWebsites
binary
AUC
0.9964
0.997
0.9966
0.9961
0.9965
0.9968
0.9961
0.9961
Satellite
binary
AUC
0.9746
0.9443
0.9821
0.9873
0.9935
0.9945
0.9908
0.9879
ada
binary
AUC
0.9227
0.9237
0.9215
0.9247
0.9055
0.9175
0.9197
0.9185
adult
binary
AUC
0.9176
0.9288
0.928
0.929
0.9143
0.9138
0.9154
0.9167
airlines
binary
AUC
0.7252
0.7301
0.7262
0.7266
0.7204
0.7192
0.7154
0.7128
albert
binary
AUC
0.7342
0.7687
0.7758
0.7846
0.7569
0.7653
0.7559
0.7499
bank-marketing
binary
AUC
0.9318
0.9364
0.9385
0.9388
0.9367
0.9354
0.9411
0.9405
blood-transfusion-service-center
binary
AUC
0.7273
0.7166
0.7503
0.759
0.7443
0.7227
0.7451
0.7303
churn
binary
AUC
0.907
0.9089
0.9131
0.9194
0.9192
0.9156
0.914
0.9168
credit-g
binary
AUC
0.791
0.7512
0.7498
0.7779
0.7527
0.7458
0.7481
0.743
jasmine
binary
AUC
0.8875
0.875
0.8596
0.873
0.8516
0.8542
0.8606
0.8579
kc1
binary
AUC
0.8163
0.8154
0.7904
0.8069
0.7972
0.7984
0.7979
0.8062
kick
binary
AUC
0.7699
0.7855
0.7708
0.786
0.7771
0.7735
0.7773
0.7775
kr-vs-kp
binary
AUC
0.9998
0.9988
0.9997
0.9997
0.9985
0.9995
0.9989
0.9998
madeline
binary
AUC
0.9275
0.9364
0.9176
0.938
0.7825
0.7752
0.8628
0.8923
nomao
binary
AUC
0.9946
0.9963
0.9961
0.996
0.9928
0.9923
0.9933
0.9937
numerai28 6
binary
AUC
0.5277
0.5243
0.5262
0.5263
0.5282
0.5258
0.5258
0.5266
ozone-level-8hr
binary
AUC
0.9303
0.9231
0.9259
0.9307
0.9256
0.9446
0.9277
0.9293
pc4
binary
AUC
0.9459
0.9366
0.9437
0.9425
0.9415
0.9397
0.9412
0.9433
philippine
binary
AUC
0.8498
0.8627
0.8487
0.8541
0.7934
0.802
0.8246
0.8324
phoneme
binary
AUC
0.9604
0.9563
0.9521
0.9573
0.9332
0.9428
0.9539
0.9532
porto-seguro
binary
AUC
0.63
0.6419
0.6345
0.6394
0.6358
0.634
0.6369
0.6362
qsar-biodeg
binary
AUC
0.9162
0.9091
0.9146
0.9031
0.9187
0.9181
0.9196
0.9174
sf-police-incidents
binary
AUC
0.6706
0.686
0.681
0.7158
0.6122
0.6474
0.6068
0.607
sylvine
binary
AUC
0.9838
0.9863
0.985
0.9866
0.9826
0.9811
0.9846
0.9846
wilt
binary
AUC
0.9877
0.9901
0.991
0.9811
0.9808
0.9898
0.9898
0.9941
Diabetes130US
multiclass log loss 0.8519
0.8357
0.8499
0.8355
0.8643
0.8665
0.8433
0.8489
GesturePhaseSegmentationProcessed
multiclass log loss 0.8598
0.8242
0.8328
0.7833
1.0472
0.9798
0.9604
0.9604
car
multiclass log loss 0.0504
0.3288
0.2972
0.0578
0.2856
0.0013
0.0002
0
cmc
multiclass log loss 0.9074
0.9305
0.9117
0.9237
0.9387
0.9264
0.9519
0.9449
connect-4
multiclass log loss 0.497
0.3269
0.3218
0.3719
0.3215
0.3373
0.3383
0.3537
covertype
multiclass log loss 0.1824
0.0889
0.0915
0.109
0.1346
0.1264
0.1373
0.2386
dna
multiclass log loss 0.1487
0.0989
0.1102
0.1182
0.1484
0.1489
0.1279
0.131
eucalyptus
multiclass log loss 0.7119
0.7358
0.7493
0.7476
0.7189
0.7247
0.7481
0.7305
first-order-theorem-proving
multiclass log loss 1.0671
1.0664
1.0849
1.0858
1.2051
1.1899
1.212
1.1831
helena
multiclass log loss 2.7036
2.5968
2.6022
2.5647
2.5305
2.513
2.5355
2.5407
jannis
multiclass log loss 0.7072
0.6731
0.6807
0.6764
0.6694
0.6555
0.662
0.6603
jungle chess 2pcs raw endgame complete multiclass log loss 0.3169
0.2299
0.2257
0.2335
0.2097
0.0475
0.012
0.0122
mfeat-factors
multiclass log loss 0.1636
0.1201
0.1382
0.1114
0.1089
0.0773
0.1099
0.1094
okcupid-stem
multiclass log loss 0.5902
0.5663
0.5701
0.5637
0.5739
0.5694
0.5688
0.5694
segment
multiclass log loss 0.0762
0.0718
0.0714
0.067
0.0905
0.0818
0.0812
0.0932
shuttle
multiclass log loss 0.0006
0.0004
0.0005
0.0005
0.0077
0.0028
0.0013
0.0013
steel-plates-fault
multiclass log loss 0.5287
0.4937
0.4912
0.4834
0.6348
0.5823
0.568
0.5536
vehicle
multiclass log loss 0.4972
0.4555
0.5123
0.5383
0.3649
0.4504
0.4303
0.4256
volkert
multiclass log loss 0.9181
0.8078
0.8199
0.7951
0.801
0.8266
0.7847
0.8004
wine-quality-white
multiclass log loss 0.803
0.793
0.8602
0.8198
0.9771
0.9703
0.9789
0.9708
yeast
multiclass log loss 1.02
1.0213
1.0999
1.0018
1.054
1.0349
1.0156
1.0155
Airlines DepDelay 10M
regression RMSE
28.9108
28.577
28.5797
28.7851
28.7342
30.1429
28.7435
28.809
Allstate Claims Severity
regression RMSE
1939.89
1887.014
1885.37
1866.698
1977.002
1892.888
1885.936
1905.3
Brazilian houses
regression RMSE
5285.2022 4488.908
8505.7592 9491.7438 16486.544 3859.9434 8264.7402 8201.4656
Buzzinsocialmedia Twitter
regression RMSE
179.265
241.4524
200.1286
229.5252
168.3526
177.9844
162.3476
173.2894
MIP-2016-regression
regression RMSE
765.0452
800.3702
829.5368
823.6524
2377.88
3903.15
871.013
878.175
Mercedes Benz Greener Manufacturing
regression RMSE
8.9261
8.6234
8.7048
8.6512
9.0556
8.7859
8.6845
8.7014
Moneyball
regression RMSE
24.4026
23.0216
24.5429
22.8522
22.0157
23.1796
21.5883
21.8534
OnlineNewsPopularity
regression RMSE
11464.464 11364.592 11397.174 11410.652 11378.526 11478.684 11379.368 11365.422
SAT11-HAND-runtime-regression
regression RMSE
1139.12
1067.37
968.7284
1100.046
1079.6472 1166.976
1034.3146 1032.3848
Yolanda
regression RMSE
9.229
8.6079
8.7664
8.701
8.6134
8.7159
8.6318
8.7462
abalone
regression RMSE
2.1789
2.1927
2.2062
2.2103
2.1402
2.1496
2.1335
2.142
black friday
regression RMSE
3503.918
3452.056
3452.454
3463.792
3573.808
3592.846
3500.544
3513.162
boston
regression RMSE
3.3039
3.0809
3.3606
2.945
3.3487
3.4282
3.4638
2.8631
colleges
regression RMSE
0.1426
0.1381
0.1407
0.1397
0.1537
0.1529
0.147
0.1451
diamonds
regression RMSE
544.454
534.047
521.6772
517.6136
593.0908
549.5522
520.3338
517.9442
elevators
regression RMSE
0.0027
0.0022
0.0021
0.002
0.0019
0.0019
0.0018
0.0019
house 16H
regression RMSE
29691.38
28688.28
28892.28
27962.54
30915.52
29078.48
27869.3
29179
house prices nominal
regression RMSE
25655.78
21950.74
22964.88
21954.12
22389.62
23721.84
22199.78
22056.98
house sales
regression RMSE
121712.2
111883.4
110022.38 107470.58 111026.4
118402
110166.28 109626.2
nyc-taxi-green-dec-2016
regression RMSE
1.631
1.7843
1.6683
1.6148
1.5909
1.737
1.7596
1.698
pol
regression RMSE
4.6848
4.6111
4.4196
3.9429
3.6529
3.6789
2.0737
2.0981
quake
regression RMSE
0.1845
0.1896
0.1872
0.1851
0.1851
0.1825
0.1843
0.1844
sensory
regression RMSE
0.6731
0.7238
0.6924
0.6966
0.6588
0.7263
0.7803
0.8059
socmob
regression RMSE
16.2576
12.8328
11.3572
13.45
8.4355
11.0054
19.1915
19.1915
space ga
regression RMSE
0.1096
0.1036
0.1035
0.1028
0.0992
0.0982
0.1031
0.1007
tecator
regression RMSE
1.3897
0.9691
1.1218
1.6591
1.7807
1.6622
1.7329
1.2897
topo 2 1
regression RMSE
0.0302
0.03
0.0301
0.0301
0.0302
0.0306
0.0302
0.0301
us crime
regression RMSE
0.1379
0.1343
0.1372
0.1354
0.1391
0.1392
0.1351
0.1351
wine quality
regression RMSE
0.6004
0.6046
0.6261
0.5972
0.6767
0.6864
0.682
0.6761
yprop 4 1
regression RMSE
0.0295
0.0334
0.0298
0.0296
0.0791
0.0303
0.0303
0.0301
A Aghajanyan, A Gupta, A Shrivastava, X Chen, L Zettlemoyer, S Gupta, Muppet, arXiv:2101.11038Massive multitask representations with pre-finetuning. arXiv preprintAghajanyan, A., Gupta, A., Shrivastava, A., Chen, X., Zettlemoyer, L., and Gupta, S. Muppet: Massive multi- task representations with pre-finetuning. arXiv preprint arXiv:2101.11038, 2021.
. J L Ba, J R Kiros, G Hinton, arXiv:1607.06450E. Layer normalization. arXiv preprintBa, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.
| {'fraction_non_alphanumeric': 0.04819953369217927, 'fraction_numerical': 0.07560231426844363, 'mean_word_length': 4.107917371168124, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The success of self-supervised learning in computer vision and natural language processing has motivated pretraining methods on tabular data. However, most existing tabular self-supervised learning models fail to leverage information across multiple data tables and cannot generalize to new tables. In this work, we introduce XTab, a framework for cross-table pretraining of tabular transformers on datasets from various domains. We address the challenge of inconsistent column types and quantities among tables by utilizing independent featurizers and using federated learning to pretrain the shared component. Tested on 84 tabular prediction tasks from the OpenML-AutoML Benchmark (AMLB), we show that (1) XTab consistently boosts the generalizability, learning speed, and performance of multiple tabular transformers, (2) by pretraining FT-Transformer via XTab, we achieve superior performance than other state-of-the-art tabular deep learning models on various tasks such as regression, binary, and multiclass classification.', 'arxivid': '2305.06090', 'author': ['Bingzhao Zhu ', 'Xingjian Shi ', 'Nick Erickson ', 'Mu Li ', 'George Karypis ', 'Mahsa Shoaran '], 'authoraffiliation': [], 'corpusid': 258588344, 'doi': '10.48550/arxiv.2305.06090', 'github_urls': [], 'n_tokens_mistral': 23355, 'n_tokens_neox': 19782, 'n_words': 10132, 'pdfsha': 'fd27c08b7105da3412f3b285ea23b3a6dd92409b', 'pdfurls': ['https://export.arxiv.org/pdf/2305.06090v1.pdf'], 'title': ['XTab: Cross-table Pretraining for Tabular Transformers', 'XTab: Cross-table Pretraining for Tabular Transformers'], 'venue': []} |
arxiv |
Atomistic insights into the inhomogeneous nature of solute segregation to grain boundaries in magnesium
Risheng Pei
Institut für Metallkunde und Materialphysik
RWTH Aachen University
D-52056AachenGermany
Zhuocheng Xie
Institut für Metallkunde und Materialphysik
RWTH Aachen University
D-52056AachenGermany
Sangbong Yi
Institute of Materials and Process Design
Helmholtz-Zentrum Hereon
21502GeesthachtGermany
Sandra Korte-Kerzel
Institut für Metallkunde und Materialphysik
RWTH Aachen University
D-52056AachenGermany
Julien Guénolé
Arts et Métiers
Université de Lorraine
CNRS
LEM3, 57070MetzFrance
Labex Damas
Université de Lorraine
57070MetzFrance
Talal Al-Samman
Institut für Metallkunde und Materialphysik
RWTH Aachen University
D-52056AachenGermany
Atomistic insights into the inhomogeneous nature of solute segregation to grain boundaries in magnesium
In magnesium alloys with multiple substitutional elements, solute segregation at grain boundaries (GBs) has a strong impact on many important material characteristics, such as GB energy and mobility, and therefore, texture. Although it is well established that GB segregation is inhomogeneous, the variation of GB solute composition for random boundaries is still not understood. In the current study, atomic-scale experimental and simulation techniques were used to investigate the compositional inhomogeneity of six different GBs. Three-dimensional atom probe tomography results revealed that GB solute concentration of Nd in Mg varies between 2 to 5 at.%. This variation was not only seen for different GB orientations but also within the GB plane. Correlated atomistic simulations suggest that the inhomogeneous segregation behavior observed experimentally stems from local atomic rearrangements within the GBs and introduce the notion of potential excess free volume in the context of improving the prediction of per-site segregation energies.The drive towards energy saving and environmental protection is attracting significant interest from research laboratories worldwide to develop innovative and costeffective materials and processes for lightweight structural components. Owing to their high specific strength and stiffness, magnesium alloys are promising candidates to improve fuel economy and support a sustainable lower carbon vehicle technology [1-3].Compared to steel or aluminum alloys, processing of magnesium alloys poses challenges related to limited strength, sharp crystallographic textures and plastic anisotropy [4-6], which hinder their widespread commercial usage as rolled products. Overcoming these challenges requires a combination of well-informed processing and micro-alloying strategies in order to depart from basal textures and obtain a favorable alignment of basal planes with the principal deformation direction. This should be combined with grain refinement to increase the material strength, reduce the activity of deformation twinning and promote additional intergranular deformation mechanisms.Mg alloys containing small additions of rare earth (RE) elements demonstrate remarkable qualitative alterations of the sheet texture along with reduced mechanical anisotropy, and thereby, promoted formability [7][8][9][10][11][12]. The effectiveness of RE addition for texture softening depends on the type and concentration of the added RE, and can be further increased by co-addition of Zn or Mn. This is known to result in textures with a quadrupole characteristic, i.e. with a distribution of basal poles along the rolling and transverse sheet directions that are favorable for sheet metal forming [7,[13][14][15]. From a synergistic perspective, multiple solute species with smaller and larger atomic sizes than
magnesium tend to co-segregate and form local clusters in order to minimize the lattice misfit with the matrix [13][14][15]. This raises interest in the characteristics of solute interactions, and the resulting impact on the deformation and recrystallization behavior in terms of active deformation modes and grain boundary migration. Although concrete conclusions regarding the mechanisms of texture modification are still elusive, there is a consensus in the literature that co-segregation of combined solute species inhibits growth of grains with a basal texture by decreasing the grain boundary (GB) energy and mobility [13,14,16]. This effect will vary with the type of GB and segregating solute giving rise to a growth preference of certain orientations (e.g. ones with basal pole split in the sheet transverse direction) [7,14,[16][17][18][19][20][21][22][23][24].
Given the complexity and experimental limitations to study atomistic behaviors, it is prudent to utilize recent advancements in high performance computing to investigate computationally intensive problems. Atomistic simulations are a powerful complement to high-resolution experimental techniques targeting the atomic-scale behavior of GBs. For example, molecular dynamics (MD) and molecular statics (MS) have been used to compute the distribution of GB segregation energy in face-centered cubic polycrystals [25,26]. For hexagonal close-packed metals, density functional theory calculations have also been used to study solute segregation at twin boundaries and coincident site lattice (CSL) Σ7 GB in magnesium [27,28]. Another novel application of atomistic simulations is its combination with machine learning to investigate the segregation energetics of aluminum at <0001> symmetric tilt GBs as a function of GB structure and local atomic environment [29].
Despite previous extensive research on the effect of different solutes on recrystallization texture development, formal understanding of inhomogeneous GB segregation and resulting selective growth during recrystallization remains pending. In line with this issue, the present work combines advanced modeling and high-resolution characterization at the atomic-scale to shed light on the effect of GB structure on solute segregation.
The studied material was an extruded Mg-1.0 wt.%Mn-1.0 wt.% Nd alloy (hereafter, MN11) [12,30]. 3D atom probe tomography (APT) in a Local Electrode Atom Probe 4000X HR from Cameca was employed to quantify the chemical composition of six general GBs using laser-pulsing mode at a temperature of 30 K. Reconstruction of the evaporated tips was performed using the software package IVAS 3.8.2. The APT sample preparation was carried out by transmission Kikuchi diffraction (TKD)-assisted focused ion beam (FIB) milling in a FEI Helios 600i dual-beam electron microscope (Fig. S1). Correlative atomistic simulations to investigate the relationship between per-site segregation energy and the local site environment were performed using the open-source MD software package LAMMPS [31] in conjunction with the modified embedded atom method (MEAM) potential for Mg-Nd [32]. The atomistic configurations of general GBs were constructed using the open source tool Atomsk [33] based on experimentally determined crystallographic orientations of the GB plane and related grains obtained from TKD mapping and reconstructed APT tips (cf. supplementary material). misorientations, and (707 ̅ 10 ̅̅̅̅ ) and (2 ̅ 021) boundary planes, respectively. For simplification, the GBs are denoted hereafter by their misorientation angles. Given that Nd has a larger atomic radius (206 pm) than Mg (173 pm), Nd atoms in the solid solution matrix induce compressive elastic strains and therefore tend to segregate at GBs that are rich in microstructural defects. This is depicted in Fig. 1(a) by the obvious Nd enrichment at the two boundaries. The top part of the tip revealed a pure Mn precipitate smaller than 100 nm in diameter, demonstrating evident segregation of Nd atoms at the interface with the matrix. In Fig. 1(b) the obtained concentration profiles along a direction normal to the GB plane (region of interest ROIs 1 & 2) reveal that the segregation level depends on the GB type. The measured Nd peak concentrations at the two boundaries were 2.9 ± 0.2 at.% (86.5° GB) and 3.7 ± 0.2 at.% (59.3° GB). A similar trend was also seen in another reconstructed tip containing a different random GB with 11.3° [101 ̅ 0] misorientation and (11 ̅̅̅̅ 474 ̅ ) boundary plane (Fig. S2). As seen in the atom distribution maps ( Fig. S2 (a)), Nd segregation at the GB was more evident than Mn segregation. The corresponding mass spectrum is shown in Fig. S2 The segregation behavior of Nd atoms is not only influenced by the macroscopic features of GBs but also the local structural arrangement of atoms in the GB plane. Fig. 2(a) shows a magnified view of the Nd atom map in the triple junction region (ROI 3 in Fig. 1(a)). The pronounced segregation behavior of Nd atoms is displayed using a 1 at.% Nd iso-concentration surface ( Fig. 2(b)). As evidenced by the 2D concentration contour map of the same region shown in Fig. 2(c), the local segregation densities of Nd in the x-z plane of the measured tip exhibit strong variation along both GBs. The highest Nd concentration densities were observed at the triple junction and within a distance of 30 nm along 59.3° GB. As in the 59.3° and 86.5° GBs ( Fig. 1(c)), the x-z in-plane segregation in the 11.3° GB was similarly inhomogeneous with concentration density variations between 1.5 at.% and 3.5 at.% ( Fig. S2(c)). This can also be seen from the 2D Nd concentration density maps of the GB planes (x-y plane) in the three selected GBs, as shown in Fig. 2(d-f). The atomistic configurations were relaxed using the conjugate gradient (with box relaxation in the z-direction) and the FIRE algorithms [34,35] with force tolerance of 10 -8 eV/Å. A substitution region (80 Å × 80 Å × 20 Å) of Nd substitutions was considered in the center of the cylindrical setup across the GB to neglect the effect of the boundary conditions (Fig. S4). The local site environments within the substitution region represent the possible environments of the GB as indicated in the hydrostatic stress maps in Fig. S5. By swapping one Mg atom with one Nd atom near the GBs in the substitution region, the per-site segregation energies were calculated according to:
seg = ( GB + bulk X ) − ( GB X + bulk )
where bulk is the energy of the Mg bulk, bulk X the energy of the Mg bulk where one host atom is replaced by Nd solute, GB the energy of the Mg system with a GB, and GB X is the energy of the Mg system with Nd solute occupying a GB site. After each swap, an energy minimization using the FIRE algorithm was performed. The Open Visualization Tool OVITO was used to visualize the atomistic configurations, analyze the misfit dislocation networks and calculate the atomic displacement. The Open Visualization Tool OVITO [36] was used to visualize the atomistic configurations and calculate the atomic displacement.
The Dislocation Extraction Algorithm [37] was used to characterize the misfit dislocation networks. Fig. 3 shows the statistics of per-site segregation energy, which is binned according to the distance to the GB plane. Both GBs exhibit approximately symmetric distributions of segregation energy on either side of the GBs. For most GB sites at a minimum distance of 8 Å from the GB plane center, the segregation energy is close to zero. The 11.3° LAGB in Fig. 3(a) shows distinct segregation behavior compared to the 59.3° HAGB ( Fig. 3(b)).
The maximum value of the per-site segregation energies of each bin is higher for the 11.3°
LAGB than the 59.3° HAGB. For the 11.3° LAGB, the distributions of mean, median and third quartile segregation energies sharply increase when approaching the GB plane, and the deviation between mean and median in each bin is more significant than for the 59.3°
HAGB. In contrast, the mean, median and third quartile segregation energies within 2.5 Å of the GB plane of the 59.3° HAGB stay at similar levels (cf. Fig. 3(b)). LAGB in Fig. 4(a) shows more hot spots than the 59.3° HAGB (Fig. 4(b)). In the 11.3° The concentrations of GB sites with high segregation energy in our simulations as shown in Fig. 4(a, b) agree well with the experimental Nd-solute concentrations at GBs in the current work ( Fig. 2(c, d)), and previous experimental observations of solute clusters at HAGBs in Mg-RE solid solutions [24]. The atomistic origin for the inhomogeneous GB segregation observed is explained by correlating the segregation energy to the local structural features of the GB. As shown in Fig. 4(c, d), the distribution of the hot spots of studies have shown that the excess free volume is directly related to the GB energy [40] and GB segregation [41,42], and it was often treated as a macroscopic feature of the GB [43][44][45]. However, the MSD computed in this work is not related to the excess free volumes of these previous studies. During diffusion, the atoms of the GB reorganize themselves in an energetically favorable configuration that has sufficient free volume to host the solute.
The local structural reorganization measured by the MSD acts as a generator of excess free volume. Thus, the MSD is a measure of the potential of excess free volume for a given stable GB configuration, in contrast to the effective excess free volume classically considered. In this work, the strong association between the distributions of per-site segregation energy and MSD demonstrates the impact of the potential excess free volume on per-site segregation energy. The microscopic structural features of a GB, such as GB dislocations, GB disconnections and GB triple junctions, which affect the local site environment, could thus have significant effects on the local segregation behavior. As a result, these features could favor inhomogeneous segregation within the GB, as seen in Such local structural rearrangement also indicates that the widely used linear elasticity model for the atomistic modelling of high symmetric GBs based on effective excess free volume [19,20,28] may not be applicable in the study of general GBs. As shown in Fig. S7, the per-site segregation energy shows a deviation from the predicted segregation energy using the linear elasticity model, especially for sites with high segregation energies. In addition, there is almost no correlation of hot spots between the heat map of the simulated and predicted segregation energy density of the general GBs (see Fig. S6). To improve the prediction of per-site segregation energies in general GBs, existing models should account for the potential excess free volume of GB sites [46,47].
In summary, the inhomogeneous segregation behavior of Nd solute atoms at several random grain boundaries in a deformed and subsequently annealed MN11
Methods
TKD-EBSD assisted preparation of APT tips
Before the milling process, the orientation of grains of the sampled surface is characterized by electron backscatter diffraction (EBSD) performed in a FEI Helios 600i dual-beam scanning electron microscope/focused ion beam (SEM/FIB) with an operation voltage of 20 kV, as shown in Fig. S1 a. The specimens for EBSD measurements were prepared by conventional mechanical grinding and polishing followed by electro-polishing in Struers AC-2 reagent at 20 V for 90 s. The targeted grain boundaries were selected based on the EBSD orientation measurements and marked by Pt deposition before they were lifted-out (Fig. S1 b).
To guide the site-specific preparation process and ensure a proper position of the targeted grain boundary within the APT tips, transmission Kikuchi diffraction EBSD (TKD-EBSD)
in conjunction with FIB milling at 30 kV with a current of 5.5 nA was employed as shown in Fig. S1 c for thinning steps between 750 and 300 nm inner diameters. After the final low-energy milling step at 2 kV, the targeted GB was ~ 200 nm away from the top of the tip.
Atomistic simulations
The atomistic simulations were performed using the open-source MD software package LAMMPS [1]. The interatomic interactions were modeled by the modified embedded atom method (MEAM) potential for Mg-Nd by Kim et al. [2]. The material properties of the MEAM potential were benchmarked, particularly the per-site segregation energies at the selected grain boundaries. The results are in good agreement with the experimental and ab-initio data (as presented in Table S1) [3][4][5][6]. In addition, we calculated the per-site segregation intractable. An alternative way to optimize the general GB structures is to vary the deletion distance of overlapping atoms near the interface [9]. Since we focus on per-site segregation energies and local GB structures instead of global GB properties, only one deletion distance was chosen in this work. A reasonable distribution of local site environments, which samples the space of possible environments similar to the minimum energy GB is expected [9]. A schematic of the cylindrical setup is shown in Fig. S4. The top and bottom layers of the cylinder with a thickness of 1.2 nm (2 times interatomic potential cutoff) were fixed in the z-direction. The outermost layers of the cylindrical surface with a thickness of 1.2 nm were fixed in x and y directions. Periodic boundary conditions were applied in the z-direction and a vacuum layer with a thickness of 4.8 nm was imposed between the periodic images. where ( ) ( ) is the coordinate of th atom in a Mg system with a GB, ( ) ( ( )) is the coordinate of th atom in the Mg system with a Nd solute at the GB.
The predicted segregation energy was calculated using the linear elastic model [6]: where ∆ is the difference between Voronoi volume of the th GB site and the bulk ( bulk / ), is the host bulk modulus, bulk X is the volume of the bulk with the solute.
Fig. 1
1(a) presents elemental distribution maps of Mg, Mn and Nd atoms from a reconstructed APT tip containing two GB segments with 59.3° [3 ̅ 121] and 86.5° [123 ̅ 1]
Fig. 1 .
1(a) Reconstructed 3D atom distribution maps of Mg, Mn and Nd; (b) solute concentration profiles across the two measured grain boundaries, extracted from cylindrical region of interests ROI 1 and ROI 2 (diameter of 30 nm with a fixed bin size of 0.8 nm) outlined in (a); (c) Variation of the averaged Nd peak concentration in the GB plane for several measured boundaries denoted here by their misorientation angles for simplicity.
the bulk concentration (0.2 ± 0.1 at.%), the GB concentration of Nd was ~15 times higher. To quantify the solubility of Nd atoms at the GB, the in-plane Nd concentrations in the middle of the GB (dashed line in 2D concentration map, Fig. S2(d)) were averaged for each of the six measured GBs. An overview of the averaged peak solute concentrations shown in Fig. 1(c) illustrates the strong variations in the solute concentrations at the GBs with different macroscopic characters.
Fig. 2 .
2(a) Distribution map of Nd atoms, (b) 1.0 at. % Nd iso-concentration surfaces and (c) 2Dconcentration map of Nd in x-z plane of 59.3° and 86.5° GBs in ROI 3 as indicated in Fig. 1(a). Experimental 2D-concentration density maps of in-plane (x-y plane) GB segregation for (d) 11.3° GB in ROI 2 as indicated in Fig. S2(c); (e) 59.3° and (f) 86.5° GBs in the regions highlighted in (c). To understand the reason for the observed inhomogeneous GB segregation behavior, atomistic simulations on general GBs with experimentally informed characteristics (crystallographic orientations, misorientation angles and GB plane directions) were performed. Due to the low segregation trend of Mn atoms (Figs. 1 and S2), only substitutional Nd segregation at individual GB sites was investigated excluding solute-solute interactions. Two general GBs were selected for the calculations, i.e. 11.3° low angle grain boundary (LAGB) and 59.3° high angle grain boundary (HAGB). A third 86.5° HAGB was also considered, as shown in the supplementary material.
Fig. 3 .
3Atomistic simulations of substitutional Nd segregation to (a) 11.3° and (b) 59.3° GBs. Boxplots of segregation energy as a function of distance from the GB center. The data is divided into bins of 1 Å. The upper and lower whisker ends represent maximum and minimum values, respectively. The upper and lower bounds of the boxes are third and first quartiles, respectively. The horizontal lines in the boxes indicate median values. The blue dots represent the mean values.
Fig. 4
4displays heat maps of the segregation energy density indicating the in-plane distribution of per-site segregation energies within the two simulated GBs. The 11.3°
LAGB, the regions with high segregation energy density correlate well with the lines and junctions of misfit dislocation networks (Figs. 4 (a) and S6). Most misfit dislocations are of screw character, since the LAGB is a twist-like GB. This observation agrees with the atomistic simulations from Seki et al. on face-centered cubic [001] twist GBs [38, 39]. Both simulated GBs show a strong inhomogeneous distribution of segregation energy within the GBs plane.
Fig. 4 .
4Scatter plots of segregation energy density of the GBs (a, b) and mean squared displacement (MSD) of the GBs with a substitutional Nd atom with respect to the pure Mg counterparts (c, d). The substitution region was divided into 2 Å × 2 Å × 20 Å bins. Only GB sites within a distance of 10 Å from the GB center are calculated. For each bin, mean values of seg and MSD were obtained. The segregation energy density of each bin was calculated by dividing the mean seg by the volume of the bin (80 Å 3 ). The GB dislocations in 11.3° LAGB are color coded according to their local character (red: screw; blue: edge).
high mean squared displacement (MSD) values correlates well with those of high segregation energy density. For more details of the MSD method, see Supplementary Material. The MSD value at a particular GB site represents the displacements of all the atoms of the sample upon the replacement of the host element by a solute atom at the particular GB site. This indicates the magnitude of local structural rearrangement induced by the introduction of a substitutional solute. A high MSD value suggests that the GB structure can reorganize itself to adapt to the substitutional solute, especially for solute atoms with much larger atomic radius (Nd) than the host element (Mg). This local structural rearrangement renders the site favorable for substitutional segregation. Many
Fig. 2 (
2a~c) with the high segregation around the triple junction of two boundaries.
magnesium alloy was investigated by atom probe tomography. Complementary atomistic simulations of per-site segregation energies and mean square displacement of atoms revealed that the inhomogeneous segregation behavior (within the boundary planes and among the different boundaries) originates from the local atomic arrangement within the structural units of grain boundaries. Our results can contribute to improving the understanding of the role of inhomogeneous segregation on GB mobilities in magnesium alloys. de Lorraine, CNRS, Arts et Métiers, LEM3, 57070 Metz, France 4 Labex Damas, Université de Lorraine, 57070 Metz, France
Fig. S1
S1Illustration of the TKD-EBSD guided FIB milling for site-specific APT sample preparation: (a) EBSD confidence index (CI) map to select the targeted GBs; (b) secondary electron (SE) image of a targeted grain boundaries from (a) marked by Pt deposition; (c) TKD-EBSD maps during the milling process taken at 30 kV illustrating the misorientation information of grain boundary and the depth of the boundary from top of the tip.
Fig. S2
S2Reconstructed 3D APT tip of the MN11 alloy with a long segment of a detected GB milled by guidance of TKD shown in Fig. S: (a) elemental distributions of Mg, Mn and Nd; (b) atom probe mass spectrum, showing distinct peaks of Nd atoms; (c) concentration profile across the GB, extracted from a cylindrical ROI 1(diameter of 15 nm with a fixed bin size of 0.8 nm) outlined in (a); (d) atom distribution (Mg, Mn and Nd) and 2D Nd concentration along X-axis shown in the atom extracted from a cubic ROI 2 outlined in (a).
energy of Nd solute at Σ7 21.8° {123 ̅ 0}<0001> grain boundary, {101 ̅ 1} and {101 ̅ 2} twin boundaries and compared with the ab-initio calculations[6, 7]. The results shown inFig. S3prove the potential can describe the segregation trend of Nd solute at the Mg grain boundaries in reasonable agreement with the ab-initio results.The atomistic configurations of general GBs were constructed using Atomsk[8] based on the experimentally-informed crystallographic orientations of two grains and the GB plane (normal to the z-axis) obtained from TKD grain mapping and reconstructed APT tips. The detailed information the boundaries are listed inTable S2. Two crystals were merged along the z-axis and positioned in a cylindrical configuration with a diameter of 22.4 nm equal to the height. The cylinder axis was aligned with the z-axis of the crystals. The GB sitting in the middle of the sample was then lifted out. One atom from each pair of atoms within a distance of separation of 2 Å (62.5% of first nearest neighbor distance) was considered as an overlapping atom and was deleted. Since there is no periodic repeat distance for general GBs, the in-plane scanning over all possible translations of one crystal relative to the other, which is routinely applied in the study of CSL GBs, is
Fig. S4
S4(a) Schematic illustration of the cylindrical setup for the atomistic simulations. (b) Atomistic configuration of a 11.3° GB in Mg. Only atoms close to the grain boundary with hydrostatic stress | |> 0.2 GPa are shown. The half-transparent isosurface is constructed using the filtered atoms (| |> 0.2 GPa) with color-coding transferred from the atomic hydrostatic stress. The surface mesh of the simulation sample is half-transparent. (c) Zoomed-in view of the atoms in the substitution region (80 Å × 80 Å × 20 Å) in the center of the simulation sample colored by the segregation energies of corresponding GB sites. Other filtered atoms (| |> 0.2 GPa) and the isosurface are half-transparent.To characterize the structural rearrangement after each Nd substitution, the per-site mean squared displacement (MSD) was calculated by:
Fig. S5
S5Scatter plots of the atomic hydrostatic stress at the (a) 11.3°, (b) 59.3° and (c) 86.5° GBs. The data is divided into 2 Å × 2 Å bins. Only GB sites within a distance of 10 Å from the center of the GB are calculated. The substitution regions are outlined by dashed rectangles.
Fig. S6
S6Scatter plots of segregation energy density and predicted segregation energy density from the linear elastic model of different GBs: (a) and (d) 11.3°; (b) and (e) 59.3°; (c) and (f) 86.5°. The dashed lines in (a) represent misfit dislocation networks at 11.3° GB. The data is divided into 2 Å × 2 Å bins. Only grain boundary sites within a distance of 10 Å from the center of the grain boundary are calculated.
Fig. S7
S7Correlation between segregation energies and the predicted segregation energies from the linear elastic model of the (a) 11.3°, (b) 59.3° and (c) 86.5° grain boundaries.
Table S1
S1Potential properties of Mg calculated using the MEAM potential.Fig. S3 Atomic structures of (a) T-type Σ7 21.8° {123 ̅ 0}<0001> grain boundary, (b) 56.2° {101 ̅ 1}<101 ̅ 2> twin boundary and (c) 86.2° {101 ̅ 2}<101 ̅ 1> twin boundary. Atoms in different atomic layers are distinguished by color and size. (d) Per-site segregation energies at the boundaries in (a-c). Density functional theory (DFT)values are replotted using the data from[6] and[7].Properties
Experiment/ab-initio
MEAM
a0 (Å)
3.209[3]
3.209
c0 (Å)
5.211[3]
5.197
C11 (GPa)
63.5[4]
62.9
C12 (GPa)
25.9[4]
26.0
C13 (GPa)
21.7[4]
21.2
C33 (GPa)
66.5[4]
69.7
C44 (GPa)
18.4[4]
17.1
C66 (GPa)
18.8[4]
18.4
ESF_I1 (mJ/m 2 )
8.1[4]
15.1
ESF_I2 (mJ/m 2 )
21.8[5]
30.0
ETB_{10-11} (mJ/m 2 )
85.5[5]
85.6
ETB_{10-12} (mJ/m 2 )
118.1[5]
144.0
EGB_Σ7 (mJ/m 2 )
298[6]
335.2
Table S2
S2Orientation and misorientation data of the simulated grain boundaries.Mis. angle (°) Mis. axis
Orientation (Grain1)
Euler Angle
Orientation (Grain2)
Euler Angle
GB plane
(in Grain1)
GB plane
(in Grain2)
11.3
[101 ̅ 0]
(174.4 133.7 1.6)
(185.9 141.9 69.6)
(11 ̅̅̅̅ 474 ̅ )
(8 ̅ 113 ̅ 6 ̅ )
59.3
[3 ̅ 121]
(3.5 90.0 327.7)
(46.4 104.3 144.8)
(707 ̅ 10 ̅̅̅̅ )
(1 ̅ 1 ̅ 26)
79.3
[1 ̅ 2 ̅ 30]
(123.3 90.8 331.9) (196.4 141.7 243.6)
(7 ̅ 071 ̅ )
(93 ̅ 6 ̅ 11 ̅̅̅̅ )
86.5
[123 ̅ 1]
(46.4 104.3 144.8) (199.0 127.0 310.4)
(2 ̅ 021)
(1 ̅ 011)
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Single crystal elastic constants and calculated aggregate properties. G Simmons, H Wang, 34Single Crystal Elastic Constants & Calculated Aggregate PropertiesG. Simmons, H. Wang, Single crystal elastic constants and calculated aggregate properties, Single Crystal Elastic Constants & Calculated Aggregate Properties 34 (1971).
First-principles calculations of twin-boundary and stacking-fault energies in magnesium. Y Wang, L Chen, Z Liu, S N Mathaudhu, Scripta Mater. 629Y. Wang, L. Chen, Z. Liu, S.N. Mathaudhu, First-principles calculations of twin-boundary and stacking-fault energies in magnesium, Scripta Mater. 62(9) (2010) 646-649.
Atomistic simulations of the interaction of alloying elements with grain boundaries in Mg. L Huber, J Rottler, M Militzer, Acta Mater. 80L. Huber, J. Rottler, M. Militzer, Atomistic simulations of the interaction of alloying elements with grain boundaries in Mg, Acta Mater. 80 (2014) 194-204.
First-principles study of the solute segregation in twin boundaries in Mg and possible descriptors for mechanical properties. Z Pei, R Li, J F Nie, J R Morris, Mater. Des. 165Z. Pei, R. Li, J.F. Nie, J.R. Morris, First-principles study of the solute segregation in twin boundaries in Mg and possible descriptors for mechanical properties, Mater. Des. 165 (2019).
Atomsk: A tool for manipulating and converting atomic data files. P Hirel, Comput. Phys. Commun. 197P. Hirel, Atomsk: A tool for manipulating and converting atomic data files, Comput. Phys. Commun. 197 (2015) 212-219.
Ab initio modelling of solute segregation energies to a general grain boundary. L Huber, B Grabowski, M Militzer, J Neugebauer, J Rottler, Acta Mater. 132L. Huber, B. Grabowski, M. Militzer, J. Neugebauer, J. Rottler, Ab initio modelling of solute segregation energies to a general grain boundary, Acta Mater. 132 (2017) 138-148.
| {'fraction_non_alphanumeric': 0.0614148464090302, 'fraction_numerical': 0.04088578691137822, 'mean_word_length': 4.3292388933532315, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In magnesium alloys with multiple substitutional elements, solute segregation at grain boundaries (GBs) has a strong impact on many important material characteristics, such as GB energy and mobility, and therefore, texture. Although it is well established that GB segregation is inhomogeneous, the variation of GB solute composition for random boundaries is still not understood. In the current study, atomic-scale experimental and simulation techniques were used to investigate the compositional inhomogeneity of six different GBs. Three-dimensional atom probe tomography results revealed that GB solute concentration of Nd in Mg varies between 2 to 5 at.%. This variation was not only seen for different GB orientations but also within the GB plane. Correlated atomistic simulations suggest that the inhomogeneous segregation behavior observed experimentally stems from local atomic rearrangements within the GBs and introduce the notion of potential excess free volume in the context of improving the prediction of per-site segregation energies.The drive towards energy saving and environmental protection is attracting significant interest from research laboratories worldwide to develop innovative and costeffective materials and processes for lightweight structural components. Owing to their high specific strength and stiffness, magnesium alloys are promising candidates to improve fuel economy and support a sustainable lower carbon vehicle technology [1-3].Compared to steel or aluminum alloys, processing of magnesium alloys poses challenges related to limited strength, sharp crystallographic textures and plastic anisotropy [4-6], which hinder their widespread commercial usage as rolled products. Overcoming these challenges requires a combination of well-informed processing and micro-alloying strategies in order to depart from basal textures and obtain a favorable alignment of basal planes with the principal deformation direction. This should be combined with grain refinement to increase the material strength, reduce the activity of deformation twinning and promote additional intergranular deformation mechanisms.Mg alloys containing small additions of rare earth (RE) elements demonstrate remarkable qualitative alterations of the sheet texture along with reduced mechanical anisotropy, and thereby, promoted formability [7][8][9][10][11][12]. The effectiveness of RE addition for texture softening depends on the type and concentration of the added RE, and can be further increased by co-addition of Zn or Mn. This is known to result in textures with a quadrupole characteristic, i.e. with a distribution of basal poles along the rolling and transverse sheet directions that are favorable for sheet metal forming [7,[13][14][15]. From a synergistic perspective, multiple solute species with smaller and larger atomic sizes than', 'arxivid': '2201.02884', 'author': ['Risheng Pei \nInstitut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany\n', 'Zhuocheng Xie \nInstitut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany\n', 'Sangbong Yi \nInstitute of Materials and Process Design\nHelmholtz-Zentrum Hereon\n21502GeesthachtGermany\n', 'Sandra Korte-Kerzel \nInstitut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany\n', 'Julien Guénolé \nArts et Métiers\nUniversité de Lorraine\nCNRS\nLEM3, 57070MetzFrance\n\nLabex Damas\nUniversité de Lorraine\n57070MetzFrance\n', 'Talal Al-Samman \nInstitut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany\n'], 'authoraffiliation': ['Institut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany', 'Institut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany', 'Institute of Materials and Process Design\nHelmholtz-Zentrum Hereon\n21502GeesthachtGermany', 'Institut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany', 'Arts et Métiers\nUniversité de Lorraine\nCNRS\nLEM3, 57070MetzFrance', 'Labex Damas\nUniversité de Lorraine\n57070MetzFrance', 'Institut für Metallkunde und Materialphysik\nRWTH Aachen University\nD-52056AachenGermany'], 'corpusid': 254096542, 'doi': '10.1016/j.scriptamat.2023.115432', 'github_urls': [], 'n_tokens_mistral': 15089, 'n_tokens_neox': 12709, 'n_words': 7179, 'pdfsha': 'f0ddff9f8b4c39f780d46e75dee294c28fe4d5b9', 'pdfurls': ['https://export.arxiv.org/pdf/2201.02884v3.pdf'], 'title': ['Atomistic insights into the inhomogeneous nature of solute segregation to grain boundaries in magnesium', 'Atomistic insights into the inhomogeneous nature of solute segregation to grain boundaries in magnesium'], 'venue': []} |
arxiv |
Absolutely continuous copulas with prescribed support constructed by differential equations, with an application in toxicology
May 24, 2019
Oscar Björnham [email protected]
FOI CBRN Defence and Security
Mathematics Department
Niklas Brännström FOI CBRN Defence and Security
Umeå University
Leif Persson [email protected]
FOI CBRN Defence and Security
Mathematics Department
Niklas Brännström FOI CBRN Defence and Security
Umeå University
Absolutely continuous copulas with prescribed support constructed by differential equations, with an application in toxicology
May 24, 2019
A new method for constructing absolutely continuous two-dimensional copulas by differential equations is presented. The copulas are symmetric with respect to reflection in the opposite diagonal. The support of the copula density may be prescribed to arbitrary opposite symmetric hypographs of invertible functions, containing the diagonal. The method is applied to toxicological probit modeling, where new compatibility conditions for the probit parameters are derived.
Introduction and main results
This paper is motivated by the following result, which is probably well known, although we have not been able to find any explicit statement or proof: Proposition 1.1. Suppose that a, ∆ ∈ R, a > 0. Then there exists random variables X, Y satisfying Y ≤ aX + ∆ and X, Y standard normal (1) if and only if a = 1 and ∆ ≥ 0, and then if ∆ > 0, there exists X, Y with absolutely continuous joint distribution satisfying (1).
A proof is given at the end of this section. Our interests in this result comes from applications in toxicological probit modeling, accounted for in Section 7 where we prove new compatibility conditions for toxicological probit models. For simulation purposes, we are also interested in constructing absolute continuous distributions of Proposition 1.1: Problem 1.2. Given a number ∆ > 0, construct a pair of standard normal random variables X, Y with absolutely continuous joint distribution supported on y ≤ x + ∆.
This seems to be a very simple and basic problem in probability theory, but to our surprise we could not find any simple constructions in the literature. Independent standard normal X, Y have absolutely continuous joint distribution but do not fullfill the support condition, and truncating to y ≤ x+∆ yields nonnormal marginals. It is easy to construct singular solutions to the problem, the simplest being X = Y . The difficulty lies in imposing the absolute continuity. We reduce Problem 1.2 to a problem of the dependence structure, or copula of (X, Y ). Before we state our main result, let us briefly review the main facts about copulas.
A function C : [0, 1] 2 → [0, 1] is said to be a copula if C(u, 0) = C(0, v) = 0, C(u, 1) = u, C(1, v) = v and C(u 2 , v 2 ) − C(u 2 , v 1 ) − C(u 1 , v 2 ) + C(u 1 , v 1 ) ≥ 0 for all u, v, u 1 , v 1 , u 1 , v 2 ∈ [0, 1] such that u 1 ≤ u 2 , v 1 ≤ v 2 , cf. [18,Definition 2.2.2]. By Sklar's theorem ( [18,Theorem 2.3.3]), the cumulative distribution function (CDF) F X,Y of any bivariate random variable (X, Y ) is representable by the marginal CDF's F X , F Y and a copula C as F X,Y (x, y) = C(F X (x), F Y (y)).
(
This may be regarded as a change of variables X = F −1 X (U ), Y = F −1 Y (V ) such that (U, V ) has uniform marginals. The copula C is uniquely defined on Range(F X )×Range(F Y ) for all bivariate random variables (X, Y ), and if F X , F Y are continuous, C is uniquely defined on [0, 1] 2 . Morover, the partial derivatives C u , C v , C uv of a copula C(u, v) are defined almost everywhere on [0, 1] 2 ([18, Theorem 2.2.7]) and C uv ≥ 0. If C uv dudv = 1, C is said to be absolutely continuous. Copulas are common in statistical modeling, in particular mathematical finance. The main benefit of copulas is that by Sklar's theorem, the marginal statistics and dependence structure can be modeled separately. For an introduction to copulas we refer to [18], for a recent review see [9].
Returning to Problem 1.2, the half-plane {(x, y) : y ≤ x + ∆} is symmetric with respect to reflection (x, y) → (−y, −x) through the line x + y = 0. Therefore, we assume that (X, Y ) and (−Y, −X) are equal in distribution. Moreover, X, −X, Y, −Y are all identically distributed so it follows (from Theorem 2.4 below) that the copula C(u, v) of (X, Y ) is opposite symmetric, according to the following definition. Definition 1.3. A copula C is said to be opposite symmetric if
C(u, v) = C(1 − v, 1 − u) + u + v − 1(3)
for all (u, v) ∈ [0, 1] 2 .
Opposite symmetry means symmetry with respect to reflection (u, v) → (1 − v, 1 − u) in the opposite diagonal u + v = 1, and was introduced in [5]. Applying the copula transformation, using the standard normal CDF Φ:
u = Φ(x), v = Φ(y), F X,Y (x, y) = C(u, v),(4)
Problem 1.2 reduces to finding an absolutely continuous opposite symmetric copula
C(u, v) with density supported on {(u, v) ∈ [0, 1] 2 : v ≤ H(u)} where H(u) = Φ(Φ −1 (u) + ∆).(5)
Our main result is the construction of C(u, v) in the following Theorem 1.4. We want to emphasize its simplicity, involving H and its inverse explicitly. The crucial part is the evaluation of the integral in (10), which is suitable for numerical integration if not analytically integrable.
H(u) + H −1 (1 − u) = 1 and H(u) ≥ u, u ∈ [0, 1],(6)u 0 ∈ (0, 1/2) and H(u 0 ) = 1 − u 0 ,(7)u u0 dz H(z) − z < ∞, u ∈ [u 0 , 1)(8)
and
lim u 1 u u0 dz H(z) − z = ∞.(9)
Let
G(v) = exp − 1−v u0 dz H(z) − z , v ∈ [0, 1 − u 0 ],(10)K(u) = H(u) − u G(1 − u) − 1 + 2u 0 , u ∈ [u 0 , 1](11)
and
F (u) = (1 − 2u 0 )(1 − G(1 − u)), u ∈ [u 0 , 1].(12)
Define C(u, v) by
1. If 0 < u ≤ u 0 and 0 ≤ v ≤ H(u), then C(u, v) = H −1 (v) + (K(1 − v) − K(1 − H(u)))G(v).(13)
2. If 0 < u ≤ u 0 and H(u) < v ≤ 1 − u then
C(u, v) = u(14)
3. If u 0 < u < 1 and 0 ≤ v ≤ 1 − u then
C(u, v) = H −1 (v) + (K(1 − v) + F (u))G(v).(15)
4. If 0 < u < 1 and u + v > 1 then C(u, v) is defined by (3).
Then C(u, v) is an absolutely continuous opposite symmetric copula with probability density supported on v ≤ H(u).
Note that the hypograph v ≤ H(u) is opposite symmetric if and only if (6) holds true. The copula is piecewisely defined, on parts of the unit square depicted in Figure 1. Theorem 1.4 is proved at then end of Section 5. Before that, we develop a theory for construction of opposite symmetric copulas by differential equations in Section 3 and Section 5, which we believe is of interest in its own right, and gives in fact a much larger class of copulas than Theorem 1.4. In section 4 we compare our method to two other methods in the literature, Durantes and Jaworskis construction of absolutely continuous copulas with given diagonal section [6], and Jaworskis characterization of copulas using differential equations [14]. In Section 6 we adapt our differential equation method to sampling from the copula. We conclude the paper with section 7, an application in toxicological probit modeling, where new compatibility conditions for the probit coefficients are derived. Example 1.5. In this example we construct a solution to Problem 1.2 using Theorem 1.4. Let Φ be the standard normal CDF, φ(x) = Φ (x) the standard normal probability density function (PDF), ∆ > 0 and H given by (5). Then (6) is satisfied, and
H −1 (v) = Φ(Φ −1 (v) − ∆) and because of the symmetries Φ(x) + Φ(−x) = 1, Φ −1 (u) + Φ −1 (1 − u) = 0, conditionu 0 = Φ(−∆/2).(16)
Moreover, with the change of variables z = Φ(w) and the mean value theorem we obtain
u u0 dz H(z) − z = Φ −1 (u) −∆/2 φ(w)dw Φ(w + ∆) − Φ(w) = Φ −1 (u) −∆/2 φ(w)dw φ(w + θ(w)∆) = 1 √ 2π Φ −1 (u) −∆/2 exp w∆θ(w) − ∆ 2 θ(w) 2 2 dw(17)
for some function θ(w) with 0 ≤ θ(w) ≤ 1, so
1 √ 2π∆ e ∆Φ −1 (u) − e ∆ 2 /2 ≥ u u0 dz H(z) − z ≥ e −∆ 2 /2 √ 2π Φ −1 (u) + ∆ 2(18)
which proves that conditions (8) and (9) are satisfied. The function G defined by equation (10) can not be expressed in terms of special functions (to our knowledge), but can be determined by numerical integration, and C(u, v) is then determined by equations (3) and (13)- (15). The density of C is illustrated in figure 1.5. The joint PDF of (X, Y ) is given by
p(x, y) = C uv (Φ(x), Φ(y))φ(x)φ(y)(19)
and is illustrated in For ∆ ≥ 0 we can take X = Y , which gives a singular distribution supported on x = y. If ∆ > 0, Example 1.5 shows that X, Y with absolutely continuous joint distribution exists.
Symmetries and copulas
Several notions of bivariate symmetries are considered in [17]. A pair of random variables(X, Y ) are said to be exchangeable if (X, Y ) and (Y, X) are equal in distribution, and (X, Y ) is exchangeable if and only if its copula C(u, v) is a symmetric function, i.e., C(u, v) = C(v, u). Moreover, (X, Y ) is said to be
radially symmetric about (a, b) ∈ R 2 if (X − a, Y − b) and (a − X, b − Y ) are equal in distribution, or equivalently, F X,Y (a + x, b + y) = 1 − F X (a − x) − F Y (b − y) + F X,Y (a − x, b − y) (20) Also, (X, Y ) is said to be marginally symmetric about (a, b) ∈ R 2 if F X (a + x) = 1 − F X (a − x) and F Y (b + y) = 1 − F Y (b − y).(21)
The following theorem is proved in [17, Theorem 3.2]:
Theorem 2.1. Suppose (X, Y ) is marginally symmetric about (a, b) with copula C. Then (X, Y )
is radially symmetric about (a, b) if and only if C satisfies the functional equation
C(u, v) = C(1 − u, 1 − v) + u + v − 1 (22)
There is a corresponding class of bivariate random variables associated to opposite symmetric copulas, which we propose to call opposite radially symmetric variables, in accordance with the terminology in [5], and analogous to the radially symmetric variables of [17].
Definition 2.2. The bivariate random variable (X, Y ) is said to be opposite radially symmetric about (a, b) ∈ R 2 if (a + X, b + Y ) and (b − Y, a − X) are equal in distribution, or, equivalently, F X,Y (a + x, b + y) = 1 − F X (a − y) − F Y (b − x) + F X,Y (a − y, b − x). (23)
We need to replace marginal symmetry with the following analog of (20):
Definition 2.3. The bivariate random variable (X, Y ) is said to be opposite marginally symmetric about (a, b) ∈ R 2 if F X , F Y satisfy F X (a + x) = 1 − F Y (b − x) and F Y (b + y) = 1 − F X (a − y)(24)
for all x, y.
Remark. If X, Y are identically distributed and marginally symmetric about (a, a) ∈ R 2 , then (X, Y ) is opposite marginally symmetric about (a, a). There are no identically distributed opposite marginally symmetric (
X, Y ) about (a, b) if b = a, since then the common CDF F X = F Y = F would satisfy F (x) = F (x + b − a) for all x.
We have the following analog of Theorem 2.1:
Theorem 2.4. Suppose that (X, Y ) is opposite marginally symmetric about (a, b) ∈ R 2 with copula C, and suppose that F X , F Y are continuous. Then (X, Y ) is opposite radially symmetric about (a, b) if and only if C is opposite symmetric.
Proof. It follows from equations (23) and (24) that (X, Y ) is opposite radially symmetric if and only if
C(1 − F Y (b − x), 1 − F X (a − y)) = C(F X (a + x), F Y (b + y)) = 1 − F X (a − y) − F Y (b − x) + C(F X (a − y), F Y (b − x)). (25)
Since the range of F X and F Y is [0, 1] this proves the theorem.
Remark. There is an erroneous statement in [5,
Remark 1] that if C is opposite symmetric, then (X, Y ) and (1 − Y, 1 − X) are equal in distribution, i.e., (X, Y )
is opposite radially symmetric about (1/2, 1/2), but additional assumptions like opposite marginal symmetry in Theorem 2.4 is needed to draw that conclusion.
Differential equations for copulas with opposite symmetry
The following theorem provides a characterization of absolutely continuous copulas with opposite symmetry, and constitutes the basis for deriving the differential equations. We also obtain a simple formula for Kendall's τ rank correlation coefficient for opposite symmetric copulas. Kendall's τ is defined as
τ C = −1 + 4 1 0 1 0 C(u, v)dC(u, v), cf [18, chapter 5]. Theorem 3.1. Assume that p is an integrable function on [0, 1] 2 satisfying p(u, v) = p(1 − v, 1 − u)(26)
and let
C(u, v) = u 0 v 0 p(w, z)dwdz.(27)
Then
C(u, v) = C(1 − v, 1 − u) + C(u, 1) + C(1, v) − C(1, 1)(28)
and the following two conditions are equivalent:
1. C u (u, 1) = 1 for all u ∈ [0, 1]. 2. C v (1, v) = 1 for all v ∈ [0, 1].
Furthermore, if p ≥ 0 these conditions are equivalent to 3. C is an absolutely continuous opposite symmetric copula. and then if also
1 0 1 0 C u C v dudv < ∞,(29)
Kendall's τ is given by
τ C = −1 + 8 1 0 C(u, 1 − u)du(30)
Proof of Theorem 3.1. By the inclusion-exclusion principle for integrals we have
1 u 1 v p(w, z)dzdw = C(u, v) + C(1, 1) − C(u, 1) − C(1, v).(31)
By change of variables and symmetry (26) we also have
1 u 1 v p(w, z)dzdw = 1−v 0 1−u 0 p(1 − z, 1 − w)dzdw = 1−v 0 1−u 0 p(w, z)dzdw = C(1 − v, 1 − u). (32) which proves (28). Assume that C u (u, 1) = 1 for u ∈ [0, 1], it follows that C(u, 1) = u for u ∈ [0, 1]. Then (28) with u = 0 simplifies to 0 = C(1, v) − v, so C v (1, v) = 1. Similarly, C v (1, v) ≡ 1 =⇒ C u (u, 1) ≡ 1. If these conditions hold, C(u, 1) ≡ u and C(1, v) ≡ v,
which shows that C is a copula, which is absolutely continuous by equation (27), and equation (28) implies equation (3), i.e., opposite symmetry. Conversely, assuming C an absolute continuous copula satisfying (3), differentiation yields C u (u, 1) ≡ 1 and C v (1, v) ≡ 1. Suppose in addition that (29) holds true. Differentiation of (3) yields
C u (1 − v, 1 − u) = 1 − C v (u, v), C v (1 − v, 1 − u) = 1 − C u (u, v)(33)
which gives
1 0 1 0 C u C v dvdu = 1 0 1−u 0 C u C v dvdu + 1 0 1 1−u C u C v dvdu = 1 0 1−u 0 C u C v + (1 − C v )(1 − C u )dvdu = 1 0 1−u 0 1 − C u − C v dvdu = 1 2 − 2 1 0 C(u, 1 − u)du (34)
According to [18, equation (5.1.10)], equation (29) implies that τ C = 1 − 4
1 0 1 0 C u C v dudv,
which proves (30). We will now show that copulas satisfying the assumptions in Theorem 2.4, with the additional assumption of being conditionally independent on u + v ≤ 1 can be characterized by differential equations. This method is reminiscent of the well known method of separation of variables for construction of solutions to partial differential equations. This will also give a construction method for absolutely continuous copulas with given opposite diagonal section, a problem considered in [5], cf. Theorem 3.7 below. Later, we will modify the construction, restricting the copula density support to v ≤ H(u), which is required to solve Problem 1.2.
Theorem 3.2. Assume that p(u, v) = F (u)G (v) if u + v ≤ 1 F (1 − v)G (1 − u) if u + v > 1 (35)
where F (0) = G(0) = 0, G ≥ 0 and C is given by (27). Then
C u (u, v) = F (u)G(v) if u + v ≤ 1 G(1 − u)F (u) + G (1 − u)(F (u) − F (1 − v)) if u + v > 1
(36) and the following are equivalent:
1. F ≥ 0 and G(1 − u)F (u) + G (1 − u)F (u) = 1, u ∈ [0, 1] (37) 2. C(u, v)
is an absolutely continuous copula, and then
C(u, v) = F (u)G(v) if u + v ≤ 1 F (1 − v)G(1 − u) + u + v − 1 if u + v > 1 (38) Proof. Integration C u (u, v) = v 0 C uv (u, z)dz of the piecewise defined function p = C uv yields C u (u, v) = F (u)G(v) for u + v ≤ 1 and C u (u, v) = G(1 − u)F (u) + G (1 − u)(F (u) − F (1 − v)) for u + v > 1, so C u (u, 1) = G(1 − u)F (u) + G (1 − u)F (u)
. Suppose that F ≥ 0 and (37) holds true. Then p ≥ 0 and C u (u, 1) ≡ 1 so C is an absolutely continuous copula by Theorem 3.1. Conversely, suppose that C is an absolutely continuous copula. Then C uv = p ≥ 0 so F ≥ 0 by (35), and (37) holds since C u (u, 1) ≡ 1. Moreover, integration C(u, v) = u 0 C u (z, v)dz yields (38) for u + v ≤ 1, and (38) for u + v > 1 follows from Theorem 3.1.
The differential equation (37) can be solved with the integrating factor method. Moreover, a condition for F (u) ≥ 0 can be derived.
F (u) = G(1 − u) u 0 dz G(1 − z) 2 (39) Moreover, if F (u) is given by (39), then F (u) = G (1 − u) L(0) G(1) 2 + u 0 1 + L (z) G(1 − z) 2 dz (40) where L(u) = G(1 − u) G (1 − u) .(41)Finally, if u * ∈ [0, 1], L (u) ≥ −1 for u ∈ (u * , 1) and if − u * 0 1 + L (z) G(1 − z) 2 dz ≤ L(0) G(1) 2 (42) then F (u) ≥ 0 for u ∈ (0, 1).
Proof. Equation (39) is obtained by multiplying (37) with the integrating factor
1/G(1 − u) 2 . Equation (37) yields F (u) = 1 G(1 − u) − 1 L(u) F (u)(43)
and substituting (39) in (43) using (41) yields
F (u) = G (1 − u) L(u) G(1 − u) 2 − u 0 dz G(1 − z) 2(44)
and the identity d du
L(u) G(1 − u) 2 = 2 + L (u) G(1 − u) 2 (45) yields L(u) G(1 − u) 2 = L(0) G(1) 2 + u 0 2 + L (z) G(1 − z) 2 dz(46)
which proves (40). Moreover, by the assumptions,
u → − u 0 (1 + L (z))/G(1 − z) 2 dz has its maximum for u = u * , so it follows from (42) that F (u) ≥ F (u * ) ≥ 0 for u ∈ [0, 1]. Example 3.4. G(v) = v, L(u) = 1 − u, 1 + L (u) = 0, F (u) = G (1 − u)/G(1), yields the independence copula C(u, v) = uv. Example 3.5. If k ≥ 1 and G(v) = v k , then (37) has solution F (u) = (1 − u) 1−k − (1 − u) k 2k − 1(47)
and
F (u) ≥ 0 for u ∈ [0, 1], so C(u, v) = ((1 − u) 1−k − (1 − u) k )v k /(2k − 1) if u + v ≤ 1 (1 − u) k (v 1−k − v k )/(2k − 1) + u + v − 1 if u + v > 1(48)
is a one-parameter family of absolutely continuous copulas. In particular, for k = 1 we obtain the independence copula uv. For k > 1, lim u 1 F (u) = ∞. and F (u) ≥ 0 for u ∈ [0, 1], so
C(u, v) = 2 sin(πu/2) sin(πv/2)/π if u + v ≤ 1 2 cos(πu/2) cos(πv/2)/π + u + v − 1 if u + v > 1(50)
is an absolutely continuous copula.
Since the positivity conditions in Theorem 3.3 is formulated in terms of the function L, it is natural to start by specifying L satisfying (42). This is also related to the problem of constructing copulas with prescribed opposite diagonal section ω(u) = C(u, 1 − u) considered in [5]. In fact, given ω, the function L is given by the explicit formula (55) below. This is formulated in Theorem 3.7.
Theorem 3.7. Suppose that L is a positive real-valued function defined on [0, 1] such that
u 0 dz L(z) < ∞(51)
for u ∈ [0, 1) and
lim u→1− u 0 dz L(z) = ∞.(52)
Let
G(v) = exp − 1−v 0 dz L(z)(53)
and suppose that (42) holds true. Moreover, let F (u) be given by (39). Then C given by (38) is an absolutely continuous copula. Moreover, the opposite diagonal section
ω(u) ≡ C(u, 1 − u) (54) satisfies L(u) = 2ω(u) 1 − ω (u) .(55)
Proof. Clearly, because L is positive and satisfies (51) and (52), G defined by (53) is positive, G is increasing (in fact strictly increasing) and G(0) = 0. Moreover, it follows from (53) that (41) holds true. By Theorem 3.3, F (u) ≥ 0 and by Theorem 3.2, C is an absolutely continuous copula. Differentiation of
F (u)G(1 − u) = ω(u) yields F (u)G(1 − u) − F (u)G (1 − u) = ω (u)
, so in view of (37) we get
F (u)G(1 − u) = 1 + ω (u) 2(56)
and
F (u)G (1 − u) = 1 − ω (u) 2(57)
Solving for F (u) in (57), differentiating and substituting F (u) in the left hand side of (56) yields
(1 − ω (u)) G (1 − u)G(1 − u) G (1 − u) 2 − ω (u) G(1 − u) G (1 − u) = 1 + ω (u).(58)
Using (41) and the identity
G (1 − u)G(1 − u) G (1 − u) 2 = 1 + L (u)(59)
we get
(1 − ω (u))L (u) − ω (u)L(u) = 2ω (u)(60)
which is integrated to (1−ω (u))L(u) = 2ω(u)+constant. Since ω(1) = C(1, 0) = 0 and L(1) = 0 in view of (52), the integration constant is zero, which proves (55).
G(1 − u) = 1 − u 1 − au 1/(1−a)
and u * = 1/2:
L (u) = −1 + a(−1 + 2u) ≤ −1 if u ≤ 1/2, L (u) ≥ −1 if u ≥ 1/2. We obtain u * 0 1 + L (z) G(1 − z) 2 dz = 1/2 0 1 − au 1 − u 2/(1−a) a(1 − 2u)du = 1 2 F 1 1, 2 1 − a , − 2 1 − a ; 3; 1 2 , a 2
Here F 1 is the Appell series (see [10, p. 1027] for a definition), which may be represented by Picard's integral formula, cf. [4]:
F 1 (a, b, b ; c; x, y) = Γ(c) Γ(a)Γ(c − a) 1 0 t a−1 (1 − t) c−a−1 (1 − tx) −b (1 − ty) −b dt
Here, Γ denotes Euler's gamma function ([10, p. 901]). The function F 1 is available in computer algebra systems like Maple R and Mathematica R , and numerical investigation reveals that the right hand side is an increasing function of a and approaches the value 0.861485 as a → 1−. Therefore condition (42) is satisfied, so Theorem 3.3 yields an absolutely continuous copula, and (39) can be evaluated to
F (u) = uG(1 − u)F 1 1, 2 1 − a , − 2 1 − a ; 2; au, u .
When 2/(1 − a) is integer, this expression can be simplified to a finite sum of powers and logarithms, cf. [4].
Comparison with other methods
A method by Durante and Jaworski is found in [6], where absolutely continuous copulas C(u, v) with given diagonal section C(t, t) are constructed, in terms of convex combinations of singular diagonal copulas
C δ (u, v) = min u, v, δ(u) + δ(v) 2(61)
(satisfying C δ (t, t) = δ(t)). The problem with this approach for our purposes is that the constraint v ≤ H(u) imposes functional inequalities δ(H(u)) + δ(u) ≤ 2u that must be fullfilled for the δ's used in the construction. In comparison, the advantage of our differential equation method is that H is used explicitly, using only elementary calculus. Regarding copulas and differential equations, there is a characterization of all copulas by Jaworski, in terms of a certain type of weak solutions to differential equations in [14]. For comparison we give here a simplified account of his method in the special case of absolutely continuous copulas with differentiable density and sectional inverse. For fixed u ∈ [0, 1] let C(u, ·) −1 (z) denote the assumed unique solution v to the equation C(u, v) = z, i.e., C(u, C(u, ·) −1 (z)) = z for all z ∈ [0, 1], and define
C [u] (t, z) = u −1 C(ut, C(u, ·) −1 (uz))(62)
Moreover, define
F C (u, z) = ∂ ∂t C [u] (t, z) t=1 − z = C u (u, C(u, ·) −1 (uz)) − z(63)
Now suppose that for each v ∈ [0, 1], g v (u) is solution to the terminal value problem
ug v (u) = F C (u, g v (u)), u ∈ (0, 1) (64) g v (1) = v(65)
Then C can be characterized in terms of g v (u) as
C(u, v) = ug u (v)(66)
To see this, note that by the definition of F C and the product rule of differentiation, (64) is equivalent to
d du (ug v (u)) = C u (u, C(u, ·) −1 (ug v (u)))(67)
and this ODE for g v (u) is satisfied for g v (u) = C(u, v)/u, so by uniqueness of solution to (64)-(65), (66) must hold. The general result (valid for all copulas) can be found in [14, Theorems 3.1 and 3.2]. Now, applying Jaworski's characterization theorem to a copula of the form (38), we need to compute C(u, ·) −1 (z)
to obtain F C . For z ≤ 1 − u we get F (u)G(v) = z, which can be solved explicitly, yielding v = C(u, ·) −1 (z) = G −1 (z/F (u)). However, for
z > 1 − u, v = C(u, ·) −1 (z) is implicitly defined by F (1 − v)G(1 − u) + u + v − 1 = z, which
can not be solved for v in terms of F, G and their inverses. Therefore, we have not been able to use Jaworski's method to obtain equations for F, G for copulas of the type (38).
Absolutely continuous copulas with prescribed support
Here we construct absolutely continuous opposite symmetric copulas with the support of the probability measure prescribed by a constraint v ≤ H(v). The construction is simple, using elementary calculus and a piecewise definition of the copula density, similar to Theorem 3.2
Theorem 5.1. Suppose that 0 < u 0 < 1/2 and that H is a strictly increasing function defined on [0, 1], continuously differentiable on (0, u 0 ), satisfying H(u 0 ) = 1 − u 0 and satisfying the symmetry condition
H(u) + H −1 (1 − u) = 1. (68) Furthermore, suppose that F is a differentiable function defined on [u 0 , 1) such that F (u 0 ) = 0, G is a differentiable function defined on [0, 1 − u 0 ] such that G(0) = 0, G ≥ 0 and C(u, v) given by (27), where p(u, v) = G (v)/G(H(u)) if 0 < u ≤ u 0 , 0 < v ≤ H(u) 0 if 0 < u ≤ u 0 , H(u) < v ≤ 1 − u F (u)G (v) if u 0 < u < 1, 0 < v ≤ 1 − u p(1 − v, 1 − u) if 0 < u < 1, 1 − u < v < 1 (69) Furthermore, let K(u) = u u0 H (z)dz G(1 − z)(70)
for u 0 ≤ u ≤ 1. Then the following are equivalent:
1. F ≥ 0 and F (u)G(1 − u) + G (1 − u)(F (u) + K(u)) = 1 (71)
for u ∈ [u 0 , 1).
C(u, v)
is an absolutely continuous copula, and then
(a) If 0 ≤ u ≤ u 0 and 0 ≤ v ≤ H(u), then C(u, v) = H −1 (v) + (K(1 − v) − K(H −1 (1 − u)))G(v) (72) (b) If 0 ≤ u ≤ u 0 and H(u) ≤ v ≤ 1 − u then C(u, v) = u (73) (c) If u 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1 − u then C(u, v) = H −1 (v) + (K(1 − v) + F (u))G(v) (74) (d) If 0 ≤ u ≤ 1 and u + v > 1 then C(u, v) is given by (3).
Proof. The basic idea of the proof is similar to Theorem 3.2: integrate the given piecewise defined ansatz for the copula density C uv to derive C u and use Theo- u). To derive the expression in region 6, write K on the alternate form
C u = F (u)G(v), region 5: C u = F (u)G(1 − u) + (F (u) − F (1 − v))G (1 − u), region 6: C u = F (u)G(1 − u) + (F (u) + K(H −1 (v)))G (1 − u), and region 7: C u = F (u)G(1 − u) + (F (u) + K(u))G (1 −K(u) = u0 H −1 (1−u) dw G(H(w))(75)dz G(H(1 − z)) = u0 1−v dw G(H(w)) = u0 H −1 (1−H −1 (v)) dw G(H(w)) = K(H −1 (v))
in view of (68). If F ≥ 0 and (71) holds true, then p ≥ 0 by (69) and C u (u, 1) ≡ 1 by (71) since the left hand side of (71) is the expression for C u in region 7. Thus, by theorem 3.1, C is an absolutely continous copula. Conversely, if C is an absolutely continuous copula, then p = C uv ≥ 0 so F ≥ 0 and by (69), and C u (u, 1) = 1 which proves (71). The conditions C u (u, 1) ≡ 1 and C v (1, v) ≡ 1 are equivalent by Theorem 3.1. Assume now that C is an absolutely continuous copula, then C u = 1 in region 7 by (71). Integration C(u, v) = u 0 C u (z, v)dz yields the following piecewise defined function C(u, v); region 2,3,7: C = u which proves (73), region 1:
C = H −1 (v) + (K(1 − v) − K(H −1 (1 − u)))G(v)F (u) = −K(u) + G(1 − u) u u0 1 + H (z) G(1 − z) 2 dz.(76)
Moreover, if F (u) is given by (76), then
F (u) = G (1 − u) L(u 0 ) G(1 − u 0 ) 2 + u u0 1 + L (z) − H (z) G(1 − z) 2 dz (77)
where L is given by (41). Finally, if u * ∈ [u 0 , 1], L (u) − H (u) ≤ −1 for u ∈ (u 0 , u * ), L (u) − H (u) ≥ −1 for u ∈ (u * , 1) and
− u * u0 1 + L (z) − H (z) G(1 − z) 2 dz ≤ L(u 0 ) G(1 − u 0 ) 2 (78) then F (u) ≥ 0 for u ∈ (u 0 , 1).
Proof. Multiplying (71) with the integrating factor 1/G(1 − u) 2 and integrating by parts (using K(u 0 ) = 0) yields
F (u) = G(1 − u) u u0 1 − G (1 − z)K(z) G(1 − z) 2 = G(1 − u) u u0 dz G(1 − z) 2 − K(u) G(1 − u) + u u0 K (z) G(1 − z) so substituting K (z) = H (z) G(1 − z)(79)
according to (70) yields (76). Solving for F in (71):
F (u) = 1 G(1 − u) − 1 L(u) (K(u) + F (u))(80)
and substituting
K(u) + F (u) = u u0 1 + H (z) G(1 − z) 2(81)
according to (76) yields
F (u) = G (1 − u) L(u) G(1 − u) 2 − u u0 1 + H (z) G(1 − z) 2 dz(82)
The identity (45) yields
L(u) G(1 − u) 2 = L(u 0 ) G(1 − u 0 ) 2 + u u0 2 + L (z) G(1 − z) 2 dz(83)
which proves (77). Finally, by the assumptions, u → − u u0 (1+L (z)−H (z))/G(1− z) 2 dz has its maximum for u = u * , so it follows from (78) that F (u) ≥ F (u * ) ≥ 0 for u ∈ [u 0 , 1].
Example 5.3. If
H(u) = (1 − u 0 )u/u 0 if u ≤ u 0 1 − u 0 (1 − u)/(1 − u 0 ) if u > u 0 ,(84)G(v) = v k and k ≥ (1 − u 0 )/(1 − 2u 0 ), then (70) yields K(u) = ((1 − u) 1−k − (1 − u 0 ) 1−k )u 0 (1 − u 0 )(k − 1) ,(85)
(76) evaluates to 2u 0 ), in which case F (u) is positive. By theorem 5.1 we obtain a two-parameter family of absolutely continuous copulas (with parameters 0 < u 0 < 1/2 and k ≥ (1 − u 0 )/(1 − 2u 0 )), with probability density supported on v ≤ H(u). Indeed, in this example F (u) can be computed explicitly:
F (u) = (1 − 2u 0 )k − (1 − u 0 ) (2k − 1)(k − 1)(1 − u 0 ) (1 − u) 1−k − (1 − u 0 ) 1−2k (2k − 1)(1 − u 0 ) (1 − u) k + (1 − u 0 ) 1−k u 0 (k − 1)(1 − u 0 ) . (86) Moreover, L(u) = (1 − u)/k, so L (u) − H (u) = −1/k − u 0 /(1 − u 0 ) ≥ −1 if and only if k ≥ (1 − u 0 )/(1 −F (u) = ((1 − 2u 0 )k − (1 − u 0 ))(1 − u) −k + k(1 − u 0 ) 1−2k (1 − u) k−1 (2k − 1)(1 − u 0 )(87)
and is strictly positive on [u 0 , 1) if and only if the coefficient for
(1 − u) −k is positive, which is equivalent to k ≥ (1 − u 0 )/(1 − 2u 0 ).
Example 5.4. In this example we construct more solutions to Problem 1.2, using Theorem 5.2. Let k ∈ R, k > 1 and L(u) = (1 − u)/k. Then we obtain
G(v) = v k /(1 − u 0 ) k and K(u) = (1 − u 0 ) k u u0 H (z) (1 − z) k dz(88)
and
F (u) = −K(u) + (1 − u 0 ) k (1 − u) k u u0 1 + H (z) (1 − z) 2k dz(89)
where H is given by (5) and
H (z) = 1 √ 2π exp −∆ Φ −1 (u) + ∆ 2 .(90)
Since L (u) = −1/k and H decreasing we have u * satisfying the assumptions in Theorem 5.2 and determined by H (u * ) = 1 − 1/k. Solving this equation yields
u * = Φ − √ 2π ∆ 1 − 1 k − ∆ 2 .(91)
Thus, 1 − u * = Φ( √ 2π(1 − 1/k)/∆ + ∆/2), and also 1 − u 0 = Φ(∆/2), and one can show that condition (78) is equivalent to
u * u0 H (z) (1 − z) 2k dz ≤ (1 − u 0 ) 1−2k 2k − 1 + 1 − 1 k (1 − u * ) 1−2k(92)
so if k satisfies this condition, an absolutely continuous copula is obtained. We have the following analogue of Theorem 3.7. Here, given the opposite diagonal section ω, the function L is given by an integral equation (96), (97) below.
Theorem 5.5. Suppose that H, u 0 satisfies (6) and (7). Suppose also that L is a positive real-valued function defined on [u 0 , 1] such that
Let
G(v) = exp − 1−v u0 dz L(z)(95)
Moreover, let K(u) and F (u) be given by (70) and (76) and suppose that (78) holds true. Then C given by (72)-(74) and (3) is an absolutely continuous copula. Moreover, the opposite diagonal section (54) satisfies ω(u) = u for u ∈ [0, u 0 ] and
L(u) = 2ω(u) + G(1 − u)K(u) 1 − ω (u) (96) for u ∈ [u 0 , 1], where G(1 − u)K(u) = u u0 exp − u z dw L(w) H (z)dz (97)
Proof. The proof is similar to the proof of Theorem 3.7, with some additional terms involving K. More precisely, (56) and (57) are replaced by
G(1 − u)F (u) = 1 + ω (u) − G (1 − u)K(u) 2(98)
and
G (1 − u)F (u) = 1 − ω (u) − G (1 − u)K(u) 2 .(99)
Solving for F in (99), differentiating and substituting for F in the left hand side of (98) yields
F (u) = (1 − 2u 0 )G (1 − u)(101)
so F (u) satisfies (12). Solving for K in (71) yields
K(u) = 1 G (1 − u) − G(1 − u) G (1 − u) F (u) − F (u)(102)
and substituting (12) and (101) in (102) implies that K(u) satisfies (11).
Sampling
To sample from a two-dimensional copula C(u, v) we use the conditional density C u of Corollary 6.1 in the following way (cf. [18,Chap. 2.9]): First sample U, T , independently from U (0, 1). Then for each
T i , T i let V i satisfy T i = C u (U i , V i ). Then (U i , V i ) is distributed according to C(u, v).
For sampling from the copula, the following corollary is useful:
Corollary 6.1. Suppose that C(u, v)
is an absolutely continuous copula given by Theorem 5.1 and F, G, K defined accordingly. Then C u (u, v) is given by the following formulas:
1. If 0 ≤ u ≤ u 0 and 0 < v < H(u) then C u (u, v) = G(v)/G(H(u)) (103) 2. If 0 ≤ u ≤ 1 and H(u) ≤ v ≤ 1 then C u (u, v) = 1 (104) 3. If u 0 < u < 1 and 0 < v ≤ 1 − u then C u (u, v) = (1 − G (1 − u)(K(u) + F (u)))G(v)/G(1 − u) (105) 4. If u 0 < u < 1 and 1 − u < v ≤ 1 − u 0 then C u (u, v) = 1 − G (1 − u)(K(u) + F (1 − v))(106)
5. If u 0 < u < 1 and 1 − u 0 < v ≤ H(u) then
C u (u, v) = 1 − G (1 − u)(K(u) − F (1 − H(1 − v)))(107)
6. If u 0 < u < 1 and H(u) < v < 1 then C u (u, v) = 1.
Proof. Follows from the equations for C u in the proof of Theorem 5.1, and equations (68), (71).
Application to toxicological probit models
The probit model is the standard statistical method for estimating the injury outcome of a population exposed to a toxic substance. It originates from an analysis on the effect of pesticides conducted by Bliss in 1934 [2]. The methodology was later cast in a more rigid mathematic formulation by Finney [8,7]. It has since then been used frequently in toxicological assessments of the injury outcome when a population has been exposed to dangerous chemicals [16,1,3,13,15,19,11]. In short, the probit model operates as follows. The exposure concentration c(t) is integrated over time to yield probit values
Γ i (t) = α i + β i log t 0 c(t) ni dt .(108)
The fraction of the population that has attainted the injury at time t is then estimated by
Φ(Γ i (t))(109)
where α i , β i , n i are model parameters associated with the substance, and Φ is the CDF for a standard normal variable. There are often several levels of injury outcome used in toxicology, e.g., light injury, severe injury and death. These different injury levels are indexed by i = 1, 2, ... in equations (108)-(109). The fraction of the population that obtains an injury increases continuously with growing exposure due to the individual variation of the toxic susceptiblity within the population. It is believed that modeling this variation improves the quantitative toxicological risk assessment, cf. [12]. A population that is not resolved on an individual level is referred to as a macroscopic population and can be described as a density field. In contrast, a population can be described as a set of discrete individuals, referred to as agents. A model that uses this type of population representation is called a microscale model or an agent-based model. In an agent-based toxicological model, see for example [15], the overall population statistics is obtained from the set of agents that are exposed to the toxic substance. In such a setting, individual probit values Γ i (t), acquired by exposure to individual model concentrations c(t), are computed for each agent. In the transition from a macroscopic population to an agent-based population, it is convenient to distribute individual threshold values, γ i , for the probit values to all agents representing their susceptibilities. Thus, when an agent has been exposed to a concentration yielding a probit value exceeding the corresponding threshold value, the agent has acquired that injury. Every agent is attributed one threshold value for each injury level. These threshold values are drawn from a standard normal distribution to maintain the overall probability distribution for the entire population. This method implies that the injury outcome of the agent-based population approaches asymptotically that of the macrosopic population (with static populations) when the number of agents increases. An advantage with an agent-based population is that the agents may have individual properties including their movement patterns. In a dynamic simulation, each agent follows its individual spacetime path, passing through concentration fields, and thereby proceeds through some or all of the injury stages, transiting successive injury stages when the agent's increasing probit functions Γ i (t) pass their threshold values γ i . As mentioned, the individual toxic susceptibility thresholds γ i are random variables and must obey the requirement
P (γ i ≤ Γ) = Φ(Γ)(110)
We propose that the γ 1 , γ 2 , ... are modeled as a discrete time Markov process with absolutely continuous transition densities p i+1|i , so by the Markov property, the joint density p is p(γ 1 , ..., γ n ) = p 1 (γ 1 )p 2|1 (γ 2 | γ 1 )p 3|2 (γ 3 | γ 2 )...p n|n−1 (γ n | γ n−1 ).
However, there is a potential pitfall: the injury stages must be passed in the correct order. Therefore, it must be true with probability one that if an injury level is acquired, then also the previous injury level is acquired, i.e.
γ i+1 ≤ Γ i+1 (t) =⇒ γ i ≤ Γ i (t).(112)
Therefore, the transition densities p i+1|i must satisfy
p i+1|i (γ i+1 | γ i ) = 0 if γ i+1 ≤ Γ i+1 (t) and γ i > Γ i (t).(113)
This imposes a restriction on the support of the joint probability density of (γ i , γ i+1 ), which we need to investigate in order to ensure that the model is consistent. To this end, we need to relate possible values of Γ i (t), Γ i+1 (t) for all possible exposures c(t), t ≥ 0. This can be done in terms of
Γ i (t) − α i β i = log t 0 c ni dt(114)
according to the following lemma: Proof. Apply Hölder's inequality f gdt ≤ (f p dt) 1/p (g q dt) 1/q and the elementary estimate f p dt ≤ (max f ) p−1 f dt with f = c m , g = 1 and p = n/m.
The following theorems provide sufficient conditions for (112), and necessary compatibility conditions for the probit parameters α, β, n. Theorem 7.2. Assume that Γ i (t), Γ i+1 (t) are probit functions defined by (108), and n i+1 ≤ n i . Also assume that (γ i , γ i+1 ) is a bivariate random variable such that
γ i+1 − α i+1 β i+1 ≥ n i+1 n i γ i − α i β i + 1 − n i+1 n i log(t)(117)
almost surely. Then γ i+1 ≤ Γ i+1 (t) =⇒ γ i ≤ Γ i (t) almost surely. Moreover, there exists standard normal γ i , γ i+1 satisfying (117) if and only if
n i+1 β i+1 = n i β i(118)
and
∆ i ≡ α i − α i+1 − β i+1 1 − n i+1 n i log t ≥ 0,(119)
and then if ∆ i > 0 there exists (γ i , γ i+1 ) with absolutely continuous joint density.
Proof. Assume that γ i+1 ≤ Γ i+1 (t). Then we get by (114), (115) with m = n i+1 , n = n i , and (117) that 120) i.e., Γ i (t) ≥ γ i , which proves the first part. The second part follows from Proposition 1.1, since equation (117) is equivalent to equation (1) with X = −γ i , Y = −γ i+1 , a = (β i+1 n i+1 )/(β i n i ) and
n i+1 n i Γ i (t) − α i β i + 1 − n i+1 n i log(t) ≥ Γ i+1 (t) − α i+1 β i+1 ≥ γ i+1 − α i+1 β i+1 ≥ n i+1 n i γ i − α i β i + 1 − n i+1 n i log(t) (∆ = β i+1 n i+1 β i n i α i − α i+1 − β i+1 1 − n i+1 n i log(t),
and a = 1, ∆ ≥ 0 is equivalent to equations (118), (119).
Theorem 7.3. Assume that Γ i (t), Γ i+1 (t) are probit functions defined by (108), and n i+1 ≥ n i . Also assume that (γ i , γ i+1 ) is a bivariate random variable such that
γ i+1 − α i+1 β i+1 ≥ γ i − α i β i + (n i+1 − n i ) log max [0,t] c(121)
almost surely. Then γ i+1 ≤ Γ i+1 (t) =⇒ γ i ≤ Γ i (t) almost surely. Moreover, there exist standard normal γ i , γ i+1 satisfying (121) if and only if
β i+1 = β i(122)
and
∆ i ≡ α i − α i+1 − β i (n i+1 − n i ) log max [0,t] c ≥ 0,(123)
and then if ∆ i > 0 there exists (γ i , γ i+1 ) with absolutely continuous joint density.
Figure 1 :
1Parts of the unit square for piecewise definition of the copula in Theorem 1.4.
figure 1 . 5 .Figure 2 :
152Here, G(v) is computed with the MATLAB R function integral at 400 uniformly distributed grid points on [ , 1 − u 0 ], and computed at intermediate points on [ , 1 − u 0 ] by spline interpolation, where = 10 −11 . Consequently, the copula and its density is computed on [ , 1 − ] 2 . Copula density C uv (u, v) for Example 1.5, ∆ = 1. The density is discontinuous on the curve v = H(u) and tends to infinity when approaching (0, 0) or (1, 1).
Figure 3 :
3Probability density function p(x, y) for Example 1.5, ∆ = 1. The wiggles in the level curves at the upper right and lower left corners of right plot are numerical artifacts. Proof of Proposition 1.1. If (1) is satisfied then Φ((y − ∆)/a) = P {aX + ∆ ≤ y} ≤ P {Y ≤ y} = Φ(y) for all y ∈ R, which is possible only if a = 1 and ∆ ≥ 0.
Theorem 3 . 3 .
33Assume that G satisfies the assumptions of Theorem 3.2. Then F (u) satisfy (37) and F (0) = 0 if and only if
Example 3. 6 .F
6If G(v) = sin(πv/2), then (37) has solution (
Example 3. 8 .
8Assume that k ≥ 1 and let L(u) = (1 − u)/k. Then we get G(1 − u) = (1 − u) k so we recover Example 3.5. Also, L (u) = −1/k ≥ −1 so u * = 0 and and since 0 ≤ L(0) = 1/k we infer from Theorem 3.3 that an absolutely continuous copula is obtained.
Example 3. 9 .
9Assume that a ∈ [0, 1) and let L(u) = (1 − u)(1 − au). Then
rem 3 . 1 .
31By definition p(u, v) = C uv (u, v) and piecewisely defined on the regions 1-7 depicted in Figure 5 as follows; region 1: C uv = G (v)/G(H(u)), region 2,3,7: C uv = 0, region 4: C uv = F (u)G (v), region 5: C uv = F (1 − v)G (1 − u), and region 6: C uv = G (1 − u)/G(H(1 − v)). Integration yields C u (u, v) = v 0 C uv (u, z)dz, piecewisely defined as follows; region 1: C u = G(v)/G(H(u)), region 2,3: C u = 1, region 4:
(
derived by the change of variables z = 1 − H(w) = H −1 (1 − w))
Figure 4 :
4which proves (72), and region 4: C = H −1 (v) + (K(1 − v) + F (u))G(v) which proves (74). The final statement for u + v > 1 follows from Theorem 3.Subdivision of the unit square for piecewise definition of p = C uv in Theorem 5.1. Equation (71) can be solved with the integrating factor method, and a positivity condition can be derived, analogous to Theorem 3.3: Theorem 5.2. Assume that K(u) is given by (70), and G satisfies the assumptions of Theorem 5.1. Then F (u) satisfies (71) if and only if
Figure 5 :
5Copula density C uv (u, v) for Example 5.4, ∆ = 1, k = 2.The density is discontinuous on the curve v = H(u) and tends to infinity when approaching (0, 0), (1, 1) or (1, 0).
Figure 6 :
6Probability density function p(x, y) for Example 5.4, ∆ = 1, k = 2. The wiggles in the level curves at the upper right and lower left corners of right plot are numerical artifacts.
( 1
1− ω (u))(1 + L (u)) − ω (u)L(u)= 1 + ω (u) + K (u)G(1 − u) − K(u)G (1 − u) (100) which is integrated to (1 − ω (u))L(u) = 2ω(u) + G(1 − u)K(u)+constant.The equation (97) follows from (70) and (95). For each fixed z, the integrand in (97) is decreasing towards 0 as u → 1− in view of (93) and (94), so by the mononotone convergence theorem, lim u→1− G(1 − u)K(u) = 0. Hence the constant of integration is zero, which proves (96). Proof of Theorem 1.4. Since 1+L (z)−H (z) ≡ 0, L(u 0 ) = H(u 0 )−u 0 = 1−2u 0 and G(1 − u 0 ) = 1 we have by (77)
Figure 6
6illustrates sampling in Example 1.5.
Figure 7 :
7Samples from distributions in Example 1.5, sample size 10 5 .
Lemma 7. 1 .
1Assume that n ≥ m > 0 and c ≥ 0, t > 0.Moreover, the inequalities are sharp: if c(t) =constant, then equalities holds in the inequalities above.
Proof of Theorem 7.3. Assume that γ i+1 ≤ Γ i+1 (t). Then we get by (114), (116) with m = n i , n = n i+1 and (121) thati.e., Γ i (t) ≥ γ i , which proves the first part. The second part follows from Proposition 1.1, since equation(121)is equivalent to equation(1)and a = 1, ∆ ≥ 0 is equivalent to equations (122), (123).Remark. Note that if n i+1 = n i , then the compatibility conditions (118),(119)and(122), (123) in the preceding theorems involve only the probit coefficients α, β, n, not t or max c.
The 2016 Al-Mishraq sulphur plant fire: Source and health risk area estimation. O Björnham, H Grahn, P Schoenberg, B Liljedahl, A Waleij, N Niklas Brännström, Atmospheric Environment. 169O. Björnham, H. Grahn, P. von Schoenberg, B. Liljedahl, A. Waleij, and N. Niklas Brännström. The 2016 Al-Mishraq sulphur plant fire: Source and health risk area estimation. Atmospheric Environment, 169:287 -296, 2017.
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Table of integrals, series, and products. I S Gradshteyn, I M Ryzhik, With one CD-ROM (Windows, Macintosh and UNIX). Alan Jeffrey and Daniel ZwillingerAmsterdamElsevier/Academic Presseighth editionI. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, eighth edition, 2014. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).
Consequence modeling of a real rupture of toluene storage tank. H Haghnazarloo, M Parvini, M N Lotfollahi, Journal of Loss Prevention in the Process Industries. 37H. Haghnazarloo, M. Parvini, and M. N. Lotfollahi. Consequence modeling of a real rupture of toluene storage tank. Journal of Loss Prevention in the Process Industries, 37:11 -18, 2015.
Distributions of Individual Susceptibility among Humans for Toxic Effects: How Much Protection Does the Traditional Tenfold Factor Provide for What Fraction of Which Kinds of Chemicals and Effects?. D Hattis, P Banati, R Goble, 895Annals of the New York Academy of SciencesD. Hattis, P. Banati, and R. Goble. Distributions of Individual Suscepti- bility among Humans for Toxic Effects: How Much Protection Does the Traditional Tenfold Factor Provide for What Fraction of Which Kinds of Chemicals and Effects? Annals of the New York Academy of Sciences, 895(1):286-316, 1999.
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A dynamic approach for the impact of a toxic gas dispersion hazard considering human behaviour and dispersion modelling. R Lovreglio, E Ronchi, G Maragkos, T Beji, B Merci, Journal of Hazardous Materials. 318R. Lovreglio, E. Ronchi, G. Maragkos, T. Beji, and B. Merci. A dynamic approach for the impact of a toxic gas dispersion hazard considering hu- man behaviour and dispersion modelling. Journal of Hazardous Materials, 318:758 -771, 2016.
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| {'fraction_non_alphanumeric': 0.11978914074855035, 'fraction_numerical': 0.04953083816552451, 'mean_word_length': 3.302458495872267, 'pattern_counts': {'":': 0, '<': 40, '<?xml version=': 0, '>': 24, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 25, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'A new method for constructing absolutely continuous two-dimensional copulas by differential equations is presented. The copulas are symmetric with respect to reflection in the opposite diagonal. The support of the copula density may be prescribed to arbitrary opposite symmetric hypographs of invertible functions, containing the diagonal. The method is applied to toxicological probit modeling, where new compatibility conditions for the probit parameters are derived.', 'arxivid': '1905.09306', 'author': ['Oscar Björnham [email protected] \nFOI CBRN Defence and Security\nMathematics Department\nNiklas Brännström FOI CBRN Defence and Security\nUmeå University\n\n', 'Leif Persson [email protected] \nFOI CBRN Defence and Security\nMathematics Department\nNiklas Brännström FOI CBRN Defence and Security\nUmeå University\n\n'], 'authoraffiliation': ['FOI CBRN Defence and Security\nMathematics Department\nNiklas Brännström FOI CBRN Defence and Security\nUmeå University\n', 'FOI CBRN Defence and Security\nMathematics Department\nNiklas Brännström FOI CBRN Defence and Security\nUmeå University\n'], 'corpusid': 162184388, 'doi': '10.1080/03610926.2020.1864825', 'github_urls': [], 'n_tokens_mistral': 18438, 'n_tokens_neox': 16348, 'n_words': 9471, 'pdfsha': 'e268b6cf09768e8e344c6f1dcfb649abf2ba3b58', 'pdfurls': ['https://arxiv.org/pdf/1905.09306v1.pdf'], 'title': ['Absolutely continuous copulas with prescribed support constructed by differential equations, with an application in toxicology', 'Absolutely continuous copulas with prescribed support constructed by differential equations, with an application in toxicology'], 'venue': []} |
arxiv |
Ultrasonic Attenuation in the Vortex State of d-wave Superconductors
22 Dec 2002
Tribikram Gupta
Harish-Chandra Research Institute
Chhatnag Road211019JhusiAllahabadIndia
D M Gaitonde
High Pressure Physics Division
Bhabha Atomic Research Centre
400085MumbaiIndia
Ultrasonic Attenuation in the Vortex State of d-wave Superconductors
22 Dec 2002PACS numbers: 7472-h; 7460-w
We calculate the low temperature quasi-particle contribution to the ultrasonic attenuation rate in the mixed state of d-wave superconductors. Our calculation is performed within the semi-classical approximation using quasi-particle energies that are Doppler shifted, with respect to their values in the Meissner phase, by the supercurrent associated with the vortices. We find that the attenuation at low temperatures and at fields H c1 ≤ H ≪ H c2 has a temperature independent contribution which is proportional to √ H where H is the applied magnetic field. We indicate how our result in combination with the zero-field result for ultrasonic attenuation can be used to calculate one of the parameters v F , H c2 or ξ given the values for any two of them.
The discovery of high temperature superconductivity 1 in the cuprates has led to an intense theoretical effort at understanding the origins of the novel phenomena seen in these materials.
The normal state of the cuprates is highly anomalous 2 and remains a puzzle that is still not understood. However, there is evidence 3 that well defined quasi-particles exist in the superconducting state. The superconducting state is now known 4,5 to be a d-wave state whose order parameter has d x 2 −y 2 symmetry. The order parameter thus has nodes along the lines k x = ±k y in the two-dimensional Brillouin zone.
While the origin of the pairing interaction that leads to the occurence of d-wave superconductivity is not known at the present time, a great deal of progress in understanding the superconducting state can be made by focussing on the low energy nodal quasi-particles near the points where the Fermi surface intersects the k x = ±k y lines. These quasi-particles, which have an energy dispersion corresponding to Dirac fermions 6 in the Meissner phase, are the dominant low-energy excitations which determine the low temperature properties in the superconducting state.
It is possible to derive 7 the d-wave gap from a model Hamiltonian but the connection of that Hamiltonian with the underlying physics of strongly interacting fermions is unclear.
Thus for the purposes of this paper, we assume on phenomenological grounds, the existence of BCS like quasi-particles with an energy gap that has d-wave symmetry. These quasiparticles might not be the real strongly interacting electrons, to whom their connection is unclear at the present time.
Ultrasonic attenuation has historically been a very useful tool in investigating the energies of quasi-particles in conventional superconductors. Early verification 8 of the BCS prediction for the temperature dependence of the energy gap was done using this technique. Even afterwards 9,10 this has been a useful tool in the investigation of of superconductivity in heavy fermion superconductors. On the theoretical front calculations of the ultrasonic attenuation in d-wave superconductors have been carried out both in the clean 7,11,12,13 and dirty 14,15 limit in the Meissner phase. The former limit which corresponds to Ql ≫ 1 , Q being the ultrasound wave-vector and l being the electronic mean free path, is the one considered by us in this paper.
Assuming 100 MHz to be a typical frequency 9,10 (although ultrasound experiments can be carried out over a large frequency range from kHz to GHz) at which ultrasound experiments are done and taking the sound velocity to be 16 4×10 5 m/s we find that this translates into the requirement l ≫ 4µm. This restricts the applicability of our work to the cleanest samples 17 .
While ultrasonic attenuation in the vortex state has recieved attention in the past 18,19,20,21 the emphasis of earlier workers has been to study effects arising from the coupling of ultrasonic waves to vortex motion. This coupling comes from two different effects: a) The pinning potentials from impurity ions tend to drag the vortices along with them , b) The ionic current due to the ionic displacement will exert a Lorentz force on the vortices. In the clean limit(l ≥ 4µ − m) in which we work, the effects of pinning are expected to be weak. The Lorentz force coupling is in any case weak because it is suppressed by a factor of v/c where v is the ionic velocity and c is the velocity of light. We therefore ignore these processes and focus entirely on the phonon damping from the electronic quasi-particles.
In conventional s-wave superconductors, low energy electronic excitations in the vortex state are the bound states localised in the core region of the vortices 22 . However in d-wave superconductors, because of the presence of nodes in the gap function, the dominant low energy excitations are those in the far region away from the vortex cores. Their contribution is expected to be overwhelmingly larger 23 than that coming from the the cores. Further, experimental studies from STM measurements 24,25 reveal that that there are just a few bound states in the vortex cores. Thus we focus exclusively on the excitations in the far region.
To describe the low energy excitations in the mixed phase we make the use of the semiclassical approximation, first discussed 29 in the case of s-wave superconductors. This approximation has been employed for d-wave superconductors in recent times 23,26 and its regime of applicability is the range H c1 ≤ H ≪ H c2 , where H is the applied magnetic field and H c1 and H c2 are the lower critical and upper critical magnetic fields repectively. A striking success of this method has been the prediction 23 of a term linear in the temperature T whose co-efficient scales as √ H (H being the applied magnetic field). This prediction has been verified 27 and has given greater credibility to the semi-classical description. Recently this approximation has been systematized and put on a firm footing by Ramakrishnan and Rajagopal 28 who have derived it microscopically.
Using the semi-classical approximation to describe the electronic Green functions in the superconducting state we calculate the imaginary part of the electron density-density correlation function which is proportional to the inverse phonon lifetime. As the sound velocity is only weakly dependent on temperature 16 , the temperature dependence of the attenua-tion comes almost entirely from the the inverse lifetime. We find that at low tempeartures, and for parameters appropiate to the cuprates, the ultrasonic attenuation co-efficient in the mixed state has a temperature independent contribution which scales as √ H. The coefficient of this term and its dependence on the ultrasound wave-vector as well as the leading temperature corrections to it have been explicitly evaluated. These are our main results.
We now present the details of our calculation. The semi-classical approximation can be understood 29 as ignoring, in the first instance, the spatial variation of the supercurrents around a vortex outside its core region. This can be justified on the grounds that the spatial variation of the electronic wave-function has a characteristic length scale k −1 F which is much smaller than the smallest length scale associated with vortex currents ξ (the core radius).
In the cuprates the parameter (k F ξ) −1 ≃ 10 and thus the semi-classical approximation is expected to provide a reasonable description. We then proceed to evaluate the inverse phonon lifetime to first order in q where a current has been introduced by taking the energy
gap function to be ∆ q ( R i , R j ) = ∆ i,j e i q·( R i + R j ) .
Here R i and R j are neighbouring sites on a square lattice whose fermions are paired up in a singlet state and ∆ i,j = ∆(−∆) for R i − R j being in thex(ŷ) direction. The spatial dependence of q =φ/2r is now restored and the the inverse phonon lifetime is averaged over a unit-cell of the vortex lattice to get our final result for the attenuation co-efficient.
Solving the Bogulibov-deGennes equation for the mean-field d-wave superconductor to linear order in q we obtain the results:
G σ q ( R i − R j , iω n ) = 1 N k e i k·( R i − R j ) [ u 2 k− q iω n − E k− q + v 2 q− k iω n + E q− k ](1)
where
u k 2 = (1 + ǫ k /E 0 k )/2, v k 2 = (1 − ǫ k /E 0 k )/2, ǫ k = ξ k − µ,
ξ k being the band energy and µ the chemical potential, E 0 k = ǫ 2 k + ∆ 2 k , ∆ k = ∆(cos k x a − cos k y a) being the d-wave gap and E k = E 0 k + q · ∇ k ξ k being the Doppler shifted quasi-particle energy. Here
G σ q ( R i − R j , iω n ) = − β 0 dτ < c i,σ (τ )c † j,σ (0) > e iωnτ(2)
is the "normal" Green function. Similarly the anomalous Green functions are given by:
F q ( R i , R j , iω n ) = e i q·( R i + R j ) N k u k v k ( e −i k·( R i − R j ) iω n − E k − e i k·( R i − R j ) iω n + E k )(3)
and
F + q ( R i , R j , iω n ) = e −i q·( R i + R j ) N k u k v k ( e −i k·( R i − R j ) iω n − E k − e i k·( R i − R j ) iω n + E k )(4)
Here u k v k = ∆ k /2E 0 k and the Green functions F q and F + q are defined as
F q ( R i , R j , iω n ) = − β 0 dτ < c i,↓ (τ )c j,↑ (0) > e iωnτ(5)F + q ( R i , R j , iω n ) = − β 0 dτ < c † i,↑ (τ )c † j,↓ (0) > e iωnτ(6)
Using the Green functions in equations (1), (3) and (4) we compute the imaginary part of the density-density correlation function which is given by
χ " q ( Q, ω) = X 1 + X 2 + X 3 + X 4(7)
where
X1 = −2π N k [n(E k ) − n(E k+ Q )](u 2 k+ Q u 2 k − ∆ k ∆ k+ Q 4E 0 k E 0 k+ Q )δ(ω + E k − E k+ Q ) (8) X2 = −2π N k [n(E − k ) − n(E − k− Q )](v 2 k+ Q v 2 k − ∆ k ∆ k+ Q 4E 0 k E 0 k+ Q )δ(ω − E − k + E − k− Q ) (9) X3 = −2π N k [1 − n(E − k ) − n(E k+ Q )](u 2 k+ Q v 2 k + ∆ k ∆ k+ Q 4E 0 k E 0 k+ Q )δ(ω − E − k − E k+ Q ),(10)
and
X4 = 2π N k [1 − n(E k ) − n(E − k− Q )](v 2 k+ Q u 2 k + ∆ k ∆ k+ Q 4E 0 k E 0 k+ Q )δ(ω + E k + E − k− Q ).(11)
In s-wave superconductors the contribution of X 3 and X 4 are zero as the ultrasound frequency ω ≪ 2∆ and so the δ-fn condition in Eqs. (10) and (11) can never be satisfied. In the d-wave case the presence of nodes in ∆ k means that X 3 and X 4 will be finite. However, as the phase space for them is limited, their contribution is expected to be small and so we focus exclusively on X 1 and X 2 which make the dominant contribution to χ " q ( Q, ω). Now expanding to the leading order in ω we find
χ " q ( Q, ω) = 2πω N k [n ′ (E k )δ(E k − E k+ Q )(u 2 k+ Q u 2 k − ∆ k ∆ k+ Q 4E 0 k E 0 k+ Q ) + n ′ (E − k ) δ(E − k − E − k− Q )(v 2 k+ Q v 2 k − ∆ k ∆ k+ Q 4E 0 k E 0 k+ Q )](12)
We next expand χ " to leading order in the ultrasound wave-vector Q and on making the substitution k → −k in the second term in Eq. (12) we arrive at the result
χ " q ( Q, ω) = 2πω N Σ k n ′ (E 0 k + q · ∇ k ξ k ) ǫ 2 k E 0 2 k δ( Q · ∇ k E 0 k + q α Q β ∂ 2 ξ k ∂k α ∂k β )(13)
where a sum over repeated indices is implied in the arguement of the δ function.
To do the k summation in Eq. (13) we only consider 6 the fermions near the nodes.
This approximation is reasonable at low temperatures (k B T ≪ ∆). For the node near P a = (π/2a, π/2a) upon linearising the band energies and gap function, we have: ξ k ≃ v F k 1 and ∆ k ≃ v ∆ k 2 where k 1 and k 2 are co-ordinates normal and tangential respectively to the Fermi surface at the node. In terms of the co-ordinates k 1 and k 2 we find that the contribution from this node to χ "(a) q can be written as
χ "(a) q ( Q, ω) ≃ a 2 ω 2π dk 1 dk 2 n ′ [ v 2 F k 2 1 + v 2 ∆ k 2 2 + ρ 0 ] v 2 F k 2 1 v 2 F k 2 1 + v 2 ∆ k 2 2 δ[ αv F k 1 + βv ∆ k 2 v 2 F k 2 1 + v 2 ∆ k 2 2 − α 1 v F k 1 − α 2 v F k 2 ](14)
Here α = v F Q cos(π/4 − θ), β = v ∆ Q sin(π/4 − θ), α 1 = (Qqa 2 /2) cos(ψ − θ), α 2 = −(Qqa 2 /2) cos(ψ + θ), ρ 0 = qv F cos(ψ − π/4) and θ and ψ are the angles made by Q and q respectively with respect to the k 1 axis. We now introduce the polar co-ordinates v F k 1 = ρcos φ and v ∆ k 2 = ρsin φ. Then upon performing the φ integral we find χ "(a) q reduces to
χ "(a) q ( Q, ω) = a 2 ω πv F v ∆ ρc 0 dρρn ′ [ρ + ρ 0 ]f a [ρ] (15) where f a [ρ] = B 2 (A 2 +B 2 ) 3/2 and A = α − α 1 ρ , B = β − (v F /v ∆ )α 2 ρ and ρ c = √ πv F v ∆ /a is a
cutoff introduced to preserve the volume of the Brillouin zone while doing the k-integration.
Introducing the variable ρ ′ = ρ + ρ 0 we have χ "a = χ "(a1) + χ "(a2) where
χ "(a1) q ( Q, ω) = a 2 ω πv F v ∆ ρc ρ 0 dρ ′ ρ ′ n ′ [ρ ′ ]f a [ρ ′ − ρ 0 ](16)
and
χ "(a2) q ( Q, ω) = − a 2 ω πv F v ∆ ρ 0 ρc ρ 0 dρ ′ n ′ [ρ ′ ]f a [ρ ′ − ρ 0 ](17)
In writing Eqs. (16) and (17) we have made use of ρ c ≫ ρ 0 which follows from using q max = 1/2ξ and ξ ≈ 30Å, v F /a ≈ 192meV and v ∆ /a ≈ 28meV for parameters appropriate to the cuprates 6 .
In an analogous fashion we evaluate the corresponding contributions to χ " q from the nodes near P b = (−π/2a, −π/2a), P c = (−π/2a, π/2a) and P d = (π/2a, −π/2a). We first focus on the type of terms in Eq. (16). In this case we find, on expanding to linear order in q, that the contribution linear in q exactly vanishes due to cancellations from the terms coming from different nodes. We thus find χ
"(1) q = χ "(a1) q + χ "(b1) q + χ "(c1) q + χ "(d1) q
to be given by
χ "(1) q = − 2ln2ωa 2 k B T πv F v ∆ [ β 2 (α 2 + β 2 ) 3/2 + η 2 (λ 2 + η 2 ) 3/2 ](18)
to linear order in q. Once again we have used k B T ≪ qv F ≪ ρ c . Here α and β are previously defined and we have introduced the parameters λ = v F Q sin(π/4 − θ), η = v ∆ Q cos(π/4 − θ). The result obtained for this term is identical to the zero field result for the attenuation previously calculated in Ref. (12). We next turn our attention to the terms of the type written in Eq. (17). Here on adding the contributions from the nodes near P a and P b , we find to leading order in q,
χ "(a2) q + χ "(b2) q = − ωa 2 q πv ∆ cos(ψ − π/4) β 2 (α 2 + β 2 ) 3/2 tanh( qv F cos(ψ − π/4) 2k B T )(19)
Similarly the nodes near P c and P d yield the contribution
χ "(c2) q + χ "(d2) q = −( ωa 2 q πv ∆ ) cos(ψ + π/4) η 2 (λ 2 + η 2 ) 3/2 tanh( qv F cos(ψ + π/4) 2k B T )(20)
It is straightforward to see that for T → 0, Eqs. (19) and (20) reduce to (21) and χ "(c2) q +χ "(d2) q = −( ωa 2 q πv ∆ ) cos(ψ+π/4) η 2 (λ 2 + η 2 ) 3/2 [θ(cos(ψ+π/4))−θ(− cos(ψ+π/4))] (22) Eqs. (21) and (22) are also obtained directly by taking the T = 0 limit for n' in Eq. (15) and its analogues for the other 3 nodes.
χ "(a2) q +χ "(b2) q = −( ωa 2 q πv ∆ ) cos(ψ−π/4) β 2 (α 2 + β 2 ) 3/2 [θ(cos(ψ−π/4))−θ(− cos(ψ−π/4))]
To proceed further we now restore q =φ/2r corresponding to the supercurrents around a vortex where the vector potential has been ignored as the Ginzburg-Landau parameter κ ≫ 1. Assuming a vortex lattice with circular unit-cells we find that the radius of the cells R c ∼ ξ H c2 /H. We then average χ " q over one unit cell to obtain
χ " (H, T ) = Rc ξ rdr 2π 0 dψχ " q Rc ξ rdr 2π 0 dψ(23)
to obtain our final result for χ " .
In order to perform the averaging in Eq. (23) we make the approximation tanh(x) ≃ x for |x| < 1, tanh(x) ≃ 1 for x > 1 and tanh(x) ≃ −1 for x < −1. This approximation interpolates between the asymptotically exact behaviour at small and large | x |. Further it reproduces the exact result that can be directly calculated for T = 0. In that case the integral in Eq. (23) is elementary and can be done by substituting the expressions in Eqs. (21) and (22). We then obtain to leading order in the small parameter (ξk B T /v F ) 2 (H c2 /H):
χ "(2) (H, T ) = − 2ωa 2 π 2 v ∆ ξ ( H H c2 ) 1/2 [1 − 8ξ 2 9 ( k B T v F ) 2 H c2 H ][ β 2 (α 2 + β 2 ) 3/2 + η 2 (λ 2 + η 2 ) 3/2 ](24)
The assumption about the small parameter implies that the regime of validity of our to the attenuation that scales as √ H is entirely understandable as it has its origin in a finite density of states at the Fermi-energy 23 whose size is proportional to √ H and whose signature is seen in the specific heat measurements 27 .
result is for k B T ≪ v F ξ ( HH
The actual ultrasonic attenuation coefficient is related to χ " by
α S (T, H) = M( Q)χ " ( Q, T, H)(25)
where M( Q) is a constant which depends on the sound velocity and the electron-phonon matrix element and the ultrasound frequency. Then on combining Eqs. (18) and (24)
= v F πξ ( H H c2 ) 1/2 1 ln2k B T [1 − 8ξ 2 9 ( k B T v F ) 2 H c2 H ](26)
whose window of validity has been described above. This result is remarkable because it is independent of the ultrasound wave-vector Q as well as the gap increase parameter v ∆ . Thus a measurement of the ultrasonic attenuation at low temperatures and at fields above H c1 would enable a determination of any one of the parameters v F ,ξ and H c2 given a knowledge of the other two.
We now discuss some shortcomings of our work. We have assumed perfectly welldefined quasi-particles thus ignoring the incoherent spectral weight seen in photoemission experiments 3 . The semi-classical approximation used by us may not accurately describe all the physical effects due to the scattering of quasi-particles from the supercurrents in the vortex state. Our results are restricted to the clean limit and are applicable only to very clean samples.
Finally we conclude by recapitulating the main points of this paper. We have employed the semi-classical approximation to evaluate the phonon damping due to the electronic quasi-particles in a d-wave superconductor. We find that for parameters appropiate to the cuprates, in a temperature window k B T ≪ 1K, there is a temperature independent contribution to the ultrasonic attenuation whose magnitude scales as √ H.
and considering the fact that Eq. (18) contains the result for the Meissner phase (H = 0) we obtain the result: α S (T, H) − α S (T, H = 0) α S (T, H = 0)
c2 ) 1/2 together with the condition H c1 ≤ H ≪ H c2 . For parameters relevant to the cuprates H c2 /H c1 ≃ 100 at low temperatures. This restricts the temperature window to T ≪ 1K at H = H c1 . Our result for a temperature independent contribution
AcknowledgmentsOne of us (DMG) thanks T.V.Ramakrishnan for useful comments.
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arxiv |
Estimating latent linear correlations from fuzzy frequency tables
7 May 2021
Antonio Calcagnì [email protected]
University of Padova
Estimating latent linear correlations from fuzzy frequency tables
7 May 2021arXiv:2105.03309v1 [stat.ME]fuzzy frequencygeneralized natural numberspolychoric correlationsfuzzy data analysis
This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social and behavioral sciences, and are especially relevant when observed data are classified using fuzzy categories -as for socio-economic studies, clinical evaluations, content analysis, inter-rater reliability analysis -or when imprecise observations are classified into either precise or imprecise categories -as for the analysis of ratings data or fuzzy coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of Expectation-Maximization based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.
Introduction
The latent linear correlation (LLC) is a measure of bivariate association which is usually adopted when variables are measured at an ordinal level or when data are available in the form of frequency or contingency tables. Because LLCs are quite often used in analysing categorical ordered variables, they are also known as polychoric correlations [1]. Latent linear or polychoric correlations differ from other measures of association such as Goodman and Kruskal's γ or Kendall's τ in that they are based on a latent continuous parametric model according to which LLCs behave. Given a set of J variables, LLCs are computed pairwise for each pair j, k of variables by considering their joint frequencies N jk R×C = (n jk 11 , . . . , n jk rc , . . . , n jk RC ) over a R jk × C jk partition space of the variables' domain. The general idea is to map the observed counts N jk R×C to the real domain of the bivariate latent density model via the Muthen's thresholds-based approach [2], under the constraint that the volumes of the rectangles of the latent density should equal the observed frequencies. In doing so, changing the covariance parameter of the latent model will change the probability distribution over the latent rectangles and hence the probability masses over the cells of N jk R×C . Although several parametric models are available for estimating LLCs (e.g., Elliptical, Skew-Gaussian, Copula-based models. See: [3,4,5]), the standard formulation based on the Gaussian density with zero means and latent correlations R jk R×C is strong enough to be of practical use for many empirical applications (for a recent study, see [6,7]). Some of these include inter-rater agreement [8], reliability measurement [9,10,11], ordinal CFA and SEM [12,13,14], and polychoric-PCA for dimensionality reduction of discrete data [15].
Fuzzy frequency or contingency tables are of particular concern across several disciplines including social, behavioral, and health sciences. Overall, there are two main situations which give rise to fuzzy frequencies, namely when precise data are classified into imprecise categories or, in the opposite case, when fuzzy data are classified into either precise or imprecise categories. Examples of the first case may be found in studies involving socio-economic variables (e.g., income, labor flushes, employment) [16,17], images or scenes classification [18,19], content analysis [20], reliability analyses [21], evaluation of user-based experiences [22], multivariate analysis of qualitative data [23,24], spatial distributional data [25], and human-based risk assessment [26]. By contrast, examples of the second case are most common in studies involving rating scales-based variables such as satisfaction, quality, attitudes, and motivation [27,28]. What both of these situations have in common is that the R jk × C jk space constitutes a fuzzy partition and, consequently, observed counts in the classification grid are no longer natural numbers. There have been a number of studies that have tried to deal with fuzzy contingency tables and fuzzy association measures. For instance, [29] proposed some non-parametric tests generalized to the case of fuzzy data, [30] studied fuzzy hypotheses testing based on fuzzy random variables, [31] proposed a rank-sum test based on fuzzy partial ordering and introduced a modelization of fuzzy statistical significance test, [32] generalized the Goodman and Kruskal's γ measure to the case of fuzzy observations arranged into contingency tables, [33] presented the analysis of contingency tables for both fuzzy observations/crisp categories and crisp observations/fuzzy categories cases along with a fuzzy generalization of association measures based on frequencies. Although they differ in some respects, all of them generalize the analysis of contingency tables to the fuzzy case either by the Zadeh's extension principle or by the α-cuts based calculus [34].
Based on this research stream, this article focuses on estimating latent linear correlations from fuzzy frequency tables, which include both the cases of crisp observations/fuzzy categories and fuzzy observations/fuzzy or crisp categories. Unlike the aforementioned studies, we develop our results by generalizing the standard LLC problem to cope with fuzzy frequencies under the general fuzzy maximum likelihood framework [31,35]. In particular, we define the fuzzy frequency table N jk R×C in terms of fuzzy cardinality and generalized natural numbers first, and then we extend the sample space of the LLC model to deal with fuzzy countsñ jk 11 , . . . ,ñ jk rc , . . . ,ñ jk RC . In doing so, the fuzziness of the observations enters the model as a systematic and non-random component while the model's parameters are still crisp (i.e., the estimated latent correlation matrix R jk R×C is a non-fuzzy quantity). This offers an attractive solution to the problem of estimating LLCs with fuzzy information, with the additional benefit that statistical models that uses the LLCs statistic as input data (e.g., CFA, PCA, SEM) do not need any further generalization to cope with fuzzy data.
The reminder of this article is structured as follows. Section 2 introduces the concept of fuzzy frequency through fuzzy cardinalities and generalized natural numbers. Section 3 describes the fuzzy LLCs model and its characteristics in terms of parameters estimation and interpretation. Section 4 reports the results of a simulation study performed to assess the finite sample properties of the fuzzy LLCs model as compared with standard defuzzification-based estimation methods. Section 5 describes the application of the proposed method to two real case studies and section 6 concludes the article by providing final remarks and suggestions for further extensions of the current findings. All the materials like algorithms and datasets used throughout the article are available to download at https://github.com/antcalcagni/fuzzypolychoric/.
Fuzzy frequencies
Preliminaries
A fuzzy subsetà of a universal set A ⊂ R can be defined by means of its characteristic function ξ A : A → [0, 1]. It can also be expressed as a collection of crisp subsets called α-sets, i.e.à α = {x ∈ A : ξ A (x) > α} with α ∈ (0, 1]. If the α-sets ofà are all convex sets thenà is a convex fuzzy set. The support ofà is A 0 = {x ∈ A : ξ A (x) > 0} and the core is the set of all its maximal points A 1 = {x ∈ A : ξ A (x) = max z∈A ξ A (z)}. In the case max x∈A ξ A (x) = 1 thenà is a normal fuzzy set. Ifà is a normal and convex subset of R thenà is a fuzzy number (also called fuzzy interval). The quantity l(Ã) = sup A 0 −inf A 0 is the length of the support of the fuzzy setÃ. The simple cardinality of a fuzzy setà is defined as |Ã| = A ξ A (x) dx. Given two fuzzy setsÃ,B, the degree of inclusion ofà inB is ǫÃB = min x∈A ξ A (x), ξ B (x) max(1, |Ã|), with ǫÃB ∈ [0, 1]. The case ǫÃB = 1 indicates thatà is fully included inB. The class of all normal fuzzy numbers is denoted by F (R). Fuzzy numbers can conveniently be represented using parametric models that are indexed by some scalars. These include a number of shapes like triangular, trapezoidal, gaussian, and exponential fuzzy sets [36]. A relevant class of parametric fuzzy numbers are the so-called LR-fuzzy numbers [37] and their generalizations [38,11]. The trapezoidal fuzzy number is one of the most common fuzzy set used in many applications and it is parameterized using four parameters as follows:
ξ A (x) = 1 (c1,c2) (x) + x − x l c 1 − x l 1 (x l ,c1) (x) + x u − x x u − c 2 1 (c2,xu) (x)(1)
with x l , x u , c 1 , c 2 ∈ R being lower, upper bounds, and first and second modes, respectively. The symbol 1 (a,b) (x) denotes the indicator function in the interval (a, b). Interestingly, the trapezoidal fuzzy set includes the triangular (if c 1 = c 2 ) and rectangular (if x l = c 1 , c 2 = x u ) fuzzy sets as special cases. A degenerated fuzzy numberÅ is a particular fuzzy set with ξ A (c) = 1 and ξ A (x) = 0 for x = c, x ∈ A. Note that rectangular and degenerated fuzzy numbers can be adopted to represent crisp categories and crisp observations, respectively. When a probability space is defined over A, the probability of a fuzzy setà can be defined as P(Ã) = A ξ A (x)dP (Zadeh's probability). Similarly, the joint probability of two fuzzy sets is P(ÃB)
= A ξ A (x)ξ B (x)dP under the rule ξÃB(x) = ξ A (x)ξ B (x) (independence of fuzzy sets) [39].
Fuzzy granules
Let S = {Ã 1 , . . . ,Ã i , .
. . ,à I } be a sample of I fuzzy or non-precise observations withà i being a fuzzy number as defined by Eq. (1). Then the interval R(S) = [r 0 , r 1 ] ⊂ R is the range of the fuzzy sample where r 0 = min{A † 01 , . . . , A † 0I } and r 1 = max{A † 01 , . . . , A † 0I }, with A † 0i being the infimum of the support set A 0i computed for the i-th fuzzy observation. A collection G = {G 1 , . . . ,G c , . . . ,G C } of C fuzzy sets is a fuzzy partition of R(S) if the following two properties hold (i) max i=1,...,I l(à i ) ≤ min c=1,...,C l(C c ) and (ii) C c=1 ξ Gc (x) = 1 (Ruspini's partition) [40,41]. The fuzzy sets in G are also called granules of R(S). The evaluation of the amount of fuzzy observations in a granuleG c is called cardinality (scalar or fuzzy) and can be used to compute fuzzy frequencies or counts for a partition G given a sample S. Figure 1 (leftside panels) shows an example of fuzzy granulation for both fuzzy and crisp observations.
Fuzzy counts as generalized natural numbers
Letx j = {x j 1 , . . . ,x j i , . . . ,x j I } andx k = {x k 1 , . . . ,x k i , . .
. ,x k I } be two samples of fuzzy observations andg j = {g j 1 , . . . ,g j c , . . . ,g j C } andg k = {g k 1 , . . . ,g k r , . . . ,g k R } be two fuzzy partitions of the domains R(x j ) and R(x k ). Given a pair of granule (g r ,g c ), a fuzzy or imprecise count for the joint sample (x j ,x k ) is a fuzzy setñ jk rc with membership function ξ jk nrc : N 0 → [0, 1]. As it is defined over natural numbers, a fuzzy count is a finite generalized natural number for which extended operations are available (e.g., addition, multiplication) [42]. Analogously to fuzzy intervals, the class of all fuzzy counts is denoted as F (N 0 ). There are different choices for the computation of ξ jk nrc (e.g., see: [32,33,34,43,44]). In this contribution, we will follow the findings of [45] and [40] which are based on Zadeh's fuzzy counting functions [46] and fuzzy cardinalities [47]. More precisely, let ǫ jk rc = ǫ jk rc1 , . . . , ǫ jk rci , . . . , ǫ jk rcI be the vector of joint degrees of inclusion for the rc-th granule where
ǫ jk rci = min ǫ j ri , ǫ k ci ǫ j ri = min x∈R(xj) (ξ xji (x), ξ gr (x)) max(1, |x j i |) ǫ k ci = min x∈R(x k ) (ξ x ki (x), ξ gc (x)) max(1, |x k i |)
with |.| being the simple cardinality according to the definition given in Sect. 2.1. For n ∈ N 0 , the fuzzy count is as follows:
ξ jk nrc (n) = min (µ FLC (n), µ FGC (n))(2)
with µ FLC (n) and µ FGC (n) being the output of the Zadeh's fuzzy counting functions [46]. The following calculus can be used for µ FLC (n) and µ FGC (n). First, compute the square matrix of differences Z I×I = ǫ jk rc 1 T I − 1 I (ǫ jk rc ) T , with 1 I being a I × 1 vector of all ones. Then, for each i = 1, . . . , I the vector z I×1 is computed, with z i = 1 T I H(Z ,i ) and H(x) being the Heaviside step-function defined
by H(x) := {0 if x < 0 , 1 if x ≥ 0}.
The vector z = (z 1 , . . . , z i , . . . , z I ) contains the sums of the output of the Heaviside function applied column-wise on Z. Finally, for n = 0, 1, 2, . . . , I the Zadeh's counting functions are as follows:
µ FGC (n) = max H(z − n) ⊙ ǫ jk rc (3) µ FLC (n) = 1 − max H(z − n + 1) ⊙ ǫ jk rc
where ⊙ is the element-wise product whereas H(x) is the Heaviside function defined as above. Thus, the membership function ofñ jk rc is defined as the minimum between the degree of possibility that at least n elements from (x j ,x k ) are included in the rc-th granule (FGC count) and the degree of possibility that at most n elements are included in the rc-th granule (FLC count). By applying Eqs.
(2)-(3) for each pair of granules (g 1 ,g 1 ), . . . , (g r ,g c ), . . . , (g R ,g C ) one obtains the fuzzy frequency matrix N jk R×C . Note that the resulting fuzzy set ξ nrc may not be normal, i.e. max n ξ nrc (n) ≤ 1, and a post-hoc normalization should be applied if normal fuzzy sets were needed. Table 1 summarizes the algorithm for computing fuzzy frequencies for a given pair of fuzzy variables.
Finally, it is relevant to point out that Eqs.
(2)-(3) are quite general and can be applied for the cases of fuzzy observations/fuzzy categories, crisp observations/fuzzy categories, and fuzzy observations/crisp categories. In this context, crisp observations and crisp categories can be realized by means of degenerated fuzzy sets and rectangular fuzzy sets, respectively. For the special case of crisp observations/crisp categories, the resulting fuzzy set ξ nrc is degenerate. Figure 1 shows an exemplary case of fuzzy frequencies for fuzzy observations and fuzzy categories ( Figure 1-A, middle and rightmost panels) and crisp observations and fuzzy categories as well (Figure 1-B, middle and rightmost panels).
LLCs for fuzzy frequency tables
In this section we describe the statistical procedure for computing latent linear correlations when observations are in the general form of fuzzy frequencies.
Model
Let X = (X j i , X k i ) i = 1, . . . , I be a collection of pairs of continuous random variables (j, k ∈ {1, . . . , J}, j = k) following the bivariate Gaussian distribution centered at zero with correlation parameter ρ jk ∈ [−1, 1] and density
f X (x; ρ jk ) = 1 2π 1 − ρ 2 jk exp − 1 2 (x j ) 2 + (x k ) 2 − 2x j x k ρ jk 1 − ρ 2 jk(4)
for −∞ < x j < ∞ and −∞ < x k < ∞. Without loss of generality, consider the collection of fuzzy observationsỹ
= {(ỹ j 1 ,ỹ k 1 ), . . . , (ỹ j i ,ỹ k i ), . . . , (ỹ j I ,ỹ k I )} which relates to the (latent) bivariate Gaussian model in Eq. (4) via the constraint (ỹ j i ∈g j r ) ∧ (ỹ k i ∈g k c ) iff (X j i , X k i ) ∈ (τ X j r−1 , τ X j r ] × (τ X k c−1 , τ X k c ] ⊂ R 2(5)
Algorithm 1 Computing fuzzy frequencies
procedure Main(x j ,x k ,g j ,g k ) for r = 1, . . . , R and c = 1, . . . , C do ǫ j r ← DoI(x j ,g r ) ⊲ Compute degrees of inclusion ǫ k c ← DoI(x k ,g c ) ǫ jk rc ← min(ǫ j r , ǫ k c ) ⊲ Compute joint degree of inclusion Z ← (ǫ jk rc 1 T I − 1 T I ǫ jk rc ) for i = 1, . . . , I do z[i] ← 1 T I H(Z[:, i]) end for for n = 0, . . . , I do µ FGC [n] ← max(H(z − n) ⊙ ǫ jk rc ) ⊲ Fuzzy counting functions µ FLC [n] ← 1 − max(H(z − (n + 1)) ⊙ ǫ jk rc ) ξ jk nrc [n] ← min(µ FLC [n], µ FGC [n])
⊲ Compute fuzzy frequencies end for end for return ξ jk n end procedure procedure DoI(x,g)
for i = 1, . . . , I do ǫ[i] ← x min ξ x [i] (x), ξ g (x) dx max 1, x ξ x [i] (x)dx
⊲ Ratio of fuzzy cardinality end for return ǫ end procedure Note: The algorithm requires as input the I × 1 arrays of fuzzy observationsx j andx k along with the fuzzy categoriesg j andg k for the j, k-th pair of variables and returns as output the R × C array of membership functions ξ jk n = (ξ n 11 , . . . , ξ nrc , . . . , ξ n RC ) associated to each fuzzy countñ jk rc . where ∈ is intended as fuzzy membership, (g j r ,g k c ) are observed fuzzy categories or granules, and the arrays τ X j = (τ Xj 0 , . . . , τ Xj r , . . . , τ Xj R ) and τ X k = (τ X k 0 , . . . , τ X k c , . . . , τ X k C ) are thresholds of the bivariate support R 2 under the conventions τ
(A) G1 1 4 7 0 1 G2 (B)Xj 0 = τ X k 0 = −∞ and τ Xj R = τ X k C = ∞.
Note that since fuzzy numbers encompass crisp observations and crisp categories as special cases (i.e., degenerated and rectangular fuzzy numbers, respectively), the expression (5) can be used for the non fuzzy case as well. For instance, the simplest situation involving non fuzzy observations and non fuzzy categories can be obtained rewriting the left part of the constraint as (ẙ j i = r) ∧ (ẙ k i = c), which indicates that crisp observations take the indices of the categories.
The parameter space for the LLCs model is
θ = {ρ jk , τ X j , τ X k } ∈ [−1, 1] × R R−1 × R C−1
whereas the log-likelihood function takes the following form in the case of independent and identically distributed fuzzy observations [1,3]:
ln L(θ; N) = K − R r=1 C c=1 n∈N0
n ξ jk nrc (n) ln π jk rc (θ)
= K − R r=1 C c=1 n∈N0 n ξ jk nrc (n) ln τ X j r τ X j r−1 τ X k c τ X k c−1 f X (x; ρ jk ) dx j dx k(6)
where f X (x; ρ jk ) is the model's density in Eq. (4), ξ jk nrc (n) is the rc-th fuzzy count, and K is a constant term. Note that f X (x; ρ jk ) is not fuzzy in this context and its realizations represent unobserved (latent) quantities. The evaluation of (ỹ j i ∈g j r ) ∧ (ỹ k i ∈g k c ) gives raise to a collection of fuzzy countsñ jk 11 , . . . ,ñ jk rc , . . . ,ñ jk RC acting as possibilistic constraints on the unobserved non-fuzzy counts which would be observed if fuzziness was missed. As such, the expression ξ jk nrc (n rc ) ∈ [0, 1] should be interpreted as the possibility that the crisp count n rc has to occur, with ξ jk nrc (n rc ) = 1 indicating that n rc is fully possible. According to the epistemic viewpoint on fuzzy statistics [48], the sampling process is thought as being the consequence of a two-stage generation mechanism, the first of which is a random experiment and the second is a non-random fuzzification of the outcome being realized. As an example of this schema, consider the simplest case of crisp observations (e.g., income and tobacco use) that are classified by a group of raters or an automatic classification system on the basis of fuzzy categories (e.g., income levels: low, medium, high; tobacco use: none, sporadic, habitual). Stated in this way, the fuzzy frequencies associated to income and tobacco use encapsulate two sources of uncertainty, namely the random component due to the sampling process and the non-random component due to the post-sampling fuzzy classification.
Parameter estimation
To estimate θ we adopt the Olsson's two-stage approach for latent linear correlations which iteratively alternates between approximatingτ from the observed count data and maximizing Eq. (6) with respect toρ given the current thresholds [1]. In the case of fuzzy data, this procedure can be implemented using a variant of the Expectation-Maximization algorithm generalized to the case of fuzzy observations [31]. Likewise for the standard EM algorithm, the fuzzy-EM version alternates between the E-step, which requires computing the expected complete log-likelihood given the candidate θ ′ = θ (q−1) , and the M-step, which maximizes the expected complete log-likelihood w.r.t. θ (q) . More precisely, in the fuzzy-EM algorithm the complete-data log-likelihood is that obtained if the matrix of counts N jk R×C was precisely observed, namely:
ln L(θ; N) = ln I! − R r=1 C c=1 n jk rc ln π jk rc (θ) − R r=1 C c=1 ln n jk rc !(7)
Given the estimates θ ′ , the E-step for the (q)-th iteration consists of computing the Q-function via conditional expectation on the observed fuzzy counts:
Q(θ, θ ′ ) = E θ ′ ln L(θ; N) N ∝ R r=1 C c=1 E θ ′ N jk rc ñ jk rc ln π jk rc (θ) − E θ ′ ln N jk rc ! ñ jk rc(8)
The conditional expectations involve the density of a discrete random variable N rc conditioned on a fuzzy eventñ rc that, under the multinomial schema for random counts, can reasonably be modeled as Binomial [49]. Thus, using the definition of fuzzy probability, N rc |ñ rc is as follows:
p N jk rc |ñ jk rc (n; π jk rc (θ)) = P θ N jk rc ,ñ jk rc P θ ñ jk rc = ξ jk n jk (n)p N jk rc (n; π jk rc (θ))
n∈N0 n ξ jk n jk (n)p N jk rc (n; π jk rc (θ))
(9) π jk rc (θ) = τ X j r τ X j r−1 τ X k c τ X k c−1 f X (x; ρ jk ) dx j dx k(10)
where p N jk rc = Bin(n; π jk rc (θ)) and f X (x; ρ jk ) is the latent model's density in Eq. (4). Note that the quantity Iπ jk rc (θ) is the reconstructed count from the bivariate latent model given the current parameters θ ′ [50]. The linear form of the expectations in Eq. (8) is:
E θ ′ N jk rc ñ jk rc = n∈N0 n p N jk rc |ñ jk rc (n; π jk rc (θ ′ ))(11)
whereas, since it is not involved in the M-step of the algorithm, the non linear expectation is provided in Appendix A for the sake of completeness. Finally, the M-step for the (q)-th iteration requires maximizing the functional Q(θ, θ ′ ) with respect to θ. Given the filtered counts at the current iteration N jk R×C (see Eq. 11), the Olsson's two-stage estimation approach requires the estimation of thresholds from the cumulative marginals of filtered counts first:
τ (q) X j = Φ −1 A R×R N jk 1 C (12) τ (q) X k = Φ −1 A C×C ( N jk ) T 1 R(13)
where A is a lower triangular matrix of ones, 1 is a vector of appropriate order of all 1/I, and Φ is the Gaussian univariate distribution function with mean zero and unitary variance. Next, conditioned on { τ (q)
X j , τ (q)
X k }, the remaining parameter is found by solving the score equation of Q(θ, θ (q) ) numerically w.r.t. ρ jk :
U ρ jk = ∂Q ρ jk , { τ (q) X j , τ (q) X k } ∂π jk ∂π jk ∂ρ jk = 0(14)
The algorithm proceeds iteratively until the log-likelihood does not increase significantly. Table 1 summarizes the fuzzy-EM algorithm for the LLCs model.
Remarks
About the convergence of the algorithm. Given a candidate θ ′ , the fuzzy-EM starts by constructing the surrogate Q(θ, θ ′ ) that lower bounds the observed data log-likelihood ln L(θ; N) (E-step). Next, it is maximized to get the current estimates θ (q) (M-step), which is in turn used to construct a new lower bound Q(θ, θ (q) ) in the next iteration to get a new estimate θ (q+1) . The estimates in the M-step are chosen so that Q(θ, θ (q) ) ≥ Q(θ, θ ′ ), which forms the base of the monotonicity condition ln L(θ (q+1) ; N) ≥ ln L(θ (q) ; N) [51]. As for the standard case, the monotonicity of the sequence {ln L(θ (q) } q∈N implies the convergence to a stationary value, which can be global or local depending on the characteristics of the log-likelihood function and the starting point θ 0 . A sketch of the proof Algorithm 2 Olsson's two-stage approach via fuzzy-EM algorithm for j ∈ (1, . . . , J) and k ∈ (1, . . . , J), j = k, do: Table 1: Expectation-Maximization algorithm for estimating θ = (τ X j , τ X k , ρ jk ) in LLCs model with fuzzy frequency data.
q = 1 : Set θ (q) = (ρ 0 jk , τ 0 X j , τ 0 X k ), l (q) = l 0 , ǫ = 1e −09 initialization q > 1 : Compute π jk (θ (q−1) ) from Eq. (10) E-Step Compute N jk given θ (q−1) from Eq. (11) Compute ln N jk ! given θ (q−1) from Eq. (A.1) Compute { τ (q) X j , τ (q) X k } from Eqs. (12)-(13) M-Step Set θ (q) = ρ (q−1) jk , τ (q) X j , τ (q) X k Solve ∂ ∂ρ jk Q(θ, θ (q) ) = 0 w.r.t. ρ jk see Eq. (14) Set θ (q) = ρ (q) jk , τ (q) X j , τ (q) X k Evaluate l (q) = ln L(θ (q) ; N) see Eq. (7) Finalization Compute l δ = (l (q) − l (q−1) ) If l δ < ǫ, set θ = θ (q) and stop the algorithm R[j, k] = ρ (q) jk
of the monotonicity of the fuzzy-EM for the LLCs is provided in Appendix B whereas the formal equivalence between EM and fuzzy-EM is detailed in [31,35].
About the starting values of the algorithm. Suitable starting values θ 0 can be obtained by first defuzzifying the observed fuzzy frequencies matrix N jk to obtain non fuzzy counts and then applying the standard Olsson's two-stage approach [1] on defuzzified data. In general, this yields to convenient starting values. In the LLCs model, defuzzification can be performed via mean or max -based procedures as follows:n mean rc ∼ = n∈N0 nξ nrc (n)/ n∈N0 ξ nrc (n) ,n max rc = max{n ∈ N 0 : ξ nrc (n) = max z∈N0 ξ nrc (z)}, r = 1, . . . , R, c = 1, . . . , C.
About the term p Nrc|ñrc (n; π rc (θ)). The term p Nrc|ñrc represents the density of a non fuzzy random variable conditioned on fuzzy numbers and can mathematically be interpreted as the combination of two independent components, namely the random mechanism underlying the sampling process and the observer's partial knowledge (imprecision) about the sample realizations. In this sense, as it weights each fuzzy datum by the probability that it has to occur [39], p Nrc|ñrc should not be confused with the mean-based defuzzification of fuzzy numbers. A nice property of this formulation is that fuzziness vanishes when precise observations are available. Indeed, the conditional density involving a degenerated fuzzy numbern rc boils down to a degenerated discrete density p Nrc|nrc with nonzero probability masses only for those n such that ξ nrc (n) = 1. As a consequence, the fuzzy-EM procedure reduces to standard Olsson's two-stage maximum likelihood estimation. In general, there are a number of ways for plugging-in non-stochastic components of uncertainty into p Nrc|ñrc , such as those involving imprecise probability [52], conditional probability [53], belief measures [54], and random fuzzy variables [55].
About the computation of standard errors and inference. Standard errors forρ jk can be computed as a byproduct of the EM procedure [51]. In particular, the following approximation of the information matrix is required Iρ jk
∼ = I ê ρ jk = R r=1 C c=1 U (rc) ρ jk U (rc) ρ jk , with U (rc)
ρ jk being the score functional for the rc-th observation calculated atρ jk , to get the standard error σρ jk = (1/I ê ρ jk ) 1 2 . Alternatively, they can also be obtained by means of non-parametric or parametric bootstrap techniques [56]. Finally, it should be remarked that inference about ρ jk can be made based on the asymptotic results of fuzzy likelihood ratio statistics (e.g., see [57]).
About the polychoric correlation matrix R J×J . As for the standard approach used in computing polychoric correlation matrices (e.g., see: [1,58]), also in the case of fuzzy data the matrix of latent linear correlations is obtained by calculating each element ρ jk of the correlation matrix pairwise. Although this approach offers a simple and effective alternative to more challenging methods (e.g., see: [59,60]), in some circumstances it may lead to non-positive definite correlation matrices. This can be problematic, especially when such matrices are used as input of other statistical models such as factor analyses or SEMs [61]. In these cases, eigenvalue decomposition based smoothing [62], least squares [62] or Dykstra's [63] corrections constitute workable solutions to solve this issue.
Simulation study
The aim of this simulation study is twofold. First, we wish to evaluate the performances of fuzzy-EM algorithm in estimating parameters of the LLCs model and, second, to investigate whether the standard Olsson's maximum likelihood procedure performs as good as the proposed method if applied on max-based and mean-based defuzzified data. The case J = 2 has been considered for the sake of simplicity. The Monte Carlo study has been performed on a (remote) HPC machine based on 16 cpu Intel Xeon CPU E5-2630L v3 1.80 GHz,16x4 GB Ram whereas computations and analyses have been done in the R framework for statistical analyses. Data generation and procedure. Let I a , ρ 0 b , R d = C d be distinct levels of the factors I, ρ 0 , R, and C. Then, fuzzy frequency data have been generated according to the following procedure. For each r = 1, . . . , R d and c = 1, . . . , C d :
(i) Set n rc = I a π rc (see Eq. (10)) given τ 0 X j , τ 0 X k , ρ 0 b , and I a (ii) the imprecision concerning n rc was generated as follows: m 1 ∼ Gamma d (α m1 , β m1 ) where α m1 = 1 + n rc β m1 , β m1 = (n rc + n 2 rc + 4s 2 1 ) 1 2 2s 2 1 , s 1 ∼ Gamma d (α s1 , β s1 ), α s1 = 1 + m 0 β s1 , β s1 = (m 0 + m 2 0 + 4s 2 0 ) 1 2 2s 2 0 , m 0 = 1 and s 0 = 0.25, with Gamma d indicating the discrete Gamma random variable with shape and rate being reparameterized in terms of mean m and variance s (iii) the fuzzy set associated toñ rc was obtained via the following probability-possibility transformation: ξñ rc = f G d (n; α rc , β rc ) max f G d (n; α rc , β rc ), with n = {0, 1, . . . , I a }, α rc = 1 + m 1 β s1 , β s1 = 1+(m 1 +m 2 1 +4s 2 1 ) 1 2 /2s 2 1 , β rc = (m 1 +m 2 1 +4s 2 1 ) 1 2 2s 2 1 , and f G d (n; α rc , β rc ) being the discrete Gamma density normalized to one in order to mimics the behavior of a normal fuzzy set [37]. The discrete density f G d is computed as difference of survival functions of the continuous Gamma density S G (x) − S G (x + 1) [64,65].
Note that step (ii) is required in order to make crisp counts entirely imprecise so thatñ rc is no longer centered on n rc . Finally, parameters θ = {ρ, τ X j , τ X k } were estimated from the fuzzy counts N R d ×C d using the fuzzy-EM algorithm (fEM) and the standard Olsson's two-stage maximum likelihood on max-based (dML-max) and mean-based (dML-mean) defuzzified counts (see Sect. 3
.3).
Outcome measures. For each condition of the simulation design, the three methods (i.e., fEM, MLmax, ML-mean) were evaluated in terms of bias and root mean square errors. In addition, for each method thresholds were aggregated to form a scalar statistic, namely τ = 1 T R d τ X j and τ = 1 T C d τ X k (note that τ X j and τ X k are equal by design).
Results. Tables 2-5 report the results of the simulation study with regards toρ andτ for both R = C = 4 and R = C = 6 cases. We begin with the correlation parameter ρ for the case R = C = 4 (see Table 2). Considering ρ 0 = 0.15, the methods showed negligible bias in estimating ρ. However, they differed in terms of RMSE, with fEM showing lower values with increasing sample size if compared to dML-max and dML-mean. With increasing correlation length (ρ 0 > 0.15), bias of estimates as well as RMSE were more pronounced for dML-max and dML-mean. The same results were also observed for the case with R = C = 6 (see Table 3). With regards to the overall statisticτ for the threshold parameters, all the methods achieved comparable results regardless of ρ 0 . In particular, fEM showed slightly higher bias and RMSE then dML-max and dML-mean methods across R = C = 4 (see Table 4) and R = C = 6 (see Table 5) conditions. To further investigate these results, we studied average bias and variance of estimates forτ X j (orτ X k ) as a function of sample size I and ρ 0 . We found that the leftmost and rightmost thresholds tended to be slightly larger for fEM as opposed to the innermost thresholds for both R = C = 4 (see Supplementary Materials, Figure S1) and R = C = 6 conditions (see Supplementary Materials, Figure S2). Moreover, the variance of estimates for the leftmost and rightmost thresholds was higher if compared to the innermost thresholds (see Supplementary Materials, Table S2) but, as expected, it reduced with increasing sample size regardless of ρ 0 . This is not surprising given that we implemented a standard LLCs model in which no particular constraints were applied on threshold estimates, such as 1 T R dτ X j = 0 (e.g., see [66]). 1 Most importantly, according to the Gaussianity assumption underlying the LLCs model, estimated thresholds were symmetric and equidistant with respect to the fixed point zero (see Supplementary Materials, Table S1). Overall, the results suggest that fEM should be preferred over defuzzified maximum likelihood when the interest is in estimating the latent linear association ρ among pairs of variables and fuzzy frequency statistics are available. On the contrary, for those particular cases where ρ is known and the interest is in estimating the true threshold parameters, then standard Olsson's maximum likelihood method can directly be applied after defuzzifiyng observed fuzzy frequency counts. Table 2: Simulation study: Average bias and root mean square errors for ρ in the condition R = C = 4. Note that fEM is the fuzzy-EM algorithm whereas dML-max and dML-mean denote the standard maximum likelihood based on max-based and mean-based defuzzified counts.
Applications
In this section we describe the application of the proposed method to two case studies from health and natural sciences, involving the assessment of a psychotherapeutic intervention (application 1) and the evaluation of meteorological characteristics for forty Turkish cities (application 2). Note that both the applications are provided to merely illustrate the use of fuzzy LLCs model when dealing with imprecise data. Table 3: Simulation study: Average bias and root mean square errors for ρ in the condition R = C = 6. Note that fEM is the fuzzy-EM algorithm whereas dML-max and dML-mean denote the standard maximum likelihood based on max-based and mean-based defuzzified counts.
Application 1: Assessing the outcome of a therapy
Evaluating the quality of a psychotherapy session plays a central role in evidence-based medicine. A typical approach to understand the fundamentals of the therapeutic process consists in asking experts to assess the global quality and characteristics of the therapist-patient relationship through specialized instruments such as the PQS questionnaire [68]. The data thus collected generally consist either of ratings or classification of attributes made through bounded and graded scales. Because of their characteristics, these tasks often involve imprecision and vagueness that can adequately be accounted for by the fuzzy statistical modeling. In this application, we consider the assessment of a psychotherapy session by means of the PQS questionnaire. Data were originally collected by [69] an refers to I = 60 evaluations of psychotherapy on a 9-point scale over J = 3 dimensions of assessment. Given the nature of the task, the three variables were originally considered to be fuzzy, each with three trapezoidal fuzzy categories. To account for the extremes of the classification scale, two more outer categories were added so that R = C = 5 (see Table 6). Figure 2 shows the granulation based on five fuzzy categories (G 0 ,. . .,G 4 ) for each dimensions of assessment along with the corresponding crisp observations. The aim is to compute the correlation matrix for the three fuzzy variables, with the hypothesis that the higher degree of association is related to a good therapeutic outcome. The first step requires computing the fuzzy frequency matrix N 5×5 for each pair of J = 3 fuzzy variables given the crisp observed data. Next, the matrix of fuzzy counts is used to estimate the Table 4: Simulation study: Average bias and root mean square errors for the aggregated thresholds τ = 1 T R d τ X j in the condition R = C = 4. Note that fEM is the fuzzy-EM algorithm whereas dMLmax and dML-mean denote the standard maximum likelihood based on max-based and mean-based defuzzified counts. latent linear correlation matrix R 5×5 . Figure 3 shows a graphical representation of the matrix of fuzzy counts N 5×5 for one pair of variables (i.e., X 2 ,X 3 ). It contains fuzzy numbers with various degree of fuzziness and include combinations with degenerated fuzzy counts as well (i.e., G Table 7 reports the estimates of LLC coefficients. Overall, the results showed a low level of association among the three dimensions, which in turn indicated that the psychotherapy being assessed cannot be classified as having a good outcome.
Application 2: Effect of climatic variables on rainfall
Meteorological variables are generally used to assess the impact of climatic characteristics in many phenomena including human as well as non-human activities. Although often regarded as discrete or continuous measurements, these variables can benefit from fuzzy coding in some circumstances. Examples include cases in which these variables are imprecisely coded (e.g., when data are available in terms of intervals or linguistic categories) or when they are derived from a variety of sources (e.g., samples, historical databases, experts) that need to be integrated before being used for data analysis [24,70]. In this application, we consider the analysis of J = 5 meteorological variables (i.e., Table 5: Simulation study: Average bias and root mean square errors for the aggregated thresholds τ = 1 T R d τ X j in the condition R = C = 6. Note that fEM is the fuzzy-EM algorithm whereas dMLmax and dML-mean denote the standard maximum likelihood based on max-based and mean-based defuzzified counts. Table 6: Application 1: Fuzzy categories for the three variables of the assessment task. Note that each category is represented by means of trapezoidal fuzzy numbers (see Eq. 1).
X 1 X 2 X 3 x l c 1 c 2 x u x l c 1 c 2 x u x l c 1 c 2 x u r = 1
were originally coded using R = C = 3 fuzzy triangular categories (G 0 : minimum; G 1 : medium; G 2 : maximum) and membership grades ǫ
(j) 1 , ǫ (j) 2 , ǫ(j)
3 j = 1, . . . , 5 constitute the input data for the subsequent analysis. The aim is to explore the effects of climatic variables on rainfall (PRE) by means of a path analysis model. Likewise for the first application, the first step consisted in computing the fuzzy frequency matrix N 3×3 for each pair of the five climatic variables given the observed membership degrees. Then, the LLCs matrix was estimated using the fuzzy-EM algorithm. X 1 X 2 X 3 X 1 1.00000 X 2 0.06948 1.00000 X 3 0.00004 0.21762 1.00000 Table 7: Application 1: Latent linear correlation matrix estimated via Olsson's two-stage fuzzy-EM algorithm.
4 shows an example of fuzzy counts for the pair of variables PRE-HUM whereas Table 8 reports the estimated correlations for the variables involved in the study. As expected, the results showed a certain level of association among the five climatic variables. Once the LLCs matrix has been estimated, we proceeded by modeling the effects of the climatic variables on PRE via path analysis (see Figure 5). In particular, we expected that a higher humidity (HUM) increased rainfall (PRE) and that sunshine duration (SUN) decreased the levels of precipitation (PRE). Similarly, we also expected an indirect effect of altitude (ALT) on humidity (HUM) through temperatures (TEMP). The path model has been estimated on the LLCs matrix via maximum likelihood as implemented in the R library lavaan [72]. Overall, the estimated model showed a moderate Table 9: Application 2: Estimated coefficientsβ and residual variancesσ 2 ǫ for the path model depicted in Figure 5 along with the standard errorsσ 2 β of the estimates.
Conclusions
In this article we described a novel approach to estimate latent linear correlations (LLCs) when data are in the form of fuzzy frequency tables. In particular, we represented fuzzy counts in terms of generalized natural numbers first, and then we generalized the sample space of the standard LLCs model to cope with fuzzy counts while retaining its parameter space as non-fuzzy. The resulting model encapsulated both random and non-random/imprecision components in a unified statistical representation. Since the inferential interest is on estimating the latent correlation matrix of the observed variables, parameter estimation was performed via fuzzy maximum likelihood using the Expectation-Maximization algorithm. A simulation study and two real applications were developed to highlight the characteristics of the fuzzy LLCs model. Overall, the simulation results revealed that the fuzzy LLCs model showed more accurate results in estimating the true correlation matrix as opposed to standard methods which can be applied on defuzzified data. The applications showed how the proposed method can be of particular value in situations involving fuzzy classification and fuzzy coding as well.
A particular advantage of the fuzzy LLCs model is its simplicity and ability to deal with situations involving imprecise classification problems. Moreover, the proposed method works with both fuzzy observations/crisp categories and crisp observations/fuzzy categories and, as such, it includes the standard crisp observations/crisp categories as a special case. Again, the fuzzy LLCs model does not require the extension of its parametric representation to account for fuzzy frequency data and, consequently, parameter estimation and inference can be performed using the asymptotic properties of maximum likelihood theory. This is quite convenient and obviates the need of generalizing LLCsbased statistical modeling -such as structural equation models and factor analysis -to the fuzzy case. A limitation of the proposed approach is that it is based on the simplest, but still used, assumption of Gaussianity for LLCs. Although it has been proved that the assumption holds in several empirical contexts, there may be the need of LLCs based on more general probabilistic models (e.g., Skew-Gaussian, Elliptical, t, Copula-based). As a result, the problems already identified by other researchers, for instance bias in estimating the asymptotic covariance matrix of the LLCs matrix [7,66], still persist in the fuzzy case.
There are a number of further extensions to this project that can be undertaken in future research studies. For instance, the use of more general probabilistic model would extend the proposed method to handle with situations involving violations of Gaussianity assumption. Another aspect which might be interesting to investigate is the case where data need to be represented using more general fuzzy numbers (e.g., beta, exponential, gaussian), which would allow the proposed method to cope with cases requiring more flexible models to represent non-random imprecision. Finally, studying the properties of fuzzy LLCs-based statistical models like structural equation modeling or factor analysis would also constitute a research topic to be considered in a further study.
A Appendix
To compute the nonlinear expectation E θ ′ ln N jk rc ! ñ jk rc , we first approximate the factorial term via Stirling's formula:
E θ ′ ln N jk rc ! ñ jk rc = E θ ′ N jk rc ln N jk rc − N jk rc + 1 2 ln 2π ñ jk rc = E θ ′ N jk rc ln N jk rc ñ jk rc − E θ ′ N jk rc ñ jk rc = E θ ′ g N jk rc ñ jk rc − E θ ′ N jk rc ñ jk rc
with g(x) := x ln x. Next, since the non linear transformation g(.) is smooth and twice-differentiable on (0, ∞) with g ′′ (x) = 1/x, a second-order Taylor expansion around the first conditional moment E θ ′ N jk rc ñ rc can be developed to get the closed-form expression of the expectation term:
E θ ′ g(N jk rc ) ñ jk rc ∼ = g E θ ′ N jk rc ñ jk rc + Var θ ′ N jk rc ñ jk rc 2E θ ′ N jk rc ñ jk rc (A.1)
with the conditional variance being defined by
Var θ ′ N jk rc ñ jk rc = n∈N0 n − E θ ′ N jk rc ñ jk rc 2 p N jk rc |ñ jk rc (n; π jk rc (θ ′ ))
where E θ ′ N jk rc ñ jk rc is as in Eq. (11).
B Appendix
To establish monotonicity for a sequence of log-likelihood evaluations {ln L(θ (q) ;Ñ)} q∈N of the fuzzy Expectation Maximization algorithm we will follow the general results of [51], Sect. 3.2. A similar proof is also given by [73] for the case of rectangular fuzzy numbers (i.e., interval-valued data). In what follows, we will omit the indices j, k for the sake of simplicity. Given θ ′ = θ (q−1) and by rearranging Eq. (9), we get by standard calculus:
ln L(θ; N) = ln L(θ; N) − R r=1 C c=1
ln p Nrc|ñrc (n; π rc (θ))
= E θ ′ R r=1 C c=1 ln L(θ; N rc ) ñ rc − E θ ′ R r=1 C c=1
ln p Nrc|ñrc (n; π rc (θ)) ñ rc = Q(θ; θ ′ ) − S(θ; θ ′ ) (A.2)
Then, an increasing of the observed log-likelihood can be written in terms of the result (A.2) as follows:
ln L(θ (q) ; N) − ln L(θ ′ ; N) ≥ Q(θ (q) ; θ ′ )) − Q(θ ′ ; θ ′ ) − S(θ (q) ; θ ′ ) − S(θ ′ ; θ ′ )
Note that because θ (q) is chosen so that Q(θ (q) ; θ ′ )) − Q(θ ′ ; θ ′ ) ≥ 0 [51], the condition S(θ; θ ′ ) − S(θ ′ ; θ ′ ) ≤ 0 must hold for each θ. To do so, we proceed as follows:
S(θ; θ ′ ) − S(θ ′ ; θ ′ ) = R r=1 C c=1 E θ ′ ln p Nrc|ñrc (n; π rc (θ)) ñ rc − E θ ′ ln p Nrc|ñrc (n; π rc (θ ′ )) ñ rc = R r=1 C c=1 E θ ′ ln p Nrc|ñrc (n; π rc (θ)) p Nrc|ñrc (n; π rc (θ ′ )) ñ rc ≤ R r=1 C c=1 ln E θ ′ p Nrc|ñrc (n; π rc (θ)) p Nrc|ñrc (n; π rc (θ ′ )) ñ rc using Jensen's inequality ≤ R r=1 C c=1 ln n∈N0 p Nrc|ñrc (n; π rc (θ)) p Nrc|ñrc (n; π rc (θ ′ )) p Nrc|ñrc (n; π rc (θ ′ )) n ≤ R r=1 C c=1 ln n∈N0 p Nrc|ñrc (n; π rc (θ)) n ≤ R r=1 C c=1 ln 1 = 0
Hence, an increasing of ln L(θ (q) ; N) − ln L(θ ′ ; N) ≥ 0 is guaranteed as soon as Q(θ (q) ; θ ′ )) − Q(θ ′ ; θ ′ ) ≥ 0. Table S.1: Simulation study: Average mean values of τ X j for fEM algorithm. Note that, by design, τ X j = τ X k . Table S.2: Simulation study: Average variance values of τ X j for fEM algorithm. Note that, by design, τ X j = τ X k . Simulation study: Average bias for or leftmost and rightmost values of τ X j in the case R = C = 6. Note that, by design, τ X j = τ X k .
Supplementary Materials
R = C = 4 R = C =
Figure 1 :
1Examples of fuzzy granules and fuzzy counts for (A) fuzzy triangular observations and fuzzy trapezoidal categories and (B) crisp observations and fuzzy trapezoidal categories. Note that in both cases frequencies are represented as generalized natural numbers.
Design. The design of the study involved three factors, namely (i) I ∈ {150, 250, 500, 1000}, (ii) ρ 0 ∈ {0.15, 0.50, 0.85}, (iii) R = C ∈ {4, 6}, which were varied in a complete factorial design with 4 × 3 × 2 = 24 possible combinations. The threshold parameters were held fixed under the equidistance hypothesis[58], namely τ 0 X j = τ 0 X k = (−2.00, −0.66, 0.66, 2.00) for the conditions with R = C = 4 and τ 0 X j = τ 0 X k = (−2.00, −1.20, −0.40, 0.40, 1.20, 2.00) for R = C = 6. For each combination, B = 5000 samples were generated yielding to 5000 × 24 = 120000 new data and an equivalent number of parameters.
SUN: daily hours of sunshine; HUM: percentage of humidity; PRE: precipitations; ALT: altitude; MAX: maximum daily temperature) which were collected in forty cities of Turkey during 2004[71].
Figure 2 :
2Application 1: Granulation for the three fuzzy variables along with crisp observations (dashed gray lines).
Figure 3 :
3Application 1: Fuzzy frequency matrix for the pair X 2 ,X 3 . Note that each cell contains a fuzzy natural numberñ rc for a specific combination of the R × C granulation space. fit (R 2 = 0.20). The results highlighted that PRE increased as a function of HUM (β = 0.1844, σ 2 β = 0.1386) and descreased as sunshine duration increased (β = −0.3406,σ 2 β = 0.1386). Humidity was negatively related to temperature (β = −0.2161,σ 2 β = 0.1544), which was in turn negatively associated to altitude (β = −0.5577,σ 2 β = 0.1312) as expected.
Figure 4 :Figure 5 :
45Application 2: Fuzzy frequency matrix for the pair PRE, HUM. Note that each cell contains a fuzzy natural numberñ rc for a specific combination of the R × C granulation space. Application 2: Path model for the effect of the climatic variables on the response variable PRE. Note that straight lines represent direct effects whereas dotted lines indicate correlations.
Figure S. 1 :
1Simulation study: Average bias for leftmost and rightmost values of τ X j in the case R = C = 4. Note that, by design, τ X j = τ X k .
Table 8 :
8Application 2: Latent linear correlation matrix estimated via Olsson's two-stage fuzzy-EM algorithm.
6 I = 1000 -2.04243 0.00261 2.04244 -2.03466 -1.01247 -0.00002 1.01250 2.03503 .06413 0.00510 2.06411 -2.03232 -1.01273 -0.00017 1.01209 2.03153 I = 1000 -2.02622 0.00005 2.02687 -2.05360 -1.01820 0.00520 1.01829 2.05286 ρ = 0.85 I = 150 -2.09765 0.00008 2.09203 -2.02365 -1.02091 0.01715 1.01890 2.02043 I = 250 -2.05866 -0.00023 2.05640 -2.02009 -1.01721 0.00020 1.01737 2.01975 I = 500 -2.02893 -0.00021 2.02840 -2.04960 -1.01681 -0.00013 1.01626 2.04844 I = 1000 -2.04424 0.00249 2.04435 -2.04356 -1.01229 0.00263 1.01250 2.04368τ 1τ2τ3τ1τ2τ3τ4τ5
ρ = 0.15
I = 150
-2.15645 0.01727 2.15876 -2.18341 -1.00894 -0.00060 1.00818 2.18752
I = 250
-2.10902 0.02070 2.10671 -2.06642 -1.03889 0.02140 1.03830 2.06281
I = 500
-2.03274 0.00006 2.03286 -2.07933 -1.02383 0.00543 1.02429 2.07785
ρ = 0.50
I = 150
-1.99019 -0.00017 1.99206 -2.32291 -1.01693 0.00006 1.01759 2.32844
I = 250
-2.06659 -0.00030 2.06650 -2.09860 -1.05156 0.03270 1.05188 2.10372
I = 500
-2
It should be remarked that the unconstrained approach is most common in LLCs-based applications, especially when the primary interest lies in making inference about ρ. In this case, the threshold parameters play an auxiliary role as they only affect the scale of the latent variables underlying the LLCs model (e.g., see[67]).
Acknowledgments. The author wishes to acknowledge and thank Dr. Andrea Spirito for his valuable comments on various issues concerning this research study.
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| {'fraction_non_alphanumeric': 0.061813488576449915, 'fraction_numerical': 0.03601438927943761, 'mean_word_length': 4.15449398443029, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 4, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 42, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This research concerns the estimation of latent linear or polychoric correlations from fuzzy frequency tables. Fuzzy counts are of particular interest to many disciplines including social and behavioral sciences, and are especially relevant when observed data are classified using fuzzy categories -as for socio-economic studies, clinical evaluations, content analysis, inter-rater reliability analysis -or when imprecise observations are classified into either precise or imprecise categories -as for the analysis of ratings data or fuzzy coded variables. In these cases, the space of count matrices is no longer defined over naturals and, consequently, the polychoric estimator cannot be used to accurately estimate latent linear correlations. The aim of this contribution is twofold. First, we illustrate a computational procedure based on generalized natural numbers for computing fuzzy frequencies. Second, we reformulate the problem of estimating latent linear correlations from fuzzy counts in the context of Expectation-Maximization based maximum likelihood estimation. A simulation study and two applications are used to investigate the characteristics of the proposed method. Overall, the results show that the fuzzy EM-based polychoric estimator is more efficient to deal with imprecise count data as opposed to standard polychoric estimators that may be used in this context.', 'arxivid': '2105.03309', 'author': ['Antonio Calcagnì [email protected] \nUniversity of Padova\n\n', 'Antonio Calcagnì [email protected] \nUniversity of Padova\n\n'], 'authoraffiliation': ['University of Padova\n', 'University of Padova\n'], 'corpusid': 234097498, 'doi': '10.1007/s40304-022-00295-6', 'github_urls': ['https://github.com/antcalcagni/fuzzypolychoric/.'], 'n_tokens_mistral': 23521, 'n_tokens_neox': 19848, 'n_words': 11792, 'pdfsha': 'ed0c2d559678f9d02c95345789f0f785980a2d02', 'pdfurls': ['https://export.arxiv.org/pdf/2105.03309v1.pdf'], 'title': ['Estimating latent linear correlations from fuzzy frequency tables', 'Estimating latent linear correlations from fuzzy frequency tables', 'Estimating latent linear correlations from fuzzy frequency tables', 'Estimating latent linear correlations from fuzzy frequency tables'], 'venue': []} |
arxiv |
The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations
6 Oct 2019
Hongbo Fu
Research Center of Nonlinear Science
College of Mathematics and Computer Science
Institut für Mathematik
Wuhan Textile University
430073WuhanPR China
Dirk Blömker [email protected]
Universität Augsburg
86135AugsburgGermany
The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations
6 Oct 2019Preprint submitted to arXiv October 8, 2019stochastic partial differential equationsamplitude equationmultiplicative noisemulti-scale analysisbifurcationslow fast system MSC: primary 60H15secondary 70K7060H1035R60
This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation.Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern.This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple onedimensional stochastic Ginzburg-Landau equation.
Introduction
In this paper we study a class of stochastic partial differential equations (SPDEs) of the following form du(t) = [Au(t) + ε 2 Lu(t) + F(u(t))]dt + εG(u(t))dW (t), (1.1) where A is a non-positive self-adjoint operator with finite-dimensional kernel, ε 2 Lu(t) is a small deterministic perturbation with ε > 0 measuring the distance to the change of stability. The nonlinearity F is a cubic mapping, and G(u) is Hilbert-Schmidt operator with G(0) = 0 so that the constant u = 0 is a solution to equation (1.1). The noise is given via a (possibly infinite dimensional cylindrical) Wiener process W on some stochastic basis.
Our aim is to study in the limit ε → 0 the asymptotic dynamics of the solution u(t) to equation (1.1) on the natural slow time-scale of order ε −2 .
Near a change of stability of the linearized operator A + ε 2 L, a natural separation of time-scales allows the original system to be transferred into slow dynamics on a dominant pattern, which couples to dynamics on a fast time scale. A reduced equation eliminating the fast variable and characterizing the behavior of dominant modes significantly simplifies the dynamics to a stochastic differential equation (SDE). This equation identifies the amplitudes of dominant pattern and is often called amplitude equation.
Amplitude approximation plays a prominent role in qualitative analysis on the dynamics of stochastic systems near a change stability. For additive noise amplitude approximation for SPDEs has been studied in many cases starting from [8] and later [4,6,9]. See also [18,25,23,19] for related work.
For the case of SPDEs on unbounded domains the effective equation is no longer an SDE, but the reduced model is still given as an infinite dimensional SPDE. For details see [22] in the case of a simple one-dimensional noise, [7] for large domains and [2,19] for results with space-time white noise and on an unbounded domain. Here we will focus on the case of bounded domains only.
Amplitude equations can be used to qualitatively describe the dynamics close to a change of stability. In [4] amplitude equations were used to give an approximation of the infinite-dimensional invariant measure for a Swift-Hohenberg equation, while in [4,5,3] ideas were presented that would allow to approximate random attractors or random invariant manifolds via amplitude equations. See also [1,18] for results for other models with additive noise.
While many results for the approximation via amplitude equations were established for additive noise, the case of multiplicative noise is not that well studied. Only for the very special case of G(u) = u and W being a scalar Brownian-motion first results for amplitude equations were obtained in [3]. With this special case of noise the approximation of random invariant manifolds was studied in [10] In first approximation the dynamics on the dominant space is given by a variant of the amplitude equation, while for the qualitative description of a random invariant manifold, one also needs an effective equation on the infinite dimensional remainder. See also [16,24] or [20] using parameterizing manifolds introduced by [11], see also [12].
In the present paper our main contribution is the analysis in the case of general infinite-dimensional multiplicative noise. We will only treat the case with G(0) = 0, so there is no contribution by an additive noise, which would lead to a different scaling in ε of the noise.
Under some smoothness assumptions on the diffusion coefficient G and regularity conditions on the noise, we derive the amplitude equations of responding equation (1.1) and show rigorously, that it captures the essential dynamics of the dominant modes. We use the Taylor expansion of G in order to directly determine the errors bounds between the solution of (1.1) and that of the amplitude equation which is only on the dominant modes.
The organization of this paper is as follows: In Section 2, we formulate the abstract framework and some basic assumptions. Section 3 contains the main results of the paper as presented in Theorem 3.1. In Section 4, we give the proof of the main results. Section 5 is devoted to a illustrative example.
Setting and assumptions
Throughout the paper, we shall work in a separable Hilbert space H, endowed with the usual scalar product ·, · and with the corresponding norm · . Concerning the leading operator A we shall assume the following conditions. Assumption 1 (Linear Operator A). Suppose A is a self-adjoint and nonpositive operator on H with eigenvalues {−λ k } k∈N such that 0 = λ 1 ≤ · · · ≤ λ k · · · , satisfying λ k ≥ Ck m for all sufficiently large k, positive constants m and C. The associated eigenvectors {e k } ∞ k=1 form a complete orthonormal basis in H such that Ae k = −λ k e k .
By N we denote the kernel space of A, which, according to Assumption 1, has finite dimension n with basis {e 1 , · · · , e n }. By P c we denote the orthogonal projector from H onto N with respect to the inner product ·, · , and by P s the orthogonal projector from H onto the orthogonal complement N ⊥ .
One standard example is with m = 4/d is the Swift-Hohenberg operator A = −(1+∆) 2 on H = L 2 ([−π, π] d ) subject to periodic boundary conditions. Similar is the Laplacian ∆ with m = 2/d. But we could also treat more general equations and also coupled systems of SPDEs here.
Remark 2.1. Let us remark that the setting of a Hilbert space and A being a self-adjoint operator is mainly for simplicity of presentation, as many crucial properties about the H α -spaces defined below, the projections P c and P s and the semigroup e tA generated by A follow in this setting as trivial Lemmas. Otherwise we would need to formulate them as an assumption and verify them in the given application.
We can now define fractional Sobolev-spaces H α = D((1 − A) α/2 ) by using the domain of definition of fractional powers of the operator A:
Definition 2.1. For α ∈ R, we define the space H α as H α = ∞ k=1 γ k e k : ∞ k=1 γ 2 k (λ k + 1) α < ∞ , which is endowed with the norm ∞ k=1 γ k e k α = ∞ k=1 γ 2 k (λ k + 1) α 1 2 .
The operator A generates an analytic semigroup {e At } t≥0 on any space H α , defined by e At ∞ k=1 γ k e k = ∞ k=1 e −λ k ,t γ k e k , t ≥ 0, and admits the following estimate, which is a classical property for an analytic semigroup. Its proof is straightforward and omitted here.
Lemma 2.1. Under Assumption 1, for all β ≤ α, ρ ∈ (λ n , λ n+1 ], there exists a constant M > 0, which is independent of u ∈ H, such that for any t > 0 e At P s u α ≤ M t − α−β m e −ρt P s u β .
(2.1)
In addition, we impose the following conditions:
Assumption 2 (Operator L). Let L : H α → H α−β for some α ∈ R, β ∈ [0, m) be a linear continuous mapping that commutes with P c and P s .
Assumption 3 (Nonlinearity F). Assume that F : (H α ) 3 → H α−β , with α and β as in Assumption 2, is a trilinear, symmetric mapping and satisfies the following conditions, for some C > 0,
F(u, v, w) α−β ≤ C u α v α w α for all u, v, w ∈ H α . (2.2)
Moreover, we have
F c (u), u ≤ 0 for all u ∈ N , (2.3) F c (u, u, w), w ≤ 0 for all u, w ∈ N ,(2.
4)
and for some positive constants C 0 , C 1 , and C 2 we have for all u, v, w ∈ N that
F c (u, v, w) − F c (v), u ≤ −C 0 u 4 + C 1 w 4 + C 2 w 2 v 2 . (2.5)
Here, to ease notation, we use F c := P s F and we define F s , L c and L s in a similar way.
Assumption 4 (Wiener Process). Let U be a separable Hilbert space with inner product ·, · U . Let {W t } t≥0 be the cylindrical Wiener process on a stochastic base (Ω, F , F t , P) with covariance operator Q = I.
Formally, W can be written (cf. Da Prato and Zabczyk [14]) as the infinite sums
W t = k∈N B k (t)f k ,
where {B k (t)} k∈N are mutually independent real-valued Brownian motions on stochastic base (Ω, F , {F t } t≥0 , P), and {f k } k∈N is any orthonormal basis on U .
We proceed with some further notation. Let V be another separable Hilbert space with inner product ·, · V . Let L 2 (U, V ) denote the Hilbert space consisting of all Hilbert-Schmidt operators from U to V , where the inner product is denoted by ·, · L 2 (U,V ) , and the norm by · L 2 (U,V ) .
Assumption 5 (Operator G). Assume that G : H α → L 2 (U, H α ) satisfying G(0) = 0, with α as in Assumption 2, is Fréchet differentiable up to order 2 and fulfills the following conditions: For one r > 0 there exists a constant l r > 0 such that
G(u) L 2 (U,H α ) ≤ l r u α , (2.6) G ′ (u) · v L 2 (U,H α ) ≤ l r v α (2.7)
and
G ′′ (u) · (v, w) L 2 (U,H α ) ≤ l r v α w α , (2.8)
for all u, v, w ∈ H α with u α ≤ r, where we use notations G ′ (u) and G ′′ (u) denote the first and second Fréchet derivatives at point u, respectively.
Let us remark that the assumption on the second Frechet-derivative is only posed for simplicity of proofs when we bound terms like G(u)−G ′ (0)·u.
To give a meaning to problem (1.1), we adapt the concept of local mild solution as in [21]. for all t ∈ (0, τ ex ).
The proof of the existence and the uniqueness of a local mild solution is standard under our assumptions, and hence is omitted here. For locally Lipschitz nonlinearities this follows using a cut-off argument and Banach's fixed-point theorem. For details see for example [14] or [13].
Let us remark that one can choose τ ex such that with probability 1 either τ ex = ∞ or lim tրτex u(t) α = ∞.
Formal Derivation and the Main Result
We consider the local mild solution u on the slow time-scale T = ε 2 t and assume that it is small of order O(ε). Let us split it into
u(t) = εa(ε 2 t) + εψ(ε 2 t),(3.1)
with a ∈ N and ψ ∈ S. By projecting and rescaling (1.1) to the slow time scale, we obtain
da(T ) = [L c a(T ) + F c (a(T ) + ψ(T ))] dT + 1 ε G c (εa(T ) + εψ(T )) dW (T ) (3.2)
and
dψ(T ) = [ 1 ε 2 A s ψ(T ) + L s ψ(T ) + F s (a(T ) + ψ(T ))]dT + 1 ε G s (εa(T ) + εψ(T ))dW (T ), (3.3) whereW (T ) := εW (ε −2 T )
is a rescaled version of the Wiener process. These equations can be written in the integral form using the mild formulation:
a(T ) = a(0) + T 0 L c a(τ )dτ + T 0 F c (a(τ ) + ψ(τ ))dτ + 1 ε T 0 G c (εa(τ ) + εψ(τ ))dW τ (3.4)
and
ψ(T ) = e AsT ε −2 ψ(0) + T 0 e As(T −τ )ε −2 L s ψ(τ )dτ + T 0 e As(T −τ )ε −2 F s (a(τ ) + ψ(τ ))dτ + 1 ε T 0 e As(T −τ )ε −2 G s (εa(τ ) + εψ(τ ))dW τ . (3.5)
We shall see later that ψ is small as long as a is of order one. Thus by neglecting all ψ-dependent terms in (3.2) or (3.4) and expanding the diffusion we obtain the amplitude equation
db(τ ) = L c b(τ )dτ + F c (b(τ ))dτ + [G ′ c (0) · b(τ )] dW τ , b(0) = a(0). (3.6)
This is equivalent to the integral equation
b(T ) = a(0) + T 0 L c b(τ )dτ + T 0 F c (b(τ ))dτ + T 0 [G ′ c (0) · b(τ )]dW τ . (3.7)
With our main assumptions we have the following main result on the approximation by amplitude equation, which is proved later at the end of Section 4. Theorem 3.1. Let the Assumptions 1 -5 be satisfied and let u be the local mild solution of (1.1) with initial condition
u(0) = εa(0) + εψ(0),
where a(0) ∈ N , ψ(0) ∈ S and b is the solution of the amplitude equation
(3.6) with b(0) = a(0). Then for any p > 1, T 0 > 0 and all small κ ∈ (0, 1 19 ), there exists a constant C > 0 such that for u(0) α ≤ ε κ/3 we have P sup t∈[0,ε −2 T 0 ] u(t) − εb(ε 2 t) − εQ(ε 2 t) α > ε 2−19κ ≤ Cε p , where Q(T ) = e AsT ε −2 ψ(0).
Let us remark that the additional term e AsT ε −2 ψ(0) in the approximation is exponentially small after any short time of order ε by the stability of the semigroup on S. This is an attractivity result for the space N and allows for slightly bigger ψ(0).
Note moreover that we did not optimize the factor in front of the κ. Both the 19κ in the final error estimate and the −κ/3 in the bound on the initial condition are not optimal. We use κ mainly for technical reasons and think of it as being very small.
Let us finally give some remarks on straightforward extensions of the result presented here.
Remark 3.1 (Other nonlinear terms). We could add higher order terms to the SPDE like quartic or quintic, for example. Formally, they are of higher order and we do not expect to change the result very much.
Quadratic nonlinear terms B(u, u) are quite different. Formally, we obtain in the amplitude equation the additional terms 1 ε B c (a, a) and 2 ε B c (a, ψ). So either we need to change the scaling of the equation, consider smaller noise, and obtain an amplitude equation with quadratic nonlinearity, or alternatively (as B c (a, a) = 0 in many applications) we have to identify the mixed term B(a, ψ). Even if ψ is small of order O(ε), then B(a, ψ) is of order O(1) and we need to identify how ψ depends on a.
See [21] for a discussion in the case of additive noise.
Remark 3.2 (Additive noise or quadratic diffusion). The Assumption that G(0) = 0 is crucial for our result, as for additive noise one sees already in the formal calculation above, that we need a different scaling. We expect to need ε 2 G(u)dW in (1.1) which leads to an additive noise term G c (0)dW in the amplitude equation. The proofs and the final theorem should nevertheless be very similar.
If we assume that not only G(0) = 0 but also G ′ (0) = 0, then we expect
to have G(u)dW in (1.1) which leads to [G ′′ c (0) · (b, b)
]dW in the amplitude equation. Again the proofs should be similar, but for the error estimate we might need additional assumptions on the third derivative of G.
Estimates and Proof
Before proving the main results, we need to state some technical lemmas used later in the proof. Also, we need to introduce a stopping time in connection with process (a, ψ). This stopping time is equivalent to a cut-off in (1.1) at radius ε 1−κ . Also this stopping time is the reason, why we only need local solutions for the SPDE.
τ * := T 0 ∧ inf T > 0 : a(T ) α > ε −κ or ψ(T ) α > ε −κ .
Next, we will denote by Q(T ), I(T ), J(T ) and K(T ) the corresponding four terms arising in the right hand side of (3.5), respectively, that is
ψ(T ) = Q(T ) + I(T ) + J(T ) + K(T ).E sup 0≤T ≤τ * I(T ) p α ≤ Cε 2p−κp (4.2) and E sup 0≤T ≤τ * J(T ) p α ≤ Cε 2p−3κp . (4.3)
Proof. By (2.1) and definition 4.1, we first have for I
E sup 0≤T ≤τ * I(T ) p α ≤ E sup 0≤T ≤τ * T 0 e As(T −τ )ε −2 L s ψ(τ ) α dτ p ≤ Cε 2βp m E sup 0≤T ≤τ * T 0 e −ε −2 ρ(T −τ ) (T − τ ) − β m L s ψ(τ ) α−β dτ p ≤ Cε 2βp m E sup 0≤T ≤τ * T 0 e −ε −2 ρ(T −τ ) (T − τ ) − β m ψ(τ ) α dτ p ≤ Cε 2βp m sup 0≤T ≤τ * T 0 e −ε −2 ρ(T −τ ) (T − τ ) − β m ε −κ dτ p ≤ Cε 2p−κp sup 0≤T ≤τ * ε −2 ρT 0 e −r r − β m dr p ≤ Cε 2p−κp ,
so that (4.2) follows. In view of Assumption 3, Definition 4.1 and (2.1) we obtain for J,
E sup 0≤T ≤τ * J(T ) p α ≤ E sup 0≤T ≤τ * T 0 e As(T −τ )ε −2 F s (a(τ ) + ψ(τ )) α dτ p ≤ Cε 2βp m E sup 0≤T ≤τ * T 0 e −ε −2 ρ(T −τ ) (T − τ ) − β m F s (a(τ ) + ψ(τ )) α−β dτ p ≤ Cε 2βp m E sup 0≤T ≤τ * T 0 e −ε −2 ρ(T −τ ) (T − τ ) − β m a(τ ) + ψ(τ ) 3 α dτ p ≤ Cε 2βp m sup 0≤T ≤τ * T 0 e −ε −2 ρ(T −τ ) (T − τ ) − β m ε −3κ dτ p ≤ Cε 2p−3κp sup 0≤T ≤τ * ε −2 ρT 0 e −r r − β m dr p ≤ Cε 2p−3κp .
The proof of Lemma 4.1 is thus completed.
While for I and J we immediately had uniform bounds in time, for K we first establish bounds in L p ([0, τ * ], H α ).
p > 0 that E sup 0≤T ≤τ * T 0 K(τ ) p α dτ ≤ C p ε p−κp . (4.4)
Proof. Throughout this proof let λ 0 be a positive constant less than λ n+1 but close to it. For any p > 0, it holds
E sup 0≤T ≤τ * T 0 K(τ ) p α dτ = E sup 0≤T ≤τ * T 0 1 ε τ 0 e As(τ −r)ε −2 G s (εa(r) + εψ(r))dW r p α dτ ≤ E T 0 0 1 [0,τ * ] (τ ) 1 ε τ 0 e As(τ −r)ε −2 G s (εa(r) + εψ(r))dW r p α dτ ≤ E T 0 0 1 ε τ ∧τ * 0 e As(τ −r)ε −2 G s (εa(r) + εψ(r))dW r p α dτ = E T 0 0 1 ε τ 0 1 [0,τ * ] (r)e As(τ −r)ε −2 G s (εa(r) + εψ(r))dW r p α dτ = 1 ε p T 0 0 e −ε −2 λ 0 pτ E τ 0 1 [0,τ * ] (r)e (As+λ 0 I)(τ −r)ε −2 e ε −2 λ 0 r G s (εa(r) + εψ(r))dW r p α dτ.
By an application of the maximal inequality for stochastic convolutions [17] based on the Riesz-Nagy theorem (as A s + λ 0 generates a contraction semigroup on S), the condition (2.6) for G, and the definition of τ * , we obtain
E sup 0≤T ≤τ * T 0 1 ε τ 0 e As(τ −r)ε −2 G s (εa(r) + εψ(r))dW r p α dτ ≤ C T 0 0 e −ε −2 λ 0 pτ E[ τ 0 e 2ε −2 λ 0 r 1 [0,τ * ] (r) a(r) + ψ(r) 2 α dr] p 2 dτ ≤ C T 0 0 e −ε −2 λ 0 pτ [ τ 0 (ε −2κ + 1)e 2ε −2 λ 0 r dr] p 2 dτ ≤ Cε p−κp T 0 0 e −ε −2 λ 0 pτ [e 2ε −2 λ 0 τ − 1] p 2 dτ ≤ Cε (1−κ)p+2
where the constant may change from line to line, but it mainly depends on p, T 0 , the bound on G, and λ 0 .
So we have seen in the previous lemmas that ψ equals Q plus a small term. Next, let us rewrite the equation (3.2) for a as the amplitude equation plus an error term (or residual).
a(T ) = a(0) + T 0 [L c a(τ ) + F c (a(τ ))] dτ + T 0 G ′ c (0) · a(τ )dW τ + R(T ),
where the error term is given by
R(T ) = T 0 [3F c (a(τ ), ψ(τ ), ψ(τ )) + 3F c (a(τ ), a(τ ), ψ(τ )) + F c (ψ(τ ))] dτ + T 0 [ 1 ε G c (εa(τ ) + εψ(τ )) − G ′ c (0) · a(τ )]dW τ . (4.5)
Let us now start to show that R is small.
E sup 0≤T ≤τ * T 0 F c (a(τ ), ψ(τ ), ψ(τ ))dτ p α ≤ C p ε 2p−7κp + ψ(0) 2p α ε 2p−κp .
Proof. It is direct to see that by brute force expansion of the cubic using
ψ = Q + I + K + J from (4.1) that T 0 F c (a(τ ), ψ(τ ), ψ(τ ))dτ = T 0 F c (a(τ ), Q(τ ), Q(τ ))dτ + T 0 F c (a(τ ), I(τ ), I(τ ))dτ + T 0 F c (a(τ ), J(τ ), J(τ ))dτ + T 0 F c (a(τ ), K(τ ), K(τ ))dτ +2 T 0 F c (a(τ ), Q(τ ), I(τ ))dτ + 2 T 0 F c (a(τ ), Q(τ ), J(τ ))dτ +2 T 0 F c (a(τ ), Q(τ ), K(τ ))dτ + 2 T 0 F c (a(τ ), I(τ ), J(τ ))dτ +2 T 0 F c (a(τ ), I(τ ), K(τ ))dτ + 2 T 0 F c (a(τ ), J(τ ), K(τ ))dτ := 10 k=1 R 1,k (T ). (4.6)
We will estimate each term separately, which will all be very similar, as I, J, and K are small. Only for Q we need an additional averaging argument. First since all H α -norms are equivalent on N , we get
E sup 0≤T ≤τ * R 1,1 (T ) p α ≤ CE sup 0≤T ≤τ * R 1,1 (T ) p α−β ≤ CE sup 0≤T ≤τ * T 0 F c (a(τ ), Q(τ ), Q(τ )) α−β dτ p ≤ CE sup 0≤T ≤τ * T 0 a(τ ) α Q(τ ) 2 α dτ p ≤ Cε −κp T 0 0 e Asτ ε −2 ψ(0) 2 α dτ p ≤ Cε 2p−κp ψ(0) 2p α . For R 1,2 (T ), we have E sup 0≤T ≤τ * R 1,2 (T ) p α ≤ CE sup 0≤T ≤τ * R 1,2 ε (T ) p α−β ≤ CE sup 0≤T ≤τ * T 0 F c (a(τ ), I(τ ), I(τ )) α−β dτ p ≤ CE sup 0≤T ≤τ * T 0 a(τ ) α I(τ ) 2 α dτ p
Due to definition 4.1 and (4.2), we get
E sup 0≤T ≤τ * R 1,2 (T ) p α ≤ Cε −κp E sup 0≤T ≤τ * T 0 I(τ ) 2p α dτ ≤ Cε 4p−3κp .
By proceeding with analogous arguments, we can show the following results for all other terms:
E sup 0≤T ≤τ * R 1,3 (T ) p α ≤ C p ε 4p−7κp , E sup 0≤T ≤τ * R 1,4 (T ) p α ≤ C p ε 2p−3κp , E sup 0≤T ≤τ * R 1,5 (T ) p α ≤ C p ε −κp (ε 2p ψ(0) 2p α + ε 4p−2κp ), E sup 0≤T ≤τ * R 1,6 (T ) p α ≤ C p ε −κp (ε 2p ψ(0) 2p α + ε 4p−6κp ), E sup 0≤T ≤τ * R 1,7 (T ) p α ≤ C p ε −κp (ε 2p ψ(0) 2p α + ε 2p−2κp ), E sup 0≤T ≤τ * R 1,8 (T ) p α ≤ C p ε −κp (ε 4p−2κp + ε 4p−6κp ), E sup 0≤T ≤τ * R 1,9 (T ) p α ≤ C p ε −κp (ε 4p−2κp + ε 2p−2κp ), and E sup 0≤T ≤τ * R 1,10 (T ) p α ≤ C p ε −κp (ε 4p−6κp + ε 2p−2κp ).
Collecting all estimates for terms appearing in (4.6) we finish the proof.
By the same arguments which we used to derive Lemma 4.3, we are able to achieve following results:
E sup 0≤T ≤τ * T 0 F c (a(τ ), a(τ ), ψ(τ ))dτ p α ≤ C p ε p−5κp + ψ(0) p α ε 2p−2κp .E sup 0≤T ≤τ * T 0 F c (ψ(τ ))dτ p α ≤ C p ε 3p−9κp + ψ(0) 3p α ε 2p . (4.7)
Proof. As we noticed before, all norms in finite dimensional space N are equivalent. Thanks to (2.2), we get
F c (ψ(τ )) α ≤ C Q(τ ) + I(τ ) + J(τ ) + K(τ ) 3 α ≤ C Q(τ ) 3 α + I(τ ) 3 α + J(τ ) 3 α + K(τ ) 3 α .
Thus, according to the Hölder inequality, this implies
E sup 0≤T ≤τ * T 0 F c (ψ(τ ))dτ p α ≤ C p E sup 0≤T ≤τ * T 0 Q(τ ) 3 α dτ p +C p E sup 0≤T ≤τ * T 0 I(τ ) 3p α + J(τ ) 3p α + K(τ ) 3p α dτ .
It is easy to check that the first term appearing in the right side of above inequality is bounded by C p ε 2p ψ(0) 3p α for a constant C p > 0. Due to Lemma 4.1 and Lemma 4.2 we can conclude that the second term is bounded by C p ε 3p−9κp . Therefore, we finish the proof and obtain (4.7).
E sup 0≤T ≤τ * T 0 [ 1 ε G c (εa(τ ) + εψ(τ )) − G ′ c (0) · a(τ )]dW τ p α ≤ C p ε p−3κp .
Proof. Using Burkholder-Davis-Gundy inequality, we have
E sup 0≤T ≤τ * T 0 [ 1 ε G c (εa(τ ) + εψ(τ )) − G ′ c (0) · a(τ )]dW τ p α ≤ CE T 0 0 1 [0,τ * ] (τ ) 1 ε G c (εa(τ ) + εψ(τ )) − G ′ c (0) · a(τ ) 2 L 2 (U,H α ) dτ p 2 ≤ CE T 0 0 1 [0,τ * ] (τ ) 1 ε G(εa(τ ) + εψ(τ )) − G ′ (0) · a(τ ) 2 L 2 (U,H α ) dτ p 2 .
(4.8)
By using the Taylor formula, we obtain
1 ε G(εa(τ ) + εψ(τ )) − G ′ (0) · a(τ ) = 1 ε [G(0) + G ′ (0) · (εa(τ ) + εψ(τ )) + 1 2 G ′′ (z(τ )) · (εa(τ ) + εψ(τ ), εa(τ ) + εψ(τ ))] − G ′ (0) · a(τ ) = G ′ (0) · ψ(τ ) + ε 2 G ′′ (z(τ )) · (a(τ ) + ψ(τ ), a(τ ) + ψ(τ )),
where z(τ ) is a vector on the line segment connecting 0 and εa(τ ) + εψ(τ ). Now, as a consequence of the condition (2.8), we have
1 ε G(εa(τ ) + εψ(τ )) − G ′ (0) · a(τ ) 2 L 0 2 ≤ C ψ 2 α + Cε 2 a(τ ) 4 α + Cε 2 ψ(τ ) 4 α .
Therefore, if we plug the estimate above into (4.8), we get
E sup 0≤T ≤τ * T 0 [ 1 ε G c (εa(τ ) + εψ(τ )) − G ′ c (0) · a(τ )]dW τ p α ≤ CE T 0 0 1 [0,τ * ] (τ )( ψ(τ ) 2 α + ε 2 a(τ ) 4 α + ε 2 ψ(τ ) 4 α )dτ p 2 ≤ C p ε p−2κp + C p E T 0 0 1 [0,τ * ] (τ ) ψ(τ ) 2 α dτ p 2 ,
where the last estimate following from the definition of τ * . From the expression for ψ(τ ) and Hölder's inequality, we get
E T 0 0 1 [0,τ * ] (τ ) ψ(τ ) 2 α dτ p 2 ≤ C p E T 0 0 1 [0,τ * ] (τ ) Q(τ ) 2 α + I(τ ) 2 α + J(τ ) 2 α + K(τ ) 2 α dτ p 2 ≤ C p E ε p ψ(0) p α + sup 0≤τ ≤τ * I(τ ) p α + sup 0≤τ ≤τ * J(τ ) p α + τ * 0 K(τ ) p α dτ .
Recalling Lemma 4.1 and Lemma 4.2, we thus have
E sup 0≤T ≤τ * T 0 [ 1 ε G c (εa(τ ) + εψ(τ )) − G ′ c (0) · a(τ )]dW τ p α ≤ C p ε p−3κp .
Due to Lemmas 4.3, 4.4, 4.5, and 4.6, we readily obtain the following estimate for the remainder R defined in (4.5).
Lemma 4.7. Assume the setting of Lemma 4.1 and suppose furthermore that ψ(0) α ≤ ε − 1 3 κ . Then for any p > 0, there exists a constant C p > 0 such that
E sup 0≤T ≤τ * R(T ) p α ≤ C p ε p−9κp . (4.9)
In what follows, we shall consider the amplitude equation (3.6) associated with (3.2) and we show the following uniform bound on its solution b. This is crucial in order to remove the stopping time from the error estimate. Moreover note, that our assumptions do not imply global solutions for the SPDE, we rely on the existence of global solutions for the amplitude equation, which is also ensured by the following Lemma.
E sup 0≤T ≤T 0 b(T ) p α ≤ C p a(0) p α (4.10)
Proof. This proof is relatively straightforward using Itô-formula for powers of the norm. For large p > 2 define the twice continuously differentiable function
f (·) = · p : H → R. (4.11)
Directly, for any x, h ∈ H we have
f ′ (x)h = p x p−2 x, h and f ′′ (x)(h, h) = p(p − 2) x p−4 x, h x, h + p x p−2 h, h ≤ p(p − 1) x p−2 h 2 , (4.12)
so that
trace[f ′′ (b(τ ))G ′ c (0)b(τ ))(G ′ c (0)b(τ )) * ] ≤ p(p − 1) b(τ ) p−2 trace[(G ′ c (0)b(τ ))(G ′ c (0)b(τ )) * ] ≤ Cp(p − 1) b(τ ) p .
(4.13)
Applying Itô's formula [15,Theorem 2.9] and (4.13) we obtain that
b(T ) p ≤ a(0) p + p T 0 b(τ ) p−2 L c b(τ ), b(τ ) dτ +p T 0 b(τ ) p−2 F c (b(τ )), b(τ ) dτ +p T 0 b(τ ) p−2 b(τ ), G ′ c (0) · b(τ )dW τ + 1 2 Cp(p − 1) T 0 b(τ ) p dτ.
Therefore, using Assumption 2 and the bound on F from (2.3), we get
b(T ) p ≤ a(0) p + p T 0 b(τ ) p−2 b(τ ), G ′ c (0) · b(τ )dW τ +C p T 0 b(τ ) p dτ .
(4.14)
For any stopping time T ≤ T 0 , by Burkholder-Davis-Gundy inequality, we obtain
pE sup 0≤T ≤T T 0 b(τ ) p−2 b(τ ), G ′ c (0) · b(τ )dW τ ≤ 3pE T 0 ∞ k=1 b(τ ) 2p−4 b(τ ), G ′ c (0) · b(τ )e k 2 dτ 1 2 ≤ CE T 0 b(τ ) 2p dτ 1 2 ≤ CE sup 0≤T ≤T b(τ ) p T 0 b(τ ) p dτ 1 2 ≤ 1 2 E sup 0≤T ≤T b(τ ) p + C p T 0 E sup 0≤s≤τ b(s) p dτ,
where we applied Young's inequality in the final step. Therefore, using (4.14), we obtain
E sup 0≤T ≤T b(T ) p ≤ 2 a(0) p + C p T 0 E sup 0≤s≤τ b(τ ) p dτ.
Note that we need to use a stopping time T here, as initially, we do not know that the moments of b are finite. Thus we consider only the T 's which ensures this. As the equation above holds for any stopping time, we derive by using Gronwall's lemma
E sup 0≤T ≤T 0 b(τ ) p ≤ C p a(0) p .
This finishes the proof. The next step now is to remove the error from the equation for a to obtain the amplitude equation. We show an error estimate between a and the solution b of the amplitude equation. Lemma 4.9. Assume the setting of Lemma 4.1 and suppose furthermore that ψ(0) α ≤ ε − 1 3 κ . For any p > 0, there exists a constant C p > 1 such that
E sup 0≤T ≤τ * a(T ) − b(T ) p ≤ C p ε p−18κp . (4.15)
Proof. For the proof we derive an equation for the error a − b and proceed similarly than for the bound on b. But as R (defined in (4.5)) is not differentiable in the Ito-sense, we first substitute ϕ := a − R. Clearly, we have
ϕ(T ) = a(0) + T 0 L c (ϕ(τ ) + R(τ ))dτ + T 0 F c (ϕ(τ ) + R(τ ))dτ + T 0 G ′ c (0) · (ϕ(τ ) + R(τ ))dW τ .
Defining the error h :
= b − ϕ = b − a + R, we get h(T ) = T 0 L c h(τ )dτ − T 0 L c R(τ )dτ + T 0 F c (b(τ ))dτ − T 0 F c (b(τ ) − h(τ ) + R(τ ))dτ + T 0 G ′ c (0)(h(τ ) − R(τ ))dW τ .
Let f be the p-th power of the norm as in (4.11). By using again (4.13) we have
trace[f ′′ (h(τ ))(G ′ c (0)(b(τ ) − R(τ )))(G ′ c (0)(b(τ ) − R(τ ))) * ] ≤ Cp(p − 1) b(τ ) p−2 h(τ ) − R(τ ) 2 .
Applying Itô's formula and using the estimate above, we obtain
h(T ) p ≤ p T 0 h(τ ) p−2 L c h(τ ), h(τ ) dτ −p T 0 h(τ ) p−2 L c R(τ ), h(τ ) dτ +p T 0 h(τ ) p−2 F c (b(τ )) − F c (b(τ ) − h(τ ) + R(τ )), h(τ ) dτ +p T 0 h(τ ) p−2 h(τ ), G ′ c (0) · (h(τ ) − R(τ ))dW τ + 1 2 Cp(p − 1) T 0 h(τ ) p−2 h(τ ) − R(τ ) 2 dτ.
By condition (2.5) and Cauchy-Schwarz inequality, we derive
h(T ) p ≤ C p T 0 h(τ ) p dτ + C p T 0 h(τ ) p−1 R(τ ) dτ +C p T 0 h(τ ) p−2 R(τ ) 2 dτ + C p T 0 h(τ ) p−2 R(τ ) 4 dτ +C p T 0 h(τ ) p−2 R(τ ) 2 b(τ ) 2 dτ +p T 0 h(τ ) p−2 h(τ ), G ′ c (0) · (h(τ ) − R(τ ))dW τ .
Then, Young's inequality yields
h(T ) p ≤ C p T 0 h(τ ) p dτ + C p T 0 R(τ ) p dτ + C p T 0 R(τ ) 2p dτ +C p T 0 R(τ ) p b(τ ) p dτ +p T 0 h(τ ) p−2 h(τ ), G ′ c (0)[h(τ ) − R(τ )]dW τ . (4.16)
The last term on the right hand side of (4.16) is bounded as follows. By Burkholder-Davis-Gundy inequality for any T ∈ [0, T 0 ], we get
E sup 0≤T ≤τ * ∧T p T 0 h(τ ) p−2 h(τ ), G ′ c (0)[h(τ ) − R(τ )]dW τ ≤ 3pE τ * ∧T 0 h(τ ) 2p−4 h(τ ) 2 h(τ ) − R(τ ) 2 dτ 1 2 ≤ C p E τ * ∧T 0 h(τ ) 2p + h(τ ) 2p−2 R(τ ) 2 dτ 1 2 .
Using again Young's inequality implies
E sup 0≤T ≤τ * ∧T p T 0 h(τ ) p−2 h(τ ), G ′ c (0)[h(τ ) − R(τ )]dW τ ≤ C p E τ * ∧T 0 h(τ ) 2p dτ 1 2 + C p E τ * ∧T 0 R(τ ) 2p dτ 1 2 ≤ 1 2 E sup 0≤T ≤τ * ∧T h(T ) p + C p T 0 E sup 0≤r≤τ * ∧τ h(r) p dτ +C p E τ * ∧T 0 R(τ ) 2p dτ 1 2 . (4.17)
Therefore, collecting together (4.9), (4.10), (4.16) and (4.17), for any T ∈ [0, T 0 ] we obtain
E sup 0≤T ≤τ * ∧T h(T ) p ≤ C p T 0 E sup 0≤r≤τ * ∧τ h(r) p dτ +C p ε p−18κp .
Using Gronwall's lemma we can show
E sup 0≤T ≤τ * h(T ) p ≤ C p ε p−18κp + ε p−9κp a(0) p ≤ C p ε p−18κp ,
so that, in view of (4.9),
E sup 0≤T ≤τ * a(T ) − b(T ) p ≤ E sup 0≤T ≤τ * h(T ) p + E sup 0≤T ≤τ * R(T ) p ≤ C p ε p−18κp .
Remark 4.1. Notice that by Lemma 4.9, for any p > 0 and κ ∈ (0, 1 18 ), we obtain
E sup 0≤T ≤τ * a(T ) p ≤ E sup 0≤T ≤τ * a(T ) − b(T ) p + E sup 0≤T ≤τ * b(T ) p ≤ C p (1 + a(0) p ).
(4.18)
We can use this to show that a < ε −κ on [0, T 0 ] with probability almost 1.
Let us define the overall error between ε(b + Q) and u by
R(T ) := u(ε −2 T ) − εb(T ) − εQ(T ) = ε[a(T ) − b(T ) + ψ(T ) − Q(T )] = ε[a(T ) − b(T ) + I(T ) + J(T ) + K(T )]. (4.19)
We already know that I and J are uniformly small. It remains to bound K. For this we use the factorization method and start with the following stochastic integral.
E sup 0≤T ≤T 0 T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ p α ≤ C p ε p− 1 2 κ . (4.20)
Proof. For large p > 1 fix γ ∈ (0, 1/2). By the celebrated factorization method, if we set
Y γ (s) = s 0 (s − τ ) −γ e As(s−τ )ε −2 G ′ s (0) · b(τ )dW τ we have T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ = C γ T 0 (T − τ ) γ−1 e As(T −τ )ε −2 Y γ (τ )dτ
for some constant C γ > 0. Thus we obtain by Hölder inequality and the bounds on the semigroup on the space S that
T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ p α ≤ C T 0 (T − τ ) −(1−γ) e −ρ(T −τ )ε −2 Y γ (τ ) α dτ p ≤ C T 0 (T − τ ) −(1−γ)p/(p−1) e −ρ(T −τ )ε −2 p/(p−1) dτ p−1 T 0 Y γ (τ ) p α dτ ≤ Cε 2γp−2 T /ε 2 0 τ −(1−γ)p/(p−1) e −ρτ p/(p−1) dτ p−1 T 0 Y γ (τ ) p α dτ ≤ Cε 2γp−2 T 0 Y γ (τ ) p α dτ .
Note that we need to fix p ≫ 1 large or 1 ≫ γ > 0 small in order to have an integrable pole in the previous estimate. Moreover, by Burkholder-Davis-Gundy inequality, but now without the supremum in time we obtain for t ∈ [0,
T 0 ] E Y γ (t) p α ≤ CE t 0 (t − τ ) −2γ e As(t−τ )ε −2 G ′ s (0) · b(τ ) 2 L 2 (U,H α ) dτ p/2 ≤ CE t 0 (t − τ ) −2γ e −ρ(t−τ )ε −2 b(τ ) 2 dτ p/2 ≤ CE sup 0≤τ ≤T 0 b(τ ) p ε 2(1−2γ)p/2 ≤ CE sup 0≤τ ≤T 0 b(τ ) p ε (1−2γ)p .
This finally implies E sup
0≤T ≤T 0 T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ p α ≤ CT 0 E sup 0≤τ ≤T 0 b(τ ) p ε p−2 .
As we can choose p arbitrarily large, we obtain via Hölder inequality that for any smallκ > 0 there is a constant such that
E sup 0≤T ≤T 0 T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ p α ≤ CE sup 0≤τ ≤T 0 b(τ ) p ε p−κ ,
so that, thanks to (4.10), we have
E sup 0≤T ≤T 0 T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ p α ≤ C p ε p− 1 2 κ .
In order to bound K we set
M (T ) := T 0 e As(T −τ )ε −2 1 ε G s (εa(τ ) + εψ(τ )) − G ′ s (0) · b(τ ) dW τ ,
we have from (4.1)
K(T ) = M (T ) + T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ .
where we just bounded the integral on the right in Lemma 4.10. It remains to bound M . Here we proceed similarly to the previous lemma using factorization.
Lemma 4.11. Assume the setting of Lemma 4.9. For any p > 1 there exists a constant C p > 0 such that
E sup 0≤T ≤τ * M (T ) p α ≤ C p ε p−2κp . (4.21)
Proof. We can follow exactly the proof of the previous Lemma 4.10 but have to pay attention to the fact that the integrand in M is only defined for t ≤ τ * . Moreover, the integrand is due to the presence of ψ and thus K not uniformly bounded in time.
Define the integrand as
Φ(τ ) = 1 ε G s (εa(τ ) + εψ(τ )) − G ′ s (0) · b(τ )
We notice that by Taylor's formula
Φ(τ ) = G ′ (0) · ψ(τ ) + G ′ (0) · [a(τ ) − b(τ )] + ε 2 G ′′ (z(τ )) · (a(τ ) + ψ(τ ), a(τ ) + ψ(τ )),
wherez(τ ) is a vector on the line segment connecting 0 and εa(τ ) + εψ(τ ). Therefore,
Φ(τ ) L 2 (U,H α ) ≤ C ψ(τ ) α + C a(τ ) − b(τ ) α + Cε a(τ ) 2 α + Cε ψ(τ ) 2 α ,
Note that for τ ≤ τ * the right hand side above is bounded by Cε −κ uniformly in time.
We obtain following the lines of Lemma 4.10
E sup 0≤T ≤τ * M (T ) p α (4.22) ≤ E sup 0≤T ≤T 0 T 0 e As(T −τ )ε −2 1 [0,τ * ] (τ )Φ(τ )dW τ p α ≤ Cε p−κ E sup 0≤T ≤T 0 1 [0,τ * ] (τ )Φ(τ ) p L 2 (U,H α ) ≤ Cε p−κ E sup 0≤T ≤τ * Φ(τ ) p L 2 (U,H α ) so that E sup 0≤T ≤τ * M (T ) p α ≤ Cε p(1−2κ)
As a consequence of Lemmas 4.1, 4.9, and 4.11, we have the following bound on R: Lemma 4.12. Assume the setting of Lemma 4.9. For any p > 1, there exists a constant C p > 0 such that
E sup 0≤T ≤τ * R(T ) p α ≤ C p ε 2p−18κp . (4.23)
Moreover, we obtain a bound on ψ which is uniform in time:
Lemma 4.13. Assume the setting in Lemma 4.9. For any p > 1, there exists a constant C p > 0 such that
E sup 0≤T ≤τ * ψ(T ) p α ≤ C p (ε −pκ/3 + ε p−18κp ). (4.24)
Before we proceed with the final error estimate, we comment on improved bounds on ψ and M . We know by definition that
ψ(T ) = Q(T ) + T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dWE sup T ∈[0,τ * ] ψ(T ) − Q(T ) − T 0 e As(T −τ )ε −2 G ′ s (0) · b(τ )dW τ p α ≤ Cε p(2−3κ) .
(4.25) Before proving main theory, we need to construct a subset of Ω, which enjoys nearly full probability.
Definition 4.2. For κ > 0 from the definition of τ * define the set Ω * ⊂ Ω of all ω ∈ Ω such that all these estimates
sup 0≤T ≤τ * a(T ) < ε − 1 2 κ , sup 0≤T ≤τ * R(T ) < ε 2−19κ and sup 0≤T ≤τ * ψ(T ) α < ε − 1 2 κ .
hold.
Lemma 4.14. The set Ω * has approximately probability 1.
Proof. Indeed, let Ω * be as in the Definition 4.2. It easily follows that
P(Ω * ) ≥ 1 − P( sup 0≤T ≤τ * a(T ) ≥ ε − 1 2 κ ) − P( sup 0≤T ≤τ * ψ(T ) α ≥ ε − 1 2 κ ) −P( sup 0≤T ≤τ * R(T ) ≥ ε 2−19κ ).
We get by using Chebychev's inequality, (4.18), (4.20), (4.23) and (4.24)
P(Ω * ) ≥ 1 − (ε − 1 2 κ ) −q E sup [0,τ * ] a(T ) q α − (ε − 1 2 κ ) −q E sup [0,τ * ] ψ(T ) q α −(ε 2−19κ ) −q E sup [0,τ * ] R(T ) q α ≥ 1 − 2Cε κq/2 ε −qκ/3 − Cε (19κ−2)q ε (2−18κ)q ≥ 1 − Cε p
where we take for a given p the exponent q sufficiently large.
Proof of Theorem 2.1. From the definition of Ω * and τ * , we have
Ω * ⊆ sup 0≤T ≤τ * a(T ) α < ε −κ , sup 0≤T ≤τ * ψ(T ) α < ε −κ ⊆ {τ * = T 0 } ⊆ Ω.
This allows us to get on Ω * sup 0≤T ≤T 0
R(T ) α = sup 0≤T ≤τ * R(T ) α ≤ Cε 2−19κ such that P sup 0≤T ≤T 0 R(T ) α ≥ ε 2−19κ ≤ 1 − P(Ω * ) ≤ Cε p ,(4.26)
which, by recalling representation (4.19), completes the proof.
Example -Ginzburg-Landau/Allen-Cahn equation
A very simple example to illustrate the main result is the stochastic Ginzburg-Landau equation (or Allen-Cahn equation) with linear multiplicative noise on the interval D = [0, π] of the form ∂ t u = (∂ 2
x + 1)u + νε 2 u − u 3 + εu · ∂ t W (t). (5.1)
In the following we consider the Itô-representation of the the SPDE above in the Sobolev-space H 1 (D) with sufficiently smooth noise. We set A := ∂ 2 x + 1, L := νI, F := −u 3 Suppose that the equation is subjected to the Dirichlet boundary condition. Let H = L 2 ([0, π]) be the space of all square integrable real-valued functions which are defined on the interval [0, π]. In this situation the eigenvalues of −A are explicitly known to be λ k = k 2 − 1 with associated eigenvectors e k (x) = 2 π sin(kx) = δ sin(kx), k = 1, 2, · · · , and N = span{sin}. So Assumption 1 is true with m = 2.
Clearly, Assumption 2 holds true for example for any α > 1/4 and β = 0, as for the norm in H α we then have uv α ≤ C u α v α . We will fix α = 1 for simplicity.
Note that on the one-dimensional space N the H α -norm is just a multiple of the H-norm. So that for u, w ∈ N the conditions described in Assumption 3 are satisfied as follows:
F c (u), u = − π 0 u 4 (x)dx ≤ 0, F c (u, u, w), w = − π 0 u 2 (x)w 2 (x)dx ≤ 0.
In addition, condition (2.5) is true for some positive constants C 0 , C 1 and C 2 , as F is a standard cubic nonlinearity.
Define f k (x) := 1 k e k (x), k = 1, 2, · · · , such that {f k } k∈N is an orthonormal basis of H 1 . We consider in our application that W is standard cylindrical H 1 -valued Wiener process and define a covariance operator Q defined by Qf k = α k f k , k = 1, 2, · · · , satisfying trace(Q) = ∞ k=1 α k = C 0 < ∞. For the operator G defined as
G(u) • v := u · Q 1/2 v , we have G(u) 2 L 2 (H 1 ,H 1 ) = k∈N u · (Q 1 2 e k ) 2 H 1 = k∈N α k u · e k 2 H 1 ≤ C k∈N α k u 2 H 1 e k 2 H 1 = C u 2 H 1 trace(Q) ≤ C u 2 H 1 < ∞.
Therefore, G(·) : H 1 → L 2 (H 1 , H 1 ) is a Hilbert-Schmidt operator satisfying G ′ (u) · v = v · Q 1/2 and G ′′ (u) = 0, so that Assumption 5 holds. Therefore, our main theorem states that the dynamics of (5.1) can well approximated by the amplitude equation, which is for b ∈ N a stochastic ordinary differential equation of the form:
db = [νb − P c F(b)]dt + P c [G ′ (0) · b]dW .
Let us finally rewrite the the amplitude equation for the actual amplitude of b b = γ sin Clearly, as P c f = 2 π π 0 sin(y)f (y)dy sin we have P c F(b) = − 3 4 γ 3 sin. Moreover,
P c [G ′ (0) · b]dW = γ ∞ k=1 P c [sin Q 1/2 f k ]dB k = γ ∞ k=1 √ α k P c [sin f k ]dB k = γ
Definition 2.2. (Local mild solution) An H α -valued process {u(t)} t∈[0,T ] , is called a mild solution of problem (1.1) if for some stopping time τ ex we have on a set of probability 1 that τ ex > 0, u ∈ C 0 ([0, τ ex ), H α ) and u(t) = e tA u(0) + t 0 e (t−s)A [ε 2 Lu(s) + F(u(s))]ds + ε t 0 G(u(s))dW (t)
Definition 4 . 1 .
41For the N × S-valued stochastic process (a, ψ) satisfying system (3.4)-(3.5) we define, for some time T 0 > 0 and small exponent κ ∈ (0, 1 12 ), the stopping time τ * as
.
Let the Assumption 1 -Assumption 5 be satisfied. For any p > 0 and τ * from the definition 4.1, there exists a constant C > 0 such that
Lemma 4 . 2 .
42Assume the setting of Lemma 4.1. Then it holds for every
Lemma 4. 3 .
3Assume the setting of Lemma 4.1. For any p > 0, there exists a constant C p > 0 such that
Lemma 4 . 4 .
44Assume the setting of Lemma 4.1. For any p > 0, there exists a constant C > 0 such that
Lemma 4. 5 .
5Assume the setting of Lemma 4.1. For any p > 0, there exists a constant C > 0 such that
Lemma 4 . 6 .
46Assume the setting of Lemma 4.1. For any p > 0, there exists a constant C p > 0 such that
Lemma 4. 8 .
8Let the Assumptions 1 -5 be satisfied. For any p > 1, there exists a constant C p > 0 such that
Lemma 4 . 10 .
410Assume the setting of Lemma 4.9. For any p > 1, there exists a constant C p > 0 such that
τ + M (T ) + I(T ) + J(T ) As I(T ) + J(T ) = O(ε 2−3κ ) by Lemma 4.1 and both M and the stochastic integral is uniformly in time by O(ε 1−2κ ), we can show that ψ − Q is small uniformly in time. This improved bound on ψ can be used in the proof of Lemma 4.11, to show that M is smaller of order O(ε 1−2κ ). Thus we obtain
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| {'fraction_non_alphanumeric': 0.11155802930817063, 'fraction_numerical': 0.04148367178327608, 'mean_word_length': 3.0072312249520823, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 50, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 59, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant trivial solution and we study the dynamics around it for the deterministic equation being close to a bifurcation.Based on the separation of time-scales close to a change of stability, we rigorously derive an amplitude equation describing the dynamics of the bifurcating pattern.This allows us to approximate the original infinite dimensional dynamics by a simpler effective dynamics associated with the solution of the amplitude equation. To illustrate the abstract result we apply it to a simple onedimensional stochastic Ginzburg-Landau equation.', 'arxivid': '1910.02424', 'author': ['Hongbo Fu \nResearch Center of Nonlinear Science\nCollege of Mathematics and Computer Science\nInstitut für Mathematik\nWuhan Textile University\n430073WuhanPR China\n', 'Dirk Blömker [email protected] \nUniversität Augsburg\n86135AugsburgGermany\n'], 'authoraffiliation': ['Research Center of Nonlinear Science\nCollege of Mathematics and Computer Science\nInstitut für Mathematik\nWuhan Textile University\n430073WuhanPR China', 'Universität Augsburg\n86135AugsburgGermany'], 'corpusid': 203836239, 'doi': '10.1088/1361-6544/ab801e', 'github_urls': [], 'n_tokens_mistral': 19591, 'n_tokens_neox': 16906, 'n_words': 9433, 'pdfsha': '4e58b1cb7786da5e0e6fd0c00f850f3ead780e2e', 'pdfurls': ['https://arxiv.org/pdf/1910.02424v1.pdf'], 'title': ['The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations', 'The impact of multiplicative noise in SPDEs close to bifurcation via amplitude equations'], 'venue': []} |
arxiv |
A new form of Tsallis distribution based on the probabilistically independent postulate *
Du Jiulin
Department of Physics
School of Science
Tianjin University
300072TianjinChina
A new form of Tsallis distribution based on the probabilistically independent postulate *
* Project supported by the National Natural Science Foundation of China (Grant No. 10675088) † E-mail address: [email protected] 1Nonextensive statisticsTsallis distributionNonequilibrium dynamicsFokker-Planck equation
The current form of Tsallis distribution for a Hamiltonian system with an arbitrary potential is found to represent a simple isothermal situation. In this letter, the q-exponential of a sum can be applied as the product of the q-exponential based on the probabilistically independent postulate employed in nonextensive statistical mechanics. Under this framework, a new form of Tsallis distribution is suggested. It is shown that the new form of Tsallis distribution can supply the statistical description for the nonequilibrium dynamical property of the Hamiltonian system governed by an arbitrary potential, and it is found to be one potential statistical distribution for the dark matter.
Nonextensive statistical mechanics since it started in 1988 has obtained very wide applications in many interesting scientific fields. In principle, almost all the formulae and the theory using Boltzmann-Gibbs statistics so far could be generalized under this framework [1][2][3][4] (for more detail, see http://tsallis.cat.cbpf.br/biblio.htm ).
However, the problems such as under what circumstances, e.g. which class of nonextensive systems and under what physical situation, should the nonextensive statistical mechanics be used for their statistical descriptions have been long-standing [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . In particular, the problem at present appears that [16] , unexpectedly, the current form of Tsallis distribution, [ ] q H q f − − − Theoretically, the Tsallis distribution function employed in the self-gravitating collisionless system was found to be only an isothermal distribution for any q 1 ≠ [17] .
The example was reported recently in the N-body simulation for a self-gravitating system with the result that the Tsallis distribution is inconsistency generally with the dark matter halos except the isothermal parts for the polytropic index n ∞ → [18] , in which, unawarely, the Tsallis distribution function employed was actually isothermal one. On the other hand, however, one can apply the Maxwell q-distribution,
[ ] q mv q f − − − 1 1 2 2 / ) 1 ( 1 β
, to deal with some nonequilibrium property of the velocity distribution for self-gravitating and plasma systems [19,20] , where the nonextensive parameter q 1 is found to be related to the potential function ≠ ϕ (ϕ can be any one) and the temperature gradient by the formula expression,
T ∇ T q ∇ ∇ −) 1 ( ϕ .
The results therefore imply clearly that the Maxwell q-distribution can be used for the statistical description of the dynamical system governed by an arbitrary potential when it reaches at the nonequilibrium stationary-state. The applications of the Maxwell q-distribution have included the examples such as in the nuclear reaction systems [21][22][23][24] , in the astrophysical systems (see [17] [19] [2] and the references therein), in the plasmas systems [25][26][27][28][29][30] , in the solar wind theory [31][32][33] , in the non-local distributions in the solar and stellar interior [34][35][36] , and in others [37][38][39][40][41][42][1][2][3][4] . Obviously, the questions need to be replied, why can the Maxwell q-distribution be a possible statistical description for the nonequilibrium dynamical system being at the stationary-state but cannot the current form of Tsallis q-distribution? Where does the above discrepancy come from? The purpose of this work is to try a new form of the Tsallis distribution on the basis of the probabilistically independent postulate in nonextensive statistical mechanics, which may be as one reasonable scheme to solve the above discrepancy.
Tsallis proposed the q-entropy in 1988 as a generalization of the Boltzmann-Gibbs entropy [43] , given by ,
∑ − = i i q q i q P P k S ln(1)
where k is Boltzmann constant, the set {P i } are the probabilities of the microscopic configurations of the system under investigation, the parameter q is real number different from unity, the q-logarithm is defined as ), ln ln ; 0 ( 1 1 ln
1 1 x x q x x q q = > − − ≡ − (2) the inverse function, the q-exponential, is [ ] ) exp (exp ) 1 ( 1 exp 1 1 1 x x x q x q q = − + ≡ − ,(3)
if 1+(1-q)x >0 and by exp q x =0 otherwise. Thus the probability of a system at the value x i reads , a power-law distribution. When q=1, all the formulae return to be Boltzmann-Gibbs statistics.
i q i x P exp
Nonextensive statistical mechanics is founded on the basis of the q-entropy and the probabilistically independent postulate [43] . The so-called probabilistically independent postulate is that, if the probability of a system at the value x i is
i q i x P exp
and at the value x j is , and they are probabilistically independent, then the probability at the value (x
j q j x P exp i +x j ) is ) )(exp (exp j q i q j i ij x x P P P = .
The q-entropy S q ( 1) is nonextensive, namely, if a system composed of two probabilistically independent parts A and B, i.e., (the probabilistically independent postulate), then the Tsallis q-entropy of the system is
≠ q ) ( ij P P P B j A i B A ij ∀ = + ) ( ) ( ) 1 ( ) ( ) ( ) ( 1 B S A S k q B S A S B A S q q q q q − − + + = + .(4)
Clearly, only if q=1 is the entropy extensive. Under this framework, one leads to the basic form of the Tsallis distribution used so far in the nonextensive statistical mechanics,
f ~ [ ] q H q − − − 1 1 ) 1 ( 1 β ,(5)f ~q i i i r m p q − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − − ∑ 1 1 2 }) ({ 2 / ) 1 ( 1 ϕ β ,(6)
which, however, actually contravenes the original postulate of the probabilistic independence. In other words, the q-exponential of a sum cannot apply mechanically the definition Eq.(3), namely,
) 1 ( 1 exp 1 1 q i i i i q x q x − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + ≠ ∑ ∑ .(7)
But, in fact, in terms of the probabilistically independent postulate, we should express the q-exponential of a sum as the product of the q-exponential, i.e.,
, or
∏ ∑ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ i i i i x P x P ) ( ∏ ∑ = i i q i i q x x exp exp .(8)
Under this framework, the entropy and the energy are both nonextensive in the power-law q-distribution. Thus, instead of Eq. (6), a new form of Tsallis distribution for the Hamiltonian many-body system is suggested by
f ~[ ] q i i i q i m p q r q − − ∏ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − 1 1 2 1 1 2 ) 1 ( 1 }) ({ ) 1 ( 1 β βϕ .(9)
Clearly, only if taking q =1, does the Tsallis q-distribution (9) become the Boltzmann
distribution, f ~) exp( H β −
. For an ideal gas of one particle system, H = p 2 /2m, from the above function, directly one can obtain the Maxwell q-distribution function, f
~[ ] q m p q − − − 1 1 2 2 / ) 1 ( 1 β .
The new form of Tsallis q-distribution (9) is a result based on the probabilistically independent postulate employed in nonextensive statistical mechanics. With this basic postulate, the q-entropy is nonextensive not only, but also is the energy [43] . However, quite questioningly, up to now nonextensive statistical mechanics develops without taking into consideration the nonextensivity of energy.
We now search for possible dynamical property of the new form of Tsallis q-distribution Eq.(9) from a general Fokker-Planck equation. Following the lines of previous work [16] , we can assume the q-distribution Eq.
p dt dx = , ) (t F m p dx d dt dp p + − − = ζ ϕ ,(10)
where ζ is the frictional coefficient. The noise is Gaussian and it is delta-function correlated,
) ' ( 2 ) ' ( ) ( t t B t F t F p p − = δ .(11)
Then the corresponding Fokker-Planck equation to the Langevin equations is given for the noise-averaged distribution function [49,16] by
2 2 p f B f m p dx d p f m p x t f ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∂ ∂ + ∂ ∂ − = ∂ ∂ ζ ϕ ,(12)
The stationary-state solution of this Fokker-Planck equation satisfy
0 ) 1 ( 2 2 = ∂ ∂ + ∂ ∂ + + ∂ ∂ + ∂ ∂ − p f B f p p m p f dx d x f m p ζ ϕ .(13)
Equivalently, it can be written as
0 ) 1 ( ] 1 [ 2 2 1 1 1 = ∂ ∂ − + ∂ ∂ + − + ∂ ∂ + ∂ ∂ − − − − − p f Bf q f p p q m p f dx d x f m p q q q q ζ ϕ .(14)
According to Eq.(8)or Eq. (9), the new form of the Tsallis q-distribution for the above dynamical system is written as
[ ] q q q q Q R m p q q f − − − − ≡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − 1 1 1 1 1 1 2 1 1 2 ) 1 ( 1 ) 1 ( 1 β βϕ ,(15)
where one has denoted
βϕ ) 1 ( 1 q R − − ≡ , Q ≡ 1-(1-q) m p 2 2 β .(16)= + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − p qB Q mB p dx d m p p dx d Q p dx d R m β β ϕ β ζβ β ζ βϕ .(194 ) 3 )( 1 ( ) 1 ( 4 1 p m q q p dx d dx d R q m q ζ β β βϕ β β − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − [ ] 2 2 3 1 2 ) 2 ( ) 1 ( 2 ) 1 ( 2 1 p q B p dx d R q dx d q dx d ζβ β βϕ β ϕ β β − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − + + − 0 ) ( 1 = − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − β ζ ϕ β βϕ B m p dx d dx d R m .(20)
If q =1, we find 0 / = dx dβ and the fluctuation-dissipation theorem
β ζ B =
, a physical situation for thermal equilibrium known well in Boltzmann-Gibbs statistics.
If , very clearly, from Eq. (20) we can determine the following three identities to be satisfied for the Tsallis q-distribution (15). Namely, if the Tsallis distribution (15)
1 ≠ q dx d q dx d ϕ β β 2 ) 1 ( − = ,(21)0 = ζ ,(22)B = 0.(23)
The three identities can determine possible dynamics compatible with the Tsallis q-distribution function (15). As compared with the previous three identities [16] ,
) (t F dx d dt dp p + − = ϕ ,(24)
where the noise is irrelated, < )>= 0 due to B=0. Eq.(24) is the dynamical ' ( ) ( t F t F p p equation for the system governed only by the potential field ϕ . The potential can be arbitrary one, whether long-range or short-range force. The new form of Tsallis distribution can describe the nonequilibrium dynamical property for such a system as governed by the Langevin equation (24). If the potential function is the gravitational one, it describes the dynamical property of the particles evolving in a self-gravitating system, where the gravitation is the only one force among the particles, e.g. the dark matter is now just thought of such a physical situation. In other words, Eq.(24) is Langevin equation for the dark matter. Our results show that the new form of Tsallis distribution (15) can be a stationary-state solution of the Fokker-Planck equation (12) under the situation (21)-(23) and supply one statistical description for the nonequilibrium dynamical property of the system characterized by Eq. (24), so to be one potential distribution function for the dark matter distribution.
The nonextensive parameter is now given exactly by the relation,
dx d dx d q ϕ β β 2 1 = − .(25)
The nonextensivity ( 1) stands for a degree of deviation from the thermal equilibrium of the nonequilibrium dynamical system governed by an arbitrary potential field and thus has the clearly physical meaning. If take
≠ q 0 / = dx dβ
or dT/dx = 0 (thermal equilibrium ), one has q = 1 and the Tsallis q-distribution (16) becomes Boltzmann distribution. The reader might also be interested in some recent applications of nonextensive statistical mechanics, where a physical meaning of the parameter q is introduced to astrophysics [17,19] and plasmas [20] .
In the end, we would like to make remarks on the probabilistically independent problem. The probabilistic independence at the very start was as a basic postulate for nonextensive statistical mechanics [43,[1][2][3][4] . Under this postulate, the q-entropy is nonextensive, satisfying Eq.(4), and nonextensive statistical mechanics is studied and developed. On the other hand, the probabilistically independent postulate also requires the energy to be nonextensive [43] . Namely, from the probabilistic independence one also can derive the relation for the energy U, composed of two probabilistically independent parts a and b,
U(a ⊕ b)=U(a)+U(b)+(1-q)β U(a)U(b),(26)
which appears to coexist with the relation for the q-entropy, S(a ⊕ b) = S(a)+S(b) (a)S(b). Usually, the nonextensive statistics has been developed only by taking Eq.(4) for the q-entropy as the basic precondition but ignoring the coexisted Eq.(26) with it for the energy, leading to the current form of Tsallis distribution, Eq. (5) or Eq.(6). One postulate leads to two coexisted results. When the nonextensiv statistics selected one but discarded the other one, without interpretation, it had been incomplete theoretically.
+(1-q)k -1 S
In fact, from the second law of thermodynamics, e.g. dU = TdS (if the volume is fixed), we may find that it is hard to image that the entropy is nonextensive but the energy is extensive. When we use the new form of the Tsallis distribution defined by
Eq. (8) or Eq. (9), both the relation Eq.(26) for the energy's nonextensivity and the relation Eq.(4) for the entropy's nonextensivity has actually been taken into consideration.
In conclusion, we have expressed a new understanding for the q-exponential of a sum based on the probabilistically independent postulate in nonextensive statistical mechanics. Namely, the q-exponential of a sum can be applied as the product of the q-exponential by Eq. (8). Under this framework, we suggest a new form of Tsallis distribution (9), which incarnates the entropy's nonextensivity not only but the energy's nonextensivity. It is one reasonable scheme to solve the problems such as the current form of Tsallis distribution contravenes the basic postulate of the probabilistic independence, selects the entropy's nonextensivity but discards the coexisted energy's nonextensivity with it, and only stands for a simple isothermal or thermal equilibrium situation etc. The nonextensive parameter is exactly given by the relation Eq.(25) and so it has a clearly physical meaning. It ( 1 ≠ q ) stands for a degree of deviation from the thermal equilibrium of the dynamical system under an arbitrary potential field.
We show that the new Tsallis distribution (15) based on Eq.(8) can be a stationary-state solution of the Fokker-Planck equation (12). It is a physical solution with the three identities (21)-(23), so it can supply the statistical description for the nonequilibrium dynamics of the Hamiltonian systems governed by any potential when it reaches to the nonequilibrium stationary-state. If the potential is the gravitational one, it can describe the nonequilibrium dynamical property of particles evolving in a self-gravitating system, e.g. the dark matter is just such a physical situation. If the new Tsallis distribution were employed for the N-body simulation for a self-gravitating system, the results would be expected to be consistent with the dark matter halos.
is found to be only a simple isothermal or thermal equilibrium situation of the Hamiltonian systems governed by any potential, whether for long-range or short-range forces, which, of course, is among the domain of Boltzmann-Gibbs statistics. For a general Langevin equation with an arbitrary potential, it is found that there is no possible nonequilibrium dynamics that should use the Tsallis distribution for the statistical description.
the Hamiltonian H. According to Eq.(5),if the Hamiltonian of a many-body system is H
(9) to be a stationary-state solution of the Fokker-Planck equation and then search for if it is a possible physical solution compatible with the dynamical functions in the Langevin equation of a dynamical system. If it is so, then the stationary-state solution can describe the long-times dynamical behavior of such a dynamical system.We still starts with a general dynamical system of the two-variable Brownian motion of a particle, with mass m and the Hamiltonian, long-range or short-range force). The Langevin equations of the dynamical system are
If
Eq.(15) is a stationary-state solution of the Fokker-Planck equation,
for the current form of Tsallis distribution, the above identity Eq.(21) stands for one nonequilibrium dynamical property of the system. Unsatisfactorily, we determine 0 = B ζ = B = 0, do not obtain a generalized fluctuation-dissipation theorem as expected for the frictional coefficientζ and the quantity B, e.g.
.(21)-(234) represent one nonequilibrium dynamical property of the system with an arbitrary potential when it reaches to the nonequilibrium stationary-
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arxiv |
Title: Embedding bifurcations into pneumatic artificial muscle
N Akashi
Graduate School of Science
Kyoto University
KyotoJapan
Y Kuniyoshi
Graduate School of Information Science and Technology
The University of Tokyo
TokyoJapan
T Jo
Digital Engineering Division
Bridgestone Corporation
TokyoJapan
M Nishida
Digital Engineering Division
Bridgestone Corporation
TokyoJapan
R Sakurai
Digital Engineering Division
Bridgestone Corporation
TokyoJapan
Y Wakao
Advanced Materials Division
Bridgestone Corporation
TokyoJapan
K Nakajima
Graduate School of Information Science and Technology
The University of Tokyo
TokyoJapan
Title: Embedding bifurcations into pneumatic artificial muscle
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Harnessing complex body dynamics has been a long-standing challenge in robotics. Soft body dynamics is a typical example of high complexity in interacting with the environment. An increasing number of studies have reported that these dynamics can be used as a computational resource. This includes the McKibben pneumatic artificial muscle, which is a typical soft actuator. This study demonstrated that various dynamics, including periodic and chaotic dynamics, could be embedded into the pneumatic artificial muscle, with the entire bifurcation structure using the framework of physical reservoir computing. These results suggest that dynamics that are not presented in training data could be embedded by using this capability of bifurcation embeddment. This implies that it is possible to embed various qualitatively different patterns into pneumatic artificial muscle by learning specific patterns, without the need to design and learn all patterns required for the purpose. Thus, this study sheds new light on a novel pathway to simplify the robotic devices and training of the control by reducing the external pattern generators and the amount and types of training data for the control.Main Text:RESULTSPneumatic artificial muscleThis study used the McKibben PAM(Fig. 1A), which consists of a cylindrical rubber tube covered by a braided cord. This PAM is forced by a nearly constant external load. If the
INTRODUCTION
Recent studies have revealed that mechanical devices can be designed to use their body dynamics for desired information processing, such as a mechanical random number generator (1) and mechanical neural networks (2). Furthermore, the natural dynamics of mechanical bodies not designed for computation can be used as an information processing resource. The complex dynamics in soft robotic arms, which are inspired by the octopus, can be used for real-time computation, embedding a timer, and controlling the arm by employing the approach of physical reservoir computing (PRC) (3)(4)(5)(6)(7). Reservoir computing (8-10) is a recurrent neural network framework characterized by use of a highdimensional neural network with nonlinearity and memory. In PRC (11), the network is replaced with physical dynamics. There are various types of robotic bodies, such as mechanical spring-mass-dampers (12), tensegrity (13), quadruped robots (14), and fish robots (15). This suggests that body dynamics can be directly exploited for information processing and control without external memory and nonlinearity.
A pneumatic artificial muscle (PAM) is a typical soft actuator that realizes expansion/contraction or bending dynamics through air pressurization. PAM been studied since the dawn of soft robotics (16,17). The McKibben PAM (18,19) is a central component of soft machines and devices, such as wearable devices (20) and robotic arms (21), and has the advantages of high durability against impact and vibration, a high forceto-weight ratio, and low manufacturing costs. In addition, PAM has been studied as a physical reservoir. PAM length sensors can be emulated by other sensory values in the PAM using the PRC framework (22). The air pressure of a rubber tube connected to a PAM that is attached to an assistive walking device could estimate the posture of the wearer using PRC (23). Moreover, control led to the assistant timing of the walking device based on the estimated information (23). Periodic dynamics have been embedded into a robotic arm composed of PAMs with PRC closed-loop control (24).
The bifurcation structure in dynamical systems is characterized by a qualitative change in dynamics, such as periodic and chaotic, through changes in a parameter. Bifurcation structures appear in the dynamics of robot bodies (25,26). Bifurcation structures in central pattern generators have been shown to contribute to providing the capabilities of exploration and self-organized adaptation for robot control (27)(28)(29). Recently, it has been reported that artificial neural networks with rich information processing capabilities can reconstruct the entire bifurcation structure by learning a subset of dynamics included in the bifurcation structure, which we call bifurcation embedment (30)(31)(32)(33)(34). This study is the first attempt to realize bifurcation embedment into the robotic body. Embedding dynamics implies internalizing the central pattern generator into the body, which is usually externally attached to the robot. In addition, embedding bifurcation structures suggests that it is possible to control various qualitatively different patterns by learning several patterns, without the need to learn all patterns required in robot control. Such a generalization is more powerful than interpolation or extrapolation in traditional machine learning, as the properties of unseen dynamics in the bifurcation structure are qualitatively different from trained ones. Concretely, we demonstrated that periodic dynamics could be embedded into a PAM by training only chaotic dynamics, and vice versa. This study provides insight into reducing the weight and keeping the softness of robotic devices, such as wearable devices, by internalizing external pattern generators using computational capabilities in these morphologies. Furthermore, bifurcation embedment has the potential to significantly reduce the amount and types of training data for robot control.
PAM is pressurized, it expands in the radial direction and shrinks lengthwise. We used the PAM as a physical reservoir by injecting an input value as a control pressure and sensing its physical quantities. The measurement system is illustrated in Fig. 1B. Inner pressure, length, load, and electric resistance of the rubber were measured. Although traditional natural rubber has low conductivity, and it is difficult to measure its electric resistance value, this study increased the rubber conductivity from 1.0 × 10 −3 S/m to 20 S/m by mixing carbons with the rubber (35). The length of the PAM was 108 mm, with an outer diameter of 11 mm, an inner diameter of 9 mm, and an angle of braid π/6 rad in the equilibrium state (35). Fig. 1C shows the typical behaviors of each sensor value used in PRC. The input value ( ) represents the following piecewise constant periodic signal:
Dynamics
( ) = + � = � �� ,(1)= sin 2 ,(2)
where and are the input magnitude and bias, respectively, which tune the input to a suitable input range for the device. is the input interval, and is the period of input. This study used the following input parameters: A = 0.2 MPa, B, = 0.3 MPa, = 0.1 sec, and = 1.2 sec. Sensor responses are presented in Fig. 1C. Sensor value sampling timing was the timing of updating the next input signal. Measured pressure showed nearly the same behavior as control pressure. Furthermore, length and resistance exhibited different curves between the compression and extension phases and possessed the nonlinearity and hysteresis for the input. The load value barely responded to the control pressure and was likely a noise signal.
Bifurcations of electric resistance through external load
The behavior of the electric resistance of the rubber changed drastically through external load and explained the mechanics of these bifurcations. This was accomplished by focusing on changes in the rubber's thickness. The external load added to the PAM changed by 5 N from 100 N to 250 N under the periodic input signal, as depicted in Fig. 1C and represented by Eq. 1. The relationship between length and resistance is presented in Fig. 2A. Resistance response was opposed after 106 mm, which is approximately the equilibrium length of the PAM ( 0 = 108 mm). Thus, these regions can be considered to correspond to contraction and expansion phases. The length gap between 106 mm and 108 mm is considered to occur due to offset measurements. The electric resistance of a rubber tube has the same tendency as the rubber thickness (36). Rubber thickness was derived from measurements of diameter and conservation of volume. The thickness model is provided in the Materials and Methods section. The thickness of the rubber peaks at the equilibrium length is depicted in Fig. 2A. In the expansion phase, the thickness thinned out from the equilibrium thickness 0 because the inner diameter did not change. In the contraction phase, the thickness thinned out from 0 due to expansion in the radial direction. Therefore, resistance peaks at the equilibrium length as rubber thickness. Fig. 2B illustrates the response of length for applied load and pressure in the experiment. Using this method, three load regions were identified: compression phase alone, compression and extension mixing phase, and extension phase alone. The typical resistance time series of each phase is presented in Fig. 2C. Resistance in the compression phase responded to anti-phase pressure value. Resistance in the mixing phase changed to a two-peak behavior. In contrast to the compression phase, resistance in the extension phase responded to the pressure value in-phase. Fig. 2D depicts the bifurcation diagram, where local minimum values of resistance are plotted for the applied load values. Bifurcations were confirmed, revealing that the local minimum value changed from one to two at 160 N and became one again at 220 N. These bifurcation points correspond to change points of the compressing, mixing, and extension phases. Fig. 3 illustrates the computational scheme of PAM PRC. The input signal was injected as a control pressure, which is a one-dimensional value. The PAM acted as a physical reservoir by providing a nonlinear historical response to the input. We obtained reservoir variables by sensing these responses and constructing output values from a weighted sum of reservoir variables and bias term.
Computing scheme
The input signal is a piecewise constant one-dimensional signal, which is represented in Eq. 1. The nonlinearity and memory of the physical reservoir can be tuned by tuning the input magnitude and input interval in Eq. 1. The dependency of nonlinearity and PAM physical reservoir memory on input magnitude and input interval is discussed in the Supplementary Materials 2. Sensor values at time are represented in the form ( ) = � 1 ( ), … , ( )�, where M is the number of sensor values used as reservoir variables. We obtained reservoir variable , which corresponds to input , based on sensing times from input injected time to input updated time + . Eq. 2 presents ∈ ℝ +1 :
= � ( ); � + 1 � ; ⋯ ; � + − 1 � ; 1� ,(3)
where the number of samples is represented by . This multiplexing method (37), referred to as time-multiplexing, boosts the computational power of the reservoir from a small number of variables and has been widely used in PRC (6,38,39). In this study, was fixed at five in all the experiments. The output values � = � � ,1 , … , � , � ∈ ℝ were generated as follows:
� = out ⊤ ,(4)
where out ∈ ℝ ( +1) is output weight (+1 in the index means a bias term). The output weight is obtained by ridge regression:
= ( ⊤ + ) −1 ⊤ ,(5)
where = � wash +1 ⋯ wash + train � ∈ ℝ ( +1) train is the training data matrix, = � wash +1 ⋯ wash + train � ∈ ℝ train is a target data matrix corresponding, and is the ridge parameter. The numbers of washout and training data are represented as wash and train , respectively.
The open-and closed-loop settings are depicted in Fig. 3. Open-loop represents a case in which the input signal is external to the reservoir. Closed-loop represents the case in which the input signal is the output value of the reservoir one time step prior. In the closed-loop setting, input values, which are control pressures, are restricted to a certain range [ min , max ] to prevent the breakdown of the system.
= � min ( � −1 < min ) � −1 ( min ≤ � −1 ≤ max ) max ( � −1 > min ) (6)
This study used the value ( min , max ) = (0 MPa, 0.5 MPa).
Information processing capacity of a single pneumatic artificial muscle
Information-processing capabilities, that is, the nonlinearity and memory of each sensor value in the PAM, were revealed. We used the information processing capacity (IPC) (40) criteria, which describes the function that dynamical system serves for an input signal from independent and identically distributed (i.i.d.) random variables (41). Memory and linear/nonlinear transformation capability of the reservoir can be obtained by checking the bases of this function. IPC limited linear components are called memory capacity (42). where equality is established if the reservoir has echo state property (ESP). Here, ESP, which is an important property for reservoir computing, guarantees the reproducibility of the computing results (43).
First, we showed the IPCs of each single sensor value and multiplexing sensor values in the PAM, which included pressure, length, resistance, load, and all sensors combined and time-multiplexed. The IPCs are presented in Table 1. The IPC of the pressure was nearly one, and the pressure could be completely described by the input sequence. Therefore, the pressure could be a computational node that has the ESP for input. The IPCs of length and resistance were slightly lower than one, and these sensor values could nearly be described by the input sequence; however, they had few irreproducible components. The IPC of the load was nearly zero, and load moved nearly independently of the input. The IPC could be improved to approximately 10 by combining all types of sensors and using timemultiplexing. The total number of reservoir variables was 20 = 4 × 5. An IPC lower than the number of variables implied the existence of input-independent or linearly dependent components.
Further details, such as nonlinear and memory capacities of the IPCs in the PAM, were examined. Fig. 4 depicts the dependency of the IPCs on the external load. The capacities of the pressure did not change through external load. The capacities of delay zero in the length monotonically increased as external load increased from 50 N to 250 N. Therefore, a PAM with a smaller external load could produce information processing that requires more memory. However, the degree components in the resistance non-monotonically changed through the external load. In addition, the linear components monotonically increased as external load increased to 150 N, degree two components were dominant when the external load was 175 N and 200 N, and the linear components were dominant when the external load was greater than 225 N. Increasing and decreasing switches corresponded to the bifurcation points of resistance; therefore, these critical behaviors were derived from the intrinsic resistance bifurcations.
Open-loop experiments
We evaluated the performance of PAM PRC in an open-loop setting. We investigated PAM length sensor emulation (35), which is a practical task. PAM length sensor emulation is a real-world task that emulates the PAM length sensor value from the input pressure value. A laser displacement sensor, which is a standard length sensor for the PAM, is made of a rigid component that takes away the softness of the PAM. Therefore, it is a practically important method to ensure the softness of the PAM, emulate the length sensor by other sensory values, eliminate it from the PAM. Although the length dynamics of the PAM respond nonlinearly to hysteresis for the input pressure, PAM length time series can be predicted by a recurrent neural network (35,(44)(45)(46)(47)(48). Here, the input sequence in this task arose from uniformly random values and was transformed to [0, 0.5] MPa control pressure value, as ( , ) = (0.5, 0) in Eq. 1. In addition, we provided the performance of PAM PRC for a typical benchmark task in the Supplementary Materials 3.
Furthermore, we evaluated the performance of the task using the normalized mean squared error (NMSE), as follows:
NMSE = 1 eval ∑ ( � − ) 2 eval =1 2 ( ) (8)
where eval is the number of evaluation data. In the following experiments, the number of washout data was wash = 1,000, the number of training data was train = 40,000, and the number of evaluation data was eval = 9,000. We compared the performances between PAM PRC and echo state network (ENS) (9), which is a typical recurrent neural network in reservoir computing. The ENS had the same number of computational nodes as the number of reservoir variables in PAM PRC. The best hyperparameters of the ENS for each task were determined by the grid search. In addition, we compared the performance of the physical model in the length sensor emulation task. The formulations of the ESN and physical model are presented in the Materials and Methods section.
We multiplexed sensor values in the PAM using time-multiplexing = 5. In one instance, we used 15 reservoir variables (which are time multiplexed pressure, resistance, and load), whereas in another, we used 10 reservoir variables, which were time multiplexed pressure and resistance. The number of nodes in the ESN was the same as in the case of using the load, so this is 15.
The NMSEs of the physical model, ESN, PAM PRC without loads, and PAM PRC with loads were 0.0893, 0.0436, 0.0302, and 0.0294, respectively. PAM PRC outperformed the ESN and physical model. Fig. 4A illustrates the time series of the input, reservoir variable, target, and output signals. The responses of pressure and resistance time series were induced by the random input sequence; however, the load time series was independent of the inputs, as was the result of the IPCs for the load signals in Table 1. Although the load was not induced into the input signal information, cases using load signals had a higher performance than those without the load. In addition, the performance using the load was higher than the upper bound of the performance of the ENS with the sufficient number of nodes, as depicted in Fig. 4B. The length value was not only driven by the input signal but also reflected the noise or intrinsic time-varying dynamics, as the IPC length was 0.975. Only the part of the dynamics that was driven by input can be reconstructed by external machine learning, such as an ESN, which transforms only the inputs. Conversely, PAM PRC with load could treat not only explicit input signals but also implicit inputs and the internal state of the PAM, as the load was independent of the input due to the load IPC (Table 1).
There is another advantage of PAM PRC in the case of few training data. Fig. 4C present performance comparison between an ESN with 600 nodes and PAM PRC through training data. The performance of the ESN significantly decreased when the training data were less than 1,000; however, PAM PRC archives nearly the same performance when the number of training data ranged from 100 to 10,000. This could be considered a problem derived from the dimension of machine-learning networks. An ESN with 600 nodes produces overfitting when the training data are limited; however, PAM PRC can work with as few computational nodes such as 15. This does not easily produce overfitting, even with a small amount of data. Therefore, selecting a suitable reservoir for the task has advantages not only in performance and calculation costs but also with regard to learning.
Closed-loop experiments: Attractor embedding
This study analyzed closed-loop control by PRC in a single PAM. First, we analyzed the potential to embed attractors in a PAM. We focused on limit cycle, strange attractor of the logistic map, and the Rössler attractor. The limit cycle defined by Eq. 2 was a onedimensional periodic dynamic. Such rhythm dynamics are important as a central pattern generator in robot control (49). In addition, not only periodic but also chaotic oscillators play an important role in robot control. As a central pattern generator, a chaotic oscillator can derive adaptive and exploratory behaviors by its complex dynamics (28,29). The logistic map, which is a one-dimensional dynamical system with discrete time, was defined by the following equation:
y +1 = (1 − )(9)
where is a model parameter and is set as a chaotic parameter = 3.7. The embedding of logistic dynamics does not require memory, as the next step of the logistic map can be determined only by the current step of the logistic map. Furthermore, the Rössler attractor (50) of three-dimensional chaos with continuous time is defined by the following equation:
1 = − 2 − 3 2 = 1 + 2 3 = + 1 3 − 3 (10)
where , , and are model parameters and set as typical chaotic parameters: ( , , ) = (0.2, 0.2, 5.7). As chaos with continuous time does not occur unless it is a dynamical system with at least three dimensions (51), and the nonlinear term, which is essential for chaos, is only 1 3 , this model can be considered one of the simplest models in chaos with a continuous time.
In the training phase, we injected into the reservoir using an open-loop and trained the output weight using yn+1 as a teacher signal (this training scheme is referred to as teacher forcing (52)). The input range of all experiments was set to [0.1, 0.5] MPa by tuning and in Eq. 1. The input signal of the Rössler system, which is a three-dimensional system, was 1 , which was discretized by a sampling interval of 0.5 sec. Although the input is onedimensional, the attractor can be reconstructed with all dimensions due to Takens' embedding theorem (53) if the reservoir has sufficient memory and nonlinearity. Furthermore, the input interval of the PAM control pressure is = 0.1 sec in the experiments with the limit cycle and Rössler system, and = 0.2 sec in the experiments with the logistic map. The embedding of the limit cycle and Rössler system require memory; however, the embedding of the logistic map did not (the relationship between the input interval and IPC of PAM PRC is discussed in the Supplementary Materials 4). In all of the experiments, we fixed the number of washout data at wash = 1,000 and the number of training data at train = 4,000. In the prediction phase, we switched the openloop and closed-loop after 1,000 time steps. In addition, we evaluate the embedding result using the output time series, the attractor in the delayed coordinate system, and power spectra. Fig. 6 presents the results of the attractor embedding. In limit cycle embedding, the embedded attractor had similar properties as a time series attractor, and spectra as the target limit cycle. In logistic map embedding, the output time series deviated from the target signal as time passed after switching from an open-to a closed-loop due to the initial state sensitivity of chaos. However, there was an output signal close to the target attractor, and the embedding attractor captured the characteristics of the target attractor. In addition, the output signal had a broad range of spectra that was the same as the target signal, rather than a clear perk of spectra such as the limit cycle. Finally, in the Rössler attractor embedding, the output attractor could reconstruct the highest peak spectra of the Rössler attractor; however, it could not reconstruct multiple peaks, which are characteristic of chaos, that is, of the Rössler attractor.
Next, we evaluated the robustness of the attractor embedding. For this, we injected a random signal from the target signal and confirmed that the output signal could quickly return to the target attractor after switching to a closed loop. We focused on the limit cycle as a target attractor, and the input signals in the open-loop were zero and random signals. The results are depicted in Figs. 6D and 6E. The output signals quickly returned to the target attractor after switching.
Closed-loop experiments: Bifurcation embedding
We confirmed that the IPC of the resistance in the PAM could drastically change through the change in external load in an open-loop setting. Moreover, we found the change in the output signal of PAM PRC through the external load in a closed-loop setting. The following training data were used:
A) A limit cycle with a period of 1.2 sec with an external load of 100 N (same as the limit cycle in Fig. 6); B) Limit cycles with periods of 1. The results of experiments C and D revealed that periodic and chaotic dynamics could be embedded simultaneously and that one of the dynamics could be generated from learning another one of them. In experiment C, we trained chaotic dynamics in the logistic map when the external load was 100 N. The dynamics switched from chaotic dynamics to period 2 dynamics when the external load was 170 N, which corresponded to the first step of the bifurcation of resistance. As the bifurcation diagram indicates, period 2 dynamics appeared intermittently, acting as a window of the period-doubling bifurcation. In experiment D, we trained period 4 dynamics in the logistic map when the external load was 100 N. The dynamics switched to chaotic dynamics with a one-dimensional attractor in the delay coordinate and broad spectra at the external load 200 N, which corresponded to the second bifurcation of the resistance. The chaotic attractor in the delay coordinate had an alternative shape, similar to a cubic function, to the logistic map. In addition, dynamics had an unstable fixed point near +1 = , as there was a hole at the intersection of +1 = and the attractor.
These bifurcation embedding results could be useful for robotics applications. For instance, the automatic switching conducted in experiment A could be used for an emergency stop when the external load exceeds the threshold and an idling stop that transitions to a stationary state while the main power is on. This presents the possibility of internalizing adaptive behavioral control that depend on changes in the environment. In addition, the results of experiment B revealed that this switching can be turned off by explicit training on both sides of the bifurcation of the inherent dynamics. Furthermore, the results of experiments C and D demonstrated that multiple qualitatively different dynamics, including chaos, could be switched according to changes in the environment. The results of experiments C and D did not indicate the desired bifurcation structure but rather a bifurcation structure based on training data and reservoir dynamics. We present the embedding a bifurcation structure, which includes desired qualitative different signals, by training dynamics on both sides of bifurcation explicitly in the Supplementary Materials 4.
DISCUSSION
This study demonstrated that various dynamics and bifurcation structures can be embedded into a soft robotic actuator through systematic analyses of PAM PRC. These results reveal potentials, limitations, and future directions of computing using the robot body.
In the open-loop setting, PAM PRC can outperform the external ESN with a sufficient number of computational nodes by using resistance and load sensor values. This performance is believed to derive from the fact that the resistance and load sensor values reflect the PAM internal state, such as time-variant components and extra inputs, which cannot be represented by the pressure input. The evaluation of PAM IPCs should be extended to multi-inputs and time-variant form to test this hypothesis (41). External machine-learning networks can achieve the same level of performance as PAM PRC if resistance and load sensor values are injected into them. In addition, this study revealed that PAM PRC can obtain high performance of the sensor emulation from small-size training data. This aspect is an important advantage in the information processing of soft materials, as soft materials generally have lower durability than rigid materials, and their material properties can be easily changed over a long time period.
Furthermore, in the closed-loop control, we succeeded in embedding the attractor of the sinusoidal wave and logistic map but failed to embed the Rössler attractor. The dimension of the attractor is believed to have caused this failure. The sinusoidal wave and logistic map have one-dimensional attractor; however, the Rössler attractor that is embedded in a three-dimensional space has a fractal dimension ranging from two to three. The number of inputs in this study is one as the control pressure. To embed high-dimensional dynamics, such as Rössler, it is necessary to utilize the memory of the reservoir due to the Takens delay embedding theorem (53). Therefore, the failure could be because the memory of a single PAM is insufficient to reconstruct the target variables that are not injected as inputs.
In addition, we demonstrated that the PAM can be embedded qualitatively different attractors from the training attractor in the closed-loop experiments. These results suggest that bifurcations in the morphology may have a potential to be exploited to embed the bifurcation structure of the target dynamical system. However, the mechanism to embed the bifurcation structure into the reservoir has not been fully understood to date (33). Moreover, the necessity of the intrinsic bifurcations of the reservoir for the bifurcation embedding remains unknown.
This bifurcation embedding into the body suggests the strong potential of the control of robotics. For instance, if we can embed the period-doubling bifurcation in the morphology of the robot, it may be possible for the robot to generate all the arbitrary periodic dynamics and chaos underlying the Li-Yorke chaos (54) from learning only finite period patterns.
This study indicated that the body dynamics of the single PAM have high computational capability. We believe that these results can be expanded to practical situations. The structures that consist of multiple PAMs, such as a robot arm and wearable assistance suit, may have the potential to embed higher-dimensional and more complex dynamics than single PAMs. For example, if we can embed chaotic itineracy in the robot body, the robot can switch many primitive patterns autonomy and stochastically (55). Moreover, the embedded bifurcation structure could serve as an adaptive pattern switch for the environment, such as an anomaly detection and failure prevention of robot, as the bifurcation points correspond to the change points of the dynamic phase of the body dynamics, such as contraction and extension phases in the PAM.
MATERIALS AND METHODS Pneumatic artificial muscle thickness model
The thickness model of the PAM is presented in Fig. 2A. Thickness was calculated using the following equation: = − (11) where we assume that the rubber tube in the PAM is a uniform cylinder and that and are the outer and inner radius of the cylinder, respectively. Furthermore, we assumed the below linear relationship between length and outer radius, as the coefficient of determination between the length and outer radius was 0.9934, which was obtained from the 9 values of length and thickness of the rubber tube with an external load of 50 N.
= −0.3382 + 47.525 (12) where the length of the rubber tube is represented by . We obtained the inner radius r using the following equation due to the constraint of the constant volume of rubber and restriction of both ends of the tube:
= � 0 ( ≥ 0 ) � 2 − ( < 0 )(13)
where and are the volume and cross-sectional area of rubber, respectively, and 0 and 0 are the inner radius and length in equilibrium length, respectively. Fig. 2A presents the length and thickness that were calculated using the above equations.
Echo state network
The architectures of the ESN were compared with PAM PRC. The th computational node at time is represented as , the th input node is represented as , and the th output node at time is represented as � . The computational nodes and outputs of the ESN are given by
= � cp � , −1 =1 + in � , in =1 � (14) � = � , out =1(15)
where the activation function is given by , which is the hyperbolic tangent; each node of the input weight in = � , in � comprises a uniform distribution with [−1, 1], and each node of the internal weight = � , � comprises a uniform distribution with [−1, 1] and is normalized to make the spectral radius one. The coupling magnitude cp coincides with the spectral radius of cp . The bias term 0 is set as 0 = 1. The output weight out = ( ) is tuned by training. We fix in = 1 and optimize cp by grid search for the range [0, 1.2] in each task.
Dynamical model of the PAM
We estimated the length dynamics of PAM from the injected pressure and load for the control. The models of PAM dynamics have been widely investigated in previous studies (56,57). Based on these studies, we used the following length model of the PAM: ̈= − elas ( ) − fric () − pre � , ( )� + ex ( ) (16) where the displacement of the PAM length is represented as x, and the mass of the PAM is represented as ; the elastic force of the rubber, friction of the rubber, and tension of volume change by pressure are represented as elas ( ), fric (), and pre � , ( )�, respectively; and the input pressure and input load are represented as ( ) and ex ( ). Here, tension by pressure is derived from the following Schulze equation:
pre � , ( )� = 0 ( ) 4 1 sin 0 (3(1 − ) 2 cos 0 − 1)(17)
where the strain of the PAM is represented as = ( 0 − )/ 0 and the equilibrium length, inner radius, and angle of the braided code are represented as 0 , 0 , and 0 , respectively. Note that the Schulze equation assumes that the PAM is a uniform cylinder with zero thickness. When the real PAM is compressed, it is not a cylinder but a bent shape, as both ends of the PAM are fixed. The model that considers the non-uniform and bent shape of the PAM has been previously proposed in extant literature (57). We ensured the linear elasticity elas ( ) ∝ is the elasticity of the PAM. We could accurately estimate the length of the PAM by solving the equation of the equilibrium of pre ( ), elas ( ), and ex in the static state. However, in the dynamic state, in which the PAM continues to move, it is difficult to estimate the PAM dynamic, as the Schulze equation cannot consider the hysteresis depicted in Fig. 2 in the main text. The causes of the hysteresis can be considered as the effects of Coulomb and viscous frictions (58)(59)(60). Therefore, Eq. 25 can be rewritten using the following equation:
̈= − −̇− sgn() + �− pre � , ( )� + ex ( )�(18)
Here, , , , and are the parameters of the model. We optimized these parameters using grid search, and the parameters used in the Section 2.6 were ( , , , ) = (6353, 80.05, 10, 0.635). The measured length values were offset due to a measurement error; thus, we added a bias to the length-predicting value from the physical model to coincide with the average values of the measured and predicted lengths. Fig. S1. Information processing capacities through input interval. Fig. S2. Information processing capacities through input magnitude. Fig. S3. Information processing capacities through equilibrium length. Fig. S4. Information processing capacities through temperature. Fig. S5. Time series of the target and output signals in the NARMA2 task. Fig. S6. The results of the closed-loop during the chaos and period dynamics embedding. Table S1. Base experiment conditions.
Supplementary Materials
Detailed definitions and formulation of the IPC are provided in the Supplementary Materials 1. The IPC restricted the delay to ≤ and degree of polynomial functions to ≤ , as IPC[ , ]. IPC[ , ] can be decomposed by function reconstruction capacities [ ], where is an orthogonal basis function in the focusing functional space. The IPC[1, K] is referred to as memory capacity. For a number of the linear independent states of reservoir variable, the following theoretical equation holds (40
2 and 2.4 sec with external loads of 100 N and 250 N, respectively; C) The chaotic trajectory of the logistic map, where = 3.7, with an external load of 100 N (same as in the case of the logistic map in Fig. 6); D) The period 4 trajectory of the logistic map, where = 3.55, with an external load of 100 N.We confirmed the change in the output signal in closed-loop control when the external load changed by 5 N from 100 N to 250 N every at 2,000 time steps.Fig. 7depicts the results. In experiment A, the amplitude and frequency of the limit cycle continuously changed in the range from load 100 N to 200 N. However, the limit cycle structure of the output signal suddenly collapsed at an external load of 200 N, and the output signal changed to nearly static dynamics. This switching point was around the second bifurcation point of the resistance, as shown inFig. 2D. Thus, dynamics may switch, as the bifurcation of the resistance propagated to the entire dynamics of the PAM via closed-loop control. Conversely, the result of experiment B indicated that it is possible to suppress the closed-loop bifurcations. In experiment B, we trained limit cycles with different frequencies when external loads were 100 N and 250 N. The results revealed that the frequency of the closed-loop dynamics with intermediate external loads was linearly interpolated.
Fig. 1 .
1Pneumatic artificial muscles, measurement systems, and pneumatic artificial muscle dynamics. (A) No pressurized and pressurized pneumatic artificial muscles. (B) Pneumatic artificial muscle measurement systems. (C) Sensor responses for a sinusoidal wave input.
Fig. 2 .
2Intrinsic bifurcations of pneumatic artificial muscle. (A) The top graph plots length versus resistance; the bottom figures depict the schematic illustration of the thickness change. (B) Color map of the length for control load and pressure. (C) Resistance and pressure time series in four load conditions. (D) Bifurcation diagram of control load versus local minimum resistance.
Fig. 3 .
3Schematics of reservoir computing. (A) The right-hand side depicts a schematics of typical reservoir computing. The left-hand side depicts a schematics of Physical reservoir computing for pneumatic artificial muscle.
Fig. 4 .
4Information processing capacities of pneumatic artificial muscle sensory values. The bars indicate the decompositions of the IPCs through the degree components and memory components (The method for the decomposition is provided in the Supplementary Materials 1).
Fig. 5 .
5Open-loop length sensor emulation. (A) Time series of the input, reservoir variables, output, and prediction in the length sensor emulation task. The red line is the prediction signal of physical reservoir computing for pneumatic artificial muscle. (B) Normalized mean squared error through the number of echo state network nodes. (C) Normalized mean squared error using the number of training data. Echo state network nodes: n = 600.
Fig. 6 .
6Results of the closed-loop at the attractor embedding. (A) Time series of the target and physical reservoir computing output signals. (B) Attractors of the target and physical reservoir computing output. (C) Spectra of the target and physical reservoir computing output. (D) Time series and attractor in the limit cycle, embedding from zero inputs. (E) Time series and attractor in the limit cycle, embedding from a random input.
Fig. 7 .
7Bifurcation embedding using closed-loop control. (A) Training the sinusoidal wave when the external load is 100 N. (B) Training sinusoidal waves with different periods when external loads are 100 N and 250 N. (C) Training the strange attractor of the logistic map when the external load is 100 N. (D) Training Period 4 trajectory when the external load is 100 N.
All Supplementary Materials are available at: https://figshare.com/s/abdd59978c30f24f7c0a (This repository is currently set in private mode (no doi number), but available from the link.) Supplementary Material 1. Detailed calculation method of information processing capabilities. Supplementary Material 2. Dependencies of information processing capacities on controller, body, and environmental conditions. Supplementary Material 3. Nonlinear autoregressive moving average task. Supplementary Material 4. Embedding periodic and chaotic attractors into the same weight.
Table S2 .
S2Normalized mean squared errors in NARMA2 task. Supplementary Data. PAM sensory time series for IPC calculation. Movies S1. Open-loop length sensor emulation. Movies S2. Closed-loop attractor embedding. Movies S3. Closed-loop bifurcation embedding.
Tables : Table 1 .
:1Major capacities of sensors.Target function
Pressure
Length
Resistance
Load
All sensors
IPC
0.993
0.975
0.943
0.0037 9.861
1 ( )
0.873
0.369
0.509
0.0037 0.957
1 ( −1 )
0.089
0.527
0.223
0.000
0.996
1 ( −2 )
0.000
0.026
0.013
0.000
0.769
2 ( )
0.007
0.002
0.054
0.000
0.555
1 ( ) 1 ( −1 ) 0.006
0.012
0.095
0.000
0.340
2 ( ) 1 ( −1 ) 0.036
0.009
0.018
0.000
0.373
Competing interests: Authors declare that they have no competing interests.
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Acknowledgments: The results were partially obtained from the Innovative AI Chips and Next-Generation Computing Technology Development project and Development of Next-Generation Computing Technologies/Exploration of Neuromorphic Dynamics towards Future Symbiotic Society project, which were commissioned by NEDO. Funding: NA received support from JSPS KAKENHI. Grant Number JP22J01542Acknowledgments: The results were partially obtained from the Innovative AI Chips and Next- Generation Computing Technology Development project and Development of Next-Generation Computing Technologies/Exploration of Neuromorphic Dynamics towards Future Symbiotic Society project, which were commissioned by NEDO. Funding: NA received support from JSPS KAKENHI (Grant Number JP22J01542).
KN received support from JSPS KAKENHI (Grant Number JP18H05472). KN received support from JSPS KAKENHI (Grant Number JP18H05472).
N A Kn, : Methodology, Na, Tj, Mn, Y W Rs, : Investigation, R S Na, ; Visualization, Y K Kn, Tj, Mn, Y W Rs, NA, RS Funding acquisition: KN, NA Project administration: KN Supervision: KN, YK Writing -original draft: NA, KN Writing -review & editing. KN received support from JST CREST (Grant Number JPMJCR2014). Author contributions: Conceptualization:KN received support from JST CREST (Grant Number JPMJCR2014). Author contributions: Conceptualization: KN, NA Methodology: NA, TJ, MN, RS, YW, KN Investigation: NA, RS, KN Visualization: NA, RS Funding acquisition: KN, NA Project administration: KN Supervision: KN, YK Writing -original draft: NA, KN Writing -review & editing: KN, YK, TJ, MN, RS, YW
| {'fraction_non_alphanumeric': 0.050446122168840084, 'fraction_numerical': 0.026689336744244088, 'mean_word_length': 4.484088964927288, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Harnessing complex body dynamics has been a long-standing challenge in robotics. Soft body dynamics is a typical example of high complexity in interacting with the environment. An increasing number of studies have reported that these dynamics can be used as a computational resource. This includes the McKibben pneumatic artificial muscle, which is a typical soft actuator. This study demonstrated that various dynamics, including periodic and chaotic dynamics, could be embedded into the pneumatic artificial muscle, with the entire bifurcation structure using the framework of physical reservoir computing. These results suggest that dynamics that are not presented in training data could be embedded by using this capability of bifurcation embeddment. This implies that it is possible to embed various qualitatively different patterns into pneumatic artificial muscle by learning specific patterns, without the need to design and learn all patterns required for the purpose. Thus, this study sheds new light on a novel pathway to simplify the robotic devices and training of the control by reducing the external pattern generators and the amount and types of training data for the control.Main Text:RESULTSPneumatic artificial muscleThis study used the McKibben PAM(Fig. 1A), which consists of a cylindrical rubber tube covered by a braided cord. This PAM is forced by a nearly constant external load. If the', 'arxivid': '2305.03994', 'author': ['N Akashi \nGraduate School of Science\nKyoto University\nKyotoJapan\n', 'Y Kuniyoshi \nGraduate School of Information Science and Technology\nThe University of Tokyo\nTokyoJapan\n', 'T Jo \nDigital Engineering Division\nBridgestone Corporation\nTokyoJapan\n', 'M Nishida \nDigital Engineering Division\nBridgestone Corporation\nTokyoJapan\n', 'R Sakurai \nDigital Engineering Division\nBridgestone Corporation\nTokyoJapan\n', 'Y Wakao \nAdvanced Materials Division\nBridgestone Corporation\nTokyoJapan\n', 'K Nakajima \nGraduate School of Information Science and Technology\nThe University of Tokyo\nTokyoJapan\n'], 'authoraffiliation': ['Graduate School of Science\nKyoto University\nKyotoJapan', 'Graduate School of Information Science and Technology\nThe University of Tokyo\nTokyoJapan', 'Digital Engineering Division\nBridgestone Corporation\nTokyoJapan', 'Digital Engineering Division\nBridgestone Corporation\nTokyoJapan', 'Digital Engineering Division\nBridgestone Corporation\nTokyoJapan', 'Advanced Materials Division\nBridgestone Corporation\nTokyoJapan', 'Graduate School of Information Science and Technology\nThe University of Tokyo\nTokyoJapan'], 'corpusid': 258557883, 'doi': '10.48550/arxiv.2305.03994', 'github_urls': [], 'n_tokens_mistral': 18211, 'n_tokens_neox': 15734, 'n_words': 9852, 'pdfsha': 'd4546a392b5bf76ac578d389b86abbe2b75f246f', 'pdfurls': ['https://export.arxiv.org/pdf/2305.03994v2.pdf'], 'title': ['Title: Embedding bifurcations into pneumatic artificial muscle', 'Title: Embedding bifurcations into pneumatic artificial muscle'], 'venue': []} |
arxiv |
WANNIER FUNCTIONS OF ELLIPTIC ONE-GAP POTENTIAL
18 Jun 2004
E D Belokolos
V Z Enolskii
M Salerno
WANNIER FUNCTIONS OF ELLIPTIC ONE-GAP POTENTIAL
18 Jun 2004
Wannier functions of the one dimensional Schrödinger equation with elliptic one gap potential are explicitly constructed. Properties of these functions are analytically and numerically investigated. In particular we derive an expression for the amplitude of the Wannier function in the origin, a power series expansion valid in the vicinity of the origin and an asymptotic expansion characterising the decay of the Wannier function at large distances. Using these results we construct an approximate analytical expression of the Wannier function which is valid in the whole spatial domain and is in good agreement with numerical results.Date: March 22, 2022.
Introduction
The spectral analysis of Schrödinger operators with periodic potentials has been investigated since the arbour of quantum mechanics. In spite of this, it still represents a non exhausted topic of ever continuing interest. From one side, it plays a fundamental role in condensed matter physics where it provides the mathematical basis of the quantum theory of solids. From the other, Schrödinger operators with periodic and quasi-periodic potentials play an important role in the integration of the Kortweg-de Vries (KdV) equation. Eigenstates of these operators, also called Bloch functions (BF), have been extensively studied during the past years by several authors (see [19,2,10]). An expression of the BF in terms of hyperelliptic θ−functions was given in Ref. [12]. These studies were further developed in Ref. [15] where an algebro-geometric scheme for constructing solutions of non-linear equations was given in terms of the Baker−Akhiezer function. This function is uniquely defined on the Riemann surface associated with the energy spectrum and its properties are natural generalizations of the analytical properties of the BF of finite-gap potentials.
Besides BF, another set of functions which play an equally important role in condensed matter physics are the Wannier functions (WF) [16]. These functions are related to BF by a unitary transformation and form a complete set of localised orthonormal functions spanning a Bloch band. The properties of these functions were first investigated by Kohn in 1959 [13] in a classical paper in which the asymptotic decay of Wannier functions was characterised for the case of centrosymmetrical one-dimensional potentials. Since then, a large amount of work has been devoted to this topic and we mention here results only some of them. The projection operator technique was developed for construction of the Wannier functions and the Wannier functions were studied in the n-dimensional lattices [7], [8]. The localization problem for the Wannier functions was considered in a 1-dimensional case [6]. These functions were utilized with success in new practical methods for the electron energy calculation of solids (e.g. [14]) and in a number of modern calculation problems such as the photonic crystal circuits [4].The Wannier functions represent the ideal basis for constructing effective Hamiltonians of quantum problems involving spatial localizations induced by electric and magnetic fields [17], [18], [11].
In spite of this, the properties of these functions are still not fully understood and, except for simplest cases, there are no models for which the analytical expression of the WF can be explicitly given. On the other hand, the recent results achieved in the field of completely integrable systems open the possibility to investigate analytical properties of the WF. Quite interestingly, WF have not been considered in the field of finite gap potentials.
The present paper represents a first contribution in this direction. In particular, we consider WF of Schrödinger operators with one-gap potentials and use the well developed theory of elliptic functions to investigate their properties with sufficient completeness. As a result we derive: i) an exact value for the amplitude of the WF at the localization site; ii) an asymptotic expansion characterising the decay of the WF at large distances; iii) a power series expansion valid in the vicinity of the localization site. Using results ii),iii), we construct an approximate analytical representation of the WF which is valid in the whole spatial domain. These results are shown to be in very good agreement with the WF obtained by means of numerical methods.
The paper is organised as follows. In Section 2 we discuss the basic properties of the BF for one gap potentials. In particular we introduce basic definitions, discuss the basic properties of BF and derive the analytical expression of their normalization constants. Section 3 is devoted to the study of the WF. After recalling the basic definitions we derive the main results of the paper i.e. points i)-iii) listed above. In Section 4 we construct an approximate analytical expression of the WF and compare the results of our theory with WF obtained from the basic definition using numerical tools. Finally, in Section 5 we summarise the main results of the paper and briefly discuss future developments.
2. Properties of Bloch functions of one-gap potential 2.1. The Schrödinger equation with one-gap potential. In the paper we use standard notations and facts of the theory of elliptic functions. In particular we use the well known Weierstrass ℘−, σ− and ζ−functions.
The periodic elliptic one-gap potential U(x) considered in this paper, is expressed in terms of the Weierstrass ℘−function as
U(x) = −2℘(u), u = ix + ω, x ∈ R, U(x + nT ) = U(x), n ∈ N, (2.1)
where T = −2iω ′ is the period of the lattice (notice that T is real and U(x) = 2℘(ix+ω) is a smooth periodic real function). The Schrödinger equation associated with potential (2.1)
∂ 2 x Ψ(x; E) + (E − U(x))Ψ(x; E) = 0. (2.2)
As is well known (see for example [3] and references therein), this equation admits eigenfunctions Ψ(x; E) which satisfy the Bloch condition
(2.3) Ψ(x − T ; E) = e −ik(E)T Ψ(x; E),
where E is the energy given by
E = ℘(v), v = α + ω ′ , α ∈ R, (2.4)
and k(E) is the quasi-momentum given by The BF can be written in explicit form as
k(v) = ζ(v) − η ′ ω ′ v, v = α + ω ′ , α ∈ R.(2.6) Ψ(u; v) = C(v) σ(v − u) σ(v)σ(u) exp{uζ(α)},
or, alternatively, as
(2.7) Ψ(u, v) = D(v) ℘(u) − ℘(v)exp ℘ ′ (v) 2 x du ℘(u) − ℘(v) ,
where C(v), D(v) are proper normalization constants. In the following we shall use both representations for the BF. Notice that the BF considered as function of k instead of α, is periodic in the reciprocal space, with period 2 ω = iπ ω ′ . BF has the following following periodicity properties
Ψ(u + 2nω; v) = exp{2nωk(v)}Ψ(u; v), k(v) = ζ(v) − v η ω , (2.8) Ψ(u + 2n ′ ω ′ ; v) = exp{2n ′ ω ′ k ′ (v)}Ψ(u; v), k ′ (v) = ζ(v) − v η ′ ω ′ .
(2.9) 2.2. Normalization of the Bloch function of one-gap potentials. Since the normalization of the BF plays an important role in the construction of the WF (see next section), we shall show how to compute the normalization constant, although this question has been considered in Chapt. VIII of [2]. We normalise the BF according to
(2.10) 2π |Ψ(x, E)| 2 = 1, where f (x) = lim L→∞ 1 L 1 2 L − 1 2 L f (x)dx.
The following proposition is valid.
Proposition 2.1. Normalised Bloch functions of elliptic one-gap potentials are of the form
Ψ(x; α) = − i (2π) 1/2 −℘(v) − η ′ ω ′ −1/2 σ(v − u) σ(v)σ(u) exp{vη + (u − ω)ζ(v)}, (2.11) where u = ix + ω, x ∈ R; v = α + ω ′ , α ∈ R.
Proof. Let us denote the normalised BF as
Ψ(u; v) = C(v)Φ(u; v), where Φ(u; v) is the non-normalised BF Φ(u; v) = σ(v − u) σ(v)σ(u) e (u−ω)ζ(v) ,
and C(v) is the normalization constant defined by Eq. (2.10)
2π
C(v) 2 2ω ′ ω+2ω ′ ω Φ(u; v) 2 du = 1.
The complex conjugated (non-normalised) BF is
Φ(u; v) = σ(v − u) σ(v)σ(u) e −(u−ω)ζ(v) = σ(v − u) σ(v)σ(u) e −(u−ω)ζ(v) = − σ(v − 2ω ′ + u − 2ω) σ(v − 2ω ′ ) σ(u − 2ω)e (u−ω)ζ(v−2ω ′ ) = σ(v + u) σ(v)σ(u) e −(v−ω ′ )2η−(u−ω)ζ(v) ,
where the following elementary equalities were used
σ(z) = σ(z), ζ(z) = ζ(z), (u − ω) = −(u − ω), u = −u + 2ω, v = v − 2ω ′ , ζ(v − 2ω ′ ) = ζ(v) − 2η ′ , σ(v − 2ω ′ ) = −σ(v) exp(−(v − ω ′ )2η ′ ), σ(u − 2ω) = −σ(u) exp(−(u − ω)2η).
By multiplying the above expressions of Φ(u; v) and Φ(u; v) we get
Φ(u; v) 2 = σ(v − u)σ(v + u) σ 2 (v)σ 2 (u) e −(v−ω ′ )2η = [℘(u) − ℘(v)]e (v−ω ′ )2η ,
where in the last step we have used the well known formula
σ(v − u)σ(v + u) σ 2 (v)σ 2 (u) = ℘(u) − ℘(v).
The normalization condition can be then written in the form
1 = 2π C(v) 2 2ω ′ ω+2ω ′ ω [℘(u) − ℘(v)]e (v−ω ′ )2η du = 2π C(v) 2 − η ′ ω ′ − ℘(v) e −(v−ω ′ )2η , since 1 2ω ′ ω+2ω ′ ω ℘(u)du = 1 2ω ′ [ζ(ω) − ζ(ω + 2ω ′ )] = − η ′ ω ′ .
Thus we have obtained for the normalization constant the expression
(2.12) C(v) = e iθ (2π) 1/2 − η ′ ω ′ − ℘(v) −1/2 e (v−ω ′ )η ,
with an arbitrary phase factor exp(iθ), θ ∈ R. In the following we fix this factor as exp(iθ) = exp(ω ′ η − i(π/2)).
The normalised BF, Ψ(u; v), satisfy a number of useful properties under the action of symmetry operations. For centro-symmetrical potentials the transformation x → −x of the lattice corresponds to a transformation in the Jacobian u →û = −u + 2ω, and the following propositions can be proved.
Proposition 2.2. Ψ(û; v) = Ψ(−u + 2ω; v) = Ψ(u; v). Proof. Ψ(û; v) = Ψ(−u + 2ω; v) = i(2π) −1/2 −℘(v) − η ′ ω ′ −1/2 σ(v + u − 2ω) σ(v)σ(u − 2ω) exp[vη − (u − ω)ζ(v)] = i(2π) −1/2 −℘(v) − η ′ ω ′ −1/2 σ(v + u) σ(v)σ(u) exp[−vη − (u − ω)ζ(v)] = Ψ(u; v).
Similarly, that the transformation α → −α corresponds to v →v = −v + 2ω ′ , and the following proposition is valid.
Proposition 2.3. Ψ(u;v) = Ψ(u; −v + 2ω ′ ) = Ψ(u; v).
Proof.
Ψ(u;v) = Ψ(u; −v + 2ω ′ ) = −i(2π) −1/2 −℘(v) − η ′ ω ′ −1/2 σ(−v − u + 2ω ′ ) σ(−v + 2ω ′ )σ(u) × exp[(−v + 2ω ′ )η + (u − ω)ζ(v − 2ω ′ )] = −i(2π) −1/2 −℘(v) − η ′ ω ′ −1/2 σ(v + u) σ(v)σ(u) exp[−vη − (u − ω)ζ(v)] = Ψ(u; v).
The above propositions can be used to study the elementary properties of the BF, Ψ(
x; k) ≡ Ψ(u(x); v(k)), where u(x) = ix + ω and v(k) is the inverse of the function k(v) = ζ(v) − (η ′ /ω ′ )v. To this regard note that x(û) = −x(u), k(v) = −k(v).
The following two properties are easily proved.
Property 1. Ψ(−x; k) = Ψ(x; k), Proof. Ψ(−x; k) = Ψ(û, v) = Ψ(u; v) = Ψ(x; k); Property 2. Ψ(x; −k) = Ψ(x; k), Proof. Ψ(x; −k) = Ψ(u;v) = Ψ(u; v) = Ψ(x; k).
3. Analytical properties of the Wannier function of elliptic one-gap potentials 3.1. Definition and basic properties. In 1937 G. Wannier introduced a complete set of functions for an electron in a lattice structure [16]. The Wannier functions, W n (x), are defined as
(3.1) W n (x) = T 2π 1/2 π/T −π/T Ψ n (x; k)dk.
where the integral is made on the Brillouin zone. WF for the Schrödinger operator with periodic potential U(x), U(x − mT ), m ∈ Z are localised linear combinations of all the Bloch eigenstates of a given n−th spectral band. One can easily prove that if the BF is normalised according to Eq. (2.10), then the WF is normalised on the full line,
∞ −∞ |W n (x)| 2 dx = 1.
Using the translation operator one then constructs a countable set of WF: W (l)
n (x) := W n (x − lT ), l ∈ Z which is complete and forms an ortho-normal basis
∞ −∞ W (l) n (x)W (l ′ ) n ′ (x)dx = δ nn ′ δ ll ′ , l ∈ Z.
The inverse transformation allows to express a BF in terms of WF as
(3.2) Ψ n (x; k) = T 2π 1/2 ∞ l=−∞ W (l) n (x)e ilak .
In the following we shall omit the band index n since we deal only with one band. Properties of WF of one dimensional periodic potentials were studied by W. Kohn [13] where he proved that for every band there exists one and only one WF which satisfies simultaneously the following three properties 1)
W (x) = W (x); 2)W (−x) = ±W (x); 3)W (x) = O(exp(−h|x|)), where h > 0.
In the following we investigate the analytical properties of the WF for the one-gap potential in Eq. (2.1). In this case the WF is given by the formula
W (x) = T 2π 1/2 π/T −π/T Ψ(x; k)dk = T 2π 1/2 0 −π/T + π/T 0 Ψ(x; k)dk = T 2π 1 2 π/T 0 (Ψ(x; k) + Ψ(x; −k)) dk = T 2π 1/2 2Re π/T 0 Ψ(x; k)dk = Re −i √ −2iω ′ π ω+ω ′ ω ′ dk(v) dv σ(v − u) σ(v)σ(u) e vη+(u−ω)ζ(v) dv . (3.3)
Using the properties of the BF, Ψ(x; k), the following basic properties of the WF can be proved.
Proposition 3.1. W (x) = W (x).
Proof.
W (x) = 2T π 1/2 Re π/T 0 Ψ(x; k)dk = 2T π 1/2 Re π/T 0 Ψ(x; k)dk = W (x). Proposition 3.2. W (−x) = W (x).
Proof.
W (−x) = 2T π 1/2 Re π/T 0 Ψ(−x; k)dk = 2T π 1/2 Re π/T 0 Ψ(x; k)dk = 2T π 1/2 Re π/T 0 Ψ(x; k)dk = W (x).
3.2.
Power series expansion of the Wannier function at x=0. We shall construct in this section the power series expansion of the Wannier function of one gap potential.
Theorem 3.3. The Wannier function of the lower energy band for the one gap potential admits the following power series representation
(3.4) W (x) = ∞ p=0 (−1) p (2p)! W 2p x 2p ,
where the coefficients W 2p of the expansion (3.4) are given by the formula
W 2p = p l=0 M l q p,l . (3.5) Here M l = √ 2i π ω ′ e 3 + η ′ l j=0 (2j − 1)!!l! 2 j (j!) 2 (l − j)! e j 3 (e 2 − e 3 ) l−j (3.6) × F − 1 2 , j + 1 2 ; j + 1; ω ′ (e 3 − e 2 ) ω ′ e 3 + η ′ ,
where F (a, b; c; z) is the standard hypergeometric function, e 2 , e 3 are branch points of the elliptic curve and q p,l are coefficients of polynomials in ℘(v)
Q p (℘(v)) = p l=0 q p,l ℘ l (v)
defined by the recurrence
(3.7) Q p (℘(v)) = p−1 m=0 2p 2m − 2 φ m−p−1 Q m (℘(v)) with φ 0 = 2e 1 + ℘(v), φ p = 2℘ (2p) (ω).
First few coefficients of the expansion (3.4) are
W 0 = M 0 , W 2 = M 1 + 2e 1 M 0 , W 4 = M 2 + 4e 1 M 1 + (4e 2 1 + 2℘ ′′ (ω))M 0 , W 6 = M 3 + 6e 1 M 2 + (14℘ ′′ (ω) + 12e 2 1 )M 1 + (2℘ (IV ) (ω) + 28℘ ′′ (ω)e 1 + 8e 3 1 )M 0 . (3.8)
Proof. We have
(3.9) W (x) = 2T π 1 2 Re π/T 0 Ψ(x; k)dk.
Because the BF Ψ(x; k) is even in x we can write it in the form of Taylor expansion
(3.10) Ψ(u, v) = ∞ p=0 (−1) p (2p)! Ψ 2p (v)x 2p , where Ψ 2p (v) = d 2p du 2p Ψ(u, v) u=ω , p = 1, . . . .
If we substitute the Taylor expansion (3.10) to the (3.9) we obtain the expansion (3.4) with the following coefficients
(3.11) W 2p = 2T π 1/2 Re π/T 0 Ψ 2p (v)dk.
Using Schrödinger equation we obtain easily for the Ψ 2p (v) a recurrent relation
Ψ 2p (v) = p−1 l=0 2p 2l − 2 φ p−l−1 Ψ 2l (v) φ 0 = 2e 1 + ℘(v), φ p = 2℘ (2p) (ω).
The form of this relation leads to conclusion that
Ψ 2p (v) = Q p (℘(v))Ψ 0 (v), where Ψ 0 (v) = Ψ(ω; v) = 1 √ 2π ℘(v) − e 1 ℘(v) + η ′ ω ′ (3.12)
and Q p (℘(v)) are polynomials of the p-th order in ℘(v),
Q p (℘(v)) = p l=0 q p,l ℘ l (v).
Similarly to Ψ 2p (v) the polynomials Q p (℘(v)) satisfy the following recurrent relation
(3.13) Q p (℘(v)) = p−1 m=0 2p 2m − 2 φ m−p−1 Q m (℘(v)) with φ 0 = 2e 1 + ℘(v), φ p = 2℘ (2p) (ω).
In particular the first few polynomials Q p (℘(v)) are
Q 1 (℘(v)) = ℘(v) + 2e 1 , Q 2 (℘(v)) = ℘(v) 2 + 4e 1 ℘(v) + 2℘ ′′ (ω) + 4e 2 1 , Q 3 (℘(v)) = ℘(v) 3 + 6e 1 ℘(v) 2 + 14℘ ′′ (ω) + 12e 2 1 ℘(v) + 2℘ (IV ) (ω) + 28e 1 ℘ ′′ (ω) + 8e 3 1 .
(3.14)
Next we calculate the integral expressions of the coefficients W 2p . We show that the following formula is valid
M l = −2 ω ′ iπ 1/2 ω ′ 0 ℘(v) l ℘(v) + η ′ ω ′ Ψ(ω; v)dv = 2T π 1 2 ω ′ 0 ℘(v) l 1 √ 2π ℘(v) − e 1 ℘(v) + η ′ ω ′ −℘(v) − η ′ ω ′ dv = T 1 2 π ω ′ 0 ℘(v) l ℘(v) − e 1 ℘(v) + η ′ ω ′ dv.
After the substitution ℘(v) = s, the computation is reduced to the derivation of the complete elliptic integral
(3.15) M l = − √ −iω ′ π e 2 e 3 s l s + η ′ ω ′ (s − e 2 )(s − e 3 ) ds.
By introducing the new variable t = s − e 3 e 2 − e 3 , the integral M l acquires the form
(3.16) M l = − √ −iω ′ π √ e 2 − e 3 1 0 ((e 2 − e 3 )t − e 3 ) l 1 − k 2 t t(1 − t) dt, where (3.17) k = ω ′ (e 3 − e 2 ) ω ′ e 3 + η ′ ,
is the Jacobi modulus of the elliptic curve
(3.18) Y 2 = (X − e 2 )(X − e 3 ) X + η ′ ω ′ .
Using the integral representation of the hypergeometric function (see e.g. [1])
(3.19) F (a, b; c; z) = Γ(c) Γ(b)Γ(c − b) 1 0 t b−1 (1 − t) c−b−1 (1 − tz) a dt,
we obtain the required expression
M l = √ 2i π ω ′ e 3 + η ′ l j=0 (2j − 1)!!l! 2 j (j!) 2 (l − j)! e j 3 (e 2 − e 3 ) l−j × F − 1 2 , j + 1 2 ; j + 1; k 2 .
(3.20)
It is worth to note that the coefficient W 0 gives an exact value of the amplitude of WF at the localization site x = 0,
W 0 = √ 2i 2 ω ′ e 3 + η ′ F − 1 2 , 1 2 ; 1; k 2 . (3.21)
This amplitude can also be written in the alternative form
W 0 = √ 2i π ω ′ e 3 + η ′ E( k), (3.22)
where E( k) is the complete integral of the second kind depending on k. Also note that by using the relations
z d dz + b = bF (a, b + 1; c; z), (1 − z) d dz + c − a − b F (a, b; c; z) = (c − a)(c − b) c F (a, b; c + 1; z),
one can express all hyperegeometric functions F (− 1 2 , j + 1 2 ; j + 1; k 2 ) in (3.20) in terms of the derivatives of F (− 1 2 , 1 2 ; 1; k 2 ) with respect to k 2 , and therefore the whole expression can be written in terms of the complete integral E( k) and of its derivatives.
We remark that the quantities ℘ (2j) (ω), j = 1, . . . , can be computed in recurrent way, the some first of them are [1] ℘ ′′ (ω) = 3! e 2 1 −
1 2 2 · 3 g 2 , ℘ (IV ) (ω) = 5! e 3 1 − 3 2 2 · 5 g 2 e 1 − 1 2 · 5 g 3 , ℘ (V I) (ω) = 7! e 4 1 − 1 5 g 2 e 2 1 − 1 7 g 3 e 1 + 1 2 4 · 5 · 7 g 2 2 .
3.3. Asymptotic expansion of the Wannier function. In this section we obtain the asymptotic expression for the WF at x → +∞ by the steepest descent method. This method (see, e.g. [9]) permits to compute the asymptotic expression of integrals of the type
F (x) = γ f (z)exp{xS(z)}dz,
where γ is a contour in the complex plane and the functions f (z) and S(z) are holomorphic in the vicinity of γ. When a saddle point z 0 , which is defined by the equation d dz S(z 0 ) = 0, does not coincide with the edges of the contour, the asymptotic formula of the integral F (x) reads
F (x) = − 2π x d 2 S(z 0 ) dz 2 exp{xS(z 0 )} f (z 0 ) + O(x −1 ) .
In our case we must use a non-standard variant of the steepest descents method, since the exponential in the integrand will have the form xS(z) only at |x| → +∞ and f (z 0 ) = 0.
Proposition 3.4. At x → ∞ the Wannier function of the lower energy band for the one gap potential has the following asymptotic expression (3.23)
W (x) ≃ Re √ −2iω ′ π e 1 + η ′ ω ′ 1/2 σ(v − u) σ(v)σ(u) e (u−ω)ζ(v) i 2℘ ′ (v) 1/4 Γ 3 4 x 3 4
where v is a solution of the equation
(3.24) ℘(v) = − η ′ ω ′ , or k ′ (v) = 0, such that the complex number ω ′ k(v) = ω ′ (ζ(v) −(η ′ /ω ′ )v) has negative real part. This means that W (x) ≃ exp(−h|x|)|x| −3/4 , |x| → ∞, where h = |k(v)| and v is defined by the equation k ′ (v) = 0.
Proof. We use the expression of the WF for the first energy band [e 3 , e 2 ] given in Eq. (3.3). Since
σ(v − u) σ(v)σ(u) = − σ(u − v) σ(ω − v) σ(ω) σ(u) σ(ω − v) σ(v)σ(ω) = σ(v − ω) σ(v)σ(ω) exp u ω [ζ(s − v) − ζ(s)]ds = [℘(v) − e 1 ] 1/2 exp −vη + u ω [ζ(s − v) − ζ(s)]ds ,
we have that Eq. (3.3) can be rewritten as
W (x) = Re √ −2iω ′ π ω+ω ′ ω ′ [e 1 − ℘(v)] 1/2 dk(v) dv (3.25) × exp u ω [ζ(s − v) − ζ(s) + ζ(v)]ds dv .
When x → +∞ we can calculate the integral in Eq. (3.25) by the steepest descents method. To this regard we remark that the argument of the exponential, as a function of v, has saddle points which are defined by the equation
d dv u ω [ζ(s − v) − ζ(s) + ζ(v)]ds = 0,
or, in other words, by the equation , v 2 must be a period of the lattice, which in our case means that v 1 + v 2 = 2(ω + ω ′ ). It is not difficult to show that
℘(v) = − ζ(u − v) − ζ(ω − v) u − ω .v 1 = ω + ω ′ + iβ, v 2 = ω + ω ′ − iβ, β ∈ R,
i.e. the two saddle points v 1 , v 2 , are situated in the spectral gap. The periodicity of the Weierstrass function ℘(z) in the complex plane give rise to countable set V of saddle points,
V = {v 1 + 2n 1 ω ′ , v 2 + 2n 2 ω ′ : n 1 , n 2 ∈ Z}.
In order to build the proper asymptotic expression for the Wannier function W (x) we must select from this set a special saddle point which we denote by v 0 . In the neighbourhood of v 0 we have
dk(v) dv ≃ [−℘ ′ (v 0 )] 1/2 (v − v 0 ) 1/2 , u ω [ζ(s − v) − ζ(s) + ζ(v)]ds ≃ u ω [ζ(s − v 0 ) − ζ(s) + ζ(v 0 )]ds + 1 2 (v − v 0 ) 2 u ω [ζ ′′ (s − v 0 ) + ζ ′′ (v 0 )]ds = u ω [ζ(s − v 0 ) − ζ(s) + ζ(v 0 )]ds − (u − ω) 1 2 ℘ ′ (v 0 ) + ℘(u − v 0 ) − ℘(ω − v 0 ) u − ω (v − v 0 ) 2 ≃ u ω [ζ(s − v 0 ) − ζ(s) + ζ(v 0 )]ds − (u − ω) 1 2 ℘ ′ (v 0 )((v − v 0 ) 2 .
Substituting the last two expressions into the integral representation of the WF in (3.25) , we obtain
W 0 (x) ≃ Re √ −2iω ′ π e 1 + η ′ ω ′ 1/2 exp u ω [ζ(s − v 0 ) − ζ(s) + ζ(v 0 )]ds × C 0 dv[−℘ ′ (v 0 )] 1/2 (v − v 0 ) 1/2 exp − 1 2 (u − ω)℘ ′ (v 0 )(v − v 0 ) 2 = Re √ −2iω ′ π e 1 + η ′ ω ′ 1/2 σ(v 0 − u) σ(v 0 )σ(u) exp {(u − ω)ζ(v 0 )} × C 0 dr[−℘ ′ (v 0 )] 1/2 r 1/2 exp − 1 2 (u − ω)℘ ′ (v 0 )r 2 ,
where C 0 is a contour passing through the saddle point v 0 . The integral in the last expression can be calculated as
I 0 = C 0 dr[−℘ ′ (v 0 )] 1/2 r 1/2 exp − 1 2 (u − ω)℘ ′ (v 0 )r 2 = i 2℘ ′ (v 0 ) 1/4 1 x 3 4 ∞ 0 e −t t −1/4 dt = i 2℘ ′ (v 0 ) 1/4 Γ 3 4 x 3 4 . Notice that, since ℘ ′ (v 0 ) = −2[℘(v 0 )−e 1 ] 1/2 [℘(v 0 )−e 2 ] 1/2 [℘(v 0 )−e 3 ] 1/2 , and e 3 ≤ e 2 ≤ ℘(v 0 ) ≤ e 1 , we have that ℘ ′ (v 0 ) = −i|℘ ′ (v 0 )|.
For the function W 0 (x) we finally obtain
W 0 (x) ≃ Re √ −2iω ′ π e 1 + η ′ ω ′ 1/2 i 2℘ ′ (v 0 ) 1/4 Γ 3 4 x 3 4 × exp u ω [ζ(s − v 0 ) − ζ(s) + ζ(v 0 )]ds = Re √ −2iω ′ π e 1 + η ′ ω ′ 1/2 σ(v 0 − u) σ(v 0 )σ(u) e (u−ω)ζ(v 0 ) i 2℘ ′ (v 0 ) 1/4 Γ 3 4 x 3 4 .
The asymptotic behaviour of the function W 0 (x) at x → +∞ is defined by the factor
f (u, v 0 ) = σ(v 0 − u) σ(u) exp[(u − ω)ζ(v 0 )],
which satisfies the relation
f (u+2nω ′ , v 0 ) = f (u, v 0 ) exp[2nω ′ (ζ(v 0 )− η ′ ω ′ v 0 )] = f (u, v 0 ) exp[2nω ′ k(v 0 )
]. The saddle point v 0 must be chosen in such a manner that the complex number ω ′ k(v 0 ) has negative real part. From the previous considerations it follows that the point v 0 is also a saddle point. The asymptotic behaviour of the function W 0 (x) at x → −∞ is defined by this saddle point which corresponds to the complex number
ω ′ k(v 0 ) = ω ′ (ζ(v 0 ) − (η ′ /ω ′ )v 0 ) with positive real part.
It is appropriate to make here some remarks. According to the theorem
W (x) ≃ exp(−|x||k(α 0 )|)|x| −3/4 , |x| → ∞.
It is easy to understand such an asymptotic behaviour of the WF at |x| → ∞ if we take into account that
W (x) ≃ Re C (k − k 0 ) β e ikx ≃ 2 sin(βπ)Γ(1 + β)x −(1+β) e −ℑk 0 x ,
where k 0 is a branching point of the energy E(k), and that, due to a normalization constant of the wave function,
Ψ(k) ≃ (k − k 0 ) −1/4 ,
the equality β = −1/4 is valid. As far as we know this asymptotic law was mentioned for the first time in [6].
It is of interest to note also that the equation
℘(v) + η ′ ω ′ = 0 has obviously the following solution v = ± ∞ − η ′ ω ′ dx 4x 3 − g 2 x − g 3 .
More general problem to solve the equation
℘(v, ω, ω ′ ) = c(ω, ω ′ ),
is a well known mathematical problem in the theory of elliptic functions. Solution of the problem in terms of Eisenstein series is presented in the paper [5].
In the above theorem we have obtained results for the Wannier function of lower energy band. Results for the Wannier function of higher band,
W (x) = Re −i √ −2iω ′ π ω ω dk(v) dv σ(v − u) σ(v)σ(u) e vη+(u−ω)ζ(v) dv , k( ω) = ζ( ω) − η ′ ω ′ ω = π T ,(3.26)
are similar to ones presented above and as a result of that we omit appropriate considerations. Let us now discuss two limiting cases. The limit τ = ω ′ /ω → 0 corresponds to the free electron case or the empty lattice case. In this limit the energy gap is zero, e 1 = e 2 , there are no saddle points and as a result we have the well known free electron WF
W (x) = T 1/2 πx sin πx T .
The limit τ = ω ′ /ω → i∞ corresponds to the case of tightly bound electrons. In this case the width of the lower energy band is zero,
Ψ(x) = α 2 1/2 1 cosh αx , E 0 = −α 2 ,
where E 0 is the binding energy. The wave functions of higher energy bands are of a form,
Ψ(x, k) = 1 √ 2π √ k 2 + α 2 (|k| + iα tanh(αx)) e ±ikx , E = k 2 .
In the next section we shall compare our analytical results with numerical ones.
Approximate analytical expressions of Wannier function and numerical results
The results of the previous section permit to construct the following approximate expression of the WF for one-gap potentials:
W (x) = W 0 + W 2 x 2 + W 4 x 4 + W 6 x 6 , for |x| ≤ x 0 = Re √ −2iω ′ π e 1 + η ′ ω ′ 1/2 σ(v−u) σ(v)σ(u) (4.1) ×exp{(u − ω)ζ(v)} i 2℘ ′ (v) 1/4 Γ( 3 4 )
x 3 4 for |x| > x 0 , where u = ix + ω, v = v + + ω ′ , the coefficients W 0 . . . , W 6 are given by the formulae (3.8) and the point x 0 is chosen so to satisfy the normalization condition W (x) 2 = 1.
To check the validity of this expression we shall compare the WF obtained from Eq.(4.1) with the one obtained directly from the definition (3.1) by numerical methods, using the expression of the normalised BF in Proposition 2.1. In Fig.1 we show the band structure obtained for the one gap potential with parameter values e 1 = 2, e 2 = −0.5, e 3 = −1.5, while in Fig. 2 we depict the WF associated to the lower band. We see that the agreement between the analytical approximation and direct numerical calculations is excellent both in proximity of the origin and far away from it, these being the regions of validity of the corresponding expansions. In the intermediate region, however, some discrepancy appears. It is possible that for some set of potential parameters the validity regions of the two expansions (near the origin and far from the origin) can overlap at some point x 0 so that it is possible to join them into the single smooth analytical approximation (4.1) which stay close to the exact numerically curve in the whole spatial domain. Existence of the point of such a kind is possible only for one-gap potential. Such a good matching of two expansions is shown in Fig. 3 where the WF of the lower band for a potential with another set of parameters is depicted. By comparing Fig. 3 with Fig. 2 we see that the discrepancy in the intermediate region is smaller for WF which are more localised. This can be understood from the fact that a faster decay of the function (see Figs. 4,5 below) allows the asymptotic expansion to work up to points which are very close to the origin. In Fig.s 4 and 5 we show the asymptotic decay of the WF depicted in Figs. 2, 3, respectively from which we see that a stronger localization of the function corresponds to a faster asymptotic decay. The linear decay observed in the semi-log plots of these figures, is fully consistent with the exponential decay of the WF of one-gap elliptic potentials predicted by our analysis.
Conclusions
In this paper we have investigated the properties of the Wannier functions of the Schrödinger operator with one gap potentials. As a result we have derived the exact value for the amplitude of the functions in the origin, as well as, an asymptotic expansion characterising the decay of the function at large distances and a power series valid in the vicinity of the origin. Using these expansions we have constructed approximate analytical expressions of the Wannier functions and showed that they are in good agreement with the ones obtained from numerical results.
We remark that the developed approach can be generalised to the case of finite-gap potentials of more complicated type, like elliptic finitegap potentials and general finite-gap potentials. We shall discuss these problems in a future publication.
type (2.3) are called Bloch functions. Notice that the dependence of the quasi-momentum on the energy (and vice versa) arises from the elimination of the parameter α from Eqs. (2.4), (2.5).
At x → +∞ the last equation attains the form (3.24). Since ℘(v) = ℘(v) we have that if v is saddle point then v is also a saddle point. On the other hand, ℘(v) is an elliptic function of the second order, so it takes every value twice in the fundamental domain, this implying that there are two saddle points, say v 1 , v 2 , in the fundamental domain. The sum of the values v 1
Figure 1 .
1Energy bands of the one gap potential with parameters of the elliptic curve e 1 = 2, e 2 = −0.5, e 3 = −1.5 e 2 = e 3 , the appropriate wave function looks as follows,
Figure 2 .
2The Wannier function associated to the lower band of the one gap potential. The branching points of the elliptic curve are fixed as in Fig. 1. The amplitude of the function in the origin is W (0) = 0.93. The continuous curve denote the exact expression obtained from numerical calculations, the dashed line corresponds to part of the WF approximated by the asymptotic expansion, while the dotted line denotes the part obtained from the power expansion near the origin. The arrow shows the point where the two different analytical expansions are joined.
Figure 3 .
3Same as in Fig.2 but for a different set of parameters. The branching points of the elliptic curve are e 1 = 6, e 2 = −2.0, e 3 = −4.0. The value of the function in the origin is W (0) = 1.22302. The continuous curve denote the exact expression obtained from numerical calculations, the dashed line corresponds to part of the WF approximated by the asymptotic expansion, while the dotted line denotes the part obtained from the power expansion near the origin. The arrow shows the point where the two different analytical expansions are joined.
Figure 4 .
4Asymptotic decay of the Wannier function inFig.2in semi-log scale. The continuous curve represents our analytical approximation while the dotted line is obtained from direct numerical calculations of the WF.
Figure 5 .
5Asymptotic decay of the Wannier function inFig. 3in semi-log scale. The continuous curve represents our analytical approximation while the dotted line is obtained from direct numerical calculations of the WF.
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Soliton theory: inverse scattering method. V E Zakharov, S V Manakov, S P Novikov, L P R Pitaevski ; E, Caianiello, [email protected] Dipartimento di Scienze Fisiche. Nauka, Moscow; Ukraine, email36Kiev-142Institute of Magnetism, Vernadski str. via S.AllendeV. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevski. Soliton theory: inverse scattering method (in Russian), Nauka, Moscow, (1980) Institute of Magnetism, Vernadski str. 36, Kiev-142, Ukraine, e- mail: [email protected] Dipartimento di Scienze Fisiche "E.R.Caianiello", via S.Allende,
Italy, e-mail: [email protected] Dipartimento di Scienze Fisiche. ; E R Baronissi, Caianiello, Allende84081Baronissi (SA), Italy, eBaronissi (SA), Italy, e-mail: [email protected] Dipartimento di Scienze Fisiche "E.R.Caianiello", via S.Allende, 84081, Baronissi (SA), Italy, e:mail: [email protected]
| {'fraction_non_alphanumeric': 0.1090280357806293, 'fraction_numerical': 0.037381040839072906, 'mean_word_length': 3.3425167535368576, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 5, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 53, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Wannier functions of the one dimensional Schrödinger equation with elliptic one gap potential are explicitly constructed. Properties of these functions are analytically and numerically investigated. In particular we derive an expression for the amplitude of the Wannier function in the origin, a power series expansion valid in the vicinity of the origin and an asymptotic expansion characterising the decay of the Wannier function at large distances. Using these results we construct an approximate analytical expression of the Wannier function which is valid in the whole spatial domain and is in good agreement with numerical results.Date: March 22, 2022.', 'arxivid': 'cond-mat/0401440', 'author': ['E D Belokolos ', 'V Z Enolskii ', 'M Salerno '], 'authoraffiliation': [], 'corpusid': 2610960, 'doi': '10.1088/0305-4470/37/41/007', 'github_urls': [], 'n_tokens_mistral': 14080, 'n_tokens_neox': 12318, 'n_words': 6759, 'pdfsha': 'cdd1f1a9ab28360572ac462b76da03c488f8809b', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0401440v2.pdf'], 'title': ['WANNIER FUNCTIONS OF ELLIPTIC ONE-GAP POTENTIAL', 'WANNIER FUNCTIONS OF ELLIPTIC ONE-GAP POTENTIAL'], 'venue': []} |
arxiv |
TOPOLOGICAL REALIZATION OF ALGEBRAS OF QUASI-INVARIANTS, I
May 2023
Yuri Berest
Ajay C Ramadoss
TOPOLOGICAL REALIZATION OF ALGEBRAS OF QUASI-INVARIANTS, I
17May 2023
This is the first in a series of papers, where we introduce and study topological spaces that realize the algebras of quasi-invariants of finite reflection groups. Our result can be viewed as a generalization of a well-known theorem of A. Borel that realizes the ring of invariant polynomials a Weyl group W as a cohomology ring of the classifying space BG of the associated Lie group G. In the present paper, we state our realization problem for the algebras of quasi-invariants of Weyl groups and give its solution in the rank one case (for G = SU(2)). We call the resulting G-spaces Fm(G, T ) the m-quasi-flag manifolds and their Borel homotopy quotients Xm(G, T ) the spaces of m-quasi-invariants. We compute the equivariant K-theory and the equivariant (complex analytic) elliptic cohomology of these spaces and identify them with exponential and elliptic quasi-invariants of W . We also extend our construction of spaces quasi-invariants to a certain class of finite loop spaces ΩB of homotopy type of S 3 originally introduced by D. L. Rector[Rec71a]. We study the cochain spectra C * (Xm, k) associated to the spaces of quasi-invariants and show that these are Gorenstein commutative ring spectra in the sense of Dwyer, Greenlees and Iyengar [DGI06]. 1 a p-local version of the realization problem for algebras of quasi-invariants of non-crystallographic (in fact, non-Coxeter) groups defined over the p-adic numbers in terms of p-compact groups.We now give a general overview of our work, our problems and motivation.Quasi-invariants and cohomology theories. In mathematical physics, quasi-invariants naturally arise in three different flavors: rational (polynomial), trigonometric (exponential) and elliptic. Having in hand topological spaces X m (G, T ) that realize the algebras Q m (W ), it is natural to expect that the above three types of quasi-invariants correspond to three basic cohomology theories evaluated at X m (G, T ): namely, the ordinary (singular) cohomology, topological K-theory and elliptic cohomology. We will show that this is indeed the case: in fact, quasi-invariants can be defined for an arbitrary (complex-oriented generalized) cohomology theory, though in general their properties have yet to be studied.Quasi-flag manifolds. For a compact connected Lie group G, our spaces of quasi-invariants can be naturally realized as Borel homotopy quotients of certain G-spaces F m (G, T ):We call F m (G, T ) the m-quasi-flag manifold of G as in the special case m = 0, we have F 0
Introduction
Quasi-invariants are natural generalizations of classical invariant polynomials of finite reflection groups. In the case of Coxeter groups, they first appeared in mathematical physics -in the work of O. Chalykh and A. Veselov [CV90,CV93] in the early 1990s, and since then have found applications in many other areas: most notably, representation theory, algebraic geometry and combinatorics (see [FV02], [EG02], [Cha02], [BEG03], [FV03], [GW06], [BM08], [Tsu10], [BC11], [BEF20], [Gri21]). For arbitrary (complex) reflection groups, quasi-invariants were introduced in [BC11]. This last paper developed a general approach to quasi-invariants in the context of representation theory of rational double affine Hecke algebras, extending and refining the earlier results of [BEG03] in the Coxeter case. We will use [BC11] as our main reference on algebras of quasi-invariants; in particular, we will follow the notation and conventions of that paper in the present work.
We begin by recalling the definition of quasi-invariants in the Coxeter case. Let W be a finite real reflection group acting in its reflection representation V . Denote by A := {H} the set of reflection hyperplanes of W in V and write s H ∈ W for the reflection operator in H. The group W acts naturally on the polynomial algebra C[V ] and, since the s H 's generate W , the invariant polynomials p ∈ C[V ] W are determined by the equations (1.1) s H (p) = p for all H ∈ A. To define quasi-invariants we modify ('weaken') the equations (1.1) in the following way. For each reflection hyperplane H ∈ A, we choose a linear form α H ∈ V * such that H = Ker(α H ) and fix a non-negative integer m H ∈ Z + , assuming that m w(H) = m H for all w ∈ W . In other words, we choose a system of roots of W in V * , which (abusing notation) we still denote by A, and fix a W -invariant function m : A → Z + , H → m H , which values we will refer to as multiplicities of hyperplanes (or roots) in A. Now, with these extra data in hand, we replace the equations (1.1) by the following congruences in C[V ]:
(
(W ) = C[V ], while C[V ] W ⊆ Q m (W ) ⊆ C[V ]
in general. Thus, for varying m, the quasi-invariants interpolate between the W -invariants and all polynomials. Despite its simple definition, the algebras Q m (W ) have a complicated structure: they do not seem to admit a good combinatorial description, nor do they have a natural presentation in terms of generators and relations. Nevertheless, these algebras possess many remarkable properties, such as Gorenstein duality (see Theorem 2.3), and are closely related to some fundamental objects in representation theory, such as Dunkl operators and double affine Hecke algebras (see [BEG03,BC11]).
The goal of the present work is to give a topological realization of the algebras of quasi-invariants as (equivariant) cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result can be viewed as a generalization of a well-known theorem of A. Borel [Bor53] that realizes the algebra of invariant polynomials of a Weyl group W as the cohomology ring of the classifying space BG of the associated Lie group G. As the algebras Q m (W ) are defined over C, we should clarify what we really mean by "topological realization". It is a fundamental consequence of Quillen's rational homotopy theory [Qui69] that every reduced, locally finite, graded commutative algebra A defined over a field k of characteristic zero is topologically realizable, i.e. A ∼ = H * (X, k) for some (simply-connected) space X. When equipped with cohomological grading, the algebras Q m (W ) have all the above-listed properties (cf. Lemma 2.2); hence, the natural question: For which values of m the Q m (W )'s are realizable, has an immediate answer: for all m. A more interesting (and much less obvious) question is whether one can realize quasi-invariants topologically as a diagram of algebras {Q m (W )} (indexed by m) together with natural structure that these algebras carry (e.g., W -action). It is one of the objectives of this work to formulate a realization problem for the algebras of quasi-invariants in a precise (axiomatic) form by selecting a list of the desired properties. In the present paper, we state this problem for the classical Weyl groups (i.e., the crystallographic Coxeter groups over R or C) in terms of classifying spaces of compact Lie groups (see Section 2.4); in our subsequent paper, we will try to formulate (W ), introduced and studied in [BC11]. As observed in [BC11], for integer m, the action of C[V ] ⋊ W on Q m (W ) naturally extends to the rational double affine Hecke (a.k.a. Cherednik) algebra H m (W ) associated to (W, m). We will show that the topological construction of the quasi-flag manifolds F m (G, T ) generalizes to half-integer values of m, although at the expense of producing spaces equipped only with T -action. By [BC11], we get then an action of H m+1 (W ) on the T -equivariant cohomology of F m+ 1 2 (G, T ). This phenomenon seems to generalize to other cohomology theories, defining, in particular, an action of trigonometric (resp., non-degenerate) DAHA on T -equivariant K-theoretic (resp, elliptic) quasi-invariants. Constructing these actions algebraically and giving them a topological explanation is an interesting problem that we leave for the future.
Topological refinements. The realization of algebras of quasi-invariants raises many natural questions regarding topological analogues ('refinements') of basic properties that these algebras possess. A general framework to deal with such questions is provided by stable homotopy theory. Indeed, our spaces of quasi-invariants X m (G, T ) are closely related to the classifying spaces of compact Lie groups, and the latter have been studied extensively in recent years by means of stable homotopy theory (see, e.g., [DGI06], [BG14], [Gre18], [Gre20], [BCHV21]). From this perspective, the main object of study is the mapping spectrum (1.4) C * (X, k) := Map (Σ ∞ X + , Hk)
called the cochain spectrum of a topological space X. As its notation suggests, C * (X, k) is a commutative ring spectrum that -for an arbitrary commutative ring k -plays the same role as the usual (differential graded) k-algebra of cochains on X in the case when k is a field of characteristic zero. In particular, the (stable) homotopy groups of the spectrum (1.4) are isomorphic to the singular cohomology groups of the space X:
π −i [C * (X, k)] ∼ = H i (X, k)
The ring spectrum (1.4) thus refines (in a homotopy-theoretic sense) the cohomology ring H * (X, k). For example, if G is a compact connected Lie group and k is a field of characteristic 0, the Borel Theorem mentioned above identifies H * (BG, k) with the algebra k[V ] W of invariant polynomials of W . The cochain spectrum C * (BG, k) of the classifying space BG can thus be viewed as a refinement of the algebra k[V ] W . In the same manner, we will regard the cochain spectra C * (X m (G, T ), k) of our spaces X m (G, T ) as homotopy-theoretic refinements of the algebras of quasi-invariants Q m (W ). The point is that the known algebraic properties of Q m (W ) should have topological analogues for C * (X m (G, T ), k). For example, one of the main theorems about quasi-invariants (see Theorem 2.3) says that the (graded) algebras Q m (W ) defined over C are Gorenstein if W is a Coxeter group. It is therefore natural to expect that the corresponding ring spectra C * (X m (G, T ), k) are also Gorenstein -but now in a topological sense [DGI06] and over an arbitrary field k. We will show that this expectation is indeed correct, at least in the rank one case (see Theorem 7.1 and Theorem 7.2), and the spectra of quasi-invariants have a number of other interesting properties. Our results are only first steps in this direction, and many natural questions motivated, in particular, by representation theory have yet to be answered.
Homotopy Lie groups. The spaces of quasi-invariants of compact Lie groups, X m (G, T ), can be constructed functorially in a purely homotopy-theoretic way. In the rank one case, we use to this end the so-called fibre-cofibre construction -a classical (though not very well-known) construction in homotopy theory introduced by T. Ganea [Gan65]. A generalization of Ganea's construction allows us to define the analogues of X m (G, T ) for certain finite loop spaces closely related to compact Lie groups, and perhaps most interestingly, for p-compact groups -p-local analogues of finite loop spaces also known as homotopy Lie groups. In this last case, the classical Weyl groups are replaced by pseudo-reflection groups defined over the field Q p of p-adic numbers. It is well known that all such pseudo-reflection groups can be realized as complex reflection groups (see [CE74]), and we thus provide realizations of algebras of quasi-invariants of complex reflection groups defined in [BC11], albeit in a p-local setting. The simplest exotic examples are the rank one p-compact groupsŜ 2n−1 p , called the Sullivan spheres, whose 'Weyl groups' are the cyclic groups W = Z/n of order n > 2 such that n | (p − 1). These examples are already quite rich: we will treat them in a separate paper.
We divide our work into three parts. The present paper (Part I) focuses entirely on the 'global' rank one case: here, we define and study the spaces of quasi-invariants for the Lie group G = SU (2) and for a certain class of finite loop spaces ΩB of homotopy type of S 3 known as Rector spaces. In Part II, we formulate a p-local version of the realization problem for algebras of quasi-invariants defined over Q p and give its solution in the 'local' rank one case: namely, for the p-compact groups associated with Sullivan spheresŜ 2n−1 p . In Part III, we then use the spaces introduced in Part I and Part II as 'building blocks' for constructing spaces of quasi-invariants for arbitrary compact connected Lie groups and for 'generic' p-compact groups related to Clark-Ewing spaces.
Contents of the present paper. We now describe in more detail the results of the present paper. In Section 2, after reviewing basic facts about quasi-invariants, we state our realization problem for Weyl groups in the classical framework of compact connected Lie groups. As mentioned above, we take an axiomatic approach: the properties that we choose to characterize the topological spaces of quasi-invariants are modeled on properties of algebraic varieties of quasi-invariants introduced and studied in [BEG03]. In fact, our main axioms (QI 1 )-(QI 5 ) in Section 2.4 are natural homotopy-theoretic analogues of basic geometric properties of the varieties of quasi-invariants listed in Section 2.2.
In Section 3, we give a solution of our realization problem for G = SU (2) (see Theorem 3.9). To this end, as mentioned above, we employ the Ganea fibre-cofibre construction. This construction plays an important role in abstract homotopy theory (specifically, in the theory of LS-categories and related work on the celebrated Ganea Conjecture in algebraic topology, see e.g. [CLOT03] and Example 3.2 below). However, we could not find any applications of it in Lie theory or classical homotopy theory of compact Lie groups (perhaps, with the exception of the simple (folklore) Example 3.3). We therefore regard Proposition 3.7 and Theorem 3.9 that describe the Ganea tower of the Borel maximal torus fibration of a compact connected Lie group as original contributions of the present paper. The G-spaces F m (G, T ) that we call the m-quasi-flag manifolds of G are defined to be the homotopy fibres of iterated (level m) fibrations in this Ganea tower (see Definition 3.10). In Section 3.4 and Section 3.5, we describe some basic properties of the G-spaces F m (G, T ). First, we compute the T -equivariant cohomology of F m (G, T ) (see Proposition 3.14) and identify it with a module of 'nonsymmetric' (CW -valued) quasi-invariants (see Corollary 3.16). In this way, we provide a topological interpretation of generalized quasi-invariants introduced in [BC11]. Then, in Section 3.5, we define natural analogues of the classical Demazure (divided difference) operators for our quasi-flag manifolds F m (G, T ). Our construction is purely topological (see Proposition 3.19): it generalizes the Bressler-Evans construction of the divided difference operators for the classical flag manifolds F 0 (G, T ) given in [BE90].
In Section 4, we extend our topological construction of spaces of quasi-invariants to a large class of finite loop spaces ΩB called the Rector spaces (or fake Lie groups of type SU (2)). These remarkable loop spaces were originally constructed in [Rec71a] as examples of nonstandard ('exotic') deloopings of S 3 . Our construction does not apply to all Rector spaces, but only to those that accept homotopically nontrivial maps from CP ∞ . These last spaces admit a beautiful arithmetic characterization discovered by D. Yau in [Yau02]. We show that the 'fake' spaces of quasiinvariants, X m (ΩB, T ), associated to the Rector-Yau spaces have the same rational cohomology as our 'genuine' spaces of quasi-invariants, X m (G, T ), constructed in Section 3 (see Theorem 4.7); however, in general, they are homotopically non-equivalent (see Corollary 5.12).
In Section 5, we compute the G-equivariant (topological) K-theory K * G (F m ) of the spaces F m = F m (G, T ) and identify it with Q m (W ), the exponential quasi-invariants of the Weyl group W = Z/2Z (see Theorem 5.6). Then, we relate Q) by constructing explicitly the G-equivariant Chern character map
K * G (F m ) to the (completed) G-equivariant cohomology H * G (F m , Q) := ∞ k=0 H k G (F m ,(1.5) ch G (F ) : K * G (F m ) → H * G (F m , Q)
We show that (1.5) factors through the natural map K * G (F m ) → K * (X m ) to the Borel Gequivariant K-theory K * (X m ) = K * (EG × G F m ) of F m , inducing an isomorphism upon rationalization (see Proposition 5.8):
K * (X m ) Q ∼ = H * G (F m , Q) ∼ = Q m (W ) .
In this way, we link topologically the exponential and the usual (polynomial) quasi-invariants of W . In Section 5, we also compute the K-theory of 'fake' spaces of quasi-invariants associated to the Rector-Yau loop spaces ΩB (see Theorem 5.10). The result of this computation has an important consequence -Corollary 5.12 -that provides a numerical K-theoretic invariant N B distinguishing the spaces X m (ΩB, T ) up to homotopy equivalence for different B's.
In Section 6, we compute the T -equivariant Ell * T (F m ) and G-equivariant Ell * G (F m ) complex analytic elliptic cohomology of F m (see Theorem 6.3 and Theorem 6.6, respectively). We express the result in two ways: geometrically (as coherent sheaves on a given Tate elliptic curve E) and analytically (in terms of Θ-functions and q-difference equations). We also compute the spaces (graded modules) of global sections of the elliptic cohomology sheaves of F m with twisted coefficients:
Ell * T (E, L) := ∞ n=0 H 0 an (E, Ell * T (F m ) ⊗ L n ) and Ell * G (E, L) := Ell * T (E, L) W ,
where L n stands for the n-th tensor power of the Looijenga bundle L on the elliptic curve E, a canonical W -equivariant line bundle originally introduced and studied in [Loo77]. This computation (see Theorem 6.7) is inspired by results of [Gan14], and technically, it is perhaps the most interesting cohomological computation of the paper. Finally, in Section 7, we prove that our spaces of quasi-invariants X m (G, T ) are Gorenstein in the sense of stable homotopy theory: more precisely, the associated commutative ring spectra C * (X m , k) (see (1.4)) are orientable Gorenstein (relative to k) and satisfies the Gorenstein duality of shift a = 1 − 4m (see Theorem 7.1). This result should be viewed as a homotopy-theoretic analogue of Theorem 2.3 on Gorenstein property of algebras of quasi-invariants. We also prove the analogous result (see Theorem 7.2) for the 'fake' spaces of quasi-invariants X m (ΩB, T ), although under the additional assumption that k = F p for some prime p.
This work brings together ideas and techniques from parts of algebra and topology that are (still) fairly distant from each other. To make it accessible to readers with different background we included two appendices. In Appendix A, we briefly review Milnor's classical construction of classifying spaces of topological groups in terms of iterated joins. As it should be clear from results of Section 3, our construction of spaces of quasi-invariants can be viewed as a generalization of Milnor's construction. In Appendix B, we collect basic definitions from stable homotopy theory concerning regularity and duality properties of commutative ring spectra. This material is needed to understand our motivation and results in Section 7 that were greatly inspired by the beautiful paper [DGI06]. All in all, we tried to give references to all essential facts that we are using, even when these facts are considered to be obvious or well known by experts.
Acknowledgements. We would like to thank Oleg Chalykh and Pavel Etingof for many interesting discussions, questions and comments related to the subject of this paper. We are particularly grateful to O. Chalykh for clarifying to us his definition of quasi-invariants in the elliptic case (see Remark 6.8). The work of the first author was partially supported by NSF grant DMS 1702372 and the Simons Collaboration Grant 712995. The second author was partially supported by NSF grant DMS 1702323.
Realization problem
In this section, we state our topological realization problem for algebras of quasi-invariant polynomials of Weyl groups in terms of classifying spaces of compact connected Lie groups.
2.1. Quasi-invariants of finite reflection groups. We recall the general definition of quasiinvariants from [BC11]. Let V be a finite-dimensional vector space over C, and let W be a finite subgroup of GL(V ) generated by pseudoreflections. We recall that an element s ∈ GL(V ) is a pseudoreflection if it has finite order n s > 1 and acts as the identity on some hyperplane H s in V . We let A = {H s } denote the set of all hyperplanes corresponding to the pseudoreflections of W and observe that W acts naturally on A by permutations. The (pointwise) stabilizer W H of each H ∈ A in W is a cyclic subgroup of order n H ≥ 2 that depends only on the orbit of H in A. The characters of W H then also form a cyclic group of order n H generated by the determinant character det :
GL(V ) → C * of GL(V ) restricted to W H . We write e H, i := 1 n H w∈W H (det w) −i w , i = 0, 1, . . . , n H − 1 ,
for the corresponding idempotents in the group algebra CW H ⊆ CW . In general, Q(W ) is not an algebra: for arbitrary W and m ∈ M(W ), the subspace of quasiinvariant polynomials may not be closed under multiplication in C[V ]. In Part II of our work, we will give necessary and sufficient conditions (on W and m) that ensure the multiplicativity property of Q m (W ). In the present paper, we simply restrict our attention to Coxeter groups, i.e. the finite subgroups W of GL(V ) generated by real reflections. In this case the conditions (2.2) are equivalent to (1.2) and the above definition of quasi-invariants reduces to the original definition of Chalykh and Veselov [CV90] given in the Introduction.
Now, let C[V ] = Sym C (V * )
Thus, from now on, we assume that W is a real finite reflection group, V being its (complexified) reflection representation.
The next lemma collects some elementary properties of quasi-invariants that follow easily from the definition (see, e.g., [BEG03]).
Lemma 2.2. Let W be an arbitrary Coxeter group. Then, for any m ∈ M(W ) ,
(1)
C[V ] W ⊂ Q m (W ) ⊆ C[V ] with Q 0 (W ) = C[V ] and ∩ m Q m (W ) = C[V ] W .
(2) Q m (W ) is a graded subalgebra of C[V ] stable under the action of W .
(3) Q m (W ) is a finite module over C[V ] W and hence a finitely generated C-subalgebra of C[V ].
We may think of quasi-invariants of W as a family of subalgebras of C[V ] interpolating between the W -invariants and all polynomials. To make this more precise we will identify the set M(W ) of multiplicities on A with the set of W -invariant functions m : A → Z + and put on this set the following natural partial order 1 :
m ′ ≥ m def ⇐⇒ m ′ α ≥ m α , ∀ α ∈ A ,
The algebras of W -quasi-invariants of varying multiplicities then form a contravariant diagram of shape M(W ) -a functor M(W ) op → CommAlg C with values in the category of commutative algebras -that we simply depict as a filtration on C[V ]:
(2.3) C[V ] = Q 0 (W ) ⊇ . . . ⊇ Q m (W ) ⊇ Q m ′ (W ) ⊇ . . . ⊇ C[V ] W
The most interesting algebraic property of quasi-invariants is given by the following theorem, the proof of which (unlike the proof of Lemma 2.2) is not elementary.
Theorem 2.3 (see [EG02], [BEG03], [FV02]). For any Coxeter group W and any multiplicity
m ∈ M(W ), Q m (W ) is a free module over C[V ] W of rank |W |. Moreover, Q m (W ) is a graded Gorenstein algebra with Gorenstein shift a = dim(V ) − 2 α∈A m α .
Remark 2.4. For m = 0 (i.e., for the polynomial ring Q 0 (W ) = C[V ]), Theorem 2.3 is a wellknown result due to C. Chevalley [Che55]. For m = 0, it was first proven in the case of dihedral groups (i.e. Coxeter groups of rank 2) in [FV02]. For arbitrary Coxeter W , Theorem 2.3 was proven (by different methods) in [EG02] and [BEG03]. It is worth mentioning that the classical arguments of [Che55] do not work for nonzero m's.
Remark 2.5. The first statement of Theorem 2.3 makes sense and holds true for an arbitrary finite pseudoreflection group W and for all multiplicities. In this generality, Theorem 2.3 was proven in [BC11] (see, loc. cit., Theorem 1.1). However, for W non-Coxeter, the module Q m (W ) may not be Gorenstein even when it is an algebra.
2.2.
Varieties of quasi-invariants. The algebraic properties of quasi-invariants can be recast geometrically. To this end, following [BEG03], we introduce the affine schemes V m (W ) := Spec Q m (W ) called the varieties of quasi-invariants of W . The schemes V m (W ) come equipped with natural projections p m : V m (W ) → V //W and form a covariant diagram (tower) over the poset M(W ):
(2.4) V = V 0 (W ) → . . . → V m (W ) π m,m ′ − −−− → V m ′ (W ) → . . .
that is dual to (2.3). The following formal properties of (2.4) hold:
(1) Each V m (W ) is a reduced irreducible scheme (of finite type over C) equipped with an algebraic W -action, all morphisms in (2.4) being W -equivariant. The morphism p 0 : V 0 (W ) → V //W coincides with the canonical projection p : V → V //W , and the triangles
V m (W ) π m,m ′ ✲ V m ′ (W ) V //W p m ′ ✛ p m ✲
commute for all m ′ ≥ m. Thus, (2.4) is a diagram of W -schemes over V //W .
1 Abusing notation, in the Coxeter case, we will often write α ∈ A instead of H ∈ A for H = Ker(α).
(2) The diagram (2.4) 'converges' to V //W in the sense that the maps p m induce (4) Each map π m,m ′ : V m → V m ′ in (2.4) is a universal homeomorphism: i.e., a finite morphism of schemes that is surjective and set-theoretically injective on closed points.
colim M alg (W ) [V m (W )] ∼ → V //W .
Remark 2.6. The first three properties in the above list are formal consequences of Lemma 2.2. In contrast, Property (4) is a nontrivial geometric fact that does not follow immediately from definitions (see [BEG03,Lemma 7.3]). We recall that a morphism of schemes f : S → T is called a universal homeomorphism if for every morphism T ′ → T the pullback map T ′ × T S → T ′ is a homeomorphism in the category of schemes. For a map of algebraic varieties f : S → T defined over C, this categorical property is known to be equivalent to the geometric property (4).
We will construct a topological analogue of the diagram (2.4), where the schemes V m (W ) are replaced by topological spaces X m (G, T ), with Properties (1)-(3) holding in a homotopy meaningful (i.e. homotopy invariant) way. The universal homeomorphisms in the category of schemes will be modeled homotopy theoretically by the classical fibre-cofibre construction.
2.3. Borel Theorem. Next, we recall a fundamental result of A. Borel on cohomology of classifying spaces of compact Lie groups [Bor53]. Let G be a compact connected Lie group. Theorem 2.7 (Borel). The map p * : H * (BG, Q) → H * (BT, Q) induced by p on rational cohomology is an injective ring homomorphism whose image is precisely the subring of W invariants in H * (BT, Q) :
(2.5) H * (BG, Q) ∼ = H * (BT, Q) W .
In fact, more is true. Let V := π 1 (T ) ⊗ Q, which is Q-vector space of dimension n = rank(G). The natural action of W on T induces a group homomorphism W → Aut[π 1 (T )] that extends by linearity to a group homomorphism
(2.6) ̺ : W → GL Q (V ) .
The latter is known to be faithful, with image being a reflection subgroup of GL Q (V ) (see, e.g., [DW98,Theorem 5.16]). Furthermore, since T is a connected topological group, there is a natural isomorphism π 1 (T ) ∼ = π 2 (BT ) induced by the homotopy equivalence T ∼ → ΩBT ; combining this with the rational Hurewicz isomorphism π 2 (BT ) ⊗ Q ∼ = H 2 (BT, Q), we get a natural isomorphism of Q-vector spaces
(2.7) V ∼ = H 2 (BT, Q)
which shows that H 2 (BT, Q) carries a reflection representation of W as a Coxeter group. Dualizing (2.7) gives an isomorphism (2.8) H 2 (BT, Q) ∼ = V * which extends to an isomorphism of graded Q-algebras
(2.9) H * (BT, Q) ∼ = Sym Q (V * ) = Q[V ]
where the linear forms on V (covectors in V * ) are given cohomological degree 2 (in agreement with (2.8)). Borel's Theorem 2.7 thus identifies H * (BG, Q) with the ring Q[V ] W of polynomial invariants on the (rational) reflection representation of W .
We are now in a position to state our main problem -the realization problem for algebras of quasi-invariants of Weyl groups -in an axiomatic way.
2.4. Realization problem. Given a compact connected Lie group G with maximal torus T ⊆ G and associated Weyl group W = W G (T ), construct a diagram of spaces X m (G, T ) over the poset M(W ):
(2.10) BT = X 0 (G, T ) → . . . → X m (G, T ) π m,m ′ − −−− → X m ′ (G, T ) → . . .
together with natural maps p m : X m (G, T ) → BG, one for each m ∈ M(W ), such that
(QI 1 ) Each X m (G, T ) is a W -space (i.e.
, a CW complex equipped with an action of W ), and all maps are W -equivariant. The map p 0 : X 0 (G, T ) → BG coincides with the canonical map p : BT → BG, and for all m ′ ≥ m , the following diagrams commute up to homotopy:
X m (G, T ) π m,m ′ ✲ X m ′ (G, T ) BG p m ′ ✛ p m ✲
Thus, (2.10) is a diagram of W -spaces over BG.
(QI 2 ) The diagram (2.10) 'converges' to BG in the sense that the maps p m induce a weak homotopy equivalence of spaces:
hocolim M(W ) [X m (G, T )] ∼ → BG .
(QI 3 ) Each map p m : X m (G, T ) → BG factors naturally (in m) through the fibre inclusion into the space X m (G, T ) hW of homotopy orbits of the action of W on X m (G, T ):
X m (G, T ) p m ✲ BG X m (G, T ) hWH * (X m , Q) ⊗ C ∼ = Q m (W )
where Q m (W ) are the subalgebras of quasi-invariants in C[V ].
Remark 2.8. The first three properties of the spaces X m (G, T ) are homotopy-theoretic analogues of the corresponding geometric properties of the varieties V m (W ) listed in Section 2.2. Properties (QI 4 ) and (QI 5 ) reflect the fact that the diagram (2.10) is a topological realization of the diagram of algebras (2.3): in particular, the maps π * m,m ′ in (QI 4 ) induced by the cohomology functor correspond to the natural inclusions (2.3) of algebras Q m (W ) determined by their multiplicities.
Remark 2.9. The spaces X m (G, T ) will arise naturally as homotopy G-orbit spaces
X m (G, T ) = EG × G F m (G, T ) ,
where F m (G, T ) are the homotopy fibres of the maps p m : X m (G, T ) → BG (see Theorem 3.9). These homotopy fibres form a diagram of G-spaces
G/T = F 0 (G, T ) → . . . → F m (G, T ) → F m ′ (G, T ) → . . .
that induces the diagram (2.10). We will call F m (G, T ) the quasi-flag manifolds of the group G.
Spaces of quasi-invariants
In this section, we give a solution of our Realization Problem (see Section 2.4) in the rank one case. Our main observation (see Proposition 3.7 and Theorem 3.9) is that, for G = SU (2), the diagram of spaces (2.10) satisfying all five axioms (QI 1 )-(QI 5 ) can be obtained inductively, using the so-called 'fibre-cofibre construction' introduced in homotopy theory by T. Ganea [Gan65].
3.1. Ganea construction. First, we recall some basic definitions from topology. If f : X → Y is a map of (well) pointed spaces, its homotopy fibre is defined by
hofib * (f ) := X × Y P * (Y ) = {(x, γ) : γ(0) = * , γ(1) = f (x)} ,
where P * (Y ) := Map * (I, Y ) = {γ : I → Y , γ(0) = * } is the (based) path space over Y . Any map f : X → Y can be replaced by a fibration in the sense that it admits a factorization X ∼ → X ′ p ։ Y in Top * , where the first arrow is a weak homotopy equivalence and the second is a (Serre) fibration. The homotopy fibre is a homotopy invariant in Top * so that the pullback along a weak equivalence X ∼ → X ′ induces a weak equivalence: hofib * (f ) ∼ → hofib * (p) . On the other hand, for any fibration p : X ′ ։ Y , the natural inclusion map
p −1 ( * ) ∼ → hofib * (p) , x → (x, * )
is a (based) homotopy equivalence. Thus, the homotopy fibres of fibrations can be represented in Ho(Top * ) by usual (set-theoretic) fibres.
Dually, the homotopy cofibre of a map f : X → Y is defined by
hocof * (f ) := Y ∪ X C * (X) , where C * (X) := (X × I)/({ * } × I) ∪ (X × {1}) is the reduced cone on X. Any map f : X → Y
can be replaced by a cofibration in the sense that it admits a factorization X j ֒→ Y ′ ∼ → Y in Top * , where the first arrow is a cofibration (i.e., an injective map) in Top * and the second is (weak) homotopy equivalence. The homotopy cofibre is a homotopy invariant so that the pushout along the homotopy equivalence Y ′ ∼ → Y induces an equivalence: hocof * (j) ∼ → hocof * (f ) . On the other hand, for a cofibration j : X ֒→ Y ′ , the homotopy cofibre hocof * (j) is simply obtained by erecting the cone C * (Im j) on the image of j in Y ′ . The natural map collapsing this cone to the basepoint gives then a natural map hocof * (j) ∼ = Y ′ ∪ C * (Im j) ∼ → Y ′ /X which is a (based) homotopy equivalence. Thus, the homotopy cofibres of cofibrations can be represented in Ho(Top * ) by usual (set-theoretic) cofibres.
Formally, hofib * (f ) and hocof * (f ) can be defined in Ho(Top * ) by the following homotopy limit and homotopy colimit:
(3.1) hofib * (f ) = holim{ * → Y f ← − X} , hocof * (f ) = hocolim{ * ← X f − → Y }
The advantage of these formal definitions is that they make sense in any homotopical context: in particular, in an arbitrary pointed model category or ∞-category. Now, the Ganea construction starts with a homotopy fibration sequence with a well-pointed base
(3.2) F j − → X p − → B
and produces another homotopy fibration sequence on the same base:
(3.3) F 1 j 1 − → X 1 p 1 − → B
The space X 1 in (3.3) is defined to be the homotopy cofibre of the fibre inclusion in (3.2): X 1 := hocof * (j) . The map p 1 -called the (first) whisker map -is obtained by extending p : X → B to X 1 = X ∪ C * (F ) so that C * (F ) maps to the basepoint of B. The F 1 is then defined to be the homotopy fibre of p 1 : F 1 := hofib * (p 1 ). The above construction can be iterated ad infinitum; as a result, one gets a tower of fibration sequences over B:
(3.4) F j ✲ X p ✲ B F 1 ❄ j 1 ✲ X 1 π 0 ❄ p 1 ✲ B F 2 ❄ j 2 ✲ X 2 π 1 ❄ p 2 ✲ B . . . ❄ . . . ❄ . . .
where X m and F m are defined by (2) The whisker maps p m : X m → B induce a weak homotopy equivalence
hocolim {X π 0 − → X 1 π 1 − → X 2 → . . . → X m → . . .} ∼ → B
where the homotopy colimit is taken over the telescope diagram in the middle of (3.4).
Note that the second claim of Theorem 3.1 follows from the first by Milnor's Lemma (see A.2).
P * B π 0 − → (P * B) 1 π 1 − → (P * B) 2 π 2 − → (P * B) 3 → . . . called the Ganea tower of the space B.
The main theorem of [Gan67] asserts that if B is a normal space, its LS category cat(B) ≤ m if and only if the m-th whisker map p m : (P * B) m → B associated to the above tower splits (i.e., admits a section). Most applications of Ganea's construction in topology are related to or inspired by this observation (see, e.g., [CLOT03]).
Example 3.3 (Milnor bundles). If G is a topological group, we can apply the Ganea construction to the universal principal G-fibration G → EG → BG . In this case, the diagram (3.4) reads
G ✲ EG ✲ BG E 1 G ❄ ✲ B 1 G ❄ ✲ BG E 2 G ❄ ✲ B 2 G ❄ ✲ BG . . . ❄ . . . ❄ . . .
where E n G := G * (n+1) is the join of (n + 1) copies of the group G. The group G acts freely on E n G and hence B n G ≃ E n G/G. The induced fibration ΩBG → E n G → B n G associated to the Ganea fibration at the n-th step of the above tower is thus equivalent to Milnor's n-universal principal G-bundle G → E n G → B n G . We review the properties of such bundles in Appendix A Note that this example can be viewed as a special case of Example 3.2 if we take B = BG.
3.2.
Derived schemes of quasi-invariants. The fibre-cofibre construction is essentially formal: it can be performed in an arbitrary (pointed) model category or ∞-category. To see why this construction is relevant to our problem we will apply it first in a simple algebraic model category: the category dAff k, * of pointed derived affine schemes over a field k of characteristic 0. As a model for dAff k, * , we take the category (DGCA k ↓ k) op dual to the category of non-negatively graded commutative DG k-algebras A equipped with augmentation map A → k. Extending the standard algebro-geometric notation, we write Spec(A) for the object (affine DG scheme) in dAff k corresponding to the DG algebra A in DGCA k . Since we assume that char(k) = 0, the category DGCA k carries a natural (projective) model structure, where weak equivalences are the quasi-isomorphisms of DG algebras and fibrations are the DG algebra maps which are surjective in positive (homological) degrees (see, e.g., [BKR13,Appendix B]). The category dAff k = DGCA op k is equipped with the dual (injective) model structure. The homotopy (co)fibres of morphisms in dAff k are defined in terms of homotopy (co)limits, using formulas (3.1). Explicitly, given a morphism of pointed affine DG schemes f : Spec(A) → Spec(B) corresponding to a DG algebra homomorphism f * : B → A, its homotopy fibre and homotopy cofibre are given by
(3.6) hofib * (f ) ∼ = Spec A ⊗ L B k , hocof * (f ) ∼ = Spec B × R A k where ⊗ L B and × R
A denote the derived tensor product (homotopy pushout) and the derived direct product (homotopy pullback) in the model category DGCA k .
We apply the fibre-cofibre construction in the category dAff k, * to the canonical (algebrogeometric) quotient map p : V ։ V //W in the situation of the following simple example.
Example 3.4. Let W = Z/2Z, acting in its one-dimensional reflection representation V . Choos- ing a basis vector in V , we can identify V = C and k[V ] = k[x], with W acting on k[x] by the rule s(p)(x) = p(−x). In this case, A = {0} and m is a non-negative integer. Condition (1.2) says that p(x) is a quasi-invariant of multiplicity m iff p(x) − p(−x) is divisible by x 2m . Hence Q m (W ) is spanned by the monomials {x 2i : i ≥ 0} and {x 2i+1 : i ≥ m}, or equivalently Q m (W ) = k[x 2 ] ⊕ x 2m+1 k[x 2 ] = k[x 2 , x 2m+1 ] .
Thus, we take V to be the affine line acted upon by W = Z/2Z via the reflection at 0. Regarding V ∼ = Spec k[x] and V //W ∼ = Spec k[x 2 ] as affine (DG) schemes pointed at 0, we can compute the homotopy fibre F := hofib * (p) in dAff k, * , using formula (3.6):
(3.7) F ∼ = Spec k[x] ⊗ L k[x 2 ] k ∼ = Spec k[x] ⊗ k[x 2 ] k ∼ = Spec(k[x]/x 2 )
. Note that the second isomorphism in (3.7) is due to the fact that k[x] is a free module (and hence, a flat algebra) over k[x 2 ]. Thus, in dAff k, * , we have the fibration sequence
(3.8) F j − → V p − → V //W
where F is given by (3.7). The following simple observation, which was the starting point of the present paper, provides a motivation for our topological results in the next section.
Proposition 3.5. The fibre-cofibre construction in dAff k, * applied to the fibration (3.8) produces the tower (2.4) of varieties of quasi-invariants for the reflection representation of W = Z/2Z :
(3.9) F j ✲ V p ✲ V //W F 1 ❄ j 1 ✲ V 1 π 0 ❄ p 1 ✲ V //W F 2 ❄ j 2 ✲ V 2 π 1 ❄ p 2 ✲ V //W . . . ❄ . . . ❄ . . .
Thus, for all m ≥ 0, we have
(3.10) V m ∼ = Spec(Q m ) , F m ∼ = Spec [Q m /(x 2 )] ,
where Q m = k[x 2 , x 2m+1 ] and the maps π m , p m and j m in (3.9) correspond to the natural inclu-
sions Q m+1 ֒→ Q m , k[x 2 ] ֒→ Q m , and the projection Q m ։ Q m /(x 2 ), respectively.
Proof. The proof is an easy induction in m. For m = 0, we have already shown in (3.7) that F = F 0 , with (3.8) corresponding (i.e. dual) to the natural algebra maps k[
x 2 ] ֒→ k[x] ։ k[x]/(x 2 )
. Now, assuming that V m is given by (3.10) together with p m : V m ։ V //W corresponding to the inclusion k[x 2 ] ֒→ Q m , we compute the fibre F m in the same way as in (3.7), using formula (3.6):
F m := hofib * (p m ) ∼ = Spec Q m ⊗ L k[x 2 ] k ∼ = Spec Q m ⊗ k[x 2 ] k ∼ = Spec [Q m /(x 2 )]
Again, crucial here is the fact that Q m is a free module (and hence, a flat algebra) over k[x] W , which is a general property of quasi-invariants (see Theorem 2.3). Next, we have
(3.11) V m+1 := hocof * (j m ) ∼ = Spec Q m × R Qm/(x 2 ) k ∼ = Spec Q m × Qm/(x 2 ) k ∼ = Spec(Q m+1 )
The first isomorphism in (3.11) is the result of formula (3.6) for homotopy cofibres in dAff k, * . The second isomorphism is due to the fact that the canonical map Q m ։ Q m /(x 2 ) is surjective, and hence a fibration in the standard model structure on DGCA k (this implies that hocof * (j m ) coincides with the usual cofibre of j m in the category of affine k-schemes). Finally, the last isomorphism in (3.11) is given by the composition of canonical algebra maps
(3.12) Q m × Qm/(x 2 ) k ֒→ Q m × k ։ Q m
It is easy to see that the map (3.12) is injective, and its image is precisely
Q m+1 = k[x 2 , x 2m+3 ].
This gives an identification Q m × Qm/(x 2 ) k ∼ = Q m+1 together with the inclusion Q m+1 ֒→ Q m that defines the morphism of schemes π m : V m → V m+1 .
Remark 3.6. Proposition 3.5 does not extend directly to higher rank groups: the standard fibrecofibre construction in dAff k, * does not produce the tower of varieties of quasi-invariants, (2.4), for an arbitrary Coxeter group W (cf. Proposition 3.7 below).
Spaces of quasi-invariants of SU (2). Let G be a compact connected Lie group with a fixed maximal torus T and Weyl group
W = W G (T ). Associated to (G, T ) there is a natural fibration sequence 3 (3.13) G/T j − → BT p − → BG ,
where p is the map induced by the inclusion T ֒→ G and j is the classifying map for the principal T -bundle G → G/T .
3 If we choose a model for the universal G-bundle EG (for example, the Milnor model described in Section A) and let BG = EG/G and BT = EG/T , then (3.13) is represented by a canonical locally trivial fibre bundle G/T → EG/T ։ EG/G (see, e.g., [Hus75], Chap 4, Sect. 7).
Proposition 3.7. Assume that W is simply-laced (i.e., of ADE type). Then the fibre-cofibre construction applied to (3.13) produces a tower of fibrations (3.14)
G/T j ✲ BT p ✲ BG F 1 (G, T ) ❄ j 1 ✲ X 1 (G, T ) π 0 ❄ p 1 ✲ BG F 2 (G, T ) ❄ j 2 ✲ X 2 (G, T ) π 1 ❄ p 2 ✲ BG . . . ❄ . . . ❄ . . .
where the diagram of spaces
(3.15) BT π 0 − → X 1 (G, T ) π 1 − → X 2 (G, T ) π 2 − → . . . → X m (G, T ) πm − − → . . .
together with maps p m : X m (G, T ) → BG satisfy the first three properties (QI 1 ), (QI 2 ) and (QI 3 ) of Section 2.4.
Proof. If W is simply-laced, all reflection hyperplanes of W are in the same orbit, and the poset M(W ) consists only of constant multiplicities which we identify with Z + . By Ganea's Theorem 3.1, the homotopy fibre F m = F m (G, T ) at stage m in (3.14) can be represented by the iterated join
(3.16) F m = G/T * ΩBG * m . . . * ΩBG ≃ G/T * G * m . . . * G = G/T * E m−1 G ,
where E m−1 G is Milnor's model for the (m − 1)-universal G-bundle (see Section A). The fibre (3.16) carries a natural left (holonomy) action ΩBG × F m → F m that under the identification (3.16), corresponds to the diagonal action of G :
(3.17) G × F m → F m , g · (t 0 (g 0 T ) + t 1 g 1 + . . . + t m g m ) = t 0 (gg 0 T ) + t 1 gg 1 + . . . + t m gg m
where g, g 0 , g 1 , . . . , g m ∈ G and (t 0 , . . . , t m ) ∈ ∆ m , see (A.2). The space X m = X m (G, T ) can then be represented as the homotopy quotient
(3.18) X m = (F m ) hG = EG × G (G/T * E m−1 G) and the fibration F m → X m → BG in (3.14) is identified with the Borel fibration (3.19) F m → (F m ) hG → BG Now, the Weyl group W = N G (T )/T acts on G/T by w·(gT ) = gn −1 T , where w = nT ∈ W .
With identification (3.16), this action naturally induces a W -action on F m = G/T * E m−1 G. The latter commutes with the G-action (3.17), and hence extends to the space X m of homotopy G-orbits in F m . Explicitly, with identification (3.18), the action of W on
X m = EG × G (G/T * E m−1 G) is given by (3.20) w · (x, t 0 (g 0 T ) + t 1 g 1 + . . . + t m g m ) = (x, t 0 (g 0 n −1 T ) + t 1 g 1 + . . . + t m g m )
where x ∈ EG and w = nT ∈ W . The inclusions F m ֒→ F m+1 defined by
t 0 (g 0 T ) + t 1 g 1 + . . . + t m g m → t 0 (g 0 T ) + t 1 g 1 + . . . + t m g m + 0 e
are obviously (G × W )-equivariant, hence induce W -equivariant maps on homotopy G-quotients: π m : X m → X m+1 . The whisker maps p m : X m → BG are induced by the trivial maps F m → pt and hence are W -invariant. Thus, we have established property (P1) for the tower (3.15). Property (P2) follows directly from part (2) of Theorem 3.1. For (P3), it suffices to show that
(3.21) H * W (F m , Q) ∼ = Q Indeed, since the actions of G and W on F m commute, we have (X m ) hW = EW × W (EG × G F m ) ≃ EG × G (EW × W F m ) = EG × G (F ) hW Whence (3.22) H * W (X m , Q) ∼ = H * G ((F m ) hW , Q)
On the other hand, if (3.21) holds, the Serre spectral sequence of the Borel fibration
(F m ) hW → EG × G (F m ) hW → BG degenerates, giving an isomorphism H * G ((F m ) hW , Q) ∼ = H * (BG, Q) . Combining this last isomor- phism with (3.22) yields H * W (X m , Q) ∼ = H * (BG, Q)
, as required by (QI 3 ). Now, since F m is connected, (3.21) is equivalent to vanishing of higher cohomology:
(3.23) H n W (F m , Q) = 0 ∀ n > 0 . We prove (3.23) by induction on m. For m = 0, we have F 0 = G/T and (G/T ) hW ≃ (G/T )/W ∼ = G/N , since the action of W on G/T is free. It follows that H n W (F 0 , Q) ∼ = H n (G/N, Q)
= 0 for all n > 0 as it is well known that the space G/N is rationally acyclic for any compact connected Lie group (see [Bor67,Theorem 20.3]). Now, assume that (3.23) holds for some m ≥ 0 and consider (F m+1 ) hW = (F m * G) hW . Representing this space by homotopy colimits (see (A.3)) and using the fact that the homotopy colimits commute, we have
(F m+1 ) hW ≃ hocolim W hocolim [ F m ← F m × G → G ] ≃ hocolim hocolim W [ F m ← F m × G → G ] ≃ hocolim [ (F m ) hW ← (F m × G) hW → (G) hW ] ≃ hocolim [ (F m ) hW ← (F m ) hW × G → BW × G ]
This homotopy decomposition implies that the cohomology groups of (F m+1 ) hW and (F m ) hW are related by the following Mayer-Vietoris type long exact sequence:
H n−1 [(F m ) hW × G] → H n [(F m+1 ) hW ] → H n [(F m ) hW ] ⊕ H n [BW × G] → H n [(F m ) hW × G]
Since W is a finite, its rational cohomology vanishes in positive degrees. Hence, by Künneth
Theorem, we have H * (BW × G, Q) ∼ = H * (G, Q). Furthermore, our induction assumption (3.23) implies that H * ((F m ) hW × G, Q) ∼ = H * (G, Q)
and for each n ≥ 1, the last map in the above exact sequence is an isomorphism. Thus, for n ≥ 2, the above sequence breaks up into short exact sequences
0 → H n ((F m+1 ) hW , Q) → H n (G, Q) ∼ − → H n (G, Q) → 0 which show that H n W (F m+1 , Q) = 0 for all n ≥ 2.
On the other hand, in dimension 0 and 1, the above long exact sequence reads
H 0 ((F m ) hW , Q) ⊕ H 0 (G, Q) ։ H 0 (G, Q) → H 1 ((F m+1 ) hW , Q) → H 1 (G, Q) ∼ − → H 1 (G, Q)
where the first arrow is surjective and the last is an isomorphism. This shows that H 1 W (F m+1 , Q) also vanishes, thus finishing the induction and the proof of (QI 3 ).
Example 3.8. Let us describe the cohomology H * (X 1 , Q) of the first space X 1 = X 1 (G, T ) in the diagram (3.15) explicitly. By general properties of the Ganea construction (see Section 3.1), this space fits in the homotopy cofibration sequence
(3.24) G/T j − → BT π 0 − → X 1
Since both BT and G/T have no cohomology classes in odd dimensions and the natural map
j * : H * (BT, Q) → H * (G/T, Q)
is surjective, the long cohomology sequence associated to (3.24) reduces to the short exact sequence
(3.25) 0 →H * (X 1 , Q) π * 0 − →H * (BT, Q) j * − →H * (G/T, Q) → 0
whereH * stands for the reduced cohomology. Since X 1 is connected, (3.25) shows that the algebra map π * 0 :
H * (X 1 , Q) → H * (BT, Q) is injective, and with identification H * (BT, Q) ∼ = Q[V ] (as in (2.9)), its image being (3.26) H * (X 1 , Q) ∼ = Q + Q[V ] W + ⊂ Q[V ] , where Q[V ] W + is the ideal in Q[V ]
generated by the W -invariant polynomials of positive degrees. Formula (3.26) shows that X 1 has no odd cohomology; moreover, the map p * 1 :
H * (BG, Q) → H * (X 1 , Q) induced by the first whisker map in (3.14) is injective, and H * (X 1 , Q) is a finite module over H * (BG, Q) ∼ = Q[V ] W via p * 1 .
By Hilbert-Noether Theorem, this implies that H * (X 1 , Q) is a finitely generated graded Q-algebra, however it is not Cohen-Macaulay (and hence not Gorenstein) when dim Q (V ) ≥ 2. To see this we set R := H * (X 1 , Q), S := H * (BT, Q) and S W = H * (BG, Q) to simplify the notation. Since S is a free S W -module, the long exact sequence obtained by dualizing the short exact sequence 0 → R → S → S/R → 0 over S W yields
Ext i S W (R, S W ) ∼ = Ext i+1 S W (R/S, S W ) , ∀ i ≥ 1 Since R/S ∼ =H * (G/T, Q) by (3.25), dim Q (R/S) = |W | − 1 < ∞ . Hence Ext n S W (R/S, S W ) = 0 and therefore Ext n−1 S W (R, S W ) = 0 , where n := dim Q (V )
. It follows that when n > 1 , R is not free as a graded module over S W , and hence not Cohen-Macaulay as a graded algebra (see, e.g., [Smo72,Prop. 6.8]).
Example 3.8 shows that, unfortunately, the tower of spaces (3.15) constructed in Proposition 3.7 cannot satisfy all five axioms of our realization problem for an arbitrary compact Lie group. Indeed, if rk(G) = n ≥ 2, then (QI 5 ) already fails for
H * (X 1 (G, T ), Q), since H * (X 1 (G, T ), Q)
is not a Gorenstein algebra, while Q 1 (W ) is (see Theorem 2.3). Note, however, that in the rank one case, for G = SU (2), we still have H * (X 1 (G, T ), Q) ∼ = Q 1 (Z/2Z) by formula (3.26). The next theorem shows that this is not a coincidence.
Theorem 3.9. Assume that G = SU (2) and W = Z/2Z. Then the diagram of spaces (3.15) together with whisker maps p m produced by the fibre-cofibre construction satisfies all five properties (QI 1 )-(QI 5 ) of Section 2.4. In particular, for all m ≥ 0, there are isomorphisms of graded commutative algebras
(3.27) H * (X m (G, T ), Q) ∼ = Q m (W ) ,
where Q m (W ) is the subring of W -quasi-invariants of multiplicity m in Q[V ]. Moreover, X m (G, T ) are unique, up to rational homotopy equivalence, topological spaces realizing the algebras Q m (W ).
Proof. Properties (QI 1 )-(QI 3 ) have already been established for arbitrary G in Proposition 3.7; we need only to check (QI 4 ) and (QI 5 ). As a topological space, SU (2) is homeomorphic to S 3 and G/T = CP 1 ∼ = S 2 . Hence, applying a well-known formula for the join of spheres, we can identify the fibre (3.16):
(3.28)
F m = G/T * G * m ∼ = S 2 * (S 3 ) * m ∼ = S 4m+2 .
Thus, for G = SU (2), (3.19) is equivalent to the sphere fibration: S 4m+2 → X m → BS 3 . We will look at the Serre spectral sequence of this fibration and apply the Leray-Hirsch Theorem.
Since both the basespace and the fibre of (3.19) have no cohomology in odd dimensions, the Serre spectral sequence collapses, giving an isomorphism of graded vector spaces (see, e.g., [MT91,
Lemma III.4.5(1)]) H * (X m , Q) ∼ = H * (BG, Q) ⊗ H * (F m , Q) Then, the Leray-Hirsch Theorem (see, e.g., [MT91, Theorem III.4.2]) implies that H * (X m , Q)
is a free graded module over the algebra H * (BG, Q) = H * (BSU (2), Q), which is the rational polynomial algebra Q[c 2 ] generated by the second Chern class c 2 ∈ H 4 (BSU (2), Q) . This graded module has rank two, with
H * (BG, Q) identified with a direct summand in H * (X m , Q) under the whisker map p * m : H * (BG, Q) ֒→ H * (X m , Q). The complement of H * (BG, Q) in H * (X m , Q)
is generated by a cohomology class ξ of dimension 4m + 2 whose image under the projection
j * m : H * (X m , Q) → H * (F m , Q) ∼ = H * (S 4m+2 , Q) is the fundamental cohomology class of S 4m+2 . Thus, we have (3.29) H * (X m , Q) ∼ = Q[c 2 ] ⊕ Q[c 2 ]ξ
where |c 2 | = 4 and |ξ| = 4m + 2 . Next, we look at the homotopy cofibration sequence in (3.14)
(3.30) F m jm − − → X m πm − − → X m+1
arising from the Ganea construction. This gives a long exact sequence on (reduced) cohomology:
(3.31) . . . →H n−1 (F m , Q) →H n (X m+1 , Q) π * m − − →H n (X m , Q) j * m − − →H n (F m , Q) → . . .
Since neither F m nor X m (by (3.29)) have odd cohomology, we see immediately from (3.31) that all algebra maps π * m must be injective, i.e. property (QI 4 ) holds for (3.15). For each m ≥ 0, the composition of these maps then gives an embedding
(3.32) π * 0 π * 1 . . . π * m−1 : H * (X m , Q) ֒→ H * (X m−1 , Q) ֒→ . . . ֒→ H * (BT, Q) If we identify H * (BT, Q) = Q[x]
by choosing x ∈ H 2 (BT, Q) = H 2 (BS 1 , Q) to be the universal Euler class, which is the image of the canonical generator of H 2 (BS 1 , Z) = H 2 (K(Z, 2), Z), then the Chern class c 2 ∈ H 4 (BG, Q) maps by (3.32) to x 2 ∈ H * (BT, Q). Then, for degree reasons, the
generator ξ ∈ H 4m+2 (X m , Q) in (3.29) should map to (a scalar multiple of) x 2m+1 ∈ Q[x]. Thus the algebra homomorphism (3.32) identifies H * (X m , Q) ∼ = Q[x 2 , x 2m+1 ], which is precisely the subring Q m of W -quasi-invariants in H * (BT, Q) = Q[x]
. This gives property (QI 5 ) and completes the proof of the first part of the theorem.
The last claim of the theorem follows from Sullivan's formality theorem [Sul77]. Indeed, the algebras Q m (W ) have the presentation Q[ξ, η]/(ξ 2 − η 2m+1 ), where |η| = 4 and |ξ| = 4m + 2 (see Example 3.4). Hence, by [Sul77,Remark (v), p. 317], they are intrinsically formal. This means that, for each m ≥ 0, there is only one rational homotopy type that realizes Q m .
From now on, we will assume that G = SU (2) and T = U (1) embedded in SU (2) in the standard way as a maximal torus.
Definition 3.10. We call the G-space F m (G, T ) := G/T * E m−1 G the m-quasi-flag manifold and the associated homotopy quotient
X m (G, T ) := F m (G, T ) hG = EG × G (G/T * E m−1 G)
the space of m-quasi-invariants for G = SU (2). These spaces fit in the Borel fibration sequence
(3.33) F m (G, T ) jm − − → X m (G, T ) pm − − → BG
that generalizes the fundamental sequence (3.13).
Remark 3.11. By definition,
H * (X m (G, T ), Q) = H * G (F m (G, T ), Q) for all m ≥ 0.
With this identification, the algebra homomorphisms H * (X m , Q) → H * (BT, Q) constructed in Theorem 3.9 (see (3.32)) are induced (on G-equivariant cohomology) by the natural inclusion maps
(3.34) i 0 : G/T ֒→ F m (G, T ) , gT → 1 · (gT ) + 0 · x ,
where x ∈ E m−1 G . Note that the maps (3.34) are null-homotopic in the category Top of ordinary spaces, the null homotopy being i t : gT → (1−t)·(gT )+t·x ; however, they are not null-homotopic in the category of G-spaces and G-equivariant maps. In fact, the proof of Theorem 3.9 shows that the maps induced by (3.34) on G-equivariant cohomology are injective and hence nontrivial.
3.4. T -equivariant cohomology. Our next goal is to compute the T -equivariant cohomology of the G-spaces F m (G, T ) by restricting the G-action to the maximal torus T ⊂ G. The computation is based on the following simple observations.
Lemma 3.12. For all m ≥ 0, there is a natural T -equivariant homeomorphism (3.35) F m (G, T ) ∼ = Σ E 2m (T ) ,
where Σ stands for the unreduced suspension in Top.
Proof. First, note that G is T -equivariantly homeomorphic to the (unreduced) join of two copies of T : the required homeomorphism
(3.38) F m (G, T ) = (G/T ) * G * m ∼ = (G/T ) T * T * (2m+1) = S 0 * E 2m (T )
which is equivalent to formula (3.35).
Lemma 3.13. For all n ≥ 0, there are natural algebra isomorphisms
(3.39) H * T (ΣE n (T ) , Q) ∼ = Q[x] × Q[x]/(x n+1 ) Q[x] . Proof. We compute [ΣE n (T )] hT ≃ [hocolim( pt ← E n (T ) → pt)] hT ≃ hocolim(BT ← E n (T ) hT → BT ) (3.40) ≃ hocolim(BT ← B n (T ) → BT )
where the last equivalence follows from the fact that E n (T ) is an n-universal T -bundle, so that the T -action on E n (T ) is free and hence E n (T ) hT ≃ E n (T )/T = B n (T ) (see Section A). To complete the proof it remains to note that BT ≃ CP ∞ and B n (T ) ∼ = CP n for T = U (1), with natural map B n T → BT represented by the inclusion CP 2m ֒→ CP ∞ (see, e.g., [Sel97, Example 9.2.3]). Hence,
(3.40) shows that [Σ E n (T )] hT ≃ CP ∞
CP n CP ∞ , which, by Mayer-Vietoris sequence, yields the isomorphism (3.39).
As a consequence of Lemma 3.12 and Lemma 3.13, we get Proposition 3.14. For all multiplicities m ≥ 0, there are natural algebra isomorphisms
(3.41) H * T (F m (G, T ), Q) ∼ = Q[x] × Q[x]/(x 2m+1 ) Q[x] , where x ∈ H 2 (BT, Q) is the universal (rational) Euler class.
Remark 3.15. For m = 0, formula (3.41) is well known: it follows, for example, from a general combinatorial description of T -equivariant cohomology of equivariantly formal spaces in terms of moment graphs (see [GKM98]). In our subsequent paper, we will generalize the main localization theorem of [GKM98] to moment graphs with multiplicities, and as an application, extend the result of Proposition 3.14 to quasi-flag manifolds for an arbitrary compact connected Lie group.
Next, we recall the modules of CW -valued quasi-invariants, Q k (W ), introduced in [BC11]. In [BC11, Section 3.2], these modules are considered only for integral multiplicities k ∈ Z + ; however, their definition makes sense -in the Coxeter case -for all k ∈ 1 2 Z + (cf. [BC11, (3.8)]). We provide a natural topological interpretation of these modules.
Corollary 3.16. For all n ≥ 0, there are natural isomorphisms of Q[x] ⋊ W -modules (3.42) H * T (ΣE n (T ) , C) ∼ = Q n+1 2 (W ) .
In particular,
H * T (F m (G, T ), C) ∼ = Q m+ 1 2 (W ) for all m ≥ 0.
Proof. Under the isomorphism (3.39), the geometric action of W = Z/2Z on H * T (ΣE n (T ) , Q) corresponds to the action (p, q) → (s(q), s(p)) on the fiber product. Relative to this action, we can then define the W -equivariant map
f : Q[x] × Q[x]/(x n+1 ) Q[x] → Q[x] ⊗ QW , (p, q) → 1 2 (p + qs)
This map is obviously injective, and it is easy to see that its image is
Q[x]e 0 + Q[x]
x n+1 e 1 , where e 0 = (1 + s)/2 and e 1 = (1 − s)/2 are the idempotents in QW corresponding to the trivial and sign representations of W . Example 3.9 of [BC11] shows that Im(f ) is precisely Q n+1 2 (W ); thus, combining f with the isomorphism of Lemma 3.13 gives the required isomorphism (3.42). The last statement then follows from Proposition 3.14.
Remark 3.17. Recall that, for any compact connected Lie group G, there is a natural isomorphism
(3.43) H * G (X, Q) ∼ = H * T (X, Q) W thatH * T (F m (G, T ), C) W ∼ = e 0 Q m+ 1 2 (W ) ∼ = Q m (W ) .
Thus the isomorphism (3.27) of Theorem 3.9 can be deduced from (3.41) by (3.43).
3.5. Divided difference operators. As an application of Theorem 3.9, we give a topological construction of generalized divided difference operators associated with quasi-invariants. Recall that the classical divided difference operators ∆ α :
Q[V ] → Q[V ]
are attached to reflections s α ∈ W of a Coxeter group W by the rule (cf. [Dem73,Dem74]):
(3.44) (1 − s α )p = ∆ α (p) · α H
where α H ⊂ V * is a linear form vanishing on the reflection hyperplane H = H α . Note that (3.44) defines ∆ α uniquely up to a nonzero constant factor. The definition of quasi-invariants of Coxeter groups suggests the following natural generalization of (3.44):
(3.45) (1 − s α )p = ∆ (mα) α (p) · α 2mα+1
H To be precise, given a W -invariant multiplicity function m : A → Z + , α → m α , formulas (3.45) define unique (up to nonzero constants) linear maps
(3.46) ∆ (mα) α : Q m (W ) → Q 0 (W )
one for each reflection s α ∈ W . Note that Q 0 (W ) = Q[V ], and for m = 0, the maps (3.46) coincide with the classical divided difference operators: ∆
(0) α = ∆ α .
Definition 3.18. We call (3.46) the divided difference operators of W of multiplicity m.
When W has rank one, i.e. W is generated by a single reflection s, the corresponding map ∆ where p * m is the natural pullback map induced on cohomology by the m-th whisker map p m : X m → BG and (p m ) * is a 'wrong way' pushforward map called the Gysin homomorphism. Combining these last two maps, we get the graded linear endomorphism on H * (X m , Q) of degree −(4m + 2) :
(3.49) p * m • (p m ) * : H * (X m , Q) → H * (X m , Q)
The next proposition generalizes a well-known formula for the classical divided difference operators ∆ α (proven, for example, in [BE90]).
p * m (p m ) * (x 2k+2m+1 ) = p * m (p m ) * (x 2k · x 2m+1 ) = p * m (p m ) * (p * m (c k 2 ) · x 2m+1 ) = p * m (c k 2 ) · (p m ) * (x 2m+1 ) = κ m x 2k
Thus, up to a nonzero constant factor, we have
p * m (p m ) * (x N ) = 0 , if N = 2k x 2k , if N = 2k + 2m + 1 which agrees with the action of ∆ (m) s = 1 x 2m+1 (1 − s) on Q m (W ) = Q[x 2 , x 2m+1 ] .
'Fake' spaces of quasi-invariants
By Theorem 3.9, the spaces X m (G, T ) provide topological realizations for the algebras Q m (W ) that are unique up to rational equivalence. This raises the question whether the X m (G, T )'s are actually unique up to homotopy equivalence. In this section, we answer the above question in the negative by constructing a natural class of counterexamples related to finite loop spaces. These remarkable loop spaces -sometimes referred to as fake Lie groups -were originally constructed by D. L. Rector [Rec71a] as examples of nonstandard ('exotic') deloopings of S 3 . We will show that the rational cohomology rings of the spaces of quasi-invariants associated to Rector's spaces are isomorphic to the 'genuine' spaces of quasi-invariants X m (G, T ); however, the spaces themselves are not homotopy equivalent (in fact, as we will see in Section 5, they can be distinguished Ktheoretically). Thus, we get many different topological realizations of Q m (W ), but among these only the 'genuine' spaces of quasi-invariants X m (G, T ) satisfy all properties (QI 1 )-(QI 5 ).
4.1. Finite loop spaces. We recall the definition of a finite loop space which is a natural homotopy-theoretic generalization of a compact Lie group. An exposition of classical results as well as many interesting examples of finite loop spaces can be found in the monograph [Kan88]; for more recent developments, we refer to the survey papers [Not95], [Dwy98], and [Gro10].
Definition 4.1. A finite loop space is a pointed connected space B such that ΩB is homotopy equivalent to a finite CW-complex.
It is convenient to represent a finite loop space as a triple (X, B, e), where X is a finite CWcomplex, B is a pointed connected space, and e : X ∼ → ΩB is a homotopy equivalence. A prototypical example is (G, BG, e), where G is a compact Lie group, BG its classifying space, and e : G ∼ → ΩBG is a canonical equivalence. In general, finite loop spaces have many properties in common with compact Lie groups; however, the class of such spaces is much larger. In fact, if G is a compact connected non-abelian Lie group, there exist uncountably many homotopically distinct spaces B such that ΩB ≃ G; thus the underlying topological space of G carries uncountably many finite loop structures (see [M92]). In the case G = SU (2), this striking phenomenon was originally discovered by Rector [Rec71a] (see Theorem 4.2 below). 4.2. Fake Lie groups of type SU (2). We will work with localizations of topological spaces in the sense of D. Sullivan. A modern exposition of this classical construction can be found in [MP12]. Given a space X and a prime number p, we denote the localization of X at p by X (p) . Recall (cf. [MP12,8.5.1]) that two (nilpotent, finite type) spaces X and Y are said to be in the same genus if X (p) ≃ Y (p) for every prime p. We are interested in finite loop spaces B (see Definition 4.1) that are in the same genus as BG for some compact connected Lie group G. Such spaces (called fake Lie groups) have been studied extensively in the literature (see, e.g., [NS90]), since their original discovery in [Rec71a]. This last paper gave a complete homotopy classification of spaces in the genus of BG for G = SU (2), and proposed a simple criterion to distinguish the genuine BSU (2) among these spaces: more precisely, (2) Every combination of values of (B/p) can occur for some B. In particular, the genus of BG consists of uncountably many distinct homotopy types.
( Remark 4.4. It was a long-standing conjecture in homotopy theory (motivated in part by Theorem 4.2(4), cf. [Wil74]) that a finite loop space with a maximal torus is homotopy equivalent to the classifying space of a compact Lie group. This conjecture was eventually proved by Anderson and Grodal using the Classification Theorem of p-compact groups (see [AG09]). Thus, the existence of maximal tori provides a purely homotopy-theoretic characterization of compact Lie groups among finite loop spaces.
Even though the spaces B ≃ BG do not admit maximal tori, this does not rule out the possibility that there could exist interesting maps f : BT → B whose homotopy fibres are not finite CW complexes. In his thesis (see [Yau04]), D. Yau refined Rector's classification by describing the spaces B in the genus of BSU (2) that can occur as targets of essential (i.e., non-nullhomotopic) maps from BT . Such spaces admit a beautiful arithmetic characterization: (2) If B satisfies condition (1), then there exists a unique (up to homotopy) map p B : BT → B such that every essential map f : BT → B is homotopic to g • p B for some self-map g of B.
(3) For B = BG, the map p BG : BT → BG is induced by the maximal torus inclusion. 4 We say that a finite loop space B admits a maximal torus if there is a map p : BTn → B from the classifying space of a finite-dimensional torus with homotopy fibre being a finite CW-complex (see [Rec71b]). 5 Recall that, for a prime p, the Legendre symbol (k/p) of an integer k is defined whenever p ∤ k : for p odd, we have (k/p) = 1 (resp., −1) if k is a quadratic residue (resp., nonresidue) mod p, while for p = 2, (k/2) = 1 (resp., −1) if k is quadratic residue (resp. nonresidue) mod 8.
'Fake' spaces of quasi-invariants.
Let B be a space in the genus of BG (for G = SU (2)) that admits an essential map from BT . Theorem 4.5 shows that, for such a space, there is a natural generalization of the maximal torus: namely, the 'maximal' essential map p B : BT → B.
We let F (ΩB, T ) denote the homotopy fibre of this map and apply the Ganea construction to the associated fibration sequence:
(4.1)
F (ΩB, T ) j B ✲ BT p B ✲ B F 1 (ΩB, T ) ❄ j 1,B ✲ X 1 (ΩB, T ) π 0 ❄ p 1,B ✲ B F 2 (ΩB, T ) ❄ j 2,B ✲ X 2 (ΩB, T ) π 1 ❄ p 2,B ✲ B . . . ❄ . . . ❄ . . .
As a result, we construct a tower of spaces X m (ΩB, T ) which we will refer to as the 'fake' spaces of quasi-invariants associated to the Rector space B. Note, if B = BG, then ΩB ≃ G , and by Theorem 4.5(3), the map p B : BT → BG is the maximal torus inclusion; hence, in this case, X m (ΩB, T ) are equivalent to the 'genuine' spaces X m (G, T ) of quasi-invariants (see Definition 3.10).
To compute the cohomology of X m (ΩB, T ) we recall (cf. [Rec71a]) that any space B in the genus of BG can be represented as a (generalized) homotopy pullback:
(4.2) B = holim {p} {BG (p) r p ✲ BG (0) n p ✲ BG (0) } ,
where the indexing set {p} runs over all primes, r p denotes the natural map from the p-localization to the rationalization of BG, and the map n p is induced by multiplication by an integer n p which is relatively prime to p and such that (n p /p) = (B/p) for every p (for p = 2, one requires, in addition, that n p ≡ 1(mod 4)). Now, if a space B admits an essential map from BT , part (1) of Theorem 4.5 implies that the set of integers {n p ∈ Z : p prime} appearing in (4.2) can be chosen to be finite. Hence, for such spaces, we can define the natural number
H * (B, Q) ⊂ p * m,B ✲ H * (X m (ΩB, T ), Q) ⊂ π * m ✲ H * (BT, Q) H * (BG, Q) (p * BG ) −1 p * B ❄ ⊂ p * m ✲ H * (X m (G, T ), Q) ≀ ❄ ⊂π * m ✲ H * (BT, Q)
where the map (p * BG ) −1 p * B is given explicitly by u → N B x 2 (see Lemma 4.6). Proof. We prove part (i) by induction on m. First, note that for m = 0, (i) as well as (ii) follow from Lemma 4.6. To perform the induction we define the subalgebras Q
′ m ⊆ Q[x] for m > 0 by Q ′ 0 := Q[x] , Q ′ m := Q + N B x 2 · Q ′ m−1 , m > 0 . Clearly, Q ′ m = Q + Q · N B x 2 + . . . + Q · (N B x 2 ) m−1 + (N B x 2 ) m Q[x] .0 ✲ Q ′ m ·N B x 2 ✲ Q ′ m ✲ 0 . Since Q ′ m ⊆ Q[x] is an integral domain, Tor Q[x] i (Q, Q ′ m ) = 0 for i > 0.
The Eilenberg-Moore spectral sequence for the fibration sequence F m (ΩB, T ) → X m (ΩB, T ) → B therefore collapses to give
H * (F m (ΩB, T ), Q) ∼ = Q ′ m /(N B x 2 )
. Further, since the Eilenberg-Moore spectral sequence is multiplicative, j * m,B is the canonical quotient map. In particular, note that the cohomology of F m (ΩB, T ) is concentrated in even degree. The long exact sequence of cohomologies associated with the cofibration sequence F m (ΩB, T ) → X m (ΩB, T ) → X m+1 (ΩB, T ) yields (for n even)
H n (X m+1 [ΩB, T )] ⊂ π * m ✲H n [X m (ΩB, T )] j * m,B ✲H n [F m (ΩB, T )] ∂ ✲ ✲H n+1 [X m+1 (ΩB, T )]
Since j * m,B is surjective, we havẽ H n+1 (X m+1 (ΩB, T ), Q) = 0 for n even .
Hence,
H * (X m+1 (ΩB, T ), Q) ∼ = Q + Ker(j * m,B ) = Q + (N B x 2 ) · Q ′ m = Q ′ m+1 ,
with π * m being the inclusion Q ′ m+1 ֒→ Q ′ m . This completes the induction step, proving part (i). Part (ii) follows immediately from (i) combined with Lemma 4.6 (since p B = p m,B •π m ).
Corollary 4.8. For a fixed m ≥ 0, all spaces X m (ΩB, T ) are rationally equivalent to X m (G, T ) (and hence to each other).
Proof. This follows from Theorem 4.7 and the uniqueness part of Theorem 3.9.
In Section 5.4 (see Corollary 5.12), we will show that X m (ΩB, T ) ≃ X m (ΩB ′ , T ) whenever N B = N B ′ . Thus Theorem 4.7 provides many different topological realizations 6 for the algebras Q m (W ). However, these do not give us different solutions to our realization problem (see Section 2.4), since none of the spaces B in the genus of BG (except for BG itself) admits a maximal torus and hence none carries a natural W -action. In addition, by Ganea's Theorem 3.1, hocolim m [X m (ΩB, T )] ≃ B , which shows that property (QI 2 ) fails for X m (ΩB, T ) when B ≃ BG .
Equivariant K-theory
In this section, we compute the G-equivariant K-theory K G (F m ) of the m-quasi-flag manifold F m = F m (G, T ) associated to G = SU (2). We find that K G (F m ) is isomorphic to the ring Q m (W ) of exponential quasi-invariants of W . By the Atiyah-Segal Theorem, the (ordinary) K-theory of X m (G, T ) is then isomorphic to the completion Q m (W ) of Q m (W ) with respect to the canonical augmentation ideal of R(G). For the 'fake' spaces of quasi-invariants, X m (ΩB, T ), associated to Rector spaces, the K-theory rings K[X m (ΩB, T )] are new invariants that are not isomorphic to Q m (W ) in general and are strong enough to distinguish the X m (ΩB, T ) up to homotopy equivalence. 5.1. Equivariant K-theory. Recall that, for a compact Lie group G acting continuously on a compact topological space X, the K G (X) is defined to be the Grothendieck group of G-equivariant (complex topological) vector bundles on X. As shown in [Seg68], this construction extends to a Z/2-graded multiplicative generalized cohomology theory K * G on the category of (locally compact) G-spaces that is called the G-equivariant K-theory. We write K * G (X) := K 0 G (X) ⊕ K 1 G (X), with understanding that K 0 G (X) ∼ = K 2n G (X) and K 1 G (X) ∼ = K 2n+1 G (X) for all n ∈ Z. When G is trivial, K * G (X) coincides with the ordinary complex K-theory K * (X), while for X = pt , K * G (pt) is the representation ring R(G) of G (in particular, we have K 1 G (pt) = 0). In general, by functoriality of K * G , the trivial map X → pt gives a canonical R(G)-module structure on the ring K * G (X) for any G-space X. The ring K * G (X) has nice properties for which we refer the reader to [Seg68]. Here we only mention two technical results needed for our computations.
The first result is a well-known Künneth type formula for equivariant K-theory first studied by Hodgkin (see, e.g., [BZ00, Theorem 2.3]).
Theorem 5.1 (Hodgkin). Let G be a compact connected Lie group, such that π 1 (G) is torsion-free. Then, for any two G-spaces X and Y , there is a spectral sequence with E 2 -term E 2 * , * = Tor
R(G) * , * (K * G (X), K * G (Y )) that converges to K * G (X × Y )
, where X × Y is viewed as a G-space with the diagonal action. The second result is the following Mayer-Vietoris type formula, which is also -in one form or another -well known to experts. 6 It is tempting to conjecture that the (homotopy types of the) spaces Xm(ΩB, T ) associated with the Rector spaces B admiting an essential map from BT constitute the set of all such realizations. Unfortunately, besides Theorem 3.9(2), we do not have much evidence for this conjecture.
Lemma 5.2. Let f : U → X and g : U → Y be proper equivariant maps of G-spaces. Let
Z = hocolim(X f ← U g → Y ),
where 'hocolim' is taken in the category of G-spaces. Then, the abelian groups K * G (X), K * G (Y ) and K * G (Z) are related by the six-term exact sequence
K 0 G (Z) ✲ K 0 G (X) ⊕ K 0 G (Y ) f * − g * ✲ K 0 G (U ) K 1 G (U ) ∂ ✻ ✛ f * − g * K 1 G (X) ⊕ K 1 G (Y ) ✛ K 1 G (Z) ∂ ❄
The proof of Lemma 5.2 can be found, for example, in [JO99]. and write e λ for the elements of R(T ) corresponding to characters λ ∈T . Next, we let R ⊆T denote the root system of W determined by (G, T ) and choose a subset R + ⊂ R of positive roots in R. If s α ∈ W is the reflection in W corresponding to α ∈ R + , then the difference e λ − e sα(λ) in R(T ) is uniquely divisible by 1 − e α for any λ ∈T . Following [Dem74], we define a linear endomorphism Λ α : R(T ) → R(T ) for each α ∈ R + , such that
(5.1) (1 − s α )f = Λ α (f ) · (1 − e α ) .
The operator Λ α is an exponential analogue of the divided difference operator ∆ α introduced in Section 3.5(see (3.44)). Note that the conditions (1.2) defining the usual quasi-invariant polynomials can be written in terms of the divided difference operators as ∆ α (p) ≡ 0 mod (α) 2mα . This motivates the following definition of quasi-invariants in the exponential case.
Definition 5.
3. An element f ∈ R(T ) is called an exponential quasi-invariant of W of multiplicity m ∈ M(W ) if
(5.2) Λ α (f ) ≡ 0 mod (1 − e α 2 ) 2mα , ∀ α ∈ R + .
Remark 5.4. In general, it may happen that α 2 ∈T for some α ∈ R + , so that e α 2 ∈ R(T ). We view (5.2) as a congruence in the extended group ring Z[ 1 2T ] that naturally contains R(T ). We write Q m (W ) for the set of all f ∈ R(T ) satisfying (5.3) for a fixed multiplicity m. This set is closed under addition and multiplication in R(T ), i.e. Q m (W ) is a commutative subring of R(T ). (The latter can be easily seen from the twisted derivation property of Demazure operators: where z = e ̟ = e α 2 . Now, with these identifications, we claim that
Λ α (f 1 f 2 ) = Λ α (f 1 ) · f 2 + s α (f 1 ) · Λ α (f 2 ) that(5.4) Q m (W ) = Z ⊕ Z · (z 1/2 − z −1/2 ) 2 ⊕ Z · (z 1/2 − z −1/2 ) 4 ⊕ . . . ⊕ (z 1/2 − z −1/2 ) 2m · Z[z, z −1 ] .
Indeed, if f ∈ Z[z, z −1 ] can be written in the form (5.4), then
f − s α (f ) ∈ (z 1/2 − z −1/2 ) 2m (1 − s α ) Z[z, z −1 ] = (z 1/2 − z −1/2 ) 2m (z − z −1 ) Z[z, z −1 ] , which shows that Λ α (f ) = (1 − z 2 ) −1 (f − s α f ) is divisible by (1 − z) 2m = (1 − e α 2 ) 2m in Z[z, z −1 ]. Thus f ∈ Q m .
To see the converse denote the right-hand side of (5.4) byQ m . Note that there is a natural Q[z + z −1 ]-module decomposition
Q[z, z −1 ] ∼ = Q[z + z −1 ] ⊕ Q[z + z −1 ] · δ , where δ := z − z −1 . Writing f = p + q · δ with p, q ∈ Q[z + z −1 ], we find that f − s α (f ) = 2qδ. Thus, if f ∈ Q m then f − s α (f ) ∈ (z 1/2 − z −1/2 ) 2m (z − z −1 ) Z[z, z −1 ] and hence q ∈ (z 1/2 − z −1/2 ) 2m Q[z, z −1 ]. It follows that f ∈Q m ⊗ Q. On the other hand, (Q m ⊗ Q) ∩ Z[z, z −1 ] =Q m which implies that Q m ⊆Q m .
Let F m = F m (G, T ) be the m-quasi-flag manifold of G = SU (2) introduced in Section 3.3 (see Definition 3.10). Recall that F m is a G-space of homotopy type of a finite CW-complex. The next theorem computes the G-equivariant K-theory of F m , which is the main result of this section.
Theorem 5.6. This is a natural isomorphism of Z/2-graded commutative rings
K * G (F m ) ∼ = Q m (W ) Thus K 0 G (F m ) ∼ = Q m (W ) and K 1 G (F m ) = 0 for all m ∈ Z + . Proof. Recall that K * G (pt) = R(G) ∼ = Z[t],
where t corresponds to the 2-dimensional irreducible representation of G = SU (2). The natural map K * G (pt) → K * G (G) ∼ = K * (pt) is then identified with the projection Z[t] → Z taking t → 2 . For m = 0, by definition, we have F 0 = G/T , and hence (cf. Example 5.5)
(5.5) K * G (G/T ) ∼ = K * T (pt) = R(T ) ∼ = Z[z, z −1 ] . Thus K 0 G (F 0 ) ∼ = Z[z, z −1 ] = Q 0 (W ) and K 1 G (F 0 ) = 0 as is well known. Further, the map R(G) → R(T ) induced on G-equivariant K-theory by G/T → pt is identified with Z[t] → Z[z, z −1 ] , t → z + z −1 . Now, recall that F m+1 = F m * G, which means (5.6) F m+1 ≃ hocolim[F m ← F m × G → G] .
There is a canonical G-equivariant map F m → F m+1 which we denote by i m,m+1 , which is nontrivial (not null-homotopic) in the homotopy category of G-spaces (see Remark 3.11). Let i m,n : F m → F n denote the composite map i m,n := i n−1,n • . . . • i m,m+1 for n > m. We claim that the map i * 0,m : K * G (F m ) → K * G (G/T ) induced by i 0,m : G/T → F m is injective, and under the isomorphism (5.5), it is identified with the inclusion of Q m (W ) in Z[z, z −1 ]. We prove our claim by induction on m. For m = 0, this is just (5.5).
Assume, for some m ≥ 0, that K * G (F m ) ∼ = Q m (W ) and that the map i * 0,m :
K * G (F m ) → K * G (G/T ) is identified with the inclusion of Q m (W ) in Z[z, z −1 ] as a subring. Then the image of t ∈ K * G (pt) in K * G (F m ) ∼ = Q m (W ) is z+z −1 . Since K * G (G) ∼ = Z has the free K * G (pt) ∼ = Z[t]-module resolution 0 → Z[t] → Z[t] ·(t−2) −→ Z → 0 , the Tor-group Tor R(G) * (K * G (F m ), K * G (G)) ∼ = Tor Z[t] * (Q m , Z)
is identified with the homology of the two-term complex 0 → Q m (W )
·(z+z −1 −2)
−→ Q m (W ) → 0 , whose first homology vanishes since Q m (W ) is an integral domain. It follows that Hodgkin's spectral sequence (see Theorem 5.1) that −1 − 2). Next, applying Lemma 5.2 to the homotopy pushout (5.6), we obtain the four-term exact sequence
K * G (F m × G) ∼ = Q m (W )/(z + z −1 − 2) , and that the map K * G (F m ) → K * G (F m × G) induced by the projection F m × G → F m is the canonical quotient map π : Q m (W ) → Q m (W )/(z + z(5.7) 0 → K 0 G (F m+1 ) (i m,m+1 ,f ) * −−−−−−−→ Q m (W ) ⊕ Z i * −π * − −−− → Q m (W )/(z + z −1 − 2) ∂ − → K 1 G (F m+1 ) → 0 , where i : Z → Q m (W )
is the structure map of the ring Q m (W )) and f : G → F m+1 is the natural map associated to (5.6). It follows from (5.7) that K 1 G (F m+1 ) = 0, and
K 0 G (F m+1 ) ∼ = Ker(i * − π * ) = Z + (z + z −1 − 2) · Q m (W ) = Q m+1 (W ) . Furthermore, the inclusion Q m+1 (W ) ֒→ Q m (W ) is identified with the map i * m,m+1
. This completes the induction step, completing the proof of the theorem.
The equivariant Chern character.
Recall that the space X m = X m (G, T ) of m-quasiinvariants is defined as the homotopy G-quotient X m := EG× G F m . The Borel construction yields a natural map
(5.8) α : K * G (F m ) → K * (X m ) where K * (X) = K 0 (X) ⊕ K 1 (X)
is the (complex) topological K-theory defined by K 0 (X) = [X, BU ] and K 1 (X) = [X, U ] . Theorem 5.6 shows that K * G (F m ) is a finitely generated R(G)module for all m ∈ Z + . Hence, by Atiyah-Segal Completion Theorem [AS69], the map (5.8) extends to an isomorphism
(5.9) K * G (F m ) I G ∼ = K * (X m )
where K * G (F ) I G denotes the (adic) completion of K * G (F ) (as an R(G)-module) with respect to the augmentation ideal of R(G) defined as the kernel of the dimension function I G := Ker[dim : R(G) → Z] . If we identify R(G) ∼ = Z[z + z −1 ] as the invariant subring of R(T ) ∼ = Z[z, z −1 ] as in the proof of Theorem 5.6, then I G = (z + z −1 − 2). Thus, as a consequence of (5.9), we get Corollary 5.7. For all m ≥ 0, there is an isomorphism
K * (X m ) ∼ = Q m (W ) I
where ( Q m ) I denotes the completion of (5.4) with respect to the ideal I = (z+z −1 −2) ⊂ Z[z+z −1 ].
Next, we compute a Chern character map relating equivariant K-theory to equivariant cohomology. Recall that the Chern character of an equivariant vector bundle on a G-space F is defined as the (non-equivariant) Chern character of the associated vector bundle on EG × G F . This gives a natural map Q) . The following proposition describes the map (5.10) for F = F m (G, T ) explicitly, using the identifications of Theorem 3.9 and Theorem 5.6.
(5.10) ch G (F ) : K * G (F ) → H * G (F, Q) where H * G (F, Q) := ∞ k=0 H k G (F,
Proposition 5.8. (1) The Chern character map ch G (F m ) : K * G (F m ) → H * (X m , Q) is given by
(5.11) exp : Q m (W ) → Q m (W ) , z → ∞ n=0 x n n! , where Q m (W ) := Q m (W ) ⊗ Q[x 2 ] Q[[x 2 ]]
is the completed ring of quasi-invariants of W = Z/2Z.
(2) The map ch G (F m ) factors through (5.8) inducing an isomorphism on rational K-theory
K(X m ) Q ∼ = H * (X m , Q) ∼ = Q m (W ) Proof. For F 0 = G/T , we can identify K * G (G/T ) ∼ = R(T ) ∼ = Z[z, z −1 ] and H * G (G/T, Q) = H * (BT, Q) ∼ = Q[[x]
] as in (the proofs of) Theorem 3.9 and Theorem 5.6. With these identifications, it is well known that the equivariant Chern character is given by exponentiation (see, e.g., [FRW21, Example A.5]):
(5.12) ch G (G/T ) : K * G (G/T ) → H * (BT, Q) , z → exp(x) .K * G (pt) ⊂ ch G (pt) ✲ H * (pt, Q) K * G (F m ) ❄ ∩ ch G (F m ) ✲ H * (F m , Q) ❄ ∩ K * G (G/T ) i * 0,m ❄ ∩ ⊂ ch G (G/T ) ✲ H * (G/T, Q) i * 0,m ❄ ∩ .
where the vertical maps as well as the top and the bottom horizontal maps are injective. Hence, the map in the middle, ch G (F m ) : K * G (F m ) → H * (F m , Q), is also injective, and it is given by restriction of the exponential map (5.12). This proves the first claim of the proposition. The second claim follows from the first and Corollary 5.7. 5.4. K-theory of 'fake' spaces of quasi-invariants. In this section, we compute the K-theory of 'fake' spaces of quasi-invairants X m (ΩB, T ) constructed in Section 4.3. We will keep the notation G = SU p * B (u) = N B t 2 + higher order terms in t . is a useful K-theoretic invariant of B that depends on the Rector invariants (B/p) (see [Yau04]). Using (5.14), we define a sequence of subrings Q m (B) in Z[[t]] inductively by the rule:
(5.15) Q 0 (B) := Z[[t]] , Q m (B) := Z + p * B (u)Q m−1 (B) , m ≥ 1 . Note that there are natural inclusions Q 0 (B) ⊇ Q 1 (B) ⊇ . . . ⊇ Q m (B) ⊇ Q m+1 (B) ⊇ . . .
which are all ring homomorphisms.
I = I G is the ideal of virtual representations in K * G (pt) ∼ = R(G) of dimension 0. If we identify K * G (pt) ∼ = Z[v],
where v is the standard 2-dimensional representation of G, then I = (v − 2) , and
K * (BG) ∼ = Z[[u]], where u = v − 2. Similarly, K * (BT ) ∼ = Z[[t]]
, where t = z − 1, with z standing for the generating character of T . The naturality of (5.8) (with respect to the G-equivariant map p : G/T → pt) yields the commutative diagram
K * G (pt) ∼ = Z[u] α ✲ K * (BG) ∼ = Z[[u]] K * G (G/T ) ∼ = Z[t] p * ❄ α ✲ K * (BT ) ∼ = Z[[t]] p * B ❄ .
Since p * (v) is the restriction of v to T , we have p * (v) = z + z −1 . Hence,
p * B (u) = p * B (v − 2) = z + z −1 − 2 = (1 + t) + 1 1 + t − 2 = t 2 1 + t
It follows that Q m (BG) ∼ = Q m (W ), where the right-hand side is the completion of Q m (W ) with respect to the ideal generated by z + z −1 − 2 (cf. Corollary 5.7). Now, we state the the main result of this section. To prove Theorem 5.10 we will use an Eilenberg-Moore spectral sequence for K-theory in the following form.
Lemma 5.11. Let F → E → B be a (homotopy) fibration sequence over a base B such that K * (ΩB) is an exterior algebra in a finite number of generators of odd degrees. Then there is a multiplicative spectral sequence with E i, * 2 ∼ = Tor K * (B) i (Z, K * (E)) that strongly converges to K * (F ).
The proof of Lemma 5.11 can be found, for example, in [JO99] (see Main Theorem, Part 3).
K * [F 0 (ΩB, T )] ∼ = Z[[t]]/(p * B (u)) Next, assume that K * [X m (ΩB, T )] ∼ = Q m (B) and that K * [F m (ΩB, T )] ∼ = Q m (B)/(p * B (u)), with j * m,B being the canonical quotient map. Since X m+1 (ΩB, T ) ≃ hocolim [ pt im ←− F m (ΩB, T ) j m,B − −− → X m (ΩB, T ) ] , and since K 1 (pt) = K 1 [F m (ΩB, T )] = K 1 [X m (ΩB, T )] = 0 , Lemma 5.2 (with G trivial group)
yields the four-term exact sequence
0 → K 0 [X m+1 (ΩB, T )] (i * m , π * m ) − −−−− → Z⊕Q m (B) p * m −j * m,B − −−−−− → Q m (B)/(p * B (u)) ∂ − → K 1 (X m+1 (ΩB, T )) → 0 .
Here p m is the trivial map from F m (ΩB, T ) to the point. Since j * m,B is surjective, K 1 [X m+1 (ΩB, T )] = 0. The above six-term exact sequence also shows that K 0 [X m+1 (ΩB, T )] ∼ = Ker(p * m − j * m,B ) ⊆ Z ⊕ Q m (B) (with isomorphism given by the map (i * m , π * m )) . Projection to Q m (B) identifies this kernel with Q m+1 (B) = Z + p * B (u)Q m (B) ⊂ Q m (B) . It follows that K * [X m+1 (ΩB, T )] ∼ = Q m+1 (B) , with π * m being the inclusion of Q m+1 (B) into Q m (B). Finally, by taking the (completed) tensor product of the resolution (5.17) with Q m+1 (B), we see that Tor
K * (B) i (Z, Q m+1 (B)) is the homology of the complex 0 → Q m+1 (B) p * B (u) − −−− → Q m+1 (B) → 0
where the map in the middle is given by multiplication by the formal power series (5.14). Since
Q m+1 (B) ⊆ Z[[t]] is an integral domain, Tor K * (B) i (Z, K * (X m+1 )) = 0 for i > 0.
The spectral sequence of Lemma 5.11 associated with the fibration sequence F m+1 → X m+1 → B therefore collapses, giving
K * [F m+1 (ΩB, T )] ∼ = Q m+1 (B)/(p * B (u))
, with j * m+1,B being the canonical quotient map. This completes the induction step and thus finishes the proof of the theorem.
Theorem 5.10 allows one to distinguish spaces of quasi-invariants of the same multiplicity associated to homotopically distinct spaces in the genus of BG. First, we recall that the topological K-theory K * (X) of any space X of homotopy type of a CW complex carries a natural filtration
F 0 K * (X) ⊇ F 1 K * (X) ⊇ . . . ⊇ F n K * (X) ⊇ F n+1 K * (X) ⊇ . . .
where F n K * (X) is defined to be the kernel of the restriction map K * (X) → K * (X n−1 ) corresponding to the (n − 1)-skeleton of X. This filtration is functorial: any map f : X → X ′ of spaces, each of which has homotopy type of a CW complex, induces a morphism of filtered rings f * : K * (X ′ ) → K * (X). Moreover, by Cellular Approximation Theorem, it is independent of the CW structure in the sense that using a different CW structure on X will not change the isomorphism type of K * (X) as a filtered ring.
Corollary 5.12. Let B and B ′ be two spaces in the genus of BG admitting essential maps from BT . Assume that N B = N B ′ . Then X m (ΩB, T ) ≃ X m (ΩB ′ , T ) for any m ≥ 0. In particular, X m (ΩB, T ) is not homotopy equivalent to X m (G, T ) for any B ≃ BG.
Proof. Letπ m : BT → X m (ΩB, T ) denote the composite map π m−1 • . . . • π 0 in (4.1). By Theorem 5.10, this map induces an embedding
π * m : K * [X m (ΩB, T )] ∼ = Q m (B) ֒→ Z[[t]] ∼ = K * (BT ) which is a morphism of filtered rings. Now, recall that BT ≃ CP ∞ ; the generator t in K * (BT ) ∼ = K * (CP ∞ ) = Z[[t]]
can be taken in the form t = bξ, where ξ ∈ F 2 K 2 (BT ) and b ∈ K −2 (pt) is the Bott element (see [Yau04,Sect. 3]). Hence t ∈ F 2 K 0 (BT ), and therefore, by (5.14), we have
p * B (u) ≡ N B t 2 (mod F 5 K * (BT )) in Z[[t]]
Now, by Theorem 5.10,
K * [X m (ΩB, T )] ∼ = Q m (B) = Z + Z · p * B (u) + . . . + Z · p * B (u) m−1 + p * B (u) · Z[[t]] . Hence K * [X m (ΩB, T )]/F 5 K * [X m (ΩB, T )] ∼ = Z + Z · N B t 2 ,
where the generator N B t 2 is square zero. It follows that if p is a prime then
K * [X m (ΩB, T )]/(p, F 5 K * [X m (ΩB, T )]) ∼ = (Z/pZ) + (Z/pZ) ·N B t 2 if p ∤ N B (Z/pZ) if p | N B
where (p, F 5 K * (X m )) denotes the ideal in K * (X m ) generated by p ∈ Z and F 5 K * (X m ). This shows that X m (ΩB, T ) is not homotopy equivalent to X m (ΩB ′ , T ) unless N B = N B ′ .
Remark 5.13. The converse of Corollary 5.12 also holds true in the following sense: if two spaces B and B ′ in the genus of BG have homotopy equivalent towers of spaces of quasi-invariants {X m (ΩB, T ), π m } m≥0 and {X m (ΩB ′ , T ), π ′ m } m≥0 , then B ≃ B ′ . This simply follows from the fact that
hocolim m∈Z + X m (ΩB, T ) ≃ B ,
which is a consequence of Ganea's Theorem 3.1.
Elliptic cohomology
In this section, we compute complex analytic T -and G-equivariant elliptic cohomology of the quasi-flag manifolds F m (G, T ). We express the result in two ways: geometrically (in terms of coherent sheaves on a given elliptic curve E) and analytically (in terms of Θ-functions and qdifference equations). We also compute the spaces of global sections of the elliptic cohomology sheaves of F m (G, T ) with coefficients twisted by tensor powers of the Looijenga line bundle on E. This last computation provides a motivation for our definition of elliptic quasi-invariants of W . 6.1. Equivariant elliptic cohomology. Complex analytic elliptic cohomology was introduced by I. Grojnowski (see [Gro07]). We will follow the approach of [Gan14] that relies on earlier topological results of [And00] and [Ros01]. We begin by briefly recalling the main definitions.
Let E be an elliptic curve defined analytically over C. Given a compact connected abelian Lie group T (i.e., T ∼ = U (1) n ), we writeT := Hom(U (1), T ) for its cocharacter lattice and set
M T :=T ⊗ Z E ,
which is an abelian variety of rank n = rk(T ) defined over C. The T -equivariant elliptic cohomology is defined as a functor on the (homotopy) category of finite T -CW complexes with values in the category of (complex-analytic) coherent sheaves on M T :
(6.1) Ell * T : Ho(Top fin T ) → Coh(M T )
. This functor has the basic property that
Ell * T (T /T ′ ) ∼ = O M T ′ for any closed subgroup T ′ ⊆ T , where O M T ′ = i * O M T is the restriction of the structure sheaf of M T to M T ′ (see [Gan14, 2.1(1)]).
In particular, we have
(6.2) Ell * T (pt) ∼ = O M T Now,
if G is a compact connected Lie group with maximal torus T and Weyl group W , we define the G-equivariant elliptic cohomology of a G-space X by
(6.3) Ell * G (X) := Ell * T (X) W ,
where X is viewed as a T -space by restricting G-action (see [Gan14,3.4]). To compute the Gequivariant elliptic cohomology we thus need to compute the T -equivariant elliptic cohomology of a G-space X and take its W -invariant sections.
The coherent sheaves Ell * T (X) do not have usually many interesting global sections. To remedy this one considers a twisted version of elliptic cohomology, where the sheaves Ell * T (X) are tensored with a certain ample line bundle on M T . Recall that, if G is a simple, simply connected compact Lie group with a root system R, there is a canonical W -equivariant line bundle L on M T associated to the invariant symmetric bilinear form I on the coroot lattice of R such that I(α, α) = 2 for all roots of smallest length in R; this line bundle has I as its Chern class and has degree equal to the order of the center of G (see [Loo77]). Following [And00, Gan14], we will refer to L as the Looijenga bundle on M T and define the T -and G-equivariant elliptic cohomology of X with coefficients twisted by L by
Ell * T (X, L) := ∞ n=0 H 0 an (M T , Ell * T (X) ⊗ L n ) (6.4) Ell * G (X, L) := ∞ n=0 H 0 an (M T , Ell * T (X) ⊗ L n ) W (6.5)
where H 0 an stands for the global (holomorphic) sections of the coherent sheaf Ell * T (X) twisted by the tensor powers of L. Note that (6.4) and (6.5) are naturally graded modules over the graded commutative rings which we denote by S(E) and S(E) W , respectively. Following [Loo77], we also write S(E) −W for the subspace of S(E) consisting of all W -anti-invariant sections. The main theorem of [Loo77] asserts that S(E) W is a graded polynomial algebra generated freely by l+1 homogeneous elements, while S(E) −W is a free module over S(E) W of rank one (see [Loo77,(3.4)]). The generators of S(E) W are called the Looijenga theta functions on M T .
6.2. Elliptic cohomology of quasi-flag manifolds. In the rank one case (T = U (1)), we can identify M T = E and take for a model of E the Tate curve E q := C * /q Z with 0 < |q| < 1. The latter is defined as the quotient of the punctured line C * = C \ {0} (viewed as a complex analytic manifold) by the free action of the infinite cyclic group Z :
(6.7) C * → C * , z → q n z .
We write A := O an (C * ) for the ring of global analytic functions on C * and define A q := A ⋊ q Z to be the crossed product algebra for the action of Z on A induced by (6.7). Letting ξ be the (multiplicative) generator of Z, we can present A q as a skew-polynomial algebra A[ξ, ξ −1 ] with multiplication determined by the commutation relation ξ · a(z) = a(qz) · ξ for a(z) ∈ A. The left modules over A q can be identified with spaces of global sections of Z-equivariant quasi-coherent sheaves on C * . The natural projection π : C * → E q induces then the additive functor
(6.8) Coh(E q ) → Mod f.p. A (A q )
, F → F := H 0 an (C * , π * F) , that maps the coherent sheaves on the analytic curve E q to left A q -modules admitting a finite presentation A ⊕m → A ⊕n → M → 0 over the subalgebra A ⊂ A q . The following proposition is a well-known result that provides a convenient algebraic description of the category Coh(E q ); its proof can be found in various places (see, for example, [SV03, Thm 2.2] or [vdPR07, Sect. 2]).
Proposition 6.1. The functor (6.8) is an exact equivalence of abelian tensor categories.
We remark that the tensor structure on Coh(E q ) is the standard geometric one (defined by tensor product of sheaves of O Eq -modules), while the tensor structure on Mod f.p.
A (A q ) is defined by tensoring A q -modules over the subalgebra A with the action of A q on M 1 ⊗ A M 2 given by ξ · (m 1 ⊗ m 2 ) = (ξ · m 1 ) ⊗ (ξ · m 2 ). The vector bundles on E q correspond under (6.8) to A qmodules that are free of finite rank over A; such modules form a full subcategory of Mod f.p.
A (A q ) closed under the tensor product. The cohomology of a coherent sheaf F on E q can be computed algebraically in terms of A q -modules as invariants and coinvariants of the induced action of Z on the corresponding A-module F (see [vdPR07, Lemma 2.1]):
(6.9) H 0 an (E q , F) ∼ = Ker (ξ − id : F) , H 1 an (E q , F) ∼ = Coker (ξ − id : F ) .
where ξ is the multiplicative generator of the copy of Z in A q acting on the A q -module F.
Example 6.2. The structure sheaf O Eq of E q corresponds under (6.8) to the cyclic module O Eq = A q /A q (ξ − 1) , which can be identified as O Eq ∼ = Ae with generator e satisfying the relation ξe = e. The line bundle O Eq ([1]) corresponds to A q /A q (ξ + z) ∼ = Ae , with e satisfying ξe = −ze.
More generally, any line bundle on E q of degree d corresponds to a cyclic A q -module Ae, where the generator e satisfies the relation ξe = cz d e for some c ∈ C * (see [vdPR07, Example 2.2]).
We now proceed with computing elliptic cohomology of the spaces F m = F m (G, T ). For a fixed Tate curve E q , we first describe the T -equivariant cohomology, presenting the answer in two ways: in terms of coherent sheaves on E q and in terms of A q -modules via the equivalence (6.8).
Theorem 6.3. For all m ≥ 0, there are natural isomorphisms of coherent sheaves in Coh(E q ) (6.10)
Ell * T (F m ) ∼ = O Eq × O Eq /J 2m+1 O Eq ,
where J is the subsheaf of ideals in the structure sheaf O Eq vanishing at the identity of E q . Under the equivalence (6.8), the coherent sheaf (6.10) corresponds to the A q -module
(6.11) Ell * T (F m ) ∼ = A × A/ Θ 2m+1 A ,
where the action of A q on the fibre product is induced by the natural action of A q on A and Θ denotes the (principal) ideal in A = O an (C * ) generated by the classical theta function
(6.12) Θ(z) := (1 − z) n>0 (1 − q n z)(1 − q n z −1 ) = 1 (q; q) ∞ n∈Z q n(n−1) 2 (−z) n .
Proof. Recall that, by Lemma 3.12, there is a T -equivariant homeomorphism
F m ∼ = Σ E 2m (T ) = hocolim (pt ← E 2m (T ) → pt) ,
where E 2m (T ) = T * (2m+1) is Milnor's 2m-universal T -bundle. As equivariant K-theory, the Tequivariant elliptic cohomology is known to satisfy the Mayer-Vietoris property (see, e.g., [Ros01, Theorem 3.8]). Hence, as in Lemma 5.2, there is a six term long exact sequence of sheaves on E q :
Ell 0 T (F m ) ✲ Ell 0 T (pt) × Ell 0 T (pt) p * 1 − p * 2 ✲ Ell 0 T (E 2m (T )) Ell 1 T (E 2m T ) ✻ ✛ Ell 1 T (pt) × Ell 1 T (pt) ✛ Ell 1 T (F m ) ❄ .
where the arrow on top of the diagram is given on sections by (x 1 , x 2 ) → p * 1 (x 1 ) − p * 2 (x 2 ), with p 1 and p 2 representing two copies of the canonical map E 2m (T ) → pt. By (6.2), we know that Ell * T (pt) ∼ = O Eq ; on the other hand, by Lemma 6.4 (see below), Ell * T (E 2m T ) ∼ = O Eq /J 2m+1 , where J ⊂ O Eq is the subsheaf of sections vanishing at 1 ∈ E q . Hence, by exactness, the above commutative diagram shows that Ell 1 T (F m (G, T )) = 0 and
Ell 0 T (F m ) = Ker(p * 1 − p * 2 ) ∼ = O Eq × O Eq /J 2m+1 O Eq .
This proves the first claim of the theorem. Now, to prove the second claim we observe that the skyscraper sheaf F := i 1, * C on E q (with stalk C supported at {1}) corresponds under (6.8) to the quotient module F ∼ = A/ Θ , where the action of A q is induced by the natural action of A q on A. Indeed, F is isomorphic to the cokernel of the map O Eq → O Eq ([1]), which is given (with identifications of Example 6.2) by e → Θe. This follows from the fact that as a global analytic function on C * , Θ = Θ(z) has simple zeroes exactly at the points z = q n (n ∈ Z). Hence the ideal sheaf J ⊂ O Eq corresponds to the ideal Θ = AΘ in A, and more generally, since (6.8) is a tensor functor, J 2m+1 corresponds to Θ 2m+1 = AΘ 2m+1 for all m ≥ 0. Now, since (6.8) is an exact additive functor, it takes the fibre product
O Eq × O Eq /J 2m+1 O Eq in Coh(E q ) to the module A × A/ Θ 2m+1 A in Mod f.p.
A (A q ), thus completing the proof of the theorem.
Lemma 6.4. There are isomorphisms of sheaves Ell * T (E n T ) ∼ = O Eq /J n+1 for all n ≥ 0 . Proof. Note that T acts freely on E n (T ) := T * (n+1) . Recall (see [Ros01,Sect. 3.2]) that if X is a finite T -space, the stalk at a ∈ E of Ell * T (X) is isomorphic to H * T (X a ; C) ⊗ C[z] O C * ,1 , where O Eq,1 stands for the ring of germs of analytic functions at 1 ∈ E q . Here, X a stands for the fixed point space X Ta , where T a = Z/kZ ⊂ T if a is of finite order k in E, and T a = T if a is not of finite order in E. It follows that of T acts freely on X, the stalk Ell * T (X) a of Ell * T (X) at a vanishes for a = 1. Hence, Ell T (E n T ) a = 0 for a = 1, and for U a small neighborhood of 1 in E q , Ell * To compute the G-equivariant elliptic cohomology of F m we need to refine the result of Theorem 6.3 by taking into account the action of W = Z/2Z on Ell * T (F m ). To this end we first refine the result of Proposition 6.1. Observe that the equivalence (6.8) extends to the category of W -equivariant coherent sheaves on E q :
T (X)| U ∼ = H * T (E n T ; C) ⊗ C[x] O Eq | U , where O Eq | U(6.13) Coh W (E q ) ∼ − → Mod f.p. A (A q ⋊ W )
, where the category of A q -modules finitely presented over A is replaced by a similar category of modules over the crossed product algebra A q ⋊ W associated to the geometric action of W on C * . The algebra A q ⋊ W has the canonical presentation A ξ, ξ −1 , s , where the generators ξ, s and a(z) ∈ A are subject to the relations s · a(z) = a(z −1 ) · s , s · ξ = ξ −1 · s , ξ · a(z) = a(qz) · ξ , s 2 = 1
We let e + := (1+s)/2 denote the symmetrizing idempotent in A q ⋊W and consider the subalgebra e + (A q ⋊ W )e + of A q ⋊ W (with identity element e + ). This subalgebra can be naturally identified with the invariant subalgebra A W q of A q via the isomorphism: A W q ∼ − → e + (A q ⋊W )e + , a → e + a e + . With this identification, we can define the additive functor (6.14)
Mod(A q ⋊ W ) → Mod(A W q ) , M → e + M , that assigns to a W -equivariant A q -module its subspace of W -invariant elements viewed as a module over A W q . The next result is well known for the algebra A alg q := O alg (C * ) ⋊ q Z which is an algebraic (polynomial) analogue 7 of A q = O an (C * ) ⋊ q Z. The analytic case easily reduces to the algebraic one as A alg q is naturally a subalgebra of A q .
Lemma 6.5. The functor (6.14) is an equivalence of categories, its inverse being given by
A q ⊗ A W q (-) : Mod(A W q ) → Mod(A q ⋊ W )
Proof. Lemma can be restated by saying that the algebra A q ⋊ W is Morita equivalent to A W q . To prove this, by standard Morita theory (see [MR01,3.5.6]), it suffices to check that the idempotent e + generates the whole A q ⋊ W as its two-sided ideal. This last condition holds for A alg q ⋊ W , since A alg q ⋊ W is a simple algebra (has no proper two-sided ideals), if q is not a root of unity. But then it also holds for A q ⋊ W , since A alg q ⋊ W is a unital subalgebra of A q ⋊ W containing e + . Now, combining (6.13) with Morita equivalence (6.14), we get the equivalence
(6.15) Coh W (E q ) ∼ − → Mod f.p. A W (A W q )
, F → H 0 an (C * , π * F) W , that allows us to describe the W -equivariant coherent sheaves on E q in terms of A W q -modules. Recall that Ell * G (F m ) is defined to be the subsheaf of W -invariant sections of the coherent sheaf Ell * T (F m ) (see (6.3)). In the next theorem, we describe Ell * G (F m ) explicitly as an A W q -submodule of A, where the action of A W q on A is obtained by restricting the natural action of A q .
7 The algebra A alg q is usually referred to as a quantum Weyl algebra.
Theorem 6.6. Under the equivalence (6.15), the W -equivariant sheaf Ell * T (F m ) maps to the A W q -module representing the G-equivariant elliptic cohomology of F m :
(6.16) Ell * G (F m ) ∼ = A W + A W (Θ(z) − Θ(z −1 )) ϑ(z) 2m ⊆ A ,
where A W is the subspace of W -invariant functions in A = O an (C * ) and ϑ(z) ∈ A[z ±1/2 ] is the Jacobi theta function
(6.17) ϑ(z) := (z 1/2 − z −1/2 ) n>0 (1 − q n z)(1 − q n z −1 )
Proof. Observe that the T -space F m comes together with a natural T -equivariant map
(6.18) (G/T ) T ֒→ (G/T ) T * E 2m (T ) ∼ = F m (G, T ) ,
where (G/T ) T ⊂ G/T is the set of T -fixed points in G/T (see (3.38)). On T -equivariant elliptic cohomology, the map (6.18) induces an injective map Ell * T (F m ) ֒→ Ell * T [(G/T ) T ] , which under the isomorphism (6.10) of Theorem 6.3, corresponds to the canonical inclusion
(6.19) O Eq × O Eq /J 2m+1 O Eq ֒→ O Eq × O Eq
Now, the map (6.18) is also equivariant under the action of W which is given on (G/T ) T = S 0 simply by transposition of points. It follows that (6.19) is a morphism of W -equivariant sheaves on E q that, under equivalence (6.13), corresponds to the W -equivariant inclusion A × A/ Θ 2m+1 A ֒→ A × A , where W acts on A × A by s · (f (z), g(z)) = (g(z −1 ), f (z −1 )). As a (A q ⋊ W )-module, the product A × A is thus isomorphic to A[W ] := A ⊗ CW , where the action of A q ⋊ W is given by
a · (f (z) ⊗ w) = a(z)f (z) ⊗ w , ξ · (f (z) ⊗ w) = f (qz) ⊗ w , (6.20) s · (f (z) ⊗ w) = f (z −1 ) ⊗ sw .
Choosing a basis in CW consisting of the idempotents {e + , e − }, we can describe Ell *
T (F m ) as the (A q ⋊ W )-submodule of A[W ] (6.21) Ell * T (F m ) ∼ = A e + + A Θ(z) 2m+1 e − ,
where the isomorphism is explicitly given by (f, g) → (f + g)e + + (f − g)e − . Now, applying to (6.21) the restriction functor (6.14) and using the (obvious) algebraic identities for theta functions ϑ(z) = −z −1/2 Θ(z) and Θ(z −1 ) = −z −1 Θ(z), we get
Ell * T (F m ) W ∼ = e + A e + + e + A Θ(z) 2m+1 e − = e + A W + e + A Θ(z)ϑ(z) 2m e − = e + A W + e + A Θ(z)e − ϑ(z) 2m = e + A W + e + A e + (Θ(z) − Θ(z −1 )) ϑ(z) 2m = e + A W + A W (Θ(z) − Θ(z −1 )) ϑ(z) 2m ,
which, with our identifications Ell * G (F m ) = Ell * T (F m ) W (see (6.3)) and e + (A q ⋊ W )e + = A W q , is precisely the isomorphism (6.16). 6.3. Elliptic cohomology with twisted coefficients. The coherent sheaves Ell * T (F m ) (and a fortiori Ell * G (F m )) do not have nontrivial global sections. Indeed, by Theorem 6.3, Ell * T (F m ) fits in the short exact sequence in Coh(E q ):
(6.22) 0 → Ell * T (F m ) → O Eq ⊕ O Eq → O Eq /J 2m+1 → 0 that shows at once that H 0 an (E q , Ell * T (F m )) ∼ = C for all m ≥ 0.
With a little more work, using the long exact cohomology sequence associated to (6.22) we can also find that H 1 an (E q , Ell * T (F m )) ∼ = C 2m+2 , which -as a W -module -admits decomposition
(6.23) H 1 an (E q , Ell * T (F m )) ∼ = C ⊕(m+1) + ⊕ C ⊕(m+1) − ,
where 'C + ' and 'C − ' denote the trivial and the sign representations of W , respectively. A much richer picture emerges if we twist the elliptic cohomology sheaves Ell * T (F m ) with the Looijenga line bundle L on E q (see definitions (6.4) and (6.5)). Under the equivalence (6.8), this line bundle corresponds to the rank one free A-moduleL = A v, where the action of A q and W are determined by the relations ξ · v = q z 2 v and s · v = v (cf. Example 6.2). Since (6.8) preserves tensor products, the tensor powers L n = L ⊗n of L in Coh(E q ) correspond to the A q -modules L n = A v n with ξ · v n = q n z 2n v n and s · v n = v n . By (6.9), we can then identify the spaces of global sections of these line bundles as
(6.24) H 0 an (E q , L n ) ∼ = {f (z) ∈ A : f (qz) = q −n z −2n f (z)} , ∀ n ≥ 0 . Following [Loo77], we set (6.25) S(E) := n≥0 H 0 an (E q , L n ) ,
which, with identifications (6.24), is a graded subalgebra of A stable under the action of W . To describe this subalgebra we decompose it as the direct sum of W -invariants and anti-invariants:
(6.26) S(E) = S(E) W ⊕ S(E) −W
Then, by Looijenga Theorem (see [Loo77,(3.4)]), we know that S(E) W is a free polynomial algebra on 2 generators, while S(E) −W is a free module over S(E) W of rank one. The generators of S(E) W and S(E) −W can be explicitly given in terms of the Jacobi theta function (6.17): namely, S(E) W is generated (as an algebra) by ϑ 2 (z) and ϑ 2 (−z), which are both invariant functions in S(E) of degree 1, while S(E) −W is generated (as a module) by the function ϑ(z 2 ) which is an anti-invariant in S(E) of degree 2. Now to state our last result in this section we recall the definitions of equivariant elliptic cohomology with twisted coefficients: see formulas (6.4) and (6.5) (with M T = E q ). For X = G/T , it is well known that (see, e.g., [Gan14]): Under the identification (6.27), the composite map Ell * G (F m , L) ֒→ Ell * G (G/T, L) corresponds to the inclusion S(E) W ⊕ ϑ 2m (z) S(E) −W ֒→ S(E) , so that
(6.28) Ell * G (F m , L) ∼ = S(E) W ⊕ ϑ 2m (z) S(E) −W , where S(E) W = C[ϑ 2 (z), ϑ 2 (−z)] and S(E) −W = C[ϑ 2 (z), ϑ 2 (−z)] ϑ(z 2 ) .
Proof. We use the description of Ell * T (F m ) given in the proof of Theorem 6.6: namely, Ell *
T (F m ) = A e + + A Θ 2m+1 e − as an (A q ⋊ W )-submodule of A[W ] = A e + + A e − .
Under the equivalence (6.8), the twisted sheaves Ell * T (F m ) ⊗ L n can then be described by
(6.29) Ell * T (F m ) ⊗ A L n = A v n ⊗ e + ⊕ A Θ 2m+1 v n ⊗ e −
and we can compute their global sections using formula (6.9):
H 0 an (E q , Ell * T (F m ) ⊗ L n ) ∼ = Ker(ξ − id : Ell * T (F m ) ⊗ A L n ) ∼ = Ker(ξ − id : Av n ⊗ e + ) ⊕ Ker(ξ − id : AΘ 2m+1 v n ⊗ e − ) ∼ = H 0 an (E q , L n ) e + ⊕ (H 0 an (E q , L n ) | Θ 2m+1 ) e − , where (H 0 an (E q , L n ) | Θ 2m+1 )
denotes the subspace of all sections in H 0 an (E q , L n ) that are divisible by Θ 2m+1 under the identification (6.24). Summing up over all n ≥ 0, we find Ell * T (F m , L) ∼ = S(E) e + ⊕ (ϑ 2m+2 (z) S(E) W + ϑ 2m (z) S(E) −W ) e − Now, applying to (6.30) the restriction functor (6.14), we get
Ell * T (F m , L) ∼ = S(E) e + ⊕ (S(E) | Θ 2m+1 ) e − , where S(E) isEll * T (F m , L) W ∼ = e + S(E) e + ⊕ e + ϑ 2m+2 (z) S(E) W + ϑ 2m (z) S(E) −W e − ∼ = S(E) W ⊕ ϑ 2m (z) S(E) −W which gives (6.28) since Ell * T (F m , L) W = Ell * G (F m , L).
To complete the proof it suffices to note that the map of spaces G/T → F m induces the natural inclusion
S(E) e + ⊕ (ϑ 2m+2 (z) S(E) W + ϑ 2m (z) S(E) −W ) e − ֒→ S(E) e + ⊕ (ϑ 2 (z) S(E) W + S(E) −W ) e −
as a map representing Ell * T (F m , L) → Ell * T (G/T, L) under the isomorphism (6.30). When restricted to W -invariants this yields the inclusion
S(E) W ⊕ ϑ 2m (z) S(E) −W ֒→ S(E) W ⊕ S(E) −W = S(E)
that represents Ell * G (F m , L) ֒→ Ell * G (G/T, L). Remark 6.8. The above calculation of elliptic cohomology suggests a natural algebraic definition of quasi-invariants in the elliptic case (cf. (6.28)). This differs, however, from the definition of elliptic quasi-invariants that has already been used in the literature (see, e.g., the beautiful work of O. Chalykh on Macdonald's conjectures [Cha02]). The difference seems to be an instance of 'elliptic-elliptic' duality studied in the theory of integrable systems (see, e.g., [KS19]).
Topological Gorenstein duality
The realization of algebras of quasi-invariants raises natural questions about homotopy-theoretic analogues (refinements) of basic properties and structures associated with these algebras. In this section, we make first steps in this direction by showing that the spaces of quasi-invariants X m (G, T ) satisfy Gorenstein duality in the sense of stable homotopy theory. Our main result -Theorem 7.1 -should be viewed as a topological analogue of Theorem 2.3 on Gorensteinness of rings of quasi-invariants. For reader's convenience, we collect basic definitions from stable homotopy theory concerning duality and regularity properties of commutative ring spectra, in Appendix B. We refer to Appendix B for all unexplained notation used in this section. 7.1. Gorenstein duality of spaces of quasi-invariants. It is well known that, if X is a pointed connected topological space, the singular cochain complex C * (X, Q), computing cohomology of X with coefficients in Q, admits a commutative DG algebra model 8 . When Q is replaced by an arbitrary field k, this last fact is no longer true: in general, the cochain complex C * (X, k) is not quasi-isomorphic to any commutative DG algebra over k if char(k) = 0. A natural way to remedy this problem is to use commutative ring spectra -instead of DGAs -as models for C * (X, k). Specifically, for any commutative ring k, the cochain spectrum of the space X with coefficients in k is defined by (cf. [Man01])
(7.1) C * (X, k) := Map S (Σ ∞ X + , Hk)
where Σ ∞ X + is the suspension spectrum associated to X, Hk is the Eilenberg-MacLane spectrum of k, and Map S denotes the mapping spectrum in the category of (symmetric) spectra. By definition, (7.1) is a commutative ring spectrum with multiplication induced by the multiplication map on Hk and the diagonal map on X. In addition, following [DGI06], we introduce the chain spectrum of X:
(7.2) C * (ΩX, k) := Hk ∧ Σ ∞ (ΩX) +
which is a noncommutative ring spectrum that models the singular chain complex of the based loop space of X. Both C * (X, k) and C * (ΩX, k) are augmented k-algebras, with augmentation on C * (X, k) induced by the basepoint inclusion pt → X and on C * (ΩX, k) by the trivial map Ω → pt. For all i ∈ Z , there are natural isomorphisms
(7.3) π i [C * (X, k)] ∼ = H −i (X, k) , π i [C * (ΩX, k)] ∼ = H i (ΩX, k)
which show that C * (X, k) and C * (ΩX, k) are coconnective and connective spectra, respectively.
We are now in position to state and prove the main theorem of this section. 8 Such a model can be constructed in a functorial way, using, for example, piecewise polynomial differential forms on X defined over Q (see [BG76]).
Proof.
(1) We start with Borel fibration sequence that comes from the Ganea construction of spaces of quasi-invariants (see (3.33)):
(7.4) F m (G, T ) → X m (G, T ) pm − − → BG
To simplify the notation we set Q m := C * (F m , k) , R m := C * (X m , k) , S := C * (BG, k) .
Since F m is a finite connected complex (see (3.28)), by [DGI06,Prop. 5.3], the augmentation morphism Q m → k is cosmall (see Definition B.1). Since G is connected, the classifying space BG is simply-connected; moreover, the cohomology of BG is free, of finite type over Z, and hence, a fortiori, over any field k (see, e.g., [ (1)], we conclude from (7.5) together with our earlier observations that S → k is regular and Q m → k is cosmall that R m → k is proxy-regular. To complete the proof of part (1) it remains to note that the pair (X m , k) is of Eilenberg-Moore type for any field k. Indeed, from the fibration sequence (7.4) it follows that X m is simply-connected (since so are F m and BG); on the other hand, from the homotopy cofibration sequences (see (3.30))
F m → X m πm − − → X m+1
it follows (by induction) that X m is of finite type over k for any m ≥ 0. By construction of the Eilenberg-Moore spectral sequence, for E m = C * (ΩX m , k), we have E m ≃ Map Rm (k, k), while the equivalence R m ≃ Map Em (k, k) holds in general (see remarks in [DGI06,Sect. 4.22]). It follows that the augmented k-algebras R m and E m are both dc-complete, and then, by [DGI06,Prop. 4.17], E m is proxy-regular (since so is R m ).
(2) By the proof of Theorem 3.9, we know that (7.4) is a sphere fibration with F m ≃ S 4m+2 . Hence, F m is a Poincaré duality space of dimension 4m + 2, then its cochain spectrum Q m = C * (F m , k) satisfies Poincaré duality of dimension a = −4m − 2 (in the sense of [DGI06,8.11]). Since Q m is cosmall, by [DGI06,Prop. 8.12], we conclude that Q m is Gorenstein of shift a = −4m − 2. Further, by [DGI06, 10.2], we also know that S = C * (BG, k) is Gorenstein of dimension a = dim(G) = 3. Now, consider the morphism of cochain spectra p * m : S → R m induced by the whisker map p m : X m → BG in (7.4). We claim that R m is finitely built from S via p * m . To see this denote by E := C * (ΩBG, k) ∼ = C * (G, k) the chain spectrum of BG. Since G is simply-connected, E is a connective k-algebra with π 0 (E) ∼ = k[π 1 (G)] = k (see (7.3)). Since S is of Eilenberg-Moore type, there is an equivalence S ≃ Map E (k, k) . Furthermore, if we set M m := C * (F m , k), the action of G on F m induces a left E-module structure on M m , and by a standard Eilenberg-Moore spectral sequence argument there is an equivalence R m ≃ Map E (M m , k) . Since π * (M m ) ∼ = H * (F m , k) is finite-dimensional over k, the E-module M m is finitely built from k. Now, Proposition 3.18 of [DGI06] implies that R m ≃ Map E (M m , k) is finitely built from S ≃ Map E (k, k) as we claimed. Since R m is proxy-regular and both S and Q m are Gorenstein, it follows from [DGI06,Prop. 8.10] that R m is Gorenstein as well. The Gorenstein shift of R m can be computed from the following equivalence of k-modules induced by (7.5) (see [DGI06,Prop. 8.6]):
Map Rm (k, R m ) ≃ Map Qm (k, Map S (k, S) ∧ k Q m ) ≃ Map Qm (k, (Σ 3 k) ∧ k Q m ) ≃ Σ 3 Map Qm (k, Q m ) ≃ Σ 3 (Σ −4m−2 k) ≃ Σ 1−4m k
To complete part (2) it remains to note that, for a simply-connected space X of finite type over k, the cochain spectrum C * (X, k) is automatically orientable Gorenstein whenever it is Gorenstein. This follows from the fact that, under the above assumptions, k carries a unique action of E = Map C * (X,k) (k, k) ≃ C * (ΩX, k) (see [Gre18,Sect. 18.3] and also the proof of [BCHV21, Lemma 3.8]).
(3) follows from (2) by a standard argument. If an augmented k-algebra R is orientable Gorenstein of shift a, then
(7.6) Cell k (R) ≃ Map R (k, R) ∧ E k ≃ Σ a Map R (k, Map k (R, k)) ∧ E k ≃ Σ a Cell k [Map k (R, k)]
where the first and the last equivalences are given by (B.2) and the one in the middle is induced by (B.5). For R = C * (X, k) with π 0 (R) ∼ = H 0 (X, k) ∼ = k, we have π i Map k (R, k) = 0 for i ≪ 0. By [DGI06,Remark 3.17], the R-module Map k (R, k) is then built from k and therefore k-cellular in Mod R . Condition B.8 thus follows from (7.6). This completes the proof of the theorem. 7.2. Generalized spaces of quasi-invariants. It is natural to ask whether the result of Theorem 7.1, i.e. the topological Gorenstein property, holds for generalized ('fake') spaces of quasiinvariants introduced in Section 4. In view of Corollary 4.8, the answer is obviously affirmative when k is a field of characteristic 0. The next theorem shows that this is also true when k = F p . We keep the notation G = SU (2) and T = U (1); however, as in Section 4, we do not identify T as a maximal torus in G.
Theorem 7.2. Let B be a space in the genus of BG that admits an essential map from BT , and let X m = X m (ΩB, T ) be the space of m-quasi-invariants associated to B. Then, for any prime p, the morphism C * (X m , F p ) → F p is Gorenstein of shift a = 1 − 4m.
Proof. We give the part of the proof that differs from that of Theorem 7.1. First, observe that, for any space B in the genus of BG, we have equivalences of cochain spectra
C * (B, F p ) ≃ C * (B ∧ p , F p ) ≃ C * ((BG) ∧ p , F p ) ≃ C * (BG, F p ) , where ( − ) ∧
p denotes the F p -completion functor on pointed spaces. This follows from the fact that both B and BG are F p -good spaces (in the sense of [Bou75]) and B ∧ p ≃ (BG) ∧ p for any prime p. The above equivalences are compatible with augmentation; hence, by [DGI06, 10.2], we conclude that C * (B, F p ) → F p is a regular map, Gorenstein of shift dim(G) = 3. Now, assume that B satisfies the conditions of Theorem 4.5. Let F = F (ΩB, T ) denote the homotopy fibre of the maximal essential map p B : BT → B. Recall that this last space is not equivalent to a finite CW complex (unless B ≃ BG), and hence its cochain spectrum C * (F, F p ) need not be cosmall (as in the case of BG). Nevertheless, we claim that C * (F, F p ) → F p is always proxy-regular and satisfies the Gorenstein property of shift (−2). To see this consider the homotopy fibration sequence ΩB → F → BT associated to the map p B : BT → B. Since BT ≃ CP ∞ is of Eilenberg-Moore type (see [DGI06,4.22]), we have
C * (ΩB, F p ) ≃ C * (F, F p ) ∧ C * (BT, Fp) F p
In view of the fact that ΩB ≃ S 3 , the map C * (ΩB, F p ) → F p is cosmall, and hence, by [DGI06,Prop. 4.18], C * (F, F p ) → F p is proxy-regular. Furthermore, since F p is small over C * (BT ) = C * (BT, F p ), we have a natural equivalence of C * (BT )-modules
Map C * (BT ) (F p , C * (BT )) ∧ C * (BT ) C * (F ) ∼ − → Map C * (BT ) (F p , C * (F )) ,
which, by the proof of [DGI06, Prop. 8.6], implies that C * (F, F p ) → F p is Gorenstein of shift a = 1 + (−3) = −2.
The rest of the proof is parallel to that of Theorem 7.1. In brief, by Theorem 3.1, the fibre of the m-th Ganea fibration F m → X m → B defining the space X m = X m (ΩB, T ) has the homotopy type of Σ 4m F . Hence its cochain spectrum C * (F m , F p ) is Gorenstain of shift a = −2 − 4m. By induction, each space X m is of finite type over F p . Since C * (B, F p ) → F p is a regular Gorenstein map of shift 3, it follows from the above fibration sequence that C * (X m , F p ) → F p is Gorenstein of shift a = −2 − 4m + 3 = 1 − 4m.
with topological presentation is given by (x, t, y) = tx + (1 − t)y. The advantage of this notation is that it naturally extends to 'higher dimensions': the iterated joins of spaces
(A.2) X 0 * X 1 * . . . * X n = {t 0 x 0 + t 1 x 1 + . . . + t n x n : (t 0 , . . . , t n ) ∈ ∆ n , x i ∈ X i }/ ∼
where the equivalence relation is defined by
n i=0 t i x i ∼ n i=0 t ′ i x ′ i if and only if t i = t ′ i (for all i) and x i = x ′ i whenever t i = t ′ i > 0.
Note that, under this equivalence relation, if t i = 0 for some i, the point x i in t 0 x 0 + . . . + 0x i + . . . + t n x n ∈ X 0 * . . . * X n can be chosen arbitrarily (or simply omitted).
There is also a convenient way to represent joins by homotopy colimits. For example, it is well-known that the join of two spaces is represented by the homotopy pushout
(A.3) X * Y = hocolim[X ← X × Y → Y ]
where the maps are canonical projections and the "hocolim" is taken either in the category of pointed or unpointed spaces depending on whether we consider reduced or unreduced joins.
(A.4) X 0 * X 1 * . . . * X n = hocolim P(∆ n ) (F X )
where P(∆ n ) is the poset of all non-empty faces of the n-simplex ∆ n (ordered by reversed inclusions) and the diagram F X : P(∆ n ) → Top is defined by assigning to a face ∆ I ∈ P(∆ n ) the product of spaces i∈I X i (with indices corresponding to the vertices of ∆ I ) and to an inclusion of faces ∆ J ⊂ ∆ I the canonical projection i∈I X i → j∈J X j . It is easy to see that formula (A.4) boils down to (A.3) in case of two spaces. Now, we can describe the Milnor model. For integer n 0, we define a sequence of spaces E n G by taking the (unreduced) iterated joins of copies of G:
(A.5) E n G := G * G * . . . * G (n + 1 times) .
Each space E n G carries natural (diagonal) left and right G-actions each of which is free. We will use the right G-action E n G × G → E n G that can be written explicitly (with notation (A.2)) as (A.6) (t 0 g 0 + t 1 g 1 + . . . + t n g n ) · g = t 0 g 0 g + t 1 g 1 g + . . . + t n g n g where g 0 , . . . , g n , g ∈ G . Moreover, there are natural G-equivariant maps E n G ֒→ E n+1 G: t 0 g 0 + . . . + t n g n → t 0 g 0 + . . . + t n g n + 0 · e making {E n G} n 0 into a direct system of (right) G-spaces. We set B n G = E n G/G and define G → E n G → B n G .
The main observation of [Mil56] (see loc. cit., Theorem 3.1) is that the principal G-bundle (A.10) is n-universal in the sense that its total space is (n − 1)-connected (i.e., π i (E n G) = 0 for all i < n).
In the inductive limit, this gives Theorem A.1 (Milnor). For any topological group G the natural (quotient) map EG → BG is a numerable principal G-bundle, which is universal among all such G-bundles.
A detailed proof of Theorem A.1 can be found in [Hus75] (see Chap. 4, Theorem 11.2). We only recall one basic topological fact behind this proof that we will use repeatedly in this paper.
Lemma A.2 ([Mil56], Lemma 2.3). If each space X i in the iterated join (A.4) is (c i −1)-connected, then the space X 0 * X 1 * . . . * X n is ( c i + n − 1)-connected.
Appendix B. Duality of commutative ring spectra
In this Appendix, we collect basic definitions from stable homotopy theory concerning duality and regularity properties of commutative ring spectra. Our main references are the paper [DGI06] by Dwyer, Greenlees and Iyengar, where many concepts that we need were originally introduced, and the lecture notes of Greenlees [Gre18] that supplement [DGI06] with motivation and examples. As in [DGI06], we will work in the (stable model) category of symmetric spectra, which can be succinctly described as the category Mod S of modules 10 over the symmetric sphere spectrum S = ((S 1 ) ∧n ) n≥0 (see [HSS00]). The category Mod S is equipped with a symmetric monoidal product which is denoted as a smash A ∧ B or tensor product A ⊗ S B (depending on the context). A ring spectrum is then, by definition, an S-algebra, i.e. an S-module R given with two structure maps S → R and R ∧ R → R satisfying the usual unitality and associativity properties. We denote the category of ring spectra by Alg S . There is a natural (Eilenberg-MacLane) functor H : Alg Z → Alg S , k → Hk that embeds the category Alg Z of usual (discrete) associative rings into S-algebras by identifying a ring k with its symmetric Eilenberg-MacLane spectrum Hk = (K(k, n)) n≥0 (see [HSS00, 1.2.5]). The category Alg S can be thought of as a homotopical refinement ('thickening') of Alg Z in the same way as the category Mod S is a homotopical refinement of the category Mod Z of (discrete) abelian groups.
For a ring spectrum R ∈ Alg S , we let Mod R denote the category of left module spectra over R. This is a stable model category enriched over Mod S . The latter means that, for two R-modules A and B, there is a mapping spectrum of R-module maps A → B that we denote Map R (A, B). Moreover, if A is a right R-module and B is a left R-module, there is an associated smash product A ∧ R B defined as the (homotopy) coequalizer A ∧ R ∧ B ⇒ A ∧ B of structure maps A ∧ R → A and R ∧ B → B in Mod S . Note that both Map R (A, B) and A ∧ R B are understood as 'derived' objects in the sense that their first arguments are (replaced by) cofibrant objects in Mod R . In particular, if A and B are usual (discrete) modules over a usual (discrete) ring R, viewed as symmetric spectra via the Eilenberg-MacLane functor, then π i Map R (A, B) ∼ = Ext −i R (A, B) and π i (A ∧ R B) ∼ = Tor R i (A, B) , where π i stand for the (stable) homotopy groups of spectra. If R is a commutative ring spectrum, then both Map R (A, B) and A ∧ R B are naturally R-modules, i.e. objects in Mod R .
Next, we recall that a subcategory of a (stable) model category M is called thick if it is closed under weak equivalences, cofibration sequences (distinguished triangles) and retracts in M. Further, a subcategory of M is called localizing if it is thick and, in addition, closed under arbitrary coproducts (and hence homotopy colimits) in M. Given two objects A and B in M, we say that B is built from A if B belongs to the localizing subcategory of M generated by A, and B is finitely built from A if it belongs to the thick subcategory generated by A ([DGI06, 3.15]). Now, if M = Mod R , an R-module A is called small if it is finitely built from R in Mod R . This agrees with the usual definition of small (compact) objects in Mod R : an R-module A is small iff Map R (A, − ) commutes with arbitrary coproducts.
10 Unfortunately, the term 'S-module' in reference to spectra is very ambiguous: apart from symmetric, other popular types of spectra (e.g., orthogonal and EKMM ones) are also S-modules. A nice recent survey comparing properties and applications of different types of spectra can be found in [Dug22].
The notion of a localizing subcategory is closely related to that of cellularization. For a fixed object A ∈ Mod R , we say that a morphism f : M → N in Mod R is an A-cellular equivalence if f induces a (weak) equivalence on mapping spectra:
f * : Map R (A, M ) ∼ − → Map R (A, N )
Note that every equivalence in Mod R is automatically an A-cellular equivalence, but the converse, in general, is not true. Now, an R-module B is called A-cellular if any A-cellular equivalence f : M → N induces an equivalence Map R (B, M ) ∼ − → Map R (B, N ) . This terminology is motivated by the fact that the A-cellular modules are precisely those objects of Mod R that are built from A (see [Hir03,5.1.15]). Moreover, for any R-module B, there is a A-cellular module Cell R A (B) together with a A-equivalence in Mod R :
Cell R A (B) → B called an A-cellular approximation 11 of B. Such an approximation is determined by B uniquely up to canonical equivalence; we will use the simpler notation Cell A (B) for Cell R A (B) when the ring spectrum R is understood.
The above categorical notions can be used to impose some finiteness and regularity conditions on commutative ring spectra. First, we say that a morphism of commutative ring spectra R → k is called regular if k is small as an R-module. This definition is motivated by the fact that, in classical commutative algebra, a local Noetherian ring R with residue field k = R/m is regular iff k has a finite length resolution by f.g. free R-modules (see [Ser00]); for the associated Eilenberg-MacLane spectra, the latter means that Hk is finitely built from HR. A more flexible and technically useful condition is obtained by weakening the regularity assumption on R → k in the following way.
Definition B.1 ([DGI06], 4.6). A morphism of commutative ring spectra R → k is called proxyregular if k is a proxy-small R-module via R → k in the sense that there is a small R-module K that builds k and is finitely built from k in Mod R . Note that if K = k, then R → k is regular. On the other extreme, if K = R then R → k is called cosmall.
Let E := Map R (k, k) denote the endomorphism ring spectrum of k viewed as a left R-module via the morphism R → k. There is a standard Quillen adjunction relating right E-modules to left R-modules:
(B.1) ( -) ∧ E k : Mod E op ⇆ Mod R : Map R (k, -)
If R → k is regular, the functors (B.1) induce an equivalence between Ho(Mod E op ) and the full subcategory of Ho(Mod R ) consisting of k-cellular R-modules (see [Gre18, Theorem 6.1]).
If R → k is proxy-regular, (B.1) does not induce an equivalence in general, but the counit of this adjunction still provides a k-cellular approximation for modules in Mod R (see [ Now, we come to the key definition of a Gorenstein ring spectrum that we state under the regularity assumptions of Definition B.1 (which is a slightly less general form than in [DGI06]):
Definition B.2 (cf. [DGI06], 8.1 and 8.4). A morphism of commutative ring spectra R → k is called Gorenstein of shift a ∈ Z, if R → k is proxy-regular and there is an equivalence of k-modules
(B.4) Map R (k, R) ≃ Σ a k
where Σ denotes the suspension functor on Mod k .
We will be mostly interested in ring spectra R that are augmented k-algebras over a field k. For such algebras, we will always assume that R → k is the given augmentation morphism on R, and we will simply say that R is Gorenstein if so is R → k. The Gorenstein condition (B.4) can be slightly refined in this case. Note that, if R is a k-algebra, using the k-module structure on R, we can rewrite (B.4) in the form
(B.5) Map R (k, R) ≃ Σ a Map R (k, Map k (R, k))
Both sides of (B.5) have natural right module structures over the endomorphism ring E = Map R (k, k) but, in general, these module structures need not to agree under the equivalence (B.5). Following [DGI06] (see also [Gre18,Section 18.2])), we say that an augmented k-algebra R is orientable Gorenstein if (B.5) is an equivalence of right E-modules.
If R is a local Noetherian ring of Krull dimension d with residue field k = R/m, then R is Gorenstein (in the sense of commutative algebra) iff The last property of ring spectra that we want to review is concerned with double centralizers. Recall, for a morphism R → k, the double centralizer of R is defined to beR := Map E (k, k), where E = Map R (k, k) is the endomorphism spectrum of k in Mod R . The left multiplication on k gives a morphism of ring spectra R →R , and following [DGI06], we say Note that, in algebra, a surjective homomorphism R → k from a Noetherian commutative ring R to a field k is dc-complete iff R ∼ =R I , whereR I := lim ← − R/I n is the I-adic completion of R with respect to the ideal I = Ker(R → k). This motivates the above terminology. One can show that if R → k is dc-complete, the regularity properties of the ring spectra R and E are strongly connected (see, e.g., [DGI06,Proposition 4.17]).
( 3 )
3Each projection p m : V m → V //W factors naturally (in m) through V m //W , inducing isomorphisms of schemes V m //W ∼ = V //W for all m ∈ M(W ) .
Fix a maximal torus T ⊆ G and write N = N G (T ) for its normalizer in G. Let W := N/T be the associated Weyl group. The W acts naturally on T by conjugation: W × T → T , w · t = ntn −1 , and on the classifying space BT = EG/T via the right action of G on EG: W × BT → BT , w·[x] T = [xn −1 ] T , where w = nT ∈ W and [x] T denotes the T -orbit of x in EG. Let p : BT ։ BG denote the natural fibration, i.e. the quitient map induced by the inclusion T ֒→ G.
mapp m : X m (G, T ) hW → BG being a cohomology isomorphism: thus, for all m ∈ M(W ), we have algebra isomorphisms H * W (X m , Q) ∼ = H * (BG, Q) (QI 4 ) Each map π m,m ′ in (2.10) induces an injective homomorphism on cohomology so that the Borel homomorphism p * factors into a M(W ) op -diagram of algebra maps H * (BG, Q) ֒→ . . . ֒→ H * (X m ′ , Q) π * m,m ′ ֒→ H * (X m , Q) ֒→ . . . ֒→ H * (BT, Q) (QI 5 ) With natural identification H * (BT, Q) = Q[V ] (see (2.9)), the maps π * 0,m : H * (X m , Q) → H * (BT, Q) in (QI 4 ) induce isomorphisms
(3. 5 )
5X m := hocof * (j m−1 ) , F m := hofib * (p m ) , ∀ m ≥ 1 . Note that the horizontal arrows p m in (3.4) are whisker maps making each row F m jm − − → X m pm − − → B of the above diagram a homotopy fibration sequence. On the other hand, the vertical arrows π m are canonical maps making each triple F m jm − − → X m πm − − → X m+1 a homotopy cofibration sequence.The main observation of[Gan65] is that the homotopy fibres in (3.4) can be described explicitly in terms of iterated joins 2 of based loop spaces ΩB. More precisely, we have 2 We review the definition and basic topological properties of joins in Appendix A.
For all m ≥ 1, there are natural homotopy equivalences F m ≃ F * ΩB * . . . * ΩB (m-fold join) compatible with the fibre inclusions F m → F m+1 in (3.4).
Example 3. 2 (
2LS-categories). Recall that the LS-category of a topological space B is defined to be cat(B) := n − 1, where n is the least cardinality of an open cover {U 1 , . . . , U n } of B such that each U i is contractible as a subspace in B. Given a pointed connected space B, one applies the fibre-cofibre construction to the canonical path fibration ΩB → P * B p − → B . The result is the sequence of spaces
( 3 .
336) T * T ∼ = G can be explicitly written as tλ + (1 − t)µ → t 1/2 λ + (1 − t) 1/2 µj , where G = SU (2) is identified with the group of unit quaternions in H = C ⊕ Cj and T = U (1) with unit complex numbers. Similarly, we can define a T -equivariant homeomorphism (3.37) (G/T ) T * T ∼ = G/T where (G/T ) T denotes the set of T -fixed points in G/T . Combining (3.36) and (3.37) with natural associativity isomorphisms for joins, we get
:
in Q[V ] W thus defining a linear operator on W -quasi-invariants: Q m (W ) → Q m (W ) . The operator (3.47) has a natural topological interpretation in terms of our spaces of quasiinvariants. The proof of Theorem 3.9 shows that the basic fibration (3.33) is equivalent to a sphere fibration with fibre F m ≃ S 4m+2 . Hence, associated to (3.33) there is a Gysin long exact sequence of the form (see, e.g., [McC01, Example II.5.C]): (3.48) . . . → H n (BG, Q) p * m − − → H n (X m , Q) (pm) * −−−→ H n−4m−2 (BG, Q) → H n+1 (BG, Q) → . . .
Proposition 3. 19 .
19Under the isomorphism of Theorem 3.9, the operator (3.49) coincides with the divided difference operator (3.47) of multiplicity m: i.e.,
* m • (p m ) * Proof. Since the algebra homomorphism p * m : H * (BG, Q) → H * (X m , Q) is injective (for all m), the Gysin sequence (3.48) breaks up into short exact sequences(3.51) 0 → H * (BG, Q) p * m − − → H * (X m , Q) (pm) * −−−→ H * −4m−2 (BG, Q) → 0Now, if we identify H * (BG, Q) = Q[c 2 ] and H * (X m , Q) = Q[x 2 , x 2m+1 ] as in (the proof of) Theorem 3.9, the map p * m takes c 2 to x 2 and hence c k 2 to x 2k for all k ≥ 0. By exactness of (3.51), we then conclude that (p m ) * (x 2k ) = 0 , while (p m ) * (x 2m+1 ) = κ m , where κ m ∈ Q × is a nonzero constant. Hence, p * m (p m ) * (x 2k ) = 0 for all k ≥ 0; on the other hand, by projection formula,
Theorem 4. 2 (
2Rector). Let G = SU (2), and let B be a space in the genus of BG. Then, for each prime p, there is a homotopy invariant (B/p) ∈ {±1} called the Rector invariant of B at p, such that(1) The set {(B/p)}, where p runs over all primes, is a complete set of invariants of B in the genus of BG.
3 )
3The Rector invariant of B = BG equals 1 at all primes p. (4) The space B admits a maximal torus 4 if and only if B is homotopy equivalent to BG.
Remark 4. 3 .
3Each space B in the genus of BSU (2) defines a loop structure on S 3 , i.e. ΩB ≃ S 3 . Conversely, a uniqueness theorem of Dwyer, Miller and Wilkerson[DMW87] implies that every loop structure on S 3 belongs to the genus of BSU (2). Thus, Theorem 4.2 combined with results of[DMW87] provides a complete classification of finite loop spaces of type SU (2).
Theorem 4. 5 (
5Yau). Let G = SU (2), and let B be a space in the genus of BG. Then (1) B admits an essential map f : BT → B if and only if there is an integer k = 0 such that (B/p) = (k/p) for all but finitely many primes p, where (k/p) denotes the Legendre symbol 5 of k.
(4. 3 )
3N B := min{lcm(n p ) ∈ N : B = holim {p} (n p • r p )} , which is clearly a homotopy invariant of B. Note that N B = 1 iff B = BG; however, in general, N B does not determine the homotopy type of B (see [Yau04, (1.8)] for a counterexample).
Lemma 4. 6 .
6For any space B in the genus of BG, H * (B, Z) ∼ = Z[u] , where |u| = 4. If B admits an essential map from BT , then, with natural identification H * (BT, Z) ∼ = Z[x] as in Theorem 3.9, the map p * B :H * (B, Z) → H * (BT, Z) is given by p * B (u) = N B x 2 , where N B is defined by (4.3).Proof. The first claim can be deduced easily from the fact that ΩB ≃ S 3 by looking at the Serre spectral sequence of the path fibration ΩB → P * B → B (cf.[Rec71b, §4]). The second claim is a consequence of the last part of [Yau04, Theorem 1.7], which shows that (4.3) equals (up to sign) the degree of the map p * B on K-theory with coefficients in Z and hence on cohomology. Theorem 4.7. Let B be a space in the genus of BG admitting an essential map from BT .
All maps π m in (4.1) are injective on rational cohomology. For each m ≥ 0, the composite mapπ m = π m−1 . . . π 1 π 0 induces an embeddingH * (X m (ΩB, T ), Q) ֒→ H * (BT, Q) = Q[x] with image Q m (W ) ⊆ Q[x]. Thus, H * (X m (ΩB, T ), Q) ∼ = Q m (W ) for all m ≥ 0.(ii) For each m ≥ 0, there is an algebra isomorphismH * (X m (ΩB, T ), Q) ∼ − → H * (X m (G, T ), Q)making commutative the diagram
(
It follows that Q ′ m = Q m as subrings of Q[x] for all m. Now assume that H * (X m (ΩB, T ), Q) ∼ = Q ′ m , , and thatπ * m is the inclusion Q ′ m ֒→ Q[x]. To compute the cohomology of the fibre F m (ΩB, T ), we use the Eilenberg-Moore spectral sequence for the fibration sequence F m (ΩB, T ) → X m (ΩB, T ) → B, whose E 2 -term is H * (pt), H * (X m (ΩB, T ))) ∼ = Tor * , * Q[u] (Q, Q ′ m ) By Lemma 4.6, Tor Q[u] * , * (Q, Q ′ m ) is the (co)homology of the complex
5. 2 .
2K-theory of quasi-flag manifolds. We first introduce rings Q m (W ) of exponential quasiinvariants of a Weyl group W . Let G be a compact connected Lie group with maximal torus T and associated Weyl group W . LetT := Hom(T, U (1)) denote the character lattice and R(T ) the representation ring of T . It is well known that R(T ) ∼ = Z[T ] via the canonical map induced by taking characters of representations, and R(T ) W ∼ = R(G) via the restriction map i * : R(G) → R(T ) induced by the inclusion i : T ֒→ G (see, e.g., [Bou82, Chap. IX, Sect. 3]). Using the first isomorphism we identify R(T ) = Z[T ]
holds for all α ∈ R, see [Dem74, Sect. 5.5].) Example 5.5. We describe Q m (W ) explicitly in the case of G = SU (2) and T = U (1) the diagonal torus. In this caseT coincides with the weight lattice P (R) which is generated by the fundamental weight ̟ : T → U (1) defined by ̟ t 0 0 t −1 = t. The corresponding (simple) root is α = 2̟, and the Weyl group W = s α ∼ = Z/2Z acts onT by s α (̟) = −̟. We have (5.3) R(T ) ∼ = Z[z, z −1 ] , R(G) = R(T ) W ∼ = Z[z + z −1 ]
(2) and T = U (1) and use the identification K * (BT ) ∼ = Z[[t]] as in the previous section. Let B be a space in the genus of BG that admits an essential map from BT . By [Yau04,Proposition 2.1], there is an isomorphism of rings K * (B) ∼ = Z[[u]] , such that for any essential map f : BT → B, the induced map f * : K * (B) → K * (BT ) is given by f * (u) = deg(f )t 2 + higher order terms in t ,where the integer deg (f ) coincides (up to sign) with the degree of f in integral (co)homology in dimension 4 (cf. Lemma 4.6). In fact, by a general result ofNotbohm and Smith (see[NS90, Theorem 5.2]), the assignment f → f * gives a bijection between the homotopy classes of maps from BT to B an the λ-ring homomorphisms from K * (B) to K * (BT ):[BT, B] * ∼ = Hom λ (K * (B), K * (BT )) .Next, recall that, by Theorem 4.5, among all essential maps BT → B, there is a 'maximal' one p B : BT → B, for which deg (p B ) = N B , where N B is the integer defined by (4.3)
Example 5 . 9 .
59For B = BG, one can easily compute the power series p B (u) in an explicit form. Recall that the Atiyah-Segal completion theorem gives an isomorphism K * (BG) ∼ = K * G (pt) I , where
Theorem 5. 10 .
10There are isomorphisms of rings(5.16) K * [X m (ΩB, T )] ∼ = Q m (B) , ∀ m ≥ 0 . In particular, K 1 [X m (ΩB, T )] = 0 for all m ≥ 0. The maps π * m : K * [X m+1 (ΩB, T )] → K * [X m (ΩB,T )] induced by the Ganea maps π m in (4.1) correspond under (5.16) to the natural inclusions Q m+1 (B) ֒→ Q m (B), and hence are all injective.
(
We further claim that the ring homomorphisms j * m,B : K * [X m (ΩB, T )] → K * [F m (ΩB, T )] induced by the fibre maps j B,m in (4.1) are surjective, and with (5.16), they induce isomorphismsK * [F m (ΩB, T )] ∼ = Q m (B)/(p * B (u))We prove these facts together with the claims of Theorem 5.10 by induction on m.For m = 0, we need only to compute K * [F 0 (ΩB, T )]. This can be done using Lemma 5.11. Note that K(pt) ∼ = Z has the obvious free resolution over K * (B) ∼ = Z[[u]] Z, K * (BT )) can be identified with homology of the two-term complex 0 →Z[[t]] p * B (u) −−−→ Z[[t]] → 0 , where the map in the middle is given by the power series (5.14). Since Z[[t]]is an integral domain, it follows that Tor K * (B) i (Z, K * (BT )) = 0 for i > 0. The Eilenburg-Moore spectral sequence of Lemma 5.11 therefore collapses, giving an isomorphism
(M T , L n ) and Ell * G (pt, L) = ∞ n=0 H 0 an (M T , L n ) W
acquires the structure of a sheaf of C[x]-modules via the map C[x] → O Eq (U ), x → θ, where θ is a generator of the maximal ideal of the local ring O C * ,1 . The desired lemma therefore, follows from the fact that H * T (E n T ; C) ∼ = H * (B n T ; C) ∼ = C[x]/x n+1 (see the proof of Lemma 3.13 above).
G (G/T, L) ∼ = Ell * T (pt, L) = S(E)We extend this result to the quasi-flag manifolds F m = F m (G, T ).
Theorem 6 . 7 .
67The natural mapsG/T = F 0 (G, T ) → F 1 (G, T ) → . . . → F m−1 (G, T ) → F m (G, T ) → . . .induce injective homomorphisms on twisted elliptic cohomology: . . . ֒→ Ell * G (F m , L) ֒→ Ell * G (F m−1 , L) ֒→ . . . ֒→ Ell * G (G/T, L) .
the Looijenga ring (6.25). To compute (S(E) | Θ 2m+1 ) we observe that an element of S(E) is divisible by Θ 2m+1 in A if and only if its invariant and anti-invariant parts in S(E) W and S(E) −W are both divisible by Θ 2m+1 . Now, for f (z) ∈ S(E) W , Θ 2m+1 (z) divides f (z) if and only if ϑ 2m+2 (z) divides f (z), while for f (z) ∈ S(E) −W , Θ 2m+1 (z) divides f (z) if and only if ϑ 2m (z) ϑ(z 2 ) divides f (z). Thus (6.30)
Theorem 7. 1 .
1Let X m = X m (G, T ) be the space of m-quasi-invariants associated to G = SU (2). Let R m := C * (X m , k) and E m := C * (ΩX m , k) denote the cochain and chain spectra of X m with coefficients in an arbitrary field k. Then, for any m ≥ 0,(1)R m and E m are proxy-regular (Definition B.1) and dc-complete (Definition B.4) with Map Rm (k, k) ≃ E m and Map Em (k, k) ≃ R m (2) R m is orientable Gorenstein of shift a = 1 − 4m (Definition B.2) (3) R m satisfies Gorenstein duality of shift a = 1 − 4m (Definition B.3)
(A. 7 )
7EG := lim − → E n G and BG := lim − → B n G .By construction, the spaces EG and BG come equipped with canonical filtrationsE 0 G ֒→ . . . ֒→ E n G ֒→ E n+1 G ֒→ . . . ֒→ EG (A.8) B 0 G ֒→ . . . ֒→ B n G ֒→ B n+1 G ֒→ . . . ֒→ BG (A.9)with consecutive terms (at each level n) forming the principal G-bundles (A.10)
k (M ) ≃ Map R (k, M ) ∧ E k Moreover, for all R-modules M , there is a natural equivalence (see [Gre18, Lemma 6.6]) (B.3) Cell k (M ) ≃ Cell k (R) ∧ R M Formula (B.2) shows that when R → k is proxy-regular, the k-cellular approximation Cell k (M ) is functorial and effectively constructible in Mod R (cf. [DGI06, Definition 4.3]). 11 Cellularization is an example of a general model-categorical construction called right Bousfield localization (colocalization) with respect to an object A. In this language, A-cellular equivalences are called A-colocal equivalences, A-cellular objects are A-colocal objects, and A-cellular approximations are functorial cofibrant replacements in the A-colocal model structure on ModR (see [Hir03, 3.1.19]).
(B.6) can be written as an equivalence RHom R (k, R) ≃ Σ d k in the derived category D(R) and thus corresponds to the Gorenstein condition (B.4) of Definition B.2. In classical commutative algebra, there is another well-known characterization of Gorenstein rings in terms of local cohomology: be viewed as a special case of Grothendieck's local duality theorem. The following definition is a topological analogue of (B.7). Definition B.3. An augmented k-algebra R satisfies Gorenstein duality of shift a if there is an equivalence of R-modules(B.8) Cell k (R) ≃ Σ a Map k (R, k)While the algebraic conditions (B.6) and (B.7) are known to be equivalent, their topological analogues (B.4) and (B.8) are, in general, not (see, e.g., [BCHV21, Remark 2.11] for a counterexample). This necessitates two separate definitions for Gorensteinness of commutative ring spectra.
Definition B.4 ([DGI06], 4.16). R → k is dc-complete if R ∼ − →R is an equivalence in Alg S .
where α H denotes the principal ideal in C[V ] generated by the form α H . For each H ∈ A, the congruence (1.2) simply means that the polynomial s H (p) − p is divisible in C[V ] by the power of the linear form α H determined by the value of the multiplicity function m. It is easy to see that the set of all polynomials satisfying (1.2) (for m fixed) forms a graded subalgebra in C[V ], which we denote Q m (W ). Following[CV90], we call Q m (W ) the algebra W -quasi-invariant polynomials of multiplicity m. Note that, for m = 0, we have Q 01.2)
s H (p) ≡ p mod α H
2m H
denote the polynomial algebra of V . This algebra carries a natural W -action (extending the linear action of W on V * ) and can thus be viewed as a CW -module. We can then characterize the invariant polynomials p ∈ C[V ] W by the equations which hold for all hyperplanes H ∈ A. To define quasi-invariants we relax the equations (2.1) in the following way (cf. (1.2)). For each hyperplane H ∈ A, we fix a linear form α H ∈ V * , such that H = Ker(α H ) , and choose n H − 1 positive integers {m H,i } i=1,..., n H −1 which we refer to as multiplicities of H. We assume that m H,i = m H ′ ,i for each i whenever H and H ′ are in the same orbit of W in A. We write M(W ) := {m H,i ∈ Z + : i = 1, . . . , n H − 1} [H]∈A/W for the set of all such multiplicities regarding them as functions on the set A/W of W -orbits in A. H m H,i , i = 1, . . . , n H − 1 , for all H ∈ A. We write Q m (W ) for the subspace of all such polynomials in C[V ].(2.1)
e H,−i (p) = 0 , i = 1, . . . , n H − 1 ,
Definition 2.1 ([BC11]). A polynomial p ∈ C[V ] is called a W -quasi-invariant of multiplicity
m = {m H,i } ∈ M(W ) if it satisfies the conditions
(2.2)
e H,−i (p) ≡ 0 mod α H
n
extends the result of Borel's Theorem 2.7 to an arbitrary G-space X (see, e.g., [Hsi75, Chap III, Prop. 1]). For X = F m (G, T ), it follows from Corollary 3.16 that
MT91, III.3.17]). Therefore, in terminology of[DGI06, Sect. 4.22], the pair (BG, k) is of Eilenberg-Moore type. Since H * (ΩBG, k) ∼ = H * (G, k) is finitedimensional over k, it follows from [DGI06, 5.5(3)] that S → k is a regular morphism, i.e. k is small as an S-module. Next, since (BG, k) is of Eilenberg-Moore type, the fibration sequence (7.4) gives an equivalence of cochain spectra (see, e.g., [BCHV21, Lemma 3.7])(7.5) Q m ≃ k ∧ S R mNow, by[DGI06, Prop. 4.18
Formula (A.3) generalizes to iterated joins (see, e.g. [WZv99, Prop. 5.1])
We point out that the topological Gorenstein shifts a of Theorem 7.1 and Theorem 7.2 agree with the algebraic one of Theorem 2.3: to see this it suffices to change the standard polynomial grading on Q m (W ) to the cohomological one. by 'doubling' degrees of the generatorsRemark 7.3. We point out that the topological Gorenstein shifts a of Theorem 7.1 and Theo- rem 7.2 agree with the algebraic one of Theorem 2.3: to see this it suffices to change the standard polynomial grading on Q m (W ) to the cohomological one (by 'doubling' degrees of the generators).
This universal property determines the space BG uniquely up to homotopy, i.e. as a unique (up to unique isomorphism) object in the homotopy category Ho(Top * ) of pointed spaces. For a general 9 G, there are two classical models for the classifying space: the Milgram-Segal model [Mil67, Seg68] that defines BG as the geometric realization |B * G| of a simplicial space B * G (topological bar construction) and the Milnor model [Mil56that the join X * Y of two spaces is defined to be the space of all line segments joining points in X to points in Y : i.e., X * Y is the quotient space of X × I × Y under the identifications (x, 0, y) ∼ (x ′ , 0, y) and (x, 1, y) ∼ (x, 1, y ′ ) for all x, x ′ ∈ X and y, y ′ ∈ Y . If X and Y are both (well) pointed, it is convenient to work with a reduced version of the join obtained by collapsing to a point the line segment joining the basepoints in X and Y (i.e., by imposing on X * Y the extra identification ( * , t, * ) ∼ ( * , t ′ , * ) for all t, t ′ ∈ I). Note that inside X * Y , there are two cones CX and CY embedded via the canonical maps. A Appendix, Milnor bundles Recall that, if G is a topological group, its classifying space BG is defined to be the basespace of a principal G-bundle EG → BG that is universal among all (numerable) principal G-bundles over pointed spaces. X * Y ∼ = Σ(X ∧ Y ) ∼ = (ΣX) ∧ Y ∼ = X ∧ (ΣYSince CX and CY are both contractible in X * Y , the quotient map X * Y → Σ(X ∧ Y ) is a homotopy equivalence. in the homotopy category Ho(Top * ) of pointed spaces, we have natural isomorphisms (A.1)Appendix A. Milnor bundles Recall that, if G is a topological group, its classifying space BG is defined to be the basespace of a principal G-bundle EG → BG that is universal among all (numerable) principal G-bundles over pointed spaces. This universal property determines the space BG uniquely up to homotopy, i.e. as a unique (up to unique isomorphism) object in the homotopy category Ho(Top * ) of pointed spaces. For a general 9 G, there are two classical models for the classifying space: the Milgram- Segal model [Mil67, Seg68] that defines BG as the geometric realization |B * G| of a simplicial space B * G (topological bar construction) and the Milnor model [Mil56that the join X * Y of two spaces is defined to be the space of all line segments joining points in X to points in Y : i.e., X * Y is the quotient space of X × I × Y under the identifications (x, 0, y) ∼ (x ′ , 0, y) and (x, 1, y) ∼ (x, 1, y ′ ) for all x, x ′ ∈ X and y, y ′ ∈ Y . If X and Y are both (well) pointed, it is convenient to work with a reduced version of the join obtained by collapsing to a point the line segment joining the basepoints in X and Y (i.e., by imposing on X * Y the extra identification ( * , t, * ) ∼ ( * , t ′ , * ) for all t, t ′ ∈ I). Note that inside X * Y , there are two cones CX and CY embedded via the canonical maps CX ֒→ X * Y , (x, t) → (x, t, * ), and CY ֒→ X * Y , (y, t) → ( * , 1 − t, y). Collapsing these cones converts X * Y into the suspension of the smash product of spaces: Σ(X ∧ Y ) = (X * Y )/(CX ∨ CY ). Since CX and CY are both contractible in X * Y , the quotient map X * Y → Σ(X ∧ Y ) is a homotopy equivalence. Thus, in the homotopy category Ho(Top * ) of pointed spaces, we have natural isomorphisms (A.1) X * Y ∼ = Σ(X ∧ Y ) ∼ = (ΣX) ∧ Y ∼ = X ∧ (ΣY )
Using standard notation, we will write the points of X * Y as formal linear combinations t 0 x+t 1 y, where. ∈ X , ∈ , These are useful in practice for computing the homotopy types of joins. t 0 , t 1 ) ∈ ∆ 1 := {(t 0 , t 1 ) ∈ R 2 : t 0 + t 1 = 1, t 0 , t 1 ≥ 0}. The identification 9 For special groups (for example, classical Lie groups), there are also nice geometric models representing BG as infinite-dimensional manifolds (GrassmanniansThese are useful in practice for computing the homotopy types of joins. Using standard notation, we will write the points of X * Y as formal linear combinations t 0 x+t 1 y, where x ∈ X, y ∈ Y and (t 0 , t 1 ) ∈ ∆ 1 := {(t 0 , t 1 ) ∈ R 2 : t 0 + t 1 = 1, t 0 , t 1 ≥ 0}. The identification 9 For special groups (for example, classical Lie groups), there are also nice geometric models representing BG as infinite-dimensional manifolds (Grassmannians).
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| {'fraction_non_alphanumeric': 0.09762812490905387, 'fraction_numerical': 0.03293850586420651, 'mean_word_length': 3.2418092486976273, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 14, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 185, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This is the first in a series of papers, where we introduce and study topological spaces that realize the algebras of quasi-invariants of finite reflection groups. Our result can be viewed as a generalization of a well-known theorem of A. Borel that realizes the ring of invariant polynomials a Weyl group W as a cohomology ring of the classifying space BG of the associated Lie group G. In the present paper, we state our realization problem for the algebras of quasi-invariants of Weyl groups and give its solution in the rank one case (for G = SU(2)). We call the resulting G-spaces Fm(G, T ) the m-quasi-flag manifolds and their Borel homotopy quotients Xm(G, T ) the spaces of m-quasi-invariants. We compute the equivariant K-theory and the equivariant (complex analytic) elliptic cohomology of these spaces and identify them with exponential and elliptic quasi-invariants of W . We also extend our construction of spaces quasi-invariants to a certain class of finite loop spaces ΩB of homotopy type of S 3 originally introduced by D. L. Rector[Rec71a]. We study the cochain spectra C * (Xm, k) associated to the spaces of quasi-invariants and show that these are Gorenstein commutative ring spectra in the sense of Dwyer, Greenlees and Iyengar [DGI06]. 1 a p-local version of the realization problem for algebras of quasi-invariants of non-crystallographic (in fact, non-Coxeter) groups defined over the p-adic numbers in terms of p-compact groups.We now give a general overview of our work, our problems and motivation.Quasi-invariants and cohomology theories. In mathematical physics, quasi-invariants naturally arise in three different flavors: rational (polynomial), trigonometric (exponential) and elliptic. Having in hand topological spaces X m (G, T ) that realize the algebras Q m (W ), it is natural to expect that the above three types of quasi-invariants correspond to three basic cohomology theories evaluated at X m (G, T ): namely, the ordinary (singular) cohomology, topological K-theory and elliptic cohomology. We will show that this is indeed the case: in fact, quasi-invariants can be defined for an arbitrary (complex-oriented generalized) cohomology theory, though in general their properties have yet to be studied.Quasi-flag manifolds. For a compact connected Lie group G, our spaces of quasi-invariants can be naturally realized as Borel homotopy quotients of certain G-spaces F m (G, T ):We call F m (G, T ) the m-quasi-flag manifold of G as in the special case m = 0, we have F 0', 'arxivid': '2305.10604', 'author': ['Yuri Berest ', 'Ajay C Ramadoss '], 'authoraffiliation': [], 'corpusid': 258762465, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 63068, 'n_tokens_neox': 55419, 'n_words': 32713, 'pdfsha': '1cd84ec23df7f0444c7fb1db5cf52164cc1e0339', 'pdfurls': ['https://export.arxiv.org/pdf/2305.10604v1.pdf'], 'title': ['TOPOLOGICAL REALIZATION OF ALGEBRAS OF QUASI-INVARIANTS, I', 'TOPOLOGICAL REALIZATION OF ALGEBRAS OF QUASI-INVARIANTS, I'], 'venue': []} |
arxiv |
Size Adaptive Region Based Huffman Compression Technique
Utpal Nandi [email protected]@gmail.com
Jyotsna Kumar
Mandal #2
Dept. of Computer Sc. & Engineering
#1 Dept. of Computer Sc. & Engineering
Bankura Unnayani Institute of Engineering Bankura-722146
West Bengal#2India
University of Kalyani
Nadia -741235West BengalIndia
Size Adaptive Region Based Huffman Compression Technique
CompressionHuffman TreeFrequency Table (FT)Symbol Code Table (SCT)compression ratioregion Based Huffman (RBH)
A loss-less compression technique is proposed which uses a variable length Region formation technique to divide the input file into a number of variable length regions. Huffman codes are obtained for entire file after formation of regions. Symbols of each region are compressed one by one. Comparisons are made among proposed technique, Region Based Huffman compression technique and classical Huffman technique.The proposed technique offers better compression ratio for some files than other two.
I. INTRODUCTION
The data compression techniques reduce the amount of data required to represent a source of information to reduce the storage space requirement and the time of data transmission over network. There are two major class of data compression techniques i.e. loss-less [1][2][3][4][5][6] and lossy [6]. The loss-less techniques generate exact duplicate of the original data after compress/expand cycle. But, the lossy techniques concede a certain loss of accuracy. One of the well established loss-less technique is Huffman Coding [4,6] which is based on the frequency of elements of entire file . If an element has maximum frequency, it gets shortest code. But if we divide a file into a number of regions, it is obvious that in each region the maximum frequency element may not the maximum frequency element of entire file and has large code length. If the large codes produced by Huffman coding are used for the elements which has maximum frequency for each region ,the size of compressed file increases. In light of this Region Based Huffman (RBH) [1] coding has been introduced. The RBH coding technique divides the total input file/stream into a number of regions N. The maximum frequency elements for each region are calculated. Huffman codes are obtained based on frequency of elements for entire file/stream. Now for first region, if the code length of maximum frequency element of that region is larger than the code length of maximum frequency element of entire file/stream, the code between maximum frequency element of that region and maximum frequency element of entire file/stream is interchanged. This interchanged information is attached with the compressed file/stream. The elements of that region are compressed with the changed codes and interchanged codes are reset. Otherwise, same symbol code table is used. Similarly, all other regions are compressed repeatedly. The main problem of RBH coding is that the performance depends on the number of regions of the file and therefore also on the size of region of the file. Compression ratios of same file with different region size are not same. It is very difficult to determine the optimum region size that offers maximum compression of a file. Because, it depends on the symbols of the file. As different region contains different frequency of symbols, the compression ratios of different region are also not same. For example, let us Consider a file/stream containing the message CACBABCBCCABACBA BABACBBADBDBEB (say MSG). For the same file/stream (MSG), compression ratio is 47.9% for region size 10 and compression ratio is 45% for region size 6. Therefore, the proper region size (or number of region) must be chosen for better compression of file/stream. Modified Region Based Huffman (MRBH) [1] coding also suffers from the same problem if the optimum value of number of region does not lie in the specified range. Fixed size regions are not able to adapt its size based on symbols that offers better compression. To overcome the limitation of finding optimum region size of fixed size region based compression techniques, a technique is proposed which has the ability to adapt its region size based on symbols and termed as Size Adaptive Region Based Huffman (SARBH) coding. The proposed technique has been discussed in section II. Results have been given in section III and conclusions are drawn in section IV.
II. THE PROPOSED TECHNIQUE Most of the time, it is found that the sequence of character's ASCII values in the file are adjacent ASCII values. That is ASCII value differences among adjacent characters are not so high. The proposed algorithm uses this concept. Our aim is to group sequence of characters into regions such that the differences among the ASCII values of characters in a region do not exceed a specified value (r). Therefore, after grouping into regions, the information of each region can be preserved by storing the number of symbols, minimum ASCII value and the differences among other ASCII values of symbols in the region with the minimum ASCII value. After that each region contains ASCII values not exceeding the specified value except first two (the number of symbols and the minimum ASCII value). In this way, variable length regions ( R1 , R2 , R3 . . . Rn) of input file / stream using with a specified value(r) is formed. The frequencies of all the symbols of the input file / stream whose ASCII value lies in the range 0 to r-1 are obtained. Huffman Codes of all symbols whose ASCII value lies in the range 0 to r-1 of input file / stream is also calculated. During compression, for each region first two symbols (number of element and minimum ASCII value symbol) are kept unchanged and all other symbols (whose ASCII values lie in the range 0 to r-1) are coded by corresponding Huffman code.
For example, let us consider a file/stream containing the message -ABAABDADAAWXXZXWXY ZXXYPQPSQS PR (say MSG1). MSG1 is grouped into a number of regions with specified value(r) =16 in such a way that each region does not have two symbols with ASCII value difference greater than or equal to 16 as shown below in Fig. 1.
ABAABDADAA WXXZXWXYZXXY PQPSQSPR
Region 1 Region 2 Region 3 Information of each regions are kept by storing the number of symbol of each region, minimum ASCII value of all the symbols and ASCII value difference of all the symbols with minimum ASCII value of symbol in the corresponding region as shown below in Fig. 2. 10,65,0,1,0,0,1,3,0,3,0,0 12,87,0,1,1,3,1,0,1,2,3,1,1,2 8,80,0,1,0,3,1,3,0,2 Region 1 Region 2 Region 3 TABLE I and Huffman tree based on the frequency of numbers are obtained as shown in Fig. 3. Code of each numbers are thus obtained as given in symbol code table TABLE II. Table I Table II : SCT Number 0 1 2 3 code 0 10 110 111 During compression, for each region first two symbols are kept unchanged and all other symbols are coded by corresponding Huffman code. Therefore, the compressed message for entire file/message stream will be-10, 65,0,10,0, 0, 10,111,0,111,0,0,2,87,0,10,10,111,10,0,10,110,111,10,10,110, 8,80,0,10,0,111,10,111,0,110 IV. CONCLUSION The proposed technique eliminates some of the limitations of classical Huffman, RBH and MRBH coding techniques and offers better performance for some files over both Huffman and MRBH coding. The presented technique has also a better scope of modification. The technique can be modified by introducing the concept of region wise code interchanging. And it can also be used for image compression.
Fig. 1 :
1Symbols of variable length regions
Fig. 2 :
2variable length regions of MSG1 After formation of regions, frequencies of all the numbers in the range 0 to 15 are found as given in
Fig. 3 :
3Huffman tree based on
TABLE I :
IFT of SymbolsNumber
0
1
2
3
Frequency
11
10
3
6
. The calculation of compression ratio may be done as follows: Original message size = 30x8 bits = 180 bits, Only Compressed message size(excluding first two numbers) = 58 bits ,FrequencyTable size = 6x8 bits = 48 bits, size of first two numbers of three region=3x2x8 bits = 48bits, Total Compressed message size = ( 58 + 48+ 48 ) bits = 154 bits , Compression ratio = { ( 240 -154 ) / 240 }X100 % = 35.83 %. III. RESULTS Comparison of compression ratios among Huffman technique, MRBH coding with range 10 -25 and proposed SARBH coding have been made using five different type of files as shown in TABLE 3 with specified value(r) as 32. Graphical representation of the same is shown inFig. 4.
TABLE III :
IIIComparison of compression ratios in different techniquesFig .4. Graphical representation of compression ratios of Huffman, MRBH and SARBH coding.File Name
%Compression
Classical
Huffman
MRBH
with range 10 -25
Proposed
SARBH
Circle.java
35.48
35.50
35.80
Selection.exe
19.47
19.86
19.82
Dolly.doc
39.12
40.13
40.10
Chpst.dll
24.42
24.81
24.80
Dummy.txt
34.60
34.59
35.01
ACKNOWLEDGMENTThe authors extend sincere thanks to the department of Computer Science and Engineering and IIPC Cell, University of Kalyani, Nadia, West Bengal, India for using the infrastructure facilities for developing the technique.
Region based Huffman(RB H) Compression Technique with Code Interchange‖. U Nandi, J K , Malayasian Journal of Computer Science(MJCS), Malayasia. 232U. Nandi, J. K. Mandal, -Region based Huffman(RB H) Compression Technique with Code Interchange‖, Malayasian Journal of Computer Science(MJCS), Malayasia, Vol. 23, No. 2, pp. 111-120., September 2010.
A Compression Technique Based on Optimality of Huffman Tree (OHT)‖. J K Mandal, A Kumar, Proceedings of 12 th International Conference of IEEE on Advanced Computing and Communications -ADCOM-2004. 12 th International Conference of IEEE on Advanced Computing and Communications -ADCOM-2004Ahmedabad, IndiaJ.K. Mandal, A. Kumar, -A Compression Technique Based on Optimality of Huffman Tree (OHT)‖, in Proceedings of 12 th International Conference of IEEE on Advanced Computing and Communications -ADCOM-2004, Ahmedabad, India, pp. 589-595 , December 15-18, 2004.
Implementation of Two Data Compression Schemes‖. J K Mandal, R Gangopadhayay, Proc. First International Workshop on Telematics, NERIST, India. First International Workshop on Telematics, NERIST, IndiaJ.K. Mandal and R. Gangopadhayay, -Implementation of Two Data Compression Schemes‖, in Proc. First International Workshop on Telematics, NERIST, India, pp. 154-162,1995.
A method for the construction of minimumredundancy codes. D A Huffman, ‖ in Proceedings of the IRE. 40D.A. Huffman, -A method for the construction of minimum- redundancy codes,‖ in Proceedings of the IRE, Vol. 40, No. 9, pp. 1098-1101, September 1952.
‖ An Overview of Data Compression Techniques,‖ in IEEE Computer. H K Reglebati, H.K. Reglebati,‖ An Overview of Data Compression Techniques,‖ in IEEE Computer, pp. 71-75, April 1981.
M Nelson, The Data Compression Book‖. IndiaBPB PublicationsM. Nelson , -The Data Compression Book‖ ,ed. Second , India, BPB Publications, 2008 .
. Y Kanitkar, C project ‖, ed.Second , IndiaBPB PublicationsY. Kanitkar , -C project ‖, ed.Second , India, BPB Publications, 2002.
Technique for High-Performance Data Compression. Terry Welch, ‖ in IEEE Computer. 176Welch, Terry, -A Technique for High-Performance Data Compression,‖ in IEEE Computer, Vol. 17, No.6, pages 8-19, June 1984.
Arithmetic Coding for Data Compression. H Witten, Ian, M Neal, Radford, G Cleary, John, ‖ in Communications of the ACM. 306Witten, H. Ian, Neal, M. Radford, and Cleary, G.John, -Arithmetic Coding for Data Compression,‖ in Communications of the ACM, Vol. 30, No. 6,pp. 520-540, June 1987.
A universal algorithm for sequential data compression. J Ziv, A Lempel, ‖ in IEEE Transactions on Information Theory. 23J. Ziv, and A. Lempel, -A universal algorithm for sequential data compression,‖ in IEEE Transactions on Information Theory, Vol. 23, No. 3, pp.337-343, May 1977.
| {'fraction_non_alphanumeric': 0.04848787803049238, 'fraction_numerical': 0.037740564858785304, 'mean_word_length': 4.421860885275519, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'A loss-less compression technique is proposed which uses a variable length Region formation technique to divide the input file into a number of variable length regions. Huffman codes are obtained for entire file after formation of regions. Symbols of each region are compressed one by one. Comparisons are made among proposed technique, Region Based Huffman compression technique and classical Huffman technique.The proposed technique offers better compression ratio for some files than other two.', 'arxivid': '1403.0153', 'author': ['Utpal Nandi [email protected]@gmail.com ', 'Jyotsna Kumar ', 'Mandal #2 ', '\nDept. of Computer Sc. & Engineering\n#1 Dept. of Computer Sc. & Engineering\nBankura Unnayani Institute of Engineering Bankura-722146\nWest Bengal#2India\n', '\nUniversity of Kalyani\nNadia -741235West BengalIndia\n'], 'authoraffiliation': ['Dept. of Computer Sc. & Engineering\n#1 Dept. of Computer Sc. & Engineering\nBankura Unnayani Institute of Engineering Bankura-722146\nWest Bengal#2India', 'University of Kalyani\nNadia -741235West BengalIndia'], 'corpusid': 15335389, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3613, 'n_tokens_neox': 3084, 'n_words': 1832, 'pdfsha': '0002d0f1d9509553731fee89b9fdfc546a9f7fb1', 'pdfurls': ['https://arxiv.org/pdf/1403.0153v1.pdf'], 'title': ['Size Adaptive Region Based Huffman Compression Technique', 'Size Adaptive Region Based Huffman Compression Technique'], 'venue': []} |
arxiv |
THE ITÔ INTEGRAL WITH RESPECT TO AN INFINITE DIMENSIONAL LÉVY PROCESS: A SERIES APPROACH
Stefan Tappe
THE ITÔ INTEGRAL WITH RESPECT TO AN INFINITE DIMENSIONAL LÉVY PROCESS: A SERIES APPROACH
We present an alternative construction of the infinite dimensional Itô integral with respect to a Hilbert space valued Lévy process. This approach is based on the well-known theory of real-valued stochastic integration, and the respective Itô integral is given by a series of Itô integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the Itô integral that has been developed in the literature.2010 Mathematics Subject Classification. 60H05, 60G51. Key words and phrases. Itô integral, infinite dimensional Lévy process, covariance operator, Hilbert-Schmidt operator.The author is grateful to an anonymous referee for valuable comments and suggestions.
Introduction
The Itô integral with respect to an infinite dimensional Wiener process has been developed in [4,17,10], and for the more general case of an infinite dimensional square-integrable martingale it has been defined in [13,16]. In these references, one first constructs the Itô integral for elementary processes, and then extends it via the Itô isometry to a larger space, in which the space of elementary processes is dense.
For stochastic integrals with respect to a Wiener process, series expansions of the Itô integral have been considered, e.g., in [11,7,3]. Moreover, in the article [14], series expansions have been used in order to define the Itô integral with respect to a Wiener process for deterministic integrands with values in a Banach space. Later, in [15] this theory has been extended to general integrands with values in UMD Banach spaces.
Best to the author's knowledge, a series approach for the construction of the Itô integral with respect to an infinite dimensional Lévy process does not exist in the literature so far. The goal of the present paper is to provide such a construction, which is based on the real-valued Itô integral, see, e.g., [1,12,18], and where the Itô integral is given by a series of Itô integrals with respect to real-valued Lévy processes. This approach has the advantage that we can use results from the finite dimensional case, and it might also be beneficial for lecturers teaching students who are already aware of the real-valued Itô integral and have some background in Functional Analysis. In particular, it avoids the tedious procedure of proving that elementary processes are dense in the space of integrable processes.
In [8], the stochastic integral with respect to an infinite dimensional Lévy process is defined as a limit of Riemannian sums, and a series expansion is provided. A particular feature of [8] is that stochastic integrals are considered as L 2 -curves. The connection to the usual Itô integral for a finite dimensional Lévy process has been established in [23], see also Appendix B in [6]. Furthermore, we point out the articles [21] and [9], where the theory of stochastic integration with respect to Lévy processes has been extended to Banach spaces.
The idea to use series expansions for the definition of the stochastic integral has also been utilized in the context of cylindrical processes, see [19] for cylindrical Wiener processes and [2] for cylindrical Lévy processes.
The construction of the Itô integral, which we present in this paper, is divided into the following steps:
• For a H-valued process X (with H denoting a separable Hilbert space) and a real-valued square-integrable martingale M we define the Itô integral
X • M := k∈N X, f k H • M f k ,
where (f k ) k∈N denotes an orthonormal basis of H, and X, f k H • M denotes the real-valued Itô integral. We will show that this definition does not depend on the choice of the orthonormal basis. • Based on the just defined integral, for a 2 (H)-valued process X and a sequence (M j ) j∈N of standard Lévy processes we define the Itô integral as
j∈N X j • M j .
For this, we will ensure convergence of the series. • In the next step, let L denote a 2 λ -valued Lévy process, where 2 λ is a weighted space of sequences (cf. [5]). From the Lévy process L we can construct a sequence (M j ) j∈N of standard Lévy processes, and for a 2 (H)valued process X we define the Itô integral
X • L := j∈N X j • M j .
• Finally, let L be a general Lévy process on some separable Hilbert space U with covariance operator Q. Then, there exist sequences of eigenvalues (λ j ) j∈N and eigenvectors, which diagonalize the operator Q. Denoting by L 0 2 (H) an appropriate space of Hilbert Schmidt operators from U to H, our idea is to utilize the integral from the previous step, and to define the Itô integral for a L 0 2 (H)-valued process X as
X • L := Ψ(X) • Φ(L),
where Φ : U → 2 λ and Ψ : L 0 2 (H) → 2 (H) are isometric isomorphisms such that Φ(L) is a 2 λ -valued Lévy process. We will show that this definition does not depend on the choice of the eigenvalues and eigenvectors.
The remainder of this text is organized as follows: In Section 2 we provide the required preliminaries and notation. After that, we start with the construction of the Itô integral as outlined above. In Section 3 we define the Itô integral for Hvalued processes with respect to a real-valued square-integrable martingale, and in Section 4 we define the Itô integral for 2 (H)-valued processes with respect to a sequence of standard Lévy processes. Section 5 gives a brief overview about Lévy processes in Hilbert spaces, together with the required results. Then, in Section 6 we define the Itô integral for 2 (H)-valued processes with respect to a 2 λ -valued Lévy process, and in Section 7 we define the Itô integral in the general case, where the integrand is a L 0 2 (H)-valued process and the integrator a general Lévy process on some separable Hilbert space U . We also prove the mentioned series representation of the stochastic integral, and show that it coincides with the usual Itô integral, which has been developed in [16].
Preliminaries and notation
In this section, we provide the required preliminary results and some basic notation. Throughout this text, let (Ω, F, (F t ) t≥0 , P) be a filtered probability space satisfying the usual conditions. For the upcoming results, let E be a separable Banach space, and let T > 0 be a finite time horizon.
2.1. Definition. Let p ≥ 1 be arbitrary. Note that we have the inclusions
M p T (E) ⊂ A p T (E) ⊂ L p T (E)
. The following auxiliary result shows that these inclusions are closed.
2.3. Lemma. Let p ≥ 1 be arbitrary. Then, the following statements are true:
(1) M p T (E) is closed in A p T (E). (2) A p T (E) is closed in L p T (E). Proof. Let (M n ) n∈N ⊂ M p T (E) be a sequence and let M ∈ A p T (E) be such that M n → M in L p T (E)
. Furthermore, let τ ≤ T be a bounded stopping time. Then we have
E M τ p E ≤ E sup t∈[0,T ] M t p E < ∞,
showing that M τ ∈ L p (Ω, F τ , P; E). Furthermore, we have
E M n τ − M τ p E ≤ E sup t∈[0,T ] M n t − M t p E → 0.X, f k H • M f k (3.2)
converges unconditionally in M 2 T (H), and its value does not depend on the choice of the orthonormal basis (f k ) k∈N .
Proof. Let (f k ) k∈N be an orthonormal basis of H. For j, k ∈ N with j = k we have and, by the Cauchy-Schwarz inequality,
n k=1 x, f k H f k , h H − x, h H 2 = ∞ k=n+1 x, f k H f k , h H 2 ≤ ∞ k=1 | x, f k H | 2 ∞ k=1 | f k , h H | 2 = x 2 H h 2 H for each n ∈ N.
Therefore, by the Itô isometry for the real-valued Itô integral and Lebesgue's dominated convergence theorem together with (3.1) we obtain
h, J f H − h, X H • M 2 M 2 T (R) = lim n→∞ E T 0 n k=1 h, f k H f k , X s H − h, X s H dM s 2 = lim n→∞ E T 0 n k=1 h, f k H f k , X H − h, X H
Now, Proposition 3.1 gives rise to the following definition:
3.2. Definition. For every H-valued, predictable process X satisfying (3.1) we de- fine the Itô integral X • M = ( t 0 X s dM s ) t∈[0,T ] as X • M := k∈N X, f k H • M f k , (3.5)
where (f k ) k∈N denotes an orthonormal basis of H.
E T 0 X s dM s 2 H = E T 0 X s 2 H d M, M s .
Proof. Let (f k ) k∈N be an orthonormal basis of H. According to (3.3) we have
X, f j H • M f j , X, f k H • M f k M 2 T (H) = 0 for j = k. Thus, by Lemma 2.4 and (3.4) we obtain E T 0 X s dM s 2 H = X • M 2 M 2 T (H) = ∞ k=1 X, f k H • M f k 2 M 2 T (H) = ∞ k=1 X, f k H • M f k 2 M 2 T (H) = E T 0 X s 2 H d M, M s ,
finishing the proof.
3.5. Proposition. Let X be a H-valued simple process of the form
X = X 0 1 {0} + n i=1 X i 1 (ti,ti+1] with 0 = t 1 < . . . < t n+1 = T and F ti -measurable random variables X i : Ω → H for i = 0, . . . , n. Then, we have X • M = n i=1 X i (M ti+1 − M ti ).
Proof. Let (f k ) k∈N be an orthonormal basis of H. Then, for each k ∈ N the process X, f k is a real-valued simple process with representation
X, f k H = X 0 , f k H 1 {0} + n i=1 X i , f k H 1 (ti,ti+1]
Thus, by the definition of the real-valued Itô integral for simple processes we obtain
X • M = k∈N X, f k H • M f k = k∈N n i=1 X i , f k H (M ti+1 − M ti ) f k = n i=1 k∈N X i , f k H f k (M ti+1 − M ti ) = n i=1 X i (M ti+1 − M ti ),
finishing the proof.
3.6. Lemma. Let X be a H-valued, predictable process satisfying (3.1). Then, for every orthonormal basis
(f k ) k∈N of H we have ∞ k=1 | X, f k H | 2 • M, M = X 2 H • M, M ,
where the convergence takes place in A 1 T (R). Proof. We define the integral process
I := X 2 H • M, M (3.6)
and the sequence (I n ) n∈N of partial sums by
I n := n k=1 | X, f k H | 2 • M, M . (3.7) By (3.1) we have I ∈ A 1 T (R) and (I n ) n∈N ⊂ A 1 T (R).
Furthermore, by Lebesgue's dominated convergence theorem we have
I − I n L 1 T (R) = E sup t∈[0,T ] |I t − I n t | = E sup t∈[0,T ] t 0 ∞ k=n+1 | X s , f k | 2 d M, M s = E T 0 ∞ k=n+1 | X s , f k | 2 d M, M s → 0 for n → ∞,
which concludes the proof.
3.7. Remark. As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales X, Y ∈ M 2 T (H) there exists a (up to indistinguishability) unique real-valued, predictable process X, Y with finite variation paths and
X, Y 0 = 0 such that X, Y H − X, Y is a martingale. 3.8. Proposition. For every H-valued, predictable process X satisfying (3.1) we have X • M, X • M = X 2 H • M, M .
Proof. Let (f k ) k∈N be an orthonormal basis of H. We define the process J := X •M and the sequence (J n ) n∈N of partial sums by
J n := n k=1 X, f k H • M f k .
By Proposition 3.1 we have
J n → J in M 2 T (H). (3.8)
Defining the integral process I by (3.6) and the sequence (I n ) n∈N of partial sums by (3.7), using Lemma 3.6 we have
I n → I in A 1 T (R). (3.9)
Furthermore, we define the process M ∈ A 1 T (R) and the sequence (
M n ) n∈N ⊂ A 1 T (R) as M := J 2 H − I, M n := J n 2 H − I n , n ∈ N. Then we have (M n ) n∈N ⊂ M 1 T (R). Indeed, for each n ∈ N we have M n = n k=1 X, f k H • M f k 2 H − n k=1 | X, f k H | 2 • M, M = n k=1 X, f k H • M f k 2 H − n k=1 | X, f k H | 2 • M, M = n k=1 | X, f k H • M | 2 − | X, f k H | 2 • M, M .
For every k ∈ N the quadratic variation of the real-valued process X, f k H • M is given by
X, f k H • M, X, f k H • M = | X, f k H | 2 • M, M ,
see, e.g. [12,Thm. I.4.40.d], which shows that M n is a martingale. Since M n ∈ A 1 T (R), we deduce that M n ∈ M 1 T (R).
Next, we prove that M n → M in A 1 T (R). Indeed, since
| J 2 H − J n 2 H | ≤ J − J n 2 H + 2 J H J − J n H ,
by the Cauchy-Schwarz inequality and (3.8) we obtain
J 2 H − J n 2 H L 1 T (R) = E sup t∈[0,T ] | J t 2 H − J n t 2 H | ≤ E sup t∈[0,T ] J t − J n t 2 H + 2E sup t∈[0,T ] J t H J t − J n t H ≤ E sup t∈[0,T ] J t − J n t 2 H + 2E sup t∈[0,T ] J t 2 H 1/2 E sup t∈[0,T ] J t − J n t 2 H 1/2 = J − J n 2 L 2 T (H) + 2 J L 2 T (H) J − J n L 2 T (H) → 0.
Therefore, together with (3.9) we get
M − M n L 1 T (R) ≤ J 2 − J n 2 L 1 T (R) + I − I n L 1 T (R) → 0, showing that M n → M in A 1 T (R). Now, Lemma 2.3 yields that M ∈ M 1 T (R)
, which concludes the proof.
3.9. Theorem. Let N ∈ M 2 T (R) be another square-integrable martingale, and let X, Y be two H-valued, predictable processes satisfying (3.1) and
E T 0 Y s 2 H d N, N s < ∞.X • M, Y • N M 2 T (H) = E T 0 X s dM s , T 0 Y s dN s H = E T 0 X s dM s , T 0 Y s dN s = E T 0 X s , Y s H d M, N s = 0,
completing the proof.
The Itô integral with respect to a sequence of standard Lévy processes
In this section, we introduce the Itô integral for 2 (H)-valued processes with respect to a sequence of standard Lévy processes, which is based on the Itô integral (3.5) from the previous section. We define the space of sequences
2 (H) := (h j ) j∈N ⊂ H : ∞ j=1 h j 2 H < ∞ ,
which, equipped with the inner product
h, g 2 (H) = ∞ j=1 h j , g j H
is a separable Hilbert space.
Definition.
A sequence (M j ) j∈N of real-valued Lévy processes is called a sequence of standard Lévy processes if it consists of square-integrable martingales with M j , M k t = δ jk · t for all j, k ∈ N. Here δ jk denotes the Kronecker delta
δ jk = 1, if j = k, 0, if j = k.
For the rest of this section, let (M j ) j∈N be a sequence of standard Lévy processes.
Proposition. For every 2 (H)-valued, predictable process X with
E T 0 X s 2 2 (H) ds < ∞ (4.1) the series j∈N X j • M j (4.2)
converges unconditionally in M 2 T (H). Proof. For j, k ∈ N with j = k we have M j , M k = 0, and hence, by Proposition 3.10 we obtain
X j • M j , X k • M k M 2 T (H) = 0. (4.3)
Moreover, by the Itô isometry (Proposition 3.4) and the monotone convergence theorem we have
(4.4) ∞ j=1 X j • M j 2 M 2 T (H) = ∞ j=1 E T 0 X j s dM j s 2 H = ∞ j=1 E T 0 X j s 2 H ds = E T 0 ∞ j=1 X j s 2 H ds = E T 0 X s 2 2 (H) ds .
Thus, by (4.1) and Lemma 2.4, the series (4.2) converges unconditionally in M 2 T (H).
Therefore, for a 2 (H)-valued, predictable process X satisfying (4.1) we can define the Itô integral as the series (4.2). Proof. Using (4.3), Lemma 2.4 and identity (4.4) we obtain
E ∞ j=1 T 0 X j s dM j s 2 H = ∞ j=1 X j • M j 2 M 2 T (H) = ∞ j=1 X j • M j 2 M 2 T (H) = E T 0 X s 2 2 (H) ds ,
completing the proof.
4.5.
Proposition. Let X be a 2 (H)-valued simple process of the form
X = X 0 1 {0} + n i=1 X i 1 (ti,ti+1]
with 0 = t 1 < . . . < t n+1 = T and F ti -measurable random variables X i : Ω → 2 (H) for i = 0, . . . , n. Then we have
X • M = n i=1 j∈N X j i (M j ) ti+1 − (M j ) ti .
Proof. For each j ∈ N the process X j is a H-valued simple process having the representation
X j = X j 0 1 {0} + n i=1 X j i 1 (ti,ti+1] .
Hence, by Proposition 3.5 we obtain
X • M = j∈N X j • M j = j∈N n i=1 X j i (M j ) ti+1 − (M j ) ti = n i=1 j∈N X j i (M j ) ti+1 − (M j ) ti ,
which finishes the proof.
Lévy processes in Hilbert spaces
In this section, we provide the required results about Lévy processes in Hilbert spaces. Let U be a separable Hilbert space.
5.1.
Definition. An U -valued càdlàg, adapted process L is called a Lévy process if the following conditions are satisfied: Note that any square-integrable Lévy martingale L is indeed a martingale, that is
(1) We have L 0 = 0. (2) L t − L s is independent of F s for all s ≤ t.E[X t | F s ] = X s for all s ≤ t,
see [16,Prop. 3.25]. According to [16,Thm. 4.44], for each square-integrable Lévy martingale L there exists a unique self-adjoint, nonnegative definite trace class operator Q ∈ L(U ), called the covariance operator of L, such that for all t, s ∈ R + and u 1 ,
u 2 ∈ U we have E[ L t , u 1 U L s , u 2 U ] = (t ∧ s) Qu 1 , u 2 U .
Moreover, for all u 1 , u 2 ∈ U the angle bracket process is given by
L, u 1 U , L, u 2 U t = t Qu 1 , u 2 U , t ≥ 0, (5.1)
see [16,Thm. 4.49].
Lemma.
Let L be an U -valued square-integrable Lévy martingale with covariance operator Q, let V be another separable Hilbert space and let Φ : U → V be an isometric isomorphism. Then the process Φ(L) is a V -valued square-integrable Lévy martingale with covariance operator Q Φ := ΦQΦ −1 .
Proof. The process Φ(L) is a V -valued càdlàg, adapted process with Φ(L 0 ) = Φ(0) = 0. Let s ≤ t be arbitrary. Then the random variable Φ
(L t ) − Φ(L s ) = Φ(L t − L s ) is independent of F s , and we have Φ(L t ) − Φ(L s ) = Φ(L t − L s ) d = Φ(L t−s ), Moreover, for each t ∈ R + we have E[ Φ(L t ) 2 V ] = E[ L t 2 U ] < ∞ and E[Φ(L t )] = ΦE(L t ) = 0, showing that Φ(L) is a V -valued square-integrable Lévy martingale.
Let t, s ∈ R + and v i ∈ V , i = 1, 2 be arbitrary, and set u i := Φ −1 v i ∈ U , i = 1, 2. Then we have
E[ Φ(L t ), v 1 V Φ(L s ), v 2 V ] = E[ Φ(L t ), Φ(u 1 ) V Φ(L s ), Φ(u 2 ) V ] = E[ L t , u 1 U L s , u 2 U ] = (t ∧ s) Qu 1 , u 2 U = (t ∧ s) QΦ −1 v 1 , Φ −1 v 2 U = (t ∧ s) ΦQΦ −1 v 1 , v 2 V = (t ∧ s) Q Φ v 1 , v 2 V ,
showing that the Lévy martingale Φ(L) has the covariance operator Q Φ . Now, let Q ∈ L(U ) be a self-adjoint, positive definite trace class operator. Then there exist a sequence (λ j ) j∈N ⊂ (0, ∞) with ∞ j=1 λ j < ∞ and an orthonormal basis (e (λ) j ) j∈N of U and such that
Qe (λ) j = λ j e (λ) j for all j ∈ N.
We define the sequence of pairwise orthogonal vectors (e j ) j∈N as e j := λ j e (λ) j , j ∈ N. 5.4. Proposition. Let L be an U -valued square-integrable Lévy martingale with covariance operator Q. Then the sequence (M j ) j∈N given by
M j := 1 λ j L, e (λ) j U , j ∈ N. (5.2)
is a sequence of standard Lévy processes.
Proof. For each j ∈ N the process M j is a real-valued square-integrable Lévy martingale. By (5.1), for all j, k ∈ N we obtain
M j , M k t = 1 λ j λ k L, e (λ) j U , L, e (λ) k U t = t Qe (λ) j , e (λ) k U λ j λ k = tλ j e (λ) j , e (λ) k U λ j λ k = δ jk · t,
showing that (M j ) j∈N is a sequence of standard Lévy processes.
6. The Itô integral with respect to a 2 λ -valued Lévy process In this section, we introduce the Itô integral for 2 (H)-valued processes with respect to a 2 λ -valued Lévy process, which is based on the Itô integral (4.2) from Section 4.
Let (λ j ) j∈N ⊂ (0, ∞) be a sequence with ∞ j=1 λ j < ∞ and denote by 2 λ the weighted space of sequences
2 λ := (v j ) j∈N ⊂ R : ∞ j=1 λ j |v j | 2 < ∞ , which, equipped with the inner product v, w 2 λ = ∞ j=1 λ j v j w j
is a separable Hilbert space. Note that we have the strict inclusion 2 2 λ , where 2 denotes the space of sequences
2 = (v j ) j∈N ⊂ R : ∞ j=1 |v j | 2 < ∞ .
We denote by (g j ) j∈N the standard orthonormal basis of 2 , which is given by g 1 = (1, 0, . . .), g 2 = (0, 1, 0, . . .), . . .
Then the system (g
(λ) j ) j∈N defined as g (λ) j := g j λ j , j ∈ N (6.1)
is an orthonormal basis of 2 λ . Let Q ∈ L( 2 λ ) be a linear operator such that Qg
(λ) j = λ j g (λ) j for all j ∈ N. (6.2)
Then Q is a nuclear, self-adjoint, positive definite operator. Let L be a 2 λ -valued, square-integrable Lévy martingale with covariance operator Q. According to Proposition 5.4, the sequence (M j ) j∈N given by
M j := 1 λ j L, g (λ) j 2 λ , j ∈ N
is a sequence of standard Lévy processes. 6.2. Remark. Note that L 0 2 (H) ∼ = 2 (H), where L 0 2 (H) denotes the space of Hilbert-Schmidt operators from 2 to H. In [5], the Itô integral for L 0 2 (H)-valued processes with respect to a 2 λ -valued Wiener process has been constructed in the usual fashion (first for elementary and afterwards for general processes), and then the series representation (6.3) has been proven, see [5,Prop. 2.2.1]. Now, let (µ k ) k∈N be another sequence with ∞ k=1 µ k < ∞, and let Φ : 2 λ → 2 µ be an isometric isomorphism such that
Q Φ g (µ) k = µ k g (µ) k
for all k ∈ N. (6.4) By Lemma 5.3, the process Φ(L) is a 2 µ -valued, square integrable Lévy martingale with covariance operator Q Φ , and by Proposition 5.4, the sequence (N k ) k∈N given by
N k := 1 √ µ k Φ(L), g (µ) k 2 µ , k ∈ N
is a sequence of standard Lévy processes. Proof. Since Ψ is an isometry, by (4.1) we have
E T 0 Ψ(X s ) 2 2 (H) ds = E T 0 X s 2 2 (H) ds < ∞,
showing (6.6). Moreover, by (6.4) we have
ΦQΦ −1 g (µ) k = Q Φ g (µ) k = µ k g (µ) k for all k ∈ N,
and hence, we get
Q(Φ −1 g (µ) k ) = µ k (Φ −1 g (µ)
k ) for all k ∈ N. k ) k∈N are eigenvectors of Q with corresponding eigenvalues (λ j ) j∈N and (µ k ) k∈N . Therefore, and since Φ is an isometry, for j, k ∈ N with λ j = µ k we obtain
Φg (λ) j , g (µ) k 2 µ = g (λ) j , Φ −1 g (µ) k 2 λ = 0. (6.9)
Let h ∈ H be arbitrary. Then we have
h, X • L H = h, ∞ j=1 X j • M j H = ∞ j=1 h, X j • M j H = ∞ j=1 h, X j H • M j = ∞ j=1 1 λ j h, X H , g (λ) j 2 λ • 1 λ j L, g (λ) j 2 λ = ∞ j=1 1 λ j Φ( h, X H ), Φg (λ) j 2 µ • 1 λ j Φ(L), Φg (λ) j 2 µ = ∞ j=1 1 λ j Φ( h, X H ), Φg (λ) j 2 µ • 1 λ j ∞ k=1 Φ(L), g (µ) k 2 µ g (µ) k , Φg (λ) j 2 µ .
Since (λ j ) j∈N and (µ k ) k∈N are eigenvalues of Q, for each j ∈ N there are only finitely many k ∈ N such that λ j = µ k . Therefore, by (6.9), and since (Φ(g
(λ) j )) j∈N is an orthonormal basis of 2 µ , we obtain h, X • L H = ∞ k=1 ∞ j=1 1 λ j Φ( h, X H ), Φg (λ) j 2 µ Φg (λ) j , g (µ) k 2 µ • Φ(L), g (µ) k 2 µ = ∞ k=1 1 µ k ∞ j=1 Φ( h, X H ), Φg (λ) j 2 µ Φg (λ) j , g (µ) k 2 µ • Φ(L), g (µ) k 2 µ = ∞ k=1 1 √ µ k Φ( h, X H ), g (µ) k 2 µ • 1 √ µ k Φ(L), g (µ) k 2 µ = ∞ k=1 Φ( h, X H ) k • N k .
Thus, taking into account (6.5) gives us
h, X • L H = ∞ k=1 h, Ψ(X) k H • N k = ∞ k=1 h, Ψ(X) k • N k H = h, ∞ k=1 Ψ(X) k • N k H = h, Ψ(X) • Φ(L) H .
Since h ∈ H was arbitrary, using the separability of H as in the proof of Proposition 3.1, we arrive at (6.7).
6.4. Remark. From a geometric point of view, Theorem 6.3 says that the "angle" measured by the Itô integral is preserved under isometries.
The Itô integral with respect to a general Lévy process
In this section, we define the Itô integral with respect to a general Lévy process, which is based on the Itô integral (6.3) from the previous section.
Let U be a separable Hilbert space and let Q ∈ L(U ) be a nuclear, self-adjoint, positive definite linear operator. Then there exist a sequence (λ j ) j∈N ⊂ (0, ∞) with ∞ j=1 λ j < ∞ and an orthonormal basis (e is an eigenvector corresponding to λ j . The space U 0 := Q 1/2 (U ), equipped with the inner product u, v U0 := Q −1/2 u, Q −1/2 v U , is another separable Hilbert space and the sequence (e j ) j∈N given by e j = λ j e (λ) j , j ∈ N is an orthonormal basis of U 0 . We denote by L 0 2 (H) := L 2 (U 0 , H) the space of Hilbert-Schmidt operators from U 0 into H, which, endowed with the Hilbert-Schmidt norm
S L 0 2 (H) := ∞ j=1 Se j 2 H 1/2 , S ∈ L 0 2 (H)
itself is a separable Hilbert space. We define the isometric isomorphisms
Φ λ : U → 2 λ , Φ λ e (λ) j := g (λ) j for j ∈ N, (7.2) Ψ λ : L 0 2 (H) → 2 (H), Ψ λ (S) := Se j j∈N for S ∈ L 0 2 (H). (7.3) Recall that (g (λ)
j ) j∈N denotes the orthonormal basis of 2 λ , which we have defined in (6.1). Let L be an U -valued square-integrable Lévy martingale with covariance operator Q.
7.1. Lemma. The following statements are true:
(1) The process Φ λ (L) is a 2 λ -valued square-integrable Lévy martingale with covariance operator Q Φ λ .
(2) We have
Q Φ λ g (λ) j = λ j g (λ) j for all j ∈ N. (7.4)
Proof. By Lemma 5.3, the process Φ λ (L) is a 2 λ -valued square-integrable Lévy martingale with covariance operator Q Φ λ . Furthermore, by (7.2) and (7.1), for all j ∈ N we obtain
Q Φ λ g (λ) j = Φ λ QΦ −1 λ g (λ) j = Φ λ Qe (λ) j = Φ λ (λ j e (λ) j ) = λ j Φ λ e (λ) j = λ j g (λ)
j , showing (7.4). Now, our idea is to the define the Itô integral for a L 0 2 (H)-valued, predictable process X with E T 0 X s 2 L 0 2 (H) ds < ∞ (7.5) by setting X • L := Ψ λ (X) • Φ λ (L), (7.6) where the right-hand side of (7.6) denotes the Itô integral (6.3) from Definition 6.1. One might suspect that this definition depends on the choice of the eigenvalues (λ j ) j∈N and eigenvectors (e (λ) j ) j∈N . In order to prove that this is not the case, let (µ k ) k∈N ⊂ (0, ∞) be another sequence with ∞ k=1 µ k < ∞ and let (f Then the sequence (f k ) k∈N given by
f k = √ µ k f (µ)
k , k ∈ N is an orthonormal basis of U 0 . Analogous to (7.2) and (7.3), we define the isometric isomorphisms (1) Φ λ (L) is a 2 λ -valued Lévy process with covariance operator Q Φ λ , and we have
Φ µ : U → 2 µ , Φ µ f(Q Φ λ g (λ) j = λ j g (λ) j for all j ∈ N.
(2) Φ µ (L) is a 2 µ -valued Lévy process with covariance operator Q Φµ , and we have
Q Φµ g (µ) k = µ k g (µ) k
Thus, by Proposition 4.5, and since Φ λ is an isometry, we obtain
X| U0 • L = Ψ λ (X| U0 ) • Φ λ (L) = n i=1 j∈N Ψ λ (X i | U0 ) j (N j ) ti+1 − (N j ) ti = n i=1 j∈N X i e j Φ λ (L ti+1 − L ti ), g (λ) j 2 λ λ j = n i=1 j∈N X i e (λ) j L ti+1 − L ti , Φ −1 λ g (λ) j U = n i=1 j∈N X i e (λ) j L ti+1 − L ti , e (λ) j U = n i=1 X i j∈N L ti+1 − L ti , e (λ) j U e (λ) j = n i=1 X i (L ti+1 − L ti ),
completing the proof.
Therefore, and since the space of simple processes is dense in the space of all predictable processes satisfying (7.5), see, e.g. [16,Cor. 8.17], the Itô integral (7.6) coincides with that in [16] for every L 0 2 (H)-valued, predictable process X satisfying (7.5). In particular, for a driving Wiener process, it coincides with the Itô integral from [4,17,10].
By a standard localization argument, we can extend the definition of the Itô integral to all predictable processes X satisfying P T 0 X s 2 L 0 2 (H) ds < ∞ = 1 for all T > 0. (7.11) Since the respective spaces of predictable and adapted, measurable processes are isomorphic (see [22]), proceeding as in the [22,Sec. 3.2], we can further extend the definition of the Itô integral to all adapted, measurable processes X satisfying (7.11).
( 1 )
1We define the Lebesgue space L p T (E) := L p (Ω, F T , P; D([0, T ]; E)), where D([0, T ]; E) denotes the Skorokhod space consisting of all càdlàg functions from [0, T ] to E, equipped with the supremum norm. (2) We denote by A p T (E) the space of all E-valued adapted processes X ∈ L p T (E). (3) We denote by M p T (E) the space of all E-valued martingales M ∈ L p T (E). (4) We define the factor spaces M p T (E) := M p T (E)/N, A p T (E) := A p T (E)/N, L p T (E) := L p T (E)/N, where N ⊂ M p T (E) denotes the subspace consisting of all M ∈ M p T (E) with M = 0 up to indistinguishability.
2. 2 .
2Remark. Let us emphasize the following:
( 1 )
1Since the Skorokhod space D([0, T ]; E) equipped with the supremum norm is a Banach space, the Lebesgue space L p T (E) equipped with the standard normX L p T (E) := E X p E 1/pis a Banach space, too.(2) By the completeness of the filtration (F t ) t≥0 , adaptedness of an element X ∈ L p T (E) does not depend on the choice of the representative. This ensures that the factor space A p T (E) of adapted processes is well-defined. (3) The definition of E-valued martingales relies on the existence of conditional expectation in Banach spaces, which has been established in [4, Prop. 1.10].
According to Proposition 3.1, the Definition (3.5) of the Itô integral is independent of the choice of the orthonormal basis (f k ) k∈N , and the integral process X • M belongs to M 2 T (H). 3.3. Remark. As the proof of Proposition 3.1 shows, the components of the Itô integral X • M are pairwise orthogonal elements of the Hilbert space M 2 T (H). 3.4. Proposition. For every H-valued, predictable process X satisfying (3.1) we have the Itô isometry
X
• M, Y • N = X, Y H • M, N . + N, M + N − M − N, M − N , identity (3.11) follows from a straightforward calculation. 3.10. Proposition. Let N ∈ M 2 T (R) be another square-integrable martingale such that M, N = 0, and let X, Y be two H-valued, predictable processes satisfying (3.1) and (3.10). Then we have X • M, Y • N M 2 T (H) = 0. Proof. Using Remark 3.7, Theorem 3.9 and the hypothesis M, N = 0 we obtain
4. 3 .
3Remark. As the proof of Proposition 4.2 shows, the components of the Itô integral j∈N X j •M j are pairwise orthogonal elements of the Hilbert space M 2 T (H).4.4.Proposition. For each 2 (H)-valued, predictable process X satisfying (
( 3 )
3We have L t − L s d = L t−s for all s ≤ t.5.2.Definition. An U -valued Lévy process L with E[ L t 2 U ] < ∞ and E[L t ] = 0 for all t ≥ 0 is called a square-integrable Lévy martingale.
6. 1 .
1Definition. For every 2 (H)-valued, predictable process X satisfying (4.1) we define the Itô integral X • L := ( t 0 X s dL s ) t∈[0,T ] as X • L := j∈N X j • M j . (6.3)
6. 3 .
3Theorem. Let Ψ ∈ L( 2 (H)) be an isometric isomorphism such that h, Ψ(w) H = Φ( h, w H ) for all h ∈ H and w ∈ 2 (H). (6.5)Then for every 2 (H)-valued, predictable process X satisfying (4.
the λ j are the eigenvalues of Q, and each e
for k ∈ N,Ψ µ : L 0 2 (H) → 2 (H), Ψ µ (S) := Sf k k∈N for S ∈ L 0 2 (H).
Then, for every orthonormal basis (f k ) k∈N of H the seriesBy Doob's optional stopping theorem (which also holds true for E-valued martin-
gales, see [17, Remark 2.2.5]), it follows that
E[M τ ] = lim
n→∞
E[M n
τ ] = lim
k∈N
d M, M s = 0.Analogously, we prove thath, J g H − h, X H • M 2 M 2 T (R) = 0. Therefore, denoting byJ f ,J g ∈ M 2 T (H) representatives of J f , J g , we obtain h,J f T H = h,J g T Hfor all h ∈ H, P-almost surely.By separability of H, we deduce that h,J f T H = h,J g T H P-almost surely, for all h ∈ H.Consequently, we haveJf T =J g T P-almost surely, implying J f = J g . This proves that the value of the series (3.2) does not depend on the choice of the orthonormal basis.
Furthermore, we define the isometric isomorphisms Φ := Φ µ • Φ −1 λ : 2 λ → 2 µ and Ψ := Ψ µ • Ψ −1 λ : 2 (H) → 2 (H). The following diagram illustrates the situation:In order to show that the Itô integral (7.6) is well-defined, we have to show thatFor this, we prepare the following auxiliary result:Proof. By (7.1) and (7.7), the vectors (e (λ)Let w ∈ 2 (H) be arbitrary. By (7.3) we haveTherefore, for all h ∈ H and w ∈ 2 (H) we obtainfinishing the proof.7.3. Proposition. The following statements are true: Lemma 7.2, and yields 7.5. Proposition. For every L 0 2 (H)-valued, predictable process X satisfying (7.5) the process (ξ j ) j∈N given by ξ j := Xe j , j ∈ N is a 2 (H)-valued, predictable process, and we havewhere the right-hand side of (7.10) converges unconditionally in M 2 T (H). Proof. Since Φ λ is an isometry, for each j ∈ N we obtain
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. D Werner, deFunktionalanalysis. Springer, Berlin. Leibniz Universität HannoverInstitut für Mathematische Stochastik, Welfengarten 1, 30167 Hannover, Germany E-mail address: [email protected], D. (2007): Funktionalanalysis. Springer, Berlin. Leibniz Universität Hannover, Institut für Mathematische Stochastik, Welfengar- ten 1, 30167 Hannover, Germany E-mail address: [email protected]
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arxiv |
Anomalous Coulomb Drag in Electron-Hole Bilayers due to the Formation of Excitons
27 Jan 2016
Dmitry K Efimkin
Joint Quantum Institute and Condensed Matter Theory Center
University of Maryland
20742-4111College Park, MarylandUSA
Victor Galitski
Joint Quantum Institute and Condensed Matter Theory Center
University of Maryland
20742-4111College Park, MarylandUSA
School of Physics
Monash University
3800MelbourneVictoriaAustralia
Anomalous Coulomb Drag in Electron-Hole Bilayers due to the Formation of Excitons
27 Jan 2016
Several recent experiments have reported an anomalous temperature dependence of the Coulomb drag effect in electron-hole bilayers. Motivated by these puzzling data, we study theoretically a low-density electron-hole bilayer, where electrons and holes avoid quantum degeneracy by forming excitons. We describe the ionization-recombination crossover between the electron-hole plasma and exciton gas and calculate both the intralayer and drag resistivity as a function of temperature. The latter exhibits a minimum followed by a sharp upturn at low temperatures in a qualitative agreement with the experimental observations [see, e.g., J. A. Seamons et al., Phys. Rev. Lett. 102, 026804 (2009)]. Importantly, the drag resistivity in the proposed scenario is found to be rather insensitive to a mismatch in electron and hole concentrations in sharp contrast to the scenario of electron-hole Cooper pairing. PACS numbers: 71.35.Ee, 73.63.HsCoulomb drag effect is a sensitive probe of interactions and collective phases in bilayer systems (see Ref.[1, 2] for a review). In its usual setup, an electric current in the first layer, I drive , drags charge carriers in the other one. If the second layer is closed, the drag force is compensated by the Coulomb force induced by a voltage drop, V drag , and the drag resistivity of the bilayer ρ D = V drag /I drive is measured. If the bilayer involves two weakly-coupled Fermi liquids, the temperature dependence of the drag resistivity at low temperatures is quadratic ρ D ∼ T 2 , which is well established both theoretically[3][4][5]and experimentally[6,7]. Any deviations from that Fermi-liquid behavior can signal the appearance of collective phases or correlations in the bilayer system.
A number of recent experiments [8][9][10][11] on the electronhole GaAs/GaAlAs bilayers have observed an anomalous temperature dependence of drag resistivity at at the intermediate doping n e(h) ≈ 5 10 10 cm −2 . The Tdependence of ρ D was shown to achieve a minimum, followed by a growth and saturation at lower temperatures, which were rather insensitive to the concentrations mismatch (see also related experiments for other realizations of electron-hole bilayers [12][13][14]). This behavior cannot be attributed to interlayer exchange and correlation effects [15][16][17], that are relevant in that regime, and does not appear for electron-electron and hole-hole bilayers for similar parameters. Therefore, there is strong evidence for an excitonic origin of the effect, but its detailed understanding is still lacking.
There were a number of theoretical attempts to explain the experiments based on the Bardeen-Cooper-Schrieffer (BCS) model of electron-hole Cooper pairing [18][19][20][21], which is valid in the high doping regime and can be the origin of the dipolar superfluidity [18,22,23]. The mean-field theory predicts a jump of drag resistivity at the pairing temperature to a value comparable to a single layer resistivity [24]. The jump can be smoothed by The curves correspond to different density per layer n 0 e(h) denoted by their values in cm −2 , and axes of the inset coincide with ones of the main plot. The drag resistivity ρD achieves a minimum at TD within ionization-recombination crossover between the hightemperature regime, T ≫ Eexc, where the drag is dominated by Coulomb interactions in the electron-hole plasma, and the low-temperature regime, T ≈ Eexc, where the drag is dominated by excitons.
pairing fluctuations [25,26], which are a precursor to the paired state, and both Aslamazov-Larkin [27] and Maki-Thomson [28][29][30] contributions are important here. However, Cooper pairing and the fluctuations are very sensitive to the mismatch [31,32] in contrast to experimental observations.
Here we present an alternative theoretical scenario for the effect involving the formation of excitons, which are a bound state of spatially separated electrons and holes, with a small binding energy, E exc . For T ≫ E exc , excitons ionize to form a classical electron-hole plasma and the drag effect is dominated by the Coulomb interactions.
At low temperatures, the appearance of excitons strongly enhances the drag and single-layer resistivities, leading to the upturn in the former. The anomalous behavior is robust against the mismatch in the concentration of electrons and holes: while the magnitude of the upturn is affected by it, the temperature T D , where the drag resistivity reaches minimum is insensitive to the mismatch. Proposed scenario is valid and self-consistent at low doping, and the calculated excitonic upturn is considerable larger than the observed one. Nevertheless, our results are in a qualitative agreement with the existing experiments. The main conclusion of our work is that the picture of exciton formation is more relevant to the intermediate doping regime in experiments, than the scenario of electron-hole Cooper pairing.
Model and the excitonic crossover. The system of spatially separated electrons and holes, which can bind to form excitons, can be described by the following Hamiltonian
H = pS ǫ exc (p)b + pS b pS + ps ǫ α (p)a + αps a αps + + 1 2 pp ′ q ss ′ αα ′ V αα ′ (q)a + α,p+q,s a + α ′ ,p ′ −q,s ′ a α ′ p ′ s ′ a αps .(1)
Here a αps and b ps are annihilation operators for electrons (α = e = 1), holes (α = h = −1) and excitons with momentum p and internal degeneracy spin index s = (| ↓ , | ↑ ) and S = (| ↓↓ , | ↑↓ ), | ↓↑ , | ↑↑ ). Their dispersions are ǫ α (p) = p 2 /2m α and ǫ exc (p) = p 2 /2m exc −E exc with m exc = m e +m h and E exc being the exciton mass and its binding energy; V αα (q) = 2πe 2 /ǫq and V αᾱ (q) = −2πe 2 e −qd /ǫq are bare intralayer and interlayer Coulomb interactions with interlayer spacing d and bare dielectric permittivity ǫ. We do not specify the interaction with disorder explicitly, but assume relaxation times τ α and τ exc to be momentum independent, which implies the short-range disorder to be the dominant scattering mechanism. For all numerical calculations we use the set of parameters related to the GaAs/GaAlAs bilayer in experiments [8]: m e ≈ 0.067m 0 , m h ≈ 0.4m 0 , d ≈ 30 nm, ǫ = 12.4 with m 0 the bare electronic mass. The relaxation times τ α are parametrized by mobilities M e ≈ 2 10 6 cm 2 /Vs, and M h ≈ 3 10 5 sm 2 /Vs. The excitonic relaxation time, τ exc = m * τ e τ h /(τ e m e + τ h m h ), where the reduced mass is m * = m e m h /(m e + m h ), corresponds to the mobility M exc ≈ 3.4 10 4 cm 2 /Vs. Nevertheless, excitons being nonlocal objects are more sensitive to interlayer tunneling and other factors, so their mobility can be considerably reduced and here we use M exc ≈ 10 4 cm 2 /Vs. The effective Bohr radius, a B = 2 ǫ/e 2 m * ≈ 11.8 nm, and Rydberg energy, E B = m * e 4 /2 2 ǫ 2 = 55.4 K, give the spatial and energy scales. The exciton energy, E exc , can be considerably smaller than E B at d a B and is sensitive to screening, so here we use E exc ≈ 0.5 K, corresponding to the exciton size a exc ≈ 110 nm, as an independent parameter. The model is self-consistent if excitons weakly overlap, which corresponds to the doping n e(h) 10 10 cm −2 .
The ground state of the model is believed to be the exciton condensate that forms at the temperature T Q E exc and can coexist with the degenerate gas of electrons or holes in the presence of their concentration mismatch. However, below we focus on the ionization-recombination crossover regime T E exc , where the distributions of electrons, holes and excitons are non-degenerate. To calculate their concentrations we recall that in experiments the total concentrations of charged particles per layer n 0 α are controlled independently by electrical doping, so n exc + n α = n 0 α . Here n α and n exc are concentrations of quasiparticles. Reintroducing the grand canonical Hamiltonian,Ĥ Ω =Ĥ − α µ α (n exc +n α ), with chemical potentials µ α as Lagrange multipliers, we get the chemical potential of excitons as µ exc = µ e + µ h . The equation for concentrations can be simplified to n e n h /n * + n α = n 0 α , where the concentration n * = m * T exp[−E exc /T ]/(2π 2 ). The temperature dependencies of fermionic and excitonic concentrations are given by
n α = 1 2 δn 0 α − n * + (δn 0 α ) 2 + n 2 * + 2n * n 0 T ; n exc = 1 2 n 0 T + n * − (δn 0 α ) 2 + n 2 * + 2n * n 0 T ,(2)
where δn 0 α = n 0 α − n 0 α and n 0 T = n 0 e + n 0 h are the concen-tration mismatch and the total concentration. The temperature dependence of the concentrations is depicted in Fig. 2. At low temperatures T ≪ E exc the fraction of unbound electrons and holes is exponentially small, while within the crossover T E exc there is a long non-degenerate tail of excitons decreasing as T −1 according to n exc ≈ n 0 e n 0 h /n * . The exciton gas can be considered non-degenerate until T Q ≈ 0.3 K [33].
Phenomenology of the drag effect. In the presence of electrons, holes and excitons the conductivity tensor of the bilayer system is given by
J e J h = σ exc + σ e −σ exc − σ D −σ exc − σ D σ exc + σ h E e E h ,(3)
where σ α = n α e 2 τ α /m α and σ exc = n exc e 2 τ exc /m exc are their Drude conductivities. Excitons, being composed of electrons and holes from different layers, contribute to both diagonal and off-diagonal components of the conductivity tensor with opposite signs. The transconductivity σ D originates from the Coulomb interaction between electrons and holes and is calculated microscopically below. The drag resistivity ρ D and single layer resistivities ρ α , being the components of the inverted conductivity matrix (3), can be written in a compact way
ρ D(α) = σ D(ᾱ) + σ exc σ e σ h + (σ e + σ h )σ exc .(4)
At zero temperature the excitonic contribution dominates and they become
ρ D = α Θ αᾱ m α (n 0 α − n 0 α )e 2 τ α ; ρ α = ρ D + Θᾱ α m exc n 0 α e 2 τ exc .(5)
Here Θ αᾱ = Θ(n α − nᾱ) is the Heaviside function. If densities of electrons and holes are perfectly matched, both single layer resistivities ρ α and ρ D diverge at T = 0. This corresponds to an insulating excitonic ground state with the perfect drag effect: the relation between the electric current in a layer, induced by a current in the other layer, is I drag = −I drive . Our considerations assume T ≫ T Q , where there is a competition between σ exc and σ D , but the zero-temperature values (5) reflect the strength of the low-temperature upturn.
Electron-hole transconductivity. The transconductivity σ D can be calculated in the second order of perturbation theory in the interlayer Coulomb interaction [5] as follows
σ D = − 1 16πT q ∞ −∞ dω sinh 2 ( ω 2T ) Γ RA xe (q, ω, ω)× Γ AR xh (q, ω, ω)|U eh (q, ω)| 2 ,(6)
where U (q, ω) is the screened interlayer interaction and Γ RA xα (q, ω, ω) is the current-charge-charge nonlinear susceptibility. If the relaxation times τ α are momentum independent, as we assume here, it is given by [34]
Γ RA xα (q, ω, ω) = αq x eτ α m α Π R α2 (q, ω),(7)
where Π R α2 (q, ω) is the imaginary part of the polarization operator, which for a non-degenerate gas is given by [35]
Π R α2 = − √ πnq α T q sinh ω 2T exp −q 2 α ω 2 4T 2 q 2 − q 2 4q 2 α . (8)
Here theq α = √ 2m α T is the characteristic thermal momentum scale. For the interaction U (q, ω), the static Debye-Hückel approximation, that ignores the presence of neutral excitons, yields
U eh (q) = 2πe 2 ǫ qe −qd (q + κ e )(q + κ h ) − κ e κ h e −2qd .(9)
Here κ α (q) = κ 0 α f κ (q/q α ), κ 0 α = 2πe 2 n α /ǫT is the Debye-Hückel screening momentum and f (x) is the dimensionless function f (2x) = √ π exp[−x 2 ]Erfi(x)/2x with Erfi(x) to be the imaginary error function. The static screening approximation does not take into account possible plasmon contribution [34,36], which considerably enhances the drag effect for 0.4 T /µ 1. However, in the non-degenerate regime, the plasmons become strongly damped, and can be ignored. The integral over frequencies in Eq. (6) can be calculated explicitly and we get
σ D = √ π 32 e 2 h τ e τ h 2 q 4 d m e m h I q(10)
with momenta q d = d −1 ,q * = √ 2m * T and a dimensionless integral I q over rescaled momentum q given by
I q = ∞ 0 dxq * κ 0 e κ 0 h d 3 x 4 e −2x e − x 2 4q 2 * d 2 [(x + κ e d)(x + κ h d) − κ e κ h d 2 e −2x ] 2 . (11)
There are three different momenta q d ,q * , κ 0 α (for calculations of asymptotes we assume that κ 0 e and κ 0 h have the same order of magnitude) in the integral I q , and the characteristic momentum, transfered between electron and hole layers, is the smallest of them. If these momenta are well separated, the asymptotic behavior of the integral I q can be evaluated analytically. There are four different regimes:
I :q * ≪ q d , κ 0 α ; II + : q d ≪q * , κ 0 α ; II − : κ 0 α ≪q * ≪ q d and III : κ 0 α ≪ q d ≪q * with I : I q = √ π 2q 4 * d 2 κ 0 e κ 0 h ; II + : I q = π 4 120q * d −1 κ 0 e κ 0 h ; II − : I q = √ πq 2 * κ 0 e κ 0 h d 4 ; III : I q =q * κ 0 e κ 0 h d 3 2 .(12)
Regimes I (T < T ± 1 ) and III (T ± 2 < T ) appear at small and large temperatures. Depending on the concentration n e(h) one of II − and II + is between them (T ± 1 < T < T ± 2 ). The corresponding boundaries are given by
T + 2 = 4πE B n α a B d, T − 2 = T + 1 = E B (a B /d) 2 and T − 1 = E B (4πn α a 2 B ) 2/3
. The point at which T ± 1 = T ± 2 and the regimes II ± merge and disappear corresponds to n 12 = a B /4πd 3 = 3.1 10 9 cm −2 and T 12 = E B (a B /d) 2 ≈ 7.1 K. For the densities of interest, the momentum scales are not well separated, the range of the applicability of the asymptotes (12) is reduced to T ≪ T ± 1 and T ≫ T ± 2 . Below, we calculate I q numerically. Drag resistivity of the bilayer. First, it is instructive to analyze the dependence of drag resistivity ρ D on the temperature T ignoring the presence of excitons. It is shown in Fig. 1 T + 2 , the scattering momentum is q d and the asymptotic form is ρ D ∼ T 5/2 /n 2 e n 2 h d 5 . These two scattering regimes are usual for bilayer fermion systems (along with regimes where plasmons [34,36] and phonons [37] dominate and the hydrodynamic one [38,39]), but the latter corresponds to ρ D ∼ T 2 due to degeneracy of fermions. For the considered system, at low temperatures T T + 1 , the electrons and holes avoid degeneracy by transforming into excitons and their characteristic momentum scaleq * becomes the scattering one leading to the asymptotic behavior ρ D ∼ T 4 /n 2 e n 2 h d 2 . That regime usually does not appear in a fermionic bilayer due to quantum degeneracy of fermions.
The temperature dependence of ρ D in the presence of excitons are presented in Fig. 1-a (perfectly matched densities) and Fig. 3-a (with a mismatch). The latter is supplemented by the inset Fig. 3-b in which the dependence of ρ D on the mismatch at zero temperature is depicted. The long excitonic tail, which weakly depends on temperature, considerably enhances the drag resistivity even at high temperatures T ≫ E exc . The dependence has a clear minimum at the temperature T D , which lies within the crossover E exc T D T ± 2 . The strength of the upturn is defined by the mismatch, while the temperature T D (n 0 e , n 0 h ) smoothly increases with both its arguments and does not have any features for the matched case. This makes the minimum in the temperature dependence of ρ D shallower with increasing of both concentrations, as seen in the experiment. The excitonic contribution to the drag resistivity ρ D can be well-fitted by a combination of functions T −1 and T −2 . The former dominates at high temperatures T ≫ E exc , while the latter plays the major role at T ∼ E exc . At lower temperatures the drag resistivity saturates to a value, which depends on the imbalance of concentrations (see Fig.3-b).
Resistivity of electrons is presented in Fig.3-c and supplemented by the inset (d), where its dependence on the mismatch at T = 0 is depicted. Depending on the mismatch, its enhancement vary by an order of magnitude, while the temperature dependence is quite insensitive to it. The dependencies for the resistivity of holes are qualitatively the same.
Discussion. The proposed scenario of genuine excitonic drag effect does not assume any phase transition and/or coherence of excitons, which in our model may occur at lower temperatures T Q (Localization effects and their interplay with other ground states, not involving exciton condensation, can not be ruled out: e.g., an excitonic Bose glass [40,41] or an exotic Bose-metal phase [42], which was conjectured to exist in models involving dirty composite bosons and gapless fermionic excitations). We argue however that the upturn in ρ D is unrelated to the quantum effects including localization, but appears at the temperature T D corresponding to the ionization-recombination excitonic crossover E exc T D T ± 2 from a classical electron-hole plasma to a classical exciton gas. The exact value of the T D is non-universal and depends on the interlayer distance, quasiparticle mobilities, effective masses, etc.
For explicit calculations above, we have used a range of electron and hole concentrations, which is about an order of magnitude smaller than the ones in the published experiments to ensure that the assumptions of our model are self-consistent. In the intermediate doping regime realized in experiment so far, the excitons overlap and cannot be considered as two-particle objects anymore.
To develop a quantitative many-body theory for the Coulomb drag effect in this intermediate regime is difficult, because of complicated interplay of the Pauli blocking effects, self-consistent screening, and coexistence of excitons with degenerate gas of electrons and holes. The extrapolation of our results to this regime considerably overestimates the strength of the excitonic upturn seen in experiments. Nevertheless, the observed behavior of the drag resistivity on temperature and concentrations is qualitatively captured, and we conclude that the picture of exciton formation is more relevant to the experiments, than the scenario of electron-hole Cooper pairing and pairing fluctuations.
FIG. 1 :
1(Color online) Shown is the dependence of the drag resistivity ρD on temperature for matched concentrations of electrons and holes with excitons [Fig. 1(a)] and without excitons [inset 1(b)].
FIG. 2 :
2(Color online) The temperature dependence of concentrations of electrons ne, holes n h and excitons nexc, which are given by the Eqs. (2), for fixed total concentrations per layer n 0 e = 8 10 9 cm −2 , n 0 h = 6 10 9 cm −2 . The dependencies for other values of n 0 e(h) are qualitatively similar. At high temperatures T ≫ Eexc there is a long excitonic tail nexc ≈ n 0 e n 0 h /n * ∼ T −1 . The concentration of excitons at zero tempearture is equal to that of the minority species (holes in this case) in the limit of large temperatures.
FIG. 3 :
3(Color online) (a) and (c) The temperature dependencies of the drag resistivity ρD and the resistivity of electrons ρe in the presence of the mismatch in electron and hole concentrations. (b) and (d) The corresponding values at zero temperature, which are given by Eqs. (5). The strength of the excitonic enhancement of both ρD and ρα is defined by the mismatch, while the temperature dependencies are quite insensitive to it.
-b (for matched concentrations of electrons and holes). At high temperatures T + 2 T , the screening disappears, making κ 0 α the smallest momentum scale, and the drag resistivity decreases as ρ D ∼ T −3/2 /d. In the intermediate regime T + 1 T
q,0 b FIG. S1: a) Frequency and momentum dependence of the imaginary part of polarization operatorΠ2(q,ω); b) Momentum dependence of the real part of the polarization operatorΠ1(q,ω) in the static limit.
Supplemental Material: "Anomalous Coulomb Drag in Electron-Hole Bilayers due to the Formation of Excitons" by Dmitry K.Efimkinand Victor GalitskiThe Supplemental Material presents a calculation of the polarization operator Π(q, ω) of two-dimensional nondegenerate electron gas. The general expression for the polarization operator Π(q, ω) is given byHere ξ p = p 2 /2m − µ is the dispersion law of fermions and n F (ξ p ) ≈ exp[−βξ p ] is their Maxwell distribution function with chemical µ = T ln(π 2 n/mT ), where n is the total concentration of fermions including spin degeneracy. It is instructive to calculate the imaginary part of the polarization operator Π 2 (q, ω) at first and it can be presented as followsThe character energy and momentum of nondegenerate electron gas are ω T = T and q T = √ 2mT , and the polarization operator, which has the dimension of the density of states, has the scale n/T . As a result, introducing dimensionlessThe argument of the delta function achieves zero only if x > x 0 , where x 0 = (ω −q 2 ) 2 /4q 2 , that leads tōand finally we get the expression (8) from the main part of the paper, written in dimensionless unitsThe dependenceΠ 2 (q,ω) on momentum and frequency is presented inFig. S1-a. The valueΠ 2 (q,ω) is nonzero for arbitrary momentum and frequency and has prominent maximum atw ∼ 1 andq ∼ 1, which correspond to the thermal frequency ω T and momenta q T . The calculated dependence for nondegenerate electron gas drastically differs from one at zero temperature. The valueΠ 2 (q,ω) at zero temperatures is nonzero only within the continuum of two-particle excitations, which has sharp boundaries.In the paper we use the static screening approximation and need only real part of the polarization operatorΠ 1 (q, 0) in the static limit. It can be calculated from the imaginary part with help of Kramers-Kronig relations as followsHere Erfi(q/2) is the imaginary error function. The approximate expression interpolates the corresponding asymptotics and well describes the momentum dependence ofΠ 1 (q, 0) at arbitrary momentum. The momentum dependence of Π 1 (q, 0) is presented inFig. S1-b.
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| {'fraction_non_alphanumeric': 0.07104110160650949, 'fraction_numerical': 0.04141456290423534, 'mean_word_length': 3.5475964579380137, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 30, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Several recent experiments have reported an anomalous temperature dependence of the Coulomb drag effect in electron-hole bilayers. Motivated by these puzzling data, we study theoretically a low-density electron-hole bilayer, where electrons and holes avoid quantum degeneracy by forming excitons. We describe the ionization-recombination crossover between the electron-hole plasma and exciton gas and calculate both the intralayer and drag resistivity as a function of temperature. The latter exhibits a minimum followed by a sharp upturn at low temperatures in a qualitative agreement with the experimental observations [see, e.g., J. A. Seamons et al., Phys. Rev. Lett. 102, 026804 (2009)]. Importantly, the drag resistivity in the proposed scenario is found to be rather insensitive to a mismatch in electron and hole concentrations in sharp contrast to the scenario of electron-hole Cooper pairing. PACS numbers: 71.35.Ee, 73.63.HsCoulomb drag effect is a sensitive probe of interactions and collective phases in bilayer systems (see Ref.[1, 2] for a review). In its usual setup, an electric current in the first layer, I drive , drags charge carriers in the other one. If the second layer is closed, the drag force is compensated by the Coulomb force induced by a voltage drop, V drag , and the drag resistivity of the bilayer ρ D = V drag /I drive is measured. If the bilayer involves two weakly-coupled Fermi liquids, the temperature dependence of the drag resistivity at low temperatures is quadratic ρ D ∼ T 2 , which is well established both theoretically[3][4][5]and experimentally[6,7]. Any deviations from that Fermi-liquid behavior can signal the appearance of collective phases or correlations in the bilayer system.', 'arxivid': '1506.00305', 'author': ['Dmitry K Efimkin \nJoint Quantum Institute and Condensed Matter Theory Center\nUniversity of Maryland\n20742-4111College Park, MarylandUSA\n', 'Victor Galitski \nJoint Quantum Institute and Condensed Matter Theory Center\nUniversity of Maryland\n20742-4111College Park, MarylandUSA\n\nSchool of Physics\nMonash University\n3800MelbourneVictoriaAustralia\n'], 'authoraffiliation': ['Joint Quantum Institute and Condensed Matter Theory Center\nUniversity of Maryland\n20742-4111College Park, MarylandUSA', 'Joint Quantum Institute and Condensed Matter Theory Center\nUniversity of Maryland\n20742-4111College Park, MarylandUSA', 'School of Physics\nMonash University\n3800MelbourneVictoriaAustralia'], 'corpusid': 20537173, 'doi': '10.1103/physrevlett.116.046801', 'github_urls': [], 'n_tokens_mistral': 10076, 'n_tokens_neox': 8453, 'n_words': 5172, 'pdfsha': 'dde1d52f31c443eddb5484b0b37c1834194c5c65', 'pdfurls': ['https://arxiv.org/pdf/1506.00305v2.pdf'], 'title': ['Anomalous Coulomb Drag in Electron-Hole Bilayers due to the Formation of Excitons', 'Anomalous Coulomb Drag in Electron-Hole Bilayers due to the Formation of Excitons'], 'venue': []} |
arxiv |
p-adic l-functions and sums of powers
27 May 2006
Taekyun Kim [email protected]
Jangjeon Research Institute for Mathematical Sciences & Physics
Ju-Kong Building 103-Dong 1001-ho, 544-4 Young-chang Ri Hapcheon-Up Hapcheon-Gun Kyungnam678-802Korea
p-adic l-functions and sums of powers
27 May 2006arXiv:math/0605703v1 [math.NT] (−1) j j r as a power series in n. The coefficients are values of p-adic l-function for Euler numbers.
In this paper, we give an explicit p-adic expansion ofas a power series in n. The coefficients are values of p-adic l-function for Euler numbers.
Introduction
Let p be a fixed prime. Throughout this paper Z p , Q p , C and C p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Q p , cf. [1], [3], [6], [10]. Let v p be the normalized exponential valuation of C p with |p| p = p −vp(p) = p −1 . Kubota and Leopoldt proved the existence of meromorphic functions, L p (s, χ), defined over the p-adic number field, that serve as p-adic equivalents of the Dirichlet L-series, cf. [8], [10]. These p-adic L-functions interpolate the values L p (1 − n, χ) = − 1 n (1 − χ n (p)p n−1 )B n,χn , for n ∈ N = {1, 2, · · · }, where B n,χ denote the nth generalized Bernoulli numbers associated with the primitive Dirichlet character χ, and χ n = χw −n , with w the Teichmüller character, see [2,3,5,6,17,20]. In [14], L. C. Washington have proved the below interesting formula:
np j=1 (j,p)=1 1 j r = − ∞ k=1 −r k (pn) k L p (r + k, w 1−k−r ),
where −r k is binomial coefficient. In the recent many authors have studied qextension of Euler numbers and Bernoulli numbers (see [1,4,5,9,12,13]). These qextensions seem to be valuable and worthwhile in the areas of mathematical physics and mathematics (see [ 1,4,6,13,14,15,17,18,19]). By using q-Volkenborn integration, Kim gave the interesting properties of q-Bernoulli and Euler polynomials [8,9,10,11] and Ryoo-Kim-Agarwal have investigated the properties of the qextension of Euler numbers and polynomials by using "Mathematica package", see [14,15]. The problems to find the sums of powers of consecutive q-integers were suggested. Kim and Schlosser treated the formulae for the sums of powers of consecutive q-integers [7,11,16] and these formulae were used to give the q-extension of Washington's p-adic L-functions and sums of powers ( see [ 6,10,20]). In [11,14], we found the interesting formulae "alternating sums of powers of consecutive integers " which are related to Euler numbers and polynomials. By using these alternating sums of powers of consecutive integers, we try to construct the p-adic lfunctions and sums of powers for Euler numbers and polynomials, corresponding to Washington and Kim (see [10,20]). The purpose of this paper is to give alternating p-adic harmonic series in terms of n and p-adic l-function for Euler numbers.
A note on l-series associated with Euler numbers and polynomials
We begin with well known Euler polynomials E n (x).
Definition 1. Euler polynomials are defined by
2 e t + 1 e xt = ∞ n=1 E n (x) n! t n ,
E n (x) are called n-th Euler polynomials. For x = 0, E n = E n (0) are called Euler numbers. By the definition of Euler polynomials, we easily see that,
E l (x) = l n=0 l n E n x l−n ∈ C[x].
From the generating function of Euler polynomials F (t, x) = 2 e t +1 e xt , we derive
F (t, x) = 2e xt ∞ l=0 (−1) l e lt = 2 ∞ l=0 (−1) l e (l+x) t.(1)
For k ∈ N, we note that
d k dt k F (t, x) t=0 = 2 ∞ l=0 (−1) l d k dt k e (l+x)t t=0 = 2 ∞ l=0 (−1) l (l + x) k .(2)
Therefore we can define the Euler zeta function as follows:
Definition 2. For s ∈ C, we define Euler zeta function as
ζ E (s) = 2 ∞ l=0 (−1) l (l + x) s .(3)
By using Definition 2 and (2), we obtain the following:
Proposition 3. For k ∈ N, we have ζ E (−k, x) = E k (x).(4)For f (=odd) ∈ N, ∞ n=0 E n (x) t n n! = 2 e t + 1 e xt = 2 ∞ l=0 (−1) l e (l+x)t = 2 f −1 a=0 ∞ l=0 (−1) a+lf e (a+lf +x)t = f a=1 (−1) a 2 ∞ n=0 (−1) l e f t(l+ x+a f ) = f a=1 (−1) a ∞ n=0 E n x + a f f n t n n! = ∞ n=0 f n f a=1 (−1) a E n x + a f t n n! .
Thus we note that
E n (x) = f n f a=1 (−1) n E n x + a f ,(5)
where f (=odd) ∈ N. This (5) is so called Distribution for Euler polynomials.
2 n−1 l=0 (−1) l l m = (−1) n+1 m−1 l=0 E l n m−l m l + (−1) n+1 + 1 E m .
In particular, if n is even, then
2 n−1 l=0 (−1) l l m = − m−1 l=0 E l n m−l .
Let s be a complex variable and let a, F (=odd) be integers with 0 < a < F .
H(s, a|F ) = m≡a(F ) m>0 (−1) m m s = ∞ n=0 (−1) nF +a (a + nF ) s = (−1) a ∞ n=0 (−1) n (a + nF ) s = (−1) a ∞ n=0 (−1) n F s a F + n s = (−1) a F −s 2 2 ∞ n=1 (−1) n n + a F = (−1) a F −s 2 ζ E s, a F .(6)
Note that
H(−n, a|F ) = (−1) a F n 2 E n a F .(7)
Let χ be the primitive Dirichlet character with conductor f (=odd) ∈ N.
F χ (t) = 2 ∞ n=0 e nt χ(n)(−1) n = 2 f −1 a=0 ∞ n=0 e (a+nf )t χ(a + nf )(−1) a+nf = f −1 a=0 e at χ(a)(−1) a 2 ∞ n=0 (−1) n e nf t = 2 f −1 a=0 e at χ(a)(−1) a 1 e f t + 1 = 2 f −1 a=0 e at χ(a)(−1) a e f t + 1 = ∞ n=0 E n, χ t n n! .(8)
Thus, we can define the below generalized Euler number attached to χ. E n,χ will be called the n-th generalized Euler numbers attach to χ.
From the Definition 4, we derive the below formula:
∞ n=0 E n,χ t n n! = 2 f −1 a=0 e at χ(a)(−1) a e f t + 1 = f −1 a=0 χ(a)(−1) a 2 e f t + 1 e at = f −1 a=0 χ(a)(−1) a ∞ n=0 E n a f f n t n n! = ∞ n=0 f n f −1 a=0 χ(a)(−1) a E n a f t n n! .
By comparing the coefficients on both sides, we easily see that
E n,χ = f n f −1 a=0 χ(a)(−1) a E n a f .(9)
Definition 5. For s ∈ C, we define Dirichlet's l-function as follows:
l(s, χ) = 2 ∞ n=1 χ(n)(−1) n n s . Note that F χ (t) = 2 ∞ n=1 e nt χ(n)(−1) n = 2 f −1 a=0 (−1) a χ(a)e at e f t + 1 = ∞ n=0 E n,χ t n n! .
For k ∈ N,
E k,χ = d k dt k F χ (t) t=0 = 2 ∞ n=1 χ(n)(−1) n d k dt k e nt t=0 = 2 ∞ n=1 χ(n)(−1) n n k .(10)
By Definition 5 and (10), we easily see that l(−k, χ) = E k, χ , where k ∈ N. Therefore we obtain the following:
Proposition 6. Let k be the positive integer. Then we have
l(−k, χ) = E k,χ .(11)
Let χ be the Dirichlet character with conductor f (=odd) ∈ N. Then we note that l(s, χ) = 2 f a=1 χ(a)H(s, a|f ) (12) In Eq. (12), we give a value of l(s, x) at negative integer:
l(−n, χ) = 2 f a=1 χ(a)H(−η, a|f ) = 2 f a=1 χ(a)(−1) a f n 2 E n a f = f n f −1 a=0 χ(a)(−1) a E n a f = E n,χ .
The function H(s, a|F ) will be called partial zeta function which interpolates Euler polynomials at negative integers. The values of l(s, χ) at negative integers are algebraic, hence may be regarded as lying in an extension of Q p . We therefore look for a p-adic function which agrees with l(s, χ) at negative integers in later.
A note on p-adic l-function
We define x = x w(x) , where w(x) is the Teichmüller character. When F (=odd) is a multiple of p and (a, p) = 1, we define
H p (s, a|F ) = (−1) a 2 a −s ∞ j=0 −s j F a j E j , for s ∈ Z p .
It is easy to see that
H p (−n, a|F ) = (−1) a 2 a n n j=0 n j F a E j = (−1) a 2 F n w −n (a) n j=0 n j a F n−j E j = (−1) a 2 F n w −n (a)E n a F = w −n (a)H(−n, a|F ),
for all positive integers. Now we consider p-adic interpolation function for Euler numbers as follows;
l p (s, χ) = 2 F a=1 (a,p)=1 χ(a)H p (s, a|F ) for s ∈ Z p .
Let n be natural number. Then we have
l p (−n, χ) = 2 F n=1 (n,p)=1 χ(a)H p (−n, a|F ) = E n, χw −n − p n χw −n (p)E n, χw −n = (1 − p n χw −n (p))E n, χw −n .
In fact, we have the formula
l p (s, χ) = F a=1 (−1) a a −s χ(a) ∞ j=0 −s j F a j E j , for s ∈ Z p .
This is a p-adic analytic function and has the following properties for χ = w t .
l p (−n, w t ) = (1 − p n )E n , where n ≡ t (mod p − 1), l p (s, w t ) ∈ Z p for all s ∈ Z p , when t ≡ 0 (mod p − 1). If t ≡ 0 (mod p − 1), then l p (s 1 , w t ) ≡ l p (s 2 , w t ) (mod p) for all s 1 , s 2 ∈ Z p , l p (k, w t ) ≡ l p (k + p, w t ) (mod p).
It is easy to see that
1 r + k − 1 −r k 1 − r − k j = −1 j + k −r k + j − 1 k + j j ,
for all positive integers with r, j, k with j, k ≥ 0, j + k > 0 and r = 1 − k. Thus, we note that
1 r + k − 1 −r k 1 − r − k j = 1 r − 1 −r + 1 k + j k + j j .
Hence, we have
r r + k −r + 1 k −r − k j = −r k + j k + j j ,
where k, j are positive integers. Let F (=odd) be positive integers. Then
n−1 l=0 (−1) F l+a (F l + a) r = n−1 l=0 (−1) F l+a a −r ∞ s=0 −r s F l a s = ∞ m=0 −r m a −r F a m (−1) a n−1 l=0 (−1) l l m = ∞ m=0 −r m a −r F a m (−1) a (−1) n+1 2 m−1 l=0 E l n m−l m l + ((−1) m−1 + 1)E m ,
when n is even integer
n−1 l=0 (−1) F l+a (F l + a) r = a −r (−1) a ∞ m=0 −r m F a m (−1) m+1 2 m+1 l=0 E l n m−l m l = −a −r (−1) a ∞ s=0 −r s F a s s+1 l=0 E l n s−l s l = − ∞ s=0 −r s w −r (a) F a s (−1) a 2 a −r s−1 l=0 E l n s−l s l = ∞ k=0 ∞ l=0 −r k + l w −r (a) a F −k−l n k+l (−1) a 2 a −r E l 1 n l k + l l = ∞ k=0 ∞ l=0 r r + k −r − 1 k −r − k l a −r a F −k−l n k+l (−1) a 2 E l 1 n l = ∞ k=0 ∞ l=0 r r + k −r − 1 k w −k−r (a)(nF ) k (−1) a 2 a −k−r ∞ l=0 −r − k l E l F a = − ∞ k=0
r r + k −r − 1 k w −k−r (a)(nF ) k H p (r + k, a|F ).
For F = p, r, n(=even) ∈ N, we see that Therefore we obtain the following:
Theorem 6. Let p be an odd prime and let n(=even), r ∈ N. Then we have 2 np j=1 (j,p)=1 (−1) j j r = − ∞ k=0 r r + k −r − 1 k (pn) k l p (r + k, w −k−r ).
Definition 4 .
4Let χ be the Dirichlet character with conductor f (=odd) ∈ N. Then we define the generalized Euler numbers attached to χ as follows
) k l p (r + k, w −k−r ).
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On a p-adic interpolation function for Euler numbers and its derivetaves. K Shiratani, S Yamamoto, Mem. Fac. Sci, Kyushu Univ. 39K. Shiratani and S. Yamamoto, On a p-adic interpolation function for Euler numbers and its derivetaves, Mem. Fac. Sci, Kyushu Univ., 39(1985), 113-125.
On the two-variable Dirichlet q-L-series. Y Simsek, D Kim, S.-H Rim, Adv. Stud. Contemp. Math. (Kyungshang). 102Y. Simsek, D. Kim and S.-H. Rim, On the two-variable Dirichlet q-L-series, Adv. Stud. Contemp. Math. (Kyungshang), 10(2)(2005), 131-142
Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. H M Srivastava, T Kim, Y , Russian Journal of Mathematical Physics. 12H. M. Srivastava, T. Kim, Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series , Russian Journal of Mathematical Physics, 12(2005), 241-278
p-adic L-functions and sums of powers. L C Washington, J. Number Theory. 691L. C. Washington, p-adic L-functions and sums of powers, J. Number Theory, 69(1)(1988), 50-61.
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arxiv |
Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values
Jianfeng Wang
Department of Mathematics and Mathematical Statistics
Umeå University
901 87UmeåSESweden
Zhiyong Zhou
Department of Statistics
Zhejiang University City College
310015HangzhouChina
Jun Yu
Department of Mathematics and Mathematical Statistics
Umeå University
901 87UmeåSESweden
Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values
Compressive sensingq-ratio block sparsityq-ratio block constrained minimal singular valueConvex-concave procedure
In this paper we introduce the q-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed 2 / q and the mixed 2 / 1 norms of the reconstruction errors for stable and robust recovery using block Basis Pursuit (BBP), the block Dantzig selector (BDS) and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV based error bounds are tighter than the block restricted isotropic constant based bounds.Lasso estimator[21]. Afterwards,[19]brought in a variant of the CMSV: ω ♦ (A, s) = min z =0, z 1 / z ∞≤s Az ♦ z ∞ with · ♦ denoting a general norm, and expressed the ∞ recovery error bounds using this quantity. The latest progress concerning the CMSV can be found in[26,27].[27] generalized these two measures to a new measure called q-ratio CMSV: ρ q,s (A) = min z =0,( z 1 / z q ) q/(q−1) ≤s Az 2 z q with q ∈ (1, ∞] and established both q and 1 bounds of recovery errors.[26]investigated geometrical property of the q-ratio CMSV, which can be used to derive sufficient conditions and error bounds of signal recovery.In addition to the simple sparsity, a signal x can also possess a structure called block sparsity where the non-zero elements occur in clusters. It has been shown that using block information in CS can lead to a better signal recovery[2,7,24]. Analogue to the simple sparsity, there are block NSP and block RIP to characterize the measurement matrix in order to guarantee a successful recovery through (1)[9]. Nevertheless, they are still computationally hard to be verified for a given A. Thus it is desirable to develop a computable measure like the CMSV for recovery of simple (non-block) sparse signals.[20] proposed a new measure of the measurement matrix based on the CMSV for block sparse signal recovery and derived the mixed 2 / ∞ and 2 bounds of recovery errors. In this paper, we extend the q-ratio CMSV in [27] to q-ratio block CMSV (BCMSV) and generalize the error bounds from the mixed 2 / ∞ and 2 norms in [20] to mixed 2 / q with q ∈ (1, ∞] and mixed 2 / 1 norms.This work includes four main contributions to block sparse signal recovery in compressive sensing: (i) we establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise free block BP (BBP), and develop a convex-concave procedure to solve the corresponding non-convex problem in the condition; (ii) we introduce the q-ratio BCMSV and derive both the mixed 2 / q and the mixed 2 / 1 norms of the reconstruction errors for stable and robust recovery using the BBP, the block DS (BDS) and the group lasso in terms of the q-ratio BCMSV; (iii) we prove that for sub-gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large; (iv) we present an algorithm to compute the q-ratio BCMSV for an arbitrary measurement matrix and investigate its properties.The paper is organized as follows. In Section 2, we introduce the definitions for the q-ratio block sparsity and the q-ratio BCMSV, and present the sufficient condition for the noise free BBP recovery based on the q-ratio block sparsity and an inequality for the q-ratio BCMSV. The mixed 2 / q and the mixed 2 / 1 reconstruction errors for the BBP, the BDS and the group lasso in terms of the q-ratio BCMSV are derived in Section 3. In Section 4, the probabilistic results of the q-ratio BCMSVs for sub-gaussian random matrices are demonstrated. Section 5 is reserved for algorithms to solve the optimization problem in the sufficient condition for the noise free BBP recovery and compute the q-ratio BCMSV. The q-ratio BCMSV based bounds and the block RIC based bounds for the BBP are also compared therein. Section 6 is devoted to the conclusion. All proofs are left in the Appendix.Proof of Theorem 1. The proof procedure follows from the similar arguments in[18,19], and the procedure can be divided into two main stepsStep 1 : We first derive upper bounds of the q-ratio block sparsity of residual h =x − x for all algorithms. As x is block k-sparse, we assume that bsupp(x) = S and |S| ≤ k.For the BBP and the BDS, since x 2,1 = x + h 2,1 is the minimum among all z satisfying the constraints of BBP and BDS (including the true signal x), we have
Introduction
Compressive sensing (CS) [3,6] aims to recover an unknown sparse signal x ∈ R N from m noisy measurements y ∈ R m :
y = Ax + ,(1)
where A ∈ R m×N is a measurement matrix with m N , and ∈ R m is additive noise such that 2 ≤ ζ for some ζ ≥ 0. It has been proven that if A satisfies the (stable/robust) null space property (NSP) or restricted isometry property (RIP), (stable/robust) recovery can be achieved [8,Chapter 4 and 6]. However, it is computationally hard to verify NSP and compute the restricted isometry constant (RIC) for an arbitrarily chosen A [1, 22]. To overcome the drawback, a new class of measures for the measurement matrix has been developed during the last decade. To be specific, [18] introduced a new measure called 1 -constrained minimal singular value (CMSV): ρ s (A) = min z =0, z 2 1 / z 2 2 ≤s Az 2 z 2 and obtained the 2 recovery error bounds in terms of the proposed measure for the Basis Pursuit (BP) [5], the Dantzig selector (DS) [4], and the 2 q-ratio block sparsity and q-ratio BCMSV -definition and property
In this section, we introduce the definitions of the q-ratio block sparsity and the q-ratio BCMSV, and present their fundamental properties. A sufficient condition for block sparse signal recovery via the noise free BBP using the q-ratio block sparsity and an inequality for the q-ratio BCMSV are established.
Throughout the paper, we denote vectors by bold lower case letters or bold numbers, and matrices by upper case letters. x T denotes the transpose of a column vector x. For any vector x ∈ R N , we partition it into p blocks, each of length n, so we have x = [x T 1 , x T 2 , · · · , x T p ] T and x i ∈ R n denotes the i-th block of x. We define the mixed 2 / 0 norm x 2,0 = p i=1 1{x i = 0}, the mixed 2 / ∞ norm x 2,∞ = max 1≤i≤p x i 2 and the mixed 2 / q norm x 2,q = ( p i=1 x i q 2 ) 1/q for 0 < q < ∞. A signal x is block k-sparse if x 2,0 ≤ k.
[p] denotes the set {1, 2, · · · , p} and |S| denotes the cardinality of a set S. Furthermore, we use S c for the complement [p] \ S of a set S in [p]. The block support is defined by bsupp(x) := {i ∈ [p] : x i 2 = 0}. If S ⊂ [p], then x S is the vector coincides with x on the block indices in S and is extended to zero outside S. For any matrix A ∈ R m×N , kerA := {x ∈ R N : Ax = 0}, A T is the transpose. ·, · is the inner product function.
We first introduce the definition of the q-ratio block sparsity and its properties.
Definition 1 ( [25]). For any non-zero x ∈ R N and non-negative q / ∈ {0, 1, ∞}, the q-ratio block sparsity of x is defined as
k q (x) = x 2,1 x 2,q q q−1 .
(2)
The cases of q ∈ {0, 1, ∞} are evaluated by limits:
k 0 (x) = lim q→0 k q (x) = x 2,0(3)k 1 (x) = lim q→1 k q (x) = exp(H 1 (π(x))) (4) k ∞ (x) = lim q→∞ k q (x) = x 2,1 x 2,∞ .(5)
Here π(x) ∈ R p with entries π i (x) = x i 2 / x 2,1 and H 1 is the ordinary Shannon entropy
H 1 (π(x)) = − p i=1 π i (x) log π i (x).
This is an extension of the sparsity measures proposed in [13,14], where estimation and statistical inference via α-stable random projection method were investigated. In fact, this kind of sparsity measure is based on entropy, which measures energy of blocks of x via π i (x). Formally, we can express the q-ratio block sparsity by
k q (x) = exp(H q (π(x))) if x = 0 0 if x = 0,(6)
where H q is the Rényi entropy of order q ∈ [0, ∞] [15,23]. When q / ∈ {0, 1, ∞}, the Rényi entropy is given by H q (π(x)) = 1 1−q log( p i=1 π i (x) q ), and for the cases of q ∈ {0, 1, ∞}, the Rényi entropy is evaluated by limits and results in (3), (4) and (5), respectively. The sparsity measure k q (x) has the following basic properties (see also [13,14,25]):
• Continuity: unlike traditional block sparsity measure using the mixed 2 / 0 norm, k q (x) is continuous on R N \ {0} for all q > 0. Thus, it is stable with respect to small perturbations of a signal.
• Scale-invariance: for any c = 0, it holds that k q (cx) = k q (x). This property is in line with the common sense that the measure should not depend on absolute magnitude of a signal.
• Non-increasing with respect to q: For any q ≥ q ≥ 0, we have
x 2,1 x 2,∞ = k ∞ (x) ≤ k q (x) ≤ k q (x) ≤ k 0 (x) = x 2,0 ,
which follows from the non-increasing property of the Rényi entropy H q with respect to q.
• Range equals to [1, p]: for all x ∈ R N \ {0} with p blocks and all q ∈ [0, ∞], we have 1 ≤
x 2,1 x 2,∞ = k ∞ (x) ≤ k q (x) ≤ k 0 (x) = x 2,0 ≤ p.
Next, we present a sufficient condition for the exact recovery via the noise free BBP in terms of the q-ratio block sparsity. Recall that when the true signal x is block k-sparse, the sufficient and necessary condition for the exact recovery via the noise free BBP:
min z∈R N z 2,1 s.t. Az = Ax(7)
in terms of the block NSP of order k was given by [9,17] z S 2,1 < z S c 2,1 , ∀z ∈ kerA \ {0}, S ⊂ [p] and |S| ≤ k.
Proposition 1. If x is block k-sparse and there exists at least one q ∈ (1, ∞] such that k is strictly less than
min z∈kerA\{0} 2 q 1−q k q (z),(8)
then the unique solution to problem (7) is the true signal x.
Remark 1. This proposition is an extension of Proposition 1 in [27] from simple sparse signals to block sparse signals. In Section 5, we adopt a convex-concave procedure algorithm to solve (8) approximately.
Now we are ready to present the definition of the q-ratio BCMSV, which is developed based on the q-ratio block sparsity.
Definition 2.
For any real number s ∈ [1, p], q ∈ (1, ∞] and matrix A ∈ R m×N , the q-ratio block constrained minimal singular value (BCMSV) of A is defined as
β q,s (A) = min z =0,kq(z)≤s Az 2 z 2,q .(9)
Remark 2. For measurement matrix A with unit norm columns, it is obvious that β q,s (A) ≤ 1 since Ae i 2 = 1, e i 2,q = 1 and k q (e i ) = 1, where e i is the i-th canonical basis for R N . Moreover, when q and A are fixed, β q,s (A) is non-increasing with respect to s. Besides, it is worth noticing that the q-ratio BCMSV depends also on the block size n, we choose to not show this parameter for the sake of simplicity. Another interesting finding is that for any α ∈ R, we have β q,s (αA) = |α|β q,s (A). This fact together with Theorem 1 in Section 3 implies that in the case of adopting a measurement matrix αA, increasing the measurement energy through |α| will proportionally reduce the mixed 2 / q norm of reconstruction errors.
Comparing to the block RIP [9], there are three main advantages by using the q-ratio BCMSV:
• It is computable (see the algorithm in Section 5).
• The proof procedures and results of recovery error bounds are more concise (details in next section).
• The q-ratio BCMSV based recovery bounds are smaller (better) than the block RIC based bounds (shown in Section 5) [see also 20, 27, for another two specific examples]
As for different q, we have the following important inequality, which plays a crucial role in deriving the probabilistic behavior of β q,s (A) via the existing results established in [20].
Proposition 2. If 1 < q 2 ≤ q 1 ≤ ∞, then for any real number 1 ≤ s ≤ p 1/q withq = q 2 (q 1 −1) q 1 (q 2 −1) , we have
β q 1 ,s (A) ≥ β q 2 ,sq (A) ≥ s −q β q 1 ,sq (A).(10)
Remark 3. Let q 1 = ∞ and q 2 = 2 (thusq = 2), we have β ∞,s (A) ≥ β 2,s 2 (A) ≥ 1 s 2 β ∞,s 2 (A). If q 1 ≥ q 2 > 1, thenq = q 2 (q 1 −1) q 1 (q 2 −1) = 1 + q 1 −q 2 q 1 (q 2 −1) ≥ 1, so β q 2 ,sq (A) ≤ β q 2 ,s (A). Similarly, we have for any t ∈ [1, p] β q 2 ,t (A) ≥ 1 t β q 1 ,t (A) by letting t = sq in (10). Based on these facts, we can not obtain the monotonicity with respect to q when s and A are fixed. However, since for any z ∈ R N with p blocks, k q (z) ≤ p, it holds trivially that β q,p (A) is increasing with respect to q by using the decreasing property of the mixed 2 / q norm.
Recovery error bounds
In this section, we derive the recovery error bounds in terms of the mixed 2 / q norm and the mixed 2 / 1 norm via the q-ratio BCMSV of the measurement matrix. We focus on three renowned convex relaxation algorithms for block sparse signal recovery from (1): the BBP, the BDS and the group lasso.
BBP: min z∈R N z 2,1 s.t. y − Az 2 ≤ ζ. BDS: min z∈R N z 2,1 s.t. A T (y − Az) 2,∞ ≤ µ.
Group lasso: min
z∈R N 1 2 y − Az 2 2 + µ z 2,1 .
Here ζ and µ are parameters used in the constraints to control the noise level. We first present the following main results of recovery error bounds for the case when the true signal x is block k-sparse.
Theorem 1. Suppose x is block k-sparse. For any q ∈ (1, ∞], we have 1) If 2 ≤ ζ, then the solutionx to the BBP obeys
x − x 2,q ≤ 2ζ β q,2 q q−1 k (A) ,(11)
x
− x 2,1 ≤ 4k 1−1/q ζ β q,2 q q−1 k (A) .(12)
2) If the noise in the BDS satisfies A T 2,∞ ≤ µ, then the solutionx to the BDS obeys
x − x 2,q ≤ 4k 1−1/q β 2 q,2 q q−1 k (A) µ,(13)
x
− x 2,1 ≤ 8k 2−2/q β 2 q,2 q q−1 k (A) µ.(14)
3) If the noise in the group lasso satisfies A T 2,∞ ≤ κµ for some κ ∈ (0, 1), then the solutionx to the group lasso obeys
x − x 2,q ≤ 1 + κ 1 − κ · 2k 1−1/q β 2 q,( 2 1−κ ) q q−1 k (A) µ,(15)
x (11) and (12), then the noise free BBP (7) can uniquely recover any block k-sparse signal by letting ζ = 0.
− x 2,1 ≤ 1 + κ (1 − κ) 2 · 4k 2−2/q β 2 q,( 2 1−κ ) q q−1 k (A) µ.(16)Remark 4. Obviously, if β q,2 q q−1 k (A) = 0 in
Remark 5. The mixed 2 / q norm error bounds are generalized from the existing results in [20] (q = 2 and ∞) to any 1 < q ≤ ∞ and from [27] (simple sparse signal recovery) to block sparse signal recovery. The mixed 2 / q norm error bounds depend on the q-ratio BCMSV of the measurement matrix A, which is bounded away from zero for sub-gaussian random matrix and can be computed approximately by using a specific algorithm, which are discussed in the later sections.
Remark 6. As shown in literature, the block RIC based recovery error bounds for the BBP [9], the BDS [12] and the group lasso [10] are complicated. In contrast, as presented in this theorem, the q-ratio BCMSV based bounds are much more concise and corresponding derivations are much less complicated, which are given in the Appendix.
Next, we extend Theorem 1 to the case when the signal is block compressible, in the sense that it can be approximated by a block k-sparse signal. Given a block compressible signal x, let the mixed 2 / 1 error of the best block k-sparse approximation of
x be φ k (x) = inf z∈R N , z 2,0 =k x − z 2,1 , which measures how close
x is to the block k-sparse signal.
Theorem 2.
Suppose that x is block compressible. For any 1 < q ≤ ∞, we have 1) If 2 ≤ ζ, then the solutionx to the BBP obeys
x − x 2,q ≤ 2ζ β q,4 q q−1 k (A) + k 1/q−1 φ k (x),(17)
x
− x 2,1 ≤ 4k 1−1/q ζ β q,4 q q−1 k (A) + 4φ k (x).(18)
2) If the noise in the BDS satisfies A T 2,∞ ≤ µ, then the solutionx to the BDS obeys
x − x 2,q ≤ 8k 1−1/q β 2 q,4 q q−1 k (A) µ + k 1/q−1 φ k (x),(19)
x
− x 2,1 ≤ 16k 2−2/q β 2 q,4 q q−1 k (A) µ + 4φ k (x).(20)
3) If the noise in the group lasso satisfies A T 2,∞ ≤ κµ for some κ ∈ (0, 1), then the solutionx to the group lasso obeys
x − x 2,q ≤ 1 + κ 1 − κ · 4k 1−1/q β 2 q,( 4 1−κ ) q q−1 k (A) µ + k 1/q−1 φ k (x),(21)
x
− x 2,1 ≤ 1 + κ (1 − κ) 2 · 8k 2−2/q β 2 q,( 4 1−κ ) q q−1 k (A) µ + 4 1 − κ φ k (x).(22)
Remark 7. All the error bounds consist of two components, one is caused by the measurement error, and another one is due to the sparsity defect.
Random matrices
In this section, we study the properties of the q-ratio BCMSV of sub-gaussian random matrix. A random vector x ∈ R N is called isotropic and sub-gaussian with constant L if it holds for all u ∈ R N that E| x, u | 2 = u 2 2 and P (| x, u | ≥ t) ≤ 2 exp(− t 2 L u 2 ). Then as shown in Theorem 2 of [20], we have the following lemma.
Lemma 1 ( [20]). Suppose the rows of the scaled measurement matrix √ mA to be i.i.d isotropic and subgaussian random vectors with constant L. Then there exists constants c 1 and c 2 such that for any η > 0 and m ≥ 1 satisfying
m ≥ c 1 L 2 (sn + s log p) η 2 we have E|1 − β 2,s (A)| ≤ η and P(β 2,s (A) ≥ 1 − η) ≥ 1 − exp(−c 2 η 2 m L 4 ).
Then as a direct consequence of Proposition 2
(i.e. if 1 < q < 2, β q,s (A) ≥ s −1 β 2,s (A); if 2 ≤ q ≤ ∞, β q,s (A) ≥ β 2,s 2(q−1) q (A).
) and Lemma 1, we have the following probabilistic statements for β q,s (A).
Theorem 3. Under the assumptions and notations of Lemma 1, it holds that 1) When 1 < q < 2, there exist constants c 1 and c 2 such that for any η > 0 and m ≥ 1 satisfying m ≥ c 1 L 2 (sn + s log p) η 2
we have
E[β q,s (A)] ≥ s −1 (1 − η),(23)P β q,s (A) ≥ s −1 (1 − η) ≥ 1 − exp(−c 2 η 2 m L 4 ).(24)
2) When 2 ≤ q ≤ ∞, there exist constants c 1 and c 2 such that for any η > 0 and m ≥ 1 satisfying
m ≥ c 1 L 2 s 2(q−1) q (n + log p) η 2 we have E[β q,s (A)] ≥ 1 − η,(25)P β q,s (A) ≥ 1 − η ≥ 1 − exp(−c 2 η 2 m L 4 ).(26)
Remark 9. Theorem 3 shows that for sub-gaussian random matrix, the q-ratio BCMSV is bounded away from zero as long as the number of measurements is large enough. Sub-gaussian random matrices include Gaussian and Bernoulli ensembles.
Numerical experiments
In this section, we introduce a convex-concave method to solve the sufficient condition (8) so as to achieve the maximal block sparsity k and present an algorithm to compute the q-ratio BCMSV. We also conduct comparisons between the q-ratio BCMSV based bounds and block RIC based bounds through the BBP.
Solving the optimization problem (8)
According to Proposition 1, given a q ∈ (1, ∞] we need to solve the optimization problem (8) to obtain the maximal block sparsity k which guaranties that all block k-sparse signals can be uniquely recovered by (7). Solving (8) is equivalent to solve the problem: max z∈R N z 2,q s.t. Az = 0 and z 2,1 ≤ 1.
However, maximizing mixed 2 / q norm over a polyhedron is non-convex. Here we adopt the convexconcave procedure (CCP) (see [11] for details) to solve the problem (27) for any q ∈ (1, ∞]. The algorithm is presented as follows:
Algorithm: CCP to solve (27).
Give an initial point to z l with l = 0. Iterate 1. Linearity. Approximate z 2,q using the first order Taylor expansion
z 2,q = z l 2,q + ∇( z 2,q ) T z=z l (z − z l ) = z l 2,q + [ z l 1−q 2,q z l b q−2 2 z l ] T (z − z l ),
where z l b = [ z l 1 2 , · · · , z l 1 2 n , z l 2 2 , · · · , z l 2 2 n , · · · , z lp 2 , · · · , z lp 2 n ] with z l i 2 denoting the 2 norm of the i-th block of z l for i in [p]. 2. Maximization. Set z l+1 to be the result of
max z∈R N z l 2,q + [ z l 1−q 2,q z l b q−2 2 z l ] T (z − z l ) s.t. Az = 0, z 2,1 ≤ 1.(28)
3. Updating iteration. Let l = l + 1. until stopping criterion is satisfied and k is the largest integer smaller than z l .
We implement the algorithm to solve (27) under the following settings. Let A be either Bernoulli or Gaussian random matrix with N = 256, varying m, block size n and q. Specifically, m = 64, 128, 192, n = 1, 2, 4, 8 and q = 2, 4, 16, 128, respectively. The results are summarized in Table 1. Note that when n = 1, the algorithm (28) is identical to the one in [27]. The main findings are as follows: (i) by comparing the results between Bernoulli and Gaussian random matrices under the same settings, there is no substantial difference. Thus we can now merely focus on the left part of the table, i.e. Bernoulli random matrix part; (ii) it can be seen that the results are not monotone with respect to q (see the row with n = 4, m = 192), which verifies the conclusion in Remark 3 ; (iii) when m is the only variable, it is easy to notice that the maximal block sparsity increases as m increases; (iv) conversely, when n is the only variable, the maximal block sparsity decreases as n increases, which is in line with the main result in [16, Theorem 3.1].
Computing the q-atio BCMSVs
Computing the q-ratio BCMSV (9) is equivalent to solve
min z∈R N Az 2 s.t. z 2,1 ≤ s q−1 q , z 2,q = 1.(29)
Since the constraint set is not convex, this is a non-convex optimization problem. In order to solve (29), we use Matlab function fmincon as in [27] and define z = z + −z − with z + = max(z, 0) and z − = max(−z, 0). Consequently, (29) can be reformulated to:
min z + ,z − ∈R N (z + − z − ) T A T A(z + − z − ) s.t. z + − z − 2,1 − s q−1 q ≤ 0, z + − z − 2,q = 1, z + ≥ 0, z − ≥ 0.(30)
Due to the existence of local minima, we perform an experiment to decide a reasonable number of iterations needed to achieve the 'global' minima shown in Figure 1. In the experiment, we calculate the q-ratio BCMSV of a fixed unit norm columns Bernoulli random matrix of size 40 × 64, n = s = 4 and varying q = 2, 4, 8, respectively. 50 iterations are carried out for each q. The figure shows that after about 30 experiments, the estimate of β q,s ,β q,s , becomes convergent, so in the following experiments we repeat the algorithm 40 times and choose the smallest valueβ q,s as the 'global' minima. We test indeed to vary m, s, n, respectively, all indicate 40 is a reasonable number to be chosen (not shown).
Next, we illustrate the properties of β q,s , which have been pointed out in Remarks 2 and 3, through experiments. We set N = 64 with three different block sizes n = 1, 4, 8 (i.e. number of blocks p = 64, 16,8), three different m = 40, 50, 60, three different q = 2, 4, 8 and three different s = 2, 4, 8. Unit norm columns Bernoulli random matrices are used. Results are listed in Table 2. They are inline with the theoretical results:
(i) β q,s increases as m increases for all cases given that other parameters are fixed. (ii) β q,s decreases as s increases for most of cases given that other parameters are fixed. There are exceptions when m = 40, n = 8 with s = 4 and s = 8 under q = 4, 8, respectively. However, the difference is about 0.0002, which is possibly caused by numerical approximation.
(iii) Monotonicity of β q,s does not hold with respect to q even given that other parameters are fixed.
Comparing error bounds
Here we compare the q-ratio BCMSV based bounds against the block RIC based bounds from the BBP under different settings. The block RIC based bound is
x − x 2 ≤ 4 1 + δ 2k (A) 1 − (1 + √ 2)δ 2k (A) ζ,(31)
if A satisfies the block RIP of order 2k, i.e. the block RIC δ 2k (A) < √ 2 − 1 [7,20]. By using the Hölder's inequality, one can obtain the mixed 2/ q norm
x − x 2,q ≤ 4 1 + δ 2k (A) 1 − (1 + √ 2)δ 2k (A) k 1/q−1/2 ζ,(32)
for 0 < q ≤ 2. We compare the two bounds (32) and (12). Without loss of generality, let ζ = 1. δ 2k (A) is approximated using Monte Carlo simulations. Specifically, we randomly choose 1000 sub-matrices of A ∈ R m×N of size m×2nk to compute δ 2k (A) using the maximum of max(σ 2 max −1, 1−σ 2 min ) among all sampled sub-matrices. It turns out that this approximated block RIC is always smaller than or equal to the exact block RIC, thus the error bounds based on the exact block RIC are always larger than those based on the approximated block RIC. Therefore, it would be enough to show that the q-ratio BCMSV gives a sharper error bound than the approximated block RIC We use unit norm columns sub-matrices of a row-randomly-permuted Hadamard matrix (an orthogonal Bernoulli matrix) with N = 64, k = 1, 2, 4, n = 1, 2, q = 1.8 and a variety of m ≤ 64 to approximate the q-ratio BCMSV and the block RIC. Besides the Hadamard matrix, we also test Bernoulli random matrices and Gaussian random matrices with different configurations, which only return very fewer qualified block RICs. In the simulation results of [20], the authors showed that under all considered cases for Gaussian random matrices, δ 2k (A) > √ 2 − 1, which is coincident with our finding. Figure 2 shows that the q-ratio BCMSV based bounds are smaller than those based on the approximated block RIC. Note that when m approaches N , β q,s (A) → 1 and δ 2k (A) → 0, as a result, the q-ratio BCMSV based bounds are smaller than 2.2, while the block RIC based bounds are larger than or equal to 4.
Conclusion
In this study, we introduce the q-ratio block sparsity measure and the q-ratio BCMSV. Theoretically, through the q-ratio block sparsity measure and the q-ratio BCMSV, we (i) establish the sufficient condition for the unique noise free BBP recovery; (ii) derive both the mixed 2 / q norm and the mixed 2 / 1 norm bounds of recovery errors for the BBP, the BDS and the group lasso estimator; (iii) prove the q-ratio BCMSV is bounded away from zero if the number of measurements is relatively large for sub-gaussian random matrix. Afterwards, we use numerical experiments via two algorithms to illustrate theoretical results. In addition, we demonstrate that the q-ratio BCMSV based error bounds are much tighter than those based on block RIP through simulations.
There are still some issues left for future work. For example, analogue to the case for the q-ratio CMSV, the geometrical property of the q-ratio BCMSV can be investigated to derive sufficient conditions and error bounds for block sparse signal recovery. Error bound m k=1,n=1,q−ratio BCMSV k=2,n=1,q−ratio BCMSV k=4,n=1,q−ratio BCMSV k=1,n=2,q−ratio BCMSV k=2,n=2,q−ratio BCMSV k=4,n=2,q−ratio BCMSV k=1,n=2,block RIC k=2,n=2,block RIC k=4,n=2,block RIC
x 2,1 ≥ x 2,1 = x + h 2,1 = x S + h S 2,1 + x S c + h S c 2,1 ≥ x S 2,1 − h S 2,1 + h S c 2,1 = x 2,1 − h S 2,1 + h S c 2,1 ,
which can be simplified to h S c 2,1 ≤ h S 2,1 . Thereby, we can obtain the following inequality:
h 2,1 = h S 2,1 + h S c 2,1 ≤ 2 h S 2,1 ≤ 2k 1−1/q h S 2,q ≤ 2k 1−1/q h 2,q , ∀q ∈ (1, ∞],
which is equivalent to
k q (h) = h 2,1 h 2,q q q−1 ≤ 2 q q−1 k.
For the group lasso, since the noise satisfies A T 2,∞ ≤ κµ for κ ∈ (0, 1) andx is a solution of the group lasso, we have
1 2 Ax − y 2 2 + µ x 2,1 ≤ 1 2 Ax − y 2 2 + µ x 2,1 .
Substituting y by Ax + leads to
µ x 2,1 ≤ 1 2 2 2 − 1 2 A(x − x) − 2 2 + µ x 2,1 = 1 2 2 2 − 1 2 A(x − x) 2 2 + A(x − x), − 1 2 2 2 + µ x 2,1 ≤ A(x − x), + µ x 2,1 = x − x, A T + µ x 2,1 ≤ x − x 2,1 A T 2,∞ + µ x 2,1 ≤ κµ h 2,1 + µ x 2,1 .
The last second inequality follows by applying Cauchy-Swcharz inequality block wise and the last inequality can be written as
x 2,1 ≤ κ h 2,1 + x 2,1 .(33)
Therefore, it holds that
x 2,1 ≥ x 2,1 − κ h 2,1 = x + h S c + h S 2,1 − κ h S c + h S 2,1 ≥ x + h S c 2,1 − h S 2,1 − κ( h S c 2,1 + h S 2,1 ) = x 2,1 + (1 − κ) h S c 2,1 − (1 + κ) h S 2,1 ,
which can be simplified to
h S c 2,1 ≤ 1 + κ 1 − κ h S 2,1 .
Thus we can obtain
h 2,1 = h S c 2,1 + h S 2,1 ≤ 2 1 − κ h S 2,1 ≤ 2 1 − κ k 1−1/q h S 2,q ≤ 2 1 − κ k 1−1/q h 2,q ,
which can be reformulated by
k q (h) = h 2,1 h 2,q q q−1 ≤ 2 1 − κ q q−1 k.
Step 2 : Obtain upper bound of Ah 2 and then construct the mixed 2 / q norm and the mixed 2 / 1 norm of the recovery error vector h via the q-ratio BCMSV for each algorithm.
(i) For the BBP, since both x andx satisfy the constraint y−Az 2 ≤ ζ, by using the triangle inequality we can get
Ah 2 = A(x − x) 2 ≤ Ax − y 2 + y − Ax 2 ≤ 2ζ.(34)
Following from the definition of the q-ratio BCMSV and k q (h) ≤ 2 q q−1 k, we have
β q,2 q q−1 k (A) h 2,q ≤ Ah 2 ≤ 2ζ ⇒ h 2,q ≤ 2ζ β q,2 q q−1 k (A)
.
Furthermore, we can obtain h 2,1 ≤ 4k 1−1/q ζ β q,2 q q−1 k (A) by using the property h 2,1 ≤ 2k 1−1/q h 2,q .
(ii) Similarly for the BDS, since both x andx satisfy the constraint A T (y − Az) 2,∞ ≤ µ, we have
A T Ah 2,∞ ≤ A T (y − Ax) 2,∞ + A T (y − Ax) 2,∞ ≤ 2µ.
By applying the Cauchy-Swcharz inequality again as in Step 1, we obtain
Ah 2 2 = Ah, Ah = h, A T Ah ≤ h 2,1 A T Ah 2,∞ ≤ 2µ h 2,1 .(35)
At last, with the definition of the q-ratio BCMSV, k q (h) ≤ 2 q q−1 k and h 2,1 ≤ 2k 1−1/q h 2,q , we get the upper bounds of the mixed 2 / q norm and the mixed 2 / 1 norm for h :
β 2 q,2 q q−1 k (A) h 2 2,q ≤ Ah 2 2 ≤ 2µ h 2,1 ≤ 4µk 1−1/q h 2,q ⇒ h 2,q ≤ 4k 1−1/q β 2 q,2 q q−1 k (A) µ and h 2,1 ≤ 2k 1−1/q h 2,q ≤ 8k 2−2/q β 2 q,2 q q−1 k (A) µ.
(iii) For the group lasso, with A T 2,∞ ≤ κµ, we have
A T Ah 2,∞ ≤ A T (y − Ax) 2,∞ + A T (y − Ax) 2,∞ ≤ A T 2,∞ + A T (y − Ax) 2,∞ ≤ κµ + A T (y − Ax) 2,∞ .
Moreover, sincex is the solution of the group lasso, the optimality condition yields that
A T (y − Ax) ∈ µ∂ x 2,1 ,
where the sub-gradients in ∂ x 2,1 for the i-th block arex i / x i 2 ifx i = 0, and is some vector g satisfying g 2 ≤ 1 ifx i = 0 (which follows from the definition of sub-gradient). Thus, we have A T (y−Ax) 2,∞ ≤ µ, which leads to A T Ah 2,∞ ≤ (κ + 1)µ.
Following the inequality (35), we get Ah 2 2 ≤ (κ + 1)µ h 2,1 .
As a result, since k q (h) ≤ 2 1−κ q q−1 k and h 2,1 ≤ 2 1−κ k 1−1/q h 2,q , we can obtain
β 2 q,( 2 1−κ ) q q−1 k (A) h 2 2,q ≤ Ah 2 2 ≤ (κ + 1)µ h 2,1 ≤ µ 2(κ + 1) 1 − κ k 1−1/q h 2,q ,(37)
which is equivalent to
h 2,q ≤ k 1−1/q β 2 q,( 2 1−κ ) q q−1 k (A) · 2(κ + 1) 1 − κ µ and h 2,1 ≤ 1+κ (1−κ) 2 · 4k 2−2/q β 2 q,( 2 1−κ ) q q−1 k (A) µ.
Proof of Theorem 2. Since the infimum of φ k (x) is achieved by an block k-sparse signal z whose non-zero blocks equal to the largest k blocks, indexed by S, of x, so φ k (x) = x S c 2,1 and let h =x − x. Similar as the proof procedure for Theorem 1, the derivations also have two steps.
Step 1 : For all algorithms, bound h 2,1 via h 2,q and φ k (x).
First for the BBP and the BDS, since x 2,1 = x + h 2,1 is the minimum among all z satisfying the constraints of the BBP and the BDS, we have
x S 2,1 + x S c 2,1 = x 2,1 ≥ x 2,1 = x + h 2,1 = x S + h S 2,1 + x S c + h S c 2,1 ≥ x S 2,1 − h S 2,1 + h S c 2,1 − x S c 2,1 , which is equivalent to h S c 2,1 ≤ h S 2,1 + 2 x S c 2,1 = h S 2,1 + 2φ k (x).(38)
In consequence, we can get
h 2,1 = h S 2,1 + h S c 2,1 (39) ≤ 2 h S 2,1 + 2φ k (x) ≤ 2k 1−1/q h S 2,q + 2φ k (x) ≤ 2k 1−1/q h 2,q + 2φ k (x).(40)
As for the group lasso, by using (33), we can obtain
x S 2,1 + x S c 2,1 = x 2,1 ≥ x 2,1 − κ h 2,1 ≥ x S + x S c + h S + h S c 2,1 − κ h S + h S c 2,1 ≥ x S + h S c 2,1 − x S c 2,1 − h S 2,1 − κ h S 2,1 − κ h S c 2,1 = x S 2,1 + (1 − κ) h S c 2,1 − x S c 2,1 − (1 + κ) h S 2,1 , which points to that h S c 2,1 ≤ 1 + κ 1 − κ h S 2,1 + 2 1 − κ x S c 2,1 .
Therefore, we have
h 2,1 ≤ h S 2,1 + h S c 2,1 ≤ 2 1 − κ h S 2,1 + 2 1 − κ x S c 2,1 ≤ 2 1 − κ k 1−1/q h 2,q + 2 1 − κ φ k (x).(42)
Step 2 : Verify that the q-ratio block sparsity of h has lower bound in the form of h 2,q for each algorithm, when h 2,q is larger than the part of recovery bounds caused by the measurement error.
(i) For the BBP, we assume that h = 0 and h 2,q > Ah 2 h 2,q , which implies that k q (h) > 4 q q−1 k ⇒ h 2,1 > 4k 1−1/q h 2,q .
Combining (40), we have h 2,q < k 1/q−1 φ k (x), which completes the proof for (17). The error bound of the mixed 2 / 1 norm (18) follows immediately from (17) and (40).
(ii) As for the BDS, similarly we assume h = 0 and h 2,q > 8k 1−1/q β 2 q,4 q q−1 k (A) µ, otherwise (19) holds trivially.
As Ah 2 2 ≤ 2µ h 2,1 (see (35)
Combining (40), we have h 2,q < k 1/q−1 φ k (x), which completes the proof for (19). (20) holds as a result of (19) and (40).
(iii) For the group lasso, we assume that h = 0 and h 2,q > 1+κ 1−κ · 4k 1−1/q β 2 q,( 4 1−κ )
( 4 1−κ ) q q−1 k k q (h) 1− 1 q ⇒ k q (h) > ( 4 1 − κ ) q q−1 k ⇒ h 2,1 > 4 1 − κ k 1−1/q h 2,q .(45)
Combining (42), we have h 2,q < k 1/q−1 φ k (x), which completes the proof for (21). Consequently, (22) is obtained via (21) and (42).
Remark 8 .((
8Comparing to Theorem 1, we need stronger conditions to achieve the valid error bounds. Concisely, we require β A) > 0 for the BBP, BDS and group lasso in the block compressible case, while β A) > 0 in the block sparse case, respectively.
Figure 1 :
1q-ratio BCMSVs calculated for a Bernoulli random matrix of size 40 × 64 with n = 4, s = 4 and q = 2, 4, 8 as a function of number of experiments.
Figure 2 :
2The q-ratio BCMSV based bounds and the block RIC based bounds for Hadamard sub-matrices with N = 64, k = 1, 2, 4, n = 1, 2 and q = 1.8.Table 2: The q-ratio BCMSVs with varying m, n, p, 0.5626 0.3675 0.1502 0.5275 0.3249 0.1126 0.4926 0.2753 0.0361 8 8 0.4554 0.2147 0.0023 0.4046 0.1809 0.0021 0.3695 0.1063 0.0017
) , otherwise (17) holds trivially. Since Ah 2 ≤ 2ζ (see (34)), we have h 2,q >
) µ, otherwise (21) holds trivially. Since in this case Ah 2 2 ≤ (1 + κ)µ h 2,1 (see (36)), we have h
Table 1 :
1Maximal sparsity levels from the CCP algorithm for both Bernoulli and Gaussian random matrices with N = 256 and different combinations of n, m and q.n m
Bernoulli random matrix
Gaussian random matrix
q = 2 q = 4 q = 16 q = 128
q = 2 q = 4 q = 16 q = 128
1
64
4
4
3
2
4
4
3
3
128
12
9
6
5
13
12
7
6
192
23
22
12
10
23
20
11
10
2
64
3
3
2
2
3
3
2
2
128
9
7
5
4
9
7
5
4
192
16
16
10
8
14
14
9
8
4
64
2
2
1
1
2
2
2
2
128
5
5
4
3
5
5
3
3
192
9
10
6
5
9
10
7
6
8
64
1
1
1
1
1
1
1
1
128
3
3
2
2
2
3
3
2
192
5
6
4
4
5
6
4
4
AcknowledgementsAppendix -ProofsProof of Proposition 1. Suppose there exists z ∈ kerA \ {0} and |S| ≤ k such that z S 2,1 ≥ z S c 2,1 , then we haveIn contrast, suppose ∃ q ∈ (1, ∞] such that k < min z∈kerA\{0} 2 q 1−q k q (z), then z S 2,1 < z S c 2,1 holds for all z ∈ kerA \ {0} and |S| ≤ k, which implies that the block null space property of order k is fulfilled, thus any block k-sparse signal x can be obtained via(7).Proof of Proposition 2.(i) Prove the left hand side of (10):For any z ∈ R N \ {0} and 1 < q 2 ≤ q 1 ≤ ∞, suppose k q 1 (z) ≤ s, then we can getTherefore, we can get the left hand side of (10) through β q 1 ,s (A) = min z =0,kq 1 (z)≤sAz 2 z 2,q 2 = β q 2 ,sq (A).(ii) Verify the right hand side of (10): Suppose k q 2 (z) ≤ sq, for any z ∈ R N \ {0}, by using the non-increasing property of the q-ratio block sparsity with respect to q and q 2 ≤ q 1 ≤ ∞, we have the following two inequalities:The former inequality implies that z 2,q 2 z 2,q 1 ≤ z 2,1 z 2,∞ ≤ sq ⇒ z 2,q 1 z 2,q 2 ≥ s− q . The latter inequality implies that {z : k q 2 (z) ≤ sq} ⊆ {z : k q 1 (z) ≤ sq}.
Therefore, we can obtain the right hand side of. Therefore, we can obtain the right hand side of (10) through
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| {'fraction_non_alphanumeric': 0.07995940002255554, 'fraction_numerical': 0.049644750197361004, 'mean_word_length': 3.4442662389735363, 'pattern_counts': {'":': 0, '<': 18, '<?xml version=': 0, '>': 16, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 48, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper we introduce the q-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed 2 / q and the mixed 2 / 1 norms of the reconstruction errors for stable and robust recovery using block Basis Pursuit (BBP), the block Dantzig selector (BDS) and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV based error bounds are tighter than the block restricted isotropic constant based bounds.Lasso estimator[21]. Afterwards,[19]brought in a variant of the CMSV: ω ♦ (A, s) = min z =0, z 1 / z ∞≤s Az ♦ z ∞ with · ♦ denoting a general norm, and expressed the ∞ recovery error bounds using this quantity. The latest progress concerning the CMSV can be found in[26,27].[27] generalized these two measures to a new measure called q-ratio CMSV: ρ q,s (A) = min z =0,( z 1 / z q ) q/(q−1) ≤s Az 2 z q with q ∈ (1, ∞] and established both q and 1 bounds of recovery errors.[26]investigated geometrical property of the q-ratio CMSV, which can be used to derive sufficient conditions and error bounds of signal recovery.In addition to the simple sparsity, a signal x can also possess a structure called block sparsity where the non-zero elements occur in clusters. It has been shown that using block information in CS can lead to a better signal recovery[2,7,24]. Analogue to the simple sparsity, there are block NSP and block RIP to characterize the measurement matrix in order to guarantee a successful recovery through (1)[9]. Nevertheless, they are still computationally hard to be verified for a given A. Thus it is desirable to develop a computable measure like the CMSV for recovery of simple (non-block) sparse signals.[20] proposed a new measure of the measurement matrix based on the CMSV for block sparse signal recovery and derived the mixed 2 / ∞ and 2 bounds of recovery errors. In this paper, we extend the q-ratio CMSV in [27] to q-ratio block CMSV (BCMSV) and generalize the error bounds from the mixed 2 / ∞ and 2 norms in [20] to mixed 2 / q with q ∈ (1, ∞] and mixed 2 / 1 norms.This work includes four main contributions to block sparse signal recovery in compressive sensing: (i) we establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise free block BP (BBP), and develop a convex-concave procedure to solve the corresponding non-convex problem in the condition; (ii) we introduce the q-ratio BCMSV and derive both the mixed 2 / q and the mixed 2 / 1 norms of the reconstruction errors for stable and robust recovery using the BBP, the block DS (BDS) and the group lasso in terms of the q-ratio BCMSV; (iii) we prove that for sub-gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large; (iv) we present an algorithm to compute the q-ratio BCMSV for an arbitrary measurement matrix and investigate its properties.The paper is organized as follows. In Section 2, we introduce the definitions for the q-ratio block sparsity and the q-ratio BCMSV, and present the sufficient condition for the noise free BBP recovery based on the q-ratio block sparsity and an inequality for the q-ratio BCMSV. The mixed 2 / q and the mixed 2 / 1 reconstruction errors for the BBP, the BDS and the group lasso in terms of the q-ratio BCMSV are derived in Section 3. In Section 4, the probabilistic results of the q-ratio BCMSVs for sub-gaussian random matrices are demonstrated. Section 5 is reserved for algorithms to solve the optimization problem in the sufficient condition for the noise free BBP recovery and compute the q-ratio BCMSV. The q-ratio BCMSV based bounds and the block RIC based bounds for the BBP are also compared therein. Section 6 is devoted to the conclusion. All proofs are left in the Appendix.Proof of Theorem 1. The proof procedure follows from the similar arguments in[18,19], and the procedure can be divided into two main stepsStep 1 : We first derive upper bounds of the q-ratio block sparsity of residual h =x − x for all algorithms. As x is block k-sparse, we assume that bsupp(x) = S and |S| ≤ k.For the BBP and the BDS, since x 2,1 = x + h 2,1 is the minimum among all z satisfying the constraints of BBP and BDS (including the true signal x), we have', 'arxivid': '1908.11082', 'author': ['Jianfeng Wang \nDepartment of Mathematics and Mathematical Statistics\nUmeå University\n901 87UmeåSESweden\n', 'Zhiyong Zhou \nDepartment of Statistics\nZhejiang University City College\n310015HangzhouChina\n', 'Jun Yu \nDepartment of Mathematics and Mathematical Statistics\nUmeå University\n901 87UmeåSESweden\n'], 'authoraffiliation': ['Department of Mathematics and Mathematical Statistics\nUmeå University\n901 87UmeåSESweden', 'Department of Statistics\nZhejiang University City College\n310015HangzhouChina', 'Department of Mathematics and Mathematical Statistics\nUmeå University\n901 87UmeåSESweden'], 'corpusid': 201671375, 'doi': '10.1186/s13634-019-0653-1', 'github_urls': [], 'n_tokens_mistral': 16554, 'n_tokens_neox': 14185, 'n_words': 8585, 'pdfsha': '03d541815dfc6291ecaeb53455b6f93804b98c0f', 'pdfurls': ['https://export.arxiv.org/pdf/1908.11082v1.pdf'], 'title': ['Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values', 'Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values'], 'venue': []} |
arxiv |
Solving Rehabilitation Scheduling problems via a Two-Phase ASP approach * PAOLO DE NARDI GIUSEPPE GALATÀ
15 Mar 2023
Matteo Cardellini
Ics Maugeri
Italy Carmine Dodaro
ItalySurgiq Srl
Anna Giardini
Ics Maugeri
Italy Marco Maratea
Polytecnic of Torino
TorinoItaly
University of Genova
GenovaItaly
University of Calabria
RendeItaly
DIBRIS
University of Genova
GenovaItaly
University of Calabria
RendeItaly
IVAN PORRO
SurgiQ srlItaly
Solving Rehabilitation Scheduling problems via a Two-Phase ASP approach * PAOLO DE NARDI GIUSEPPE GALATÀ
15 Mar 202310.1017/xxxxxsubmitted xx xx xxxx; revised xx xx xxxx; accepted xx xx xxxxTPLP: Page 1-8. 1 * This paper is an extended and revised version of a conference paper appearing in the proceedings of the RuleML+RR 2021 conference (Cardellini et al. 2021). 2 Cardellini et al.Answer Set ProgrammingRehabilitation SchedulingHealthcare
A core part of the rehabilitation scheduling process consists of planning rehabilitation physiotherapy sessions for patients, by assigning proper operators to them in a certain time slot of a given day, taking into account several legal, medical and ethical requirements and optimizations, e.g., patient's preferences and operator's work balancing. Being able to efficiently solve such problem is of upmost importance, in particular after the COVID-19 pandemic that significantly increased rehabilitation's needs.In this paper, we present a two-phase solution to rehabilitation scheduling based on Answer Set Programming, which proved to be an effective tool for solving practical scheduling problems. We first present a general encoding, and then add domain specific optimizations. Results of experiments performed on both synthetic and real benchmarks, the latter provided by ICS Maugeri, show the effectiveness of our solution as well as the impact of our domain specific optimizations.Under consideration in Theory and Practice of Logic Programming (TPLP).
Introduction
The rehabilitation scheduling process consists mainly of planning daily patients' physiotherapy sessions inside a rehabilitation institute, that hereafter we refer to as Rehabilitation Scheduling Problem (RSP) (Huang et al. 2012;Huynh et al. 2018;Li and Chen 2021;Schimmelpfeng et al. 2012). Hospitals that may profitably make a practical use of such scheduling, including those managed by ICS Maugeri 1 , which will provide benchmarks in this paper, deal with up to hundreds of patients with a team of just few tens of physiotherapists; so, it is of paramount importance to be able to assign patients to operators, i.e., physiotherapists, efficiently. A recent article by Cieza et al. (2020) found that 2.41 billion people could benefit from rehabilitation services. This finding means that almost one third of the current population in the world needs rehabilitation at some point during the course of their lives due to disease or injury; further, this number is predicted to trend upward given the current demographic and health shifts. In addition, there is emerging evidence that many of the people affected by the COVID-19 pandemic have long-term consequences regardless of the disease severity or length of hospitalisation, thus further increasing the demand for rehabilitation services globally.
The RSP is subject to several constraints, i.e., legal, medical and ethical, that need to be taken into consideration in order to find a viable schedule. For example, the main constraints that have to be dealt with are the maximum capacity of rehabilitation gyms, the legal working time and rest periods for operators, and the minimum durations of physiotherapy sessions. Moreover, several preferences shall be considered, e.g., due to clinical and organizational reasons it is often best for a patient to be treated as often as possible by the same operator and at the same time slot; also, rehabilitation professionals' work balancing needs to be taken into proper account.
In this paper, we present a solution to the RSP based on Answer Set Programming (ASP) (Gelfond and Lifschitz 1991;Niemelä 1999;Baral 2003;Brewka et al. 2011), which proved to be an effective tool for solving practical scheduling problems (Gebser et al. 2018;Ricca et al. 2012;Dodaro and Maratea 2017;Dodaro et al. 2021), thanks also to the availability of efficient ASP solvers. The solution is designed as a two-phase encoding (Section 3): the first phase, called board, deals with the problem of assigning a physiotherapist to every patient considering the total working time of the physiotherapist and the minimum mandatory time of rehabilitation sessions. In the second phase, called agenda, a start and end time of every rehabilitation session is defined given the assignment among patients and physiotherapists found in the first phase. Our two-phase solution is not guaranteed to find the best possible overall solution, but has been designed in this way because: (i) it simplifies the overall encoding and its practical use, and (ii) it mimics how schedules have been computed so far (in a non-automatic way) by ICS Maugeri and gives freedom to physiotherapists' coordinators to perform any desired manual change. In fact, coordinators have specifically requested to have the possibility to manually change some patient-operator assignments (the output of the board ) before sending it to the agenda phase. Even if this manual change is seldom made, it gives the coordinators a sense of control and the possibility to introduce human knowledge and expertise in the scheduling. It is important to acknowledge that this tool is not advertised and sold as a medical device, but it is a tool for supporting the decision of the coordinators. For this reason, it is legally mandatory for the coordinators to have a more granular control (and responsibility) over the decisions. We first tested (Section 4) our encoding on real scenarios from ICS Maugeri related to the daily scheduling of neurological patients in two of their rehabilitation institutes in the North of Italy, namely Genova Nervi and Castel Goffredo. In the analysis, we decided to limit the run-time of the ASP solver clingo (Gebser et al. 2012) to only 30s (while in production the cut-off is set to 5 minutes): This narrow time limit allows for running much more experiments and having a more significant comparison with the different optimization algorithms in clingo. Then, given that ICS Maugeri is planning to instrument with automated techniques other, possibly larger, institutes in addition to Genova Nervi and Castel Goffredo, we generated a wide set of synthetic benchmarks, whose parameters are inspired by real data. We made a wide experimental evaluation, and statistically confronted synthetic and real data results using classification decision tree methods (Quinlan 1986), with the aim of predicting the behaviour of our solution on such larger institutes. Results show that the accuracy is high, so our synthetic benchmarks appear significant to indicate a possible behavior on real data coming from other institutes with other parameters and similar characteristics. As a side effect, this analysis also outlines the features of the problems that affect the results mostly. Finally, with the aim of further improving the results, and lower the still remaining percentage of instances that could not be solved, we added domain specific optimizations to our encoding (Section 5): results of the improved encoding show that we are now able to find a solution, even if not always optimal, to every instance within the time limit. The paper is completed by an informal description of the RSP in Section 2 and by discussing related work and presenting conclusions in Section 6 and 7, respectively.
Problem Description
In this section we describe the problem we face in four paragraphs. First, we present the general description of the problem, then the data that characterize the main elements of the problem, followed by the requirements of the phases. The last paragraph shows a solution schedule.
General description. The delivery of rehabilitation services is a complex task that involves many healthcare professions such as physicians, physiotherapists, speech therapists, psychologists and so on. In particular, physiotherapists are the ones who spend most of their time with patients and their sessions constitute the core of the daily agenda of the patient, around which all other commitments revolve. For this reason, this article is focused on scheduling the physiotherapy sessions in the most efficient way, optimising the overall time spent with the patient.
The agenda for the physiotherapy sessions is computed by the coordinator of the physiotherapists. This process is repeated on a daily basis in order to take into account any change in the number and type of patients to be treated, and the number of operators available. Up until recently, this computation has been performed manually by coordinators, without any decision support.
The usual scheduling practice entails two subsequent phases resulting in the computation of a board and an agenda, that we herewith describe. In short, the first phase, called board, deals with the problem of assigning a physiotherapist to every patient, keeping track of the total working time of the operator and the minimum mandatory time of rehabilitation sessions. In the second phase, called agenda, a start and end time of every rehabilitation session is computed, given the assignment among patients and operators found in the first phase.
In more details, in the board phase, we ensure that the working hours of operators are respected by counting their total working time, in minutes, and assigning patients to each operator in such a way that the cumulative time of all their sessions remains below the operator's total working time. In this phase, patient-operator assignment preferences, expressed by the coordinator before the start of the scheduling procedure, are taken into account and respected as far as possible. In the agenda phase, given an assignment found by the board, every patient-operator session is assigned a starting and ending time, respecting the more granular working hours of the operators and the times in which the patients are unavailable. At this stage, the location in which the rehabilitation session is performed is also considered. A location, either a gym or the room of the patient, is assigned to the session, according to the clinical needs of the patient. The choice of the gym is carried out by considering the maximum number of simultaneous sessions allowed inside the gym and has to be made among a subset of gyms which are located at the same floor as the room of the patient, in order to avoid elevators and stairs that can result in discomfort to patients and slowness which can quickly congest the hospital. In this phase, time preferences for each patient are also considered: in fact, plans in which the sessions are performed closer to the desired time of the patients are to be preferred to others.
Instance description. In this paragraph we describe the main elements of our problem in more details, namely patients, operators and sessions, as well as the constraints and preferences entailed by the board and agenda phases.
Patients. Patients are characterized by their:
• type (Neurological, Orthopaedic, COVID-19 Positive, COVID-19 Negative, Outpatient 2 ), • aid needs, i.e., if they need specific care or not (e.g., if they need to be lifted),
• payment status (full payer or in charge of the National Healthcare Service), • forbidden times, i.e., the time intervals when the patient cannot be scheduled, • ideal time, i.e., the preferred scheduled timeslot in which the session should take place, expressed by the coordinator, • preferred operators, i.e., the list of physiotherapists, ordered by priority, the patient can be assigned to, • overall minimum length, i.e., the minimum amount of care time that the patient is guaranteed to be scheduled, • sessions, i.e., the list of sessions to be scheduled.
Operators. Physiotherapists, which will be called operators from now on, are characterized by their:
• qualifications, i.e., patient's types the operator can treat, • operating times, i.e., the part of the operator's working times dedicated to the direct care of the patients. The operating times are usually split in morning and afternoon shifts.
Moreover, each operator has a limit on the number of patients of a specific type to treat.
Sessions. The coordinator, in accordance with the rehabilitation program set by the physician, determines the daily activities of the patient. These activities can be performed in one or two therapy sessions, in the latter case one session will be scheduled in the morning and the other one in the afternoon shift. Each session can be delivered to patients either individualized ("one-on-one" sessions) or supervised (one therapist supervising more patients at the same time, each patient carrying out their personal activity independently). It must be noted that while operators deliver one-on-one therapy to one patient, they can supervise other patients. When the operators are particularly overbooked, their one-on-one sessions can be partially converted to supervised ones. These mixed sessions can either start with a supervised part and then continue with the one-on-one part, or vice-versa, or even start and end with a supervised part with a middle one-on-one session. Obviously, an operator can supervise different patients only if their sessions are located at the same place. In the next paragraphs, when defining the agenda, Figure 1 will graphically explain the semantic of a mixed session. The characteristics of the sessions are:
• delivery mode (one-on-one, supervised), • minimum one-on-one length, i.e., the minimum length of the session guaranteed to be delivered one-on-one, • ideal overall length, i.e., the overall length of the session including the one-on-one and supervised parts, • optional status, i.e., if the session can be left out of the schedule in case of overbooked operators, • forced time, i.e., the time when the session must be scheduled; if empty, the session is placed as close as possible to the patient's preferred time, • location, i.e., the place where the session must be delivered.
Constraints of the phases. The requirements that the two phases entail are reported in the following sub-paragraphs.
Board. In the board phase, all patients are assigned to an available operator, according to the following criteria:
• compatibility between patient and operator, depending on the patient's type and operator qualifications, the patient's forced time, if any, and the operator working times, by also checking if the operator has enough time to provide the guaran-teed overall minimum length and minimum one-on-one length to each patient and session, • the patients should be fairly distributed among all available operators, taking into account their type, aid needs and payment status, • the patients should be assigned to the operators respecting as much as possible their preferred operators list, which considers primarily the choices of the coordinator and secondarily the history of the past assignments.
Agenda. The results of the board phase can be revised (e.g., in special cases, the coordinator can override the preferred operators list and force an assignment of a patient to an operator regardless of all other considerations) and, if necessary, manually modified by the coordinator. Once the coordinator is satisfied with the board, it is possible to proceed to the agenda scheduling, using the approved board as input. The criteria for the agenda phase are:
• compliance with the forced time of the session, if specified,
• two sessions of the same patient must be assigned in different shifts,
• compliance with the minimum one-on-one length of the session, • no overlap between two one-on-one sessions (or the one-on-one part of the session if mixed) assigned to the same operator, • observance of the maximum capacity of the locations (1 for each room, varying for the gyms), • respect of the minimum cumulative time that the patient should be treated among all the sessions, • respect of the one-on-one minimum session length, • compliance with the forbidden times of the patient, • sessions can only be scheduled within the working times of the operator, • the start time of each session should be as close as possible to the preferred time, either specified by the coordinator or inferred from previous schedules, • for mixed sessions, the one-on-one part should be maximized, • the largest possible number of optional sessions should be included, • the overall length, including the one-on-one and supervised parts in case of mixed sessions, should be as close as possible to the ideal overall length specified by the coordinator.
Scheduling example. As previously stated, the output of the agenda phase is a start time and duration of every session during the day. Since, in ICS Maugeri, the scheduling was already performed with a timeslot of 10 minutes, we decided to keep this discretization in place. For this reason, the sessions can start every 10 minutes in the working hours of the hospital (8AM-12AM in the morning, 1:30PM -4PM in the afternoon) and last a multiple of 10 minutes. Figure 1 shows the scheduling of the agenda in a real case scenario in the hospital of Genova Nervi. Light blue squares represent time units in which the sessions will be performed in an individual fashion, and yellow squares represent time units of sessions in which the patient will be dealt in a supervised mode. Ticks on the left side of the figure describe the period (AM = Morning, PM = Afternoon) and the number we associate to each timeslot. As it can be seen in the first column, Operator 1 (OP1) deals with the first session (S5) as a mixed session: the session starts in individual one-to-one mode with all the attention of the operator focused on the patient (P4). After 4 time slots (i.e., 40 minutes, the session minimum time) the operator moves the attention to another patient (P1 performing session S1) while the previous patient finishes the session, in the same room, on its own. In a more practical way, in the first 40 minutes the operator helps the patient in performing exercises which could not be performed alone in a correct way.
In the remaining 20 minutes, which are still very beneficial to the patient, the patient performs the exercises that can be done alone while the operator is working with another patient. Being in the same room, the operator can still intervene if the supervised patient needs correction. Some sessions (e.g., session S17 of patient P13 performed in the morning by operator O4) are performed in a complete supervised fashion since they are additional secondary sessions which are medically beneficial to the patient but not mandatory (e.g., patient P13 already performs session S16 in an individual fashion with operator O4 in the afternoon).
A Two-Phase ASP Encoding for the RSP
In the following, we assume the reader is familiar with syntax and semantics of ASP. Starting from the specifications in the previous section, here we present the ASP encoding, based on the input language of clingo (Gebser et al. 2016). For details about syntax and semantics of ASP programs, we refer the reader to (Calimeri et al. 2020).
Board encoding
Data Model. The input data is specified by means of the following atoms: • Instances of patient(P), operators(O), and type(T) represent the identifiers of patients, operators, and the different types of patients that can be visited, respectively, where P and O are numbers, whereas T is of the form value-needs-status, where value can be neurologic, orthopaedic, covid-19-positive, covid-19-negative, or outpatient; needs can be lifter or nolifter, and status can be payer or free. For instance, neurologic-lifter-payer indicates that the patient needs a neurological treatment, must be lifted, and the treatment must be paid. Moreover, a fictitious operator with ID equals to -1 is included in the list of all the operators, and it is needed to intercept all patients that cannot be assigned to other operators (like a catch-all 3 ). • Instances of operator contract(ID,TIME,MAX) represent the contract of the operator with the identifier ID, and include the quantity of time (in time units) the operator works in a day (TIME), and the maximum number of patients the operator can visit during the day (MAX). • Instances of operator limit(ID,T,VALUE) represent the maximum number of patients (VALUE) of type T the operator with identifier ID can visit. The operator with ID equals to -1 has no patients limit. • Instances of patient data(ID,T,DUR) represent the data associated to the patient with the identifier ID, and include the type of the patient (T), and the minimum cumulative time of all sessions of the patient during the day (DUR). • Instances of patient session(ID,MIN,LOC) represent a rehabilitation session that the patient with identifier ID needs to perform during the day. The session is characterized by a minimum length for the session in time units (MIN), and the location of the session (LOC). • Instances of patient preference(ID,OP,W) represent the preference of the patient with identifier ID to be treated by the operator with identifier OP, where W specifies the weight of the preference. • Similarly, instances of history preference(ID,OP,W) represent the preference of the patient based on the history of previous sessions in previous days.
The output is an assignment represented by atoms of the form assignment(OP, PAT), stating that patient PAT will be treated by operator OP.
Encoding. The related encoding is shown in Figure 2, and is described in the following.
To simplify the description, the rule appearing at line i in Figure 2 is denoted with r i . Rule r 1 ensures that each patient is assigned to exactly one operator. Rules r 2 and r 3 are used to define if the session between a patient and an operator will be performed individually in a single location (r 2 ), or it will be executed in the same location of another session (r 3 ), by creating two auxiliary atoms uniqueLocationLength(OP,PAT,DUR) and sameLocationLength(OP,PAT,DUR) that represent the duration DUR in time slots of the session between operator OP and patient PAT performed in a single or same location, respectively, used in the next rule. Rule r 4 ensures that the time required by the patients assigned to an operator does not exceed the maximum time of her/his contract. Rule r 5 ensures that each operator does not exceed the maximum number of patients to visit during the day. Rule r 6 is similar to the previous one, but in this case the limits are imposed according to the type of the patient. Weak constraints from r 7 to r 9 are then used to provide preferences among different assignments. In particular, r 7 is used to maximize the assignments that fulfil the preferences of each patient. Then, r 8 is used to minimize the number of patients that are assigned to the fictitious operator. Finally, r 9 is used to maximize the solutions that preserve assignments dictated by the history of previous sessions.
Agenda encoding
Data Model. The following atoms constitute the input data:
• Instances of patient(ID,MIN) represent a patient identified by ID, and a minimum rehabilitation session of MIN length in time units that the patient has to undertake during the day. • Instances of period(PER,OP,STA,END) define the start (STA) and end (END) time unit in the period PER (which can be morning or afternoon), which corresponds to the shift of the operator with identifier OP. • Instances of time(PER,OP,T) define the time slots T during the period PER where the operator OP works. In particular, T ranges from STA to END defined by the above atom, i.e. time is defined as time(PER,OP,STA..END):-period(PER,OP,STA,END). • Instances of location(ID,CAP,PER,STA,END) represent a location (i.e., a gym or a room), with an identifier ID, a maximum capacity of CAP, which, during the period PER, is open from the time unit STA until END. • Instances of macro location(MLOC,LOC) define that the location LOC is inside the macro-location MLOC (i.e., a floor). • Instances of session(ID,PAT,OP) represent a session between the patient PAT and the operator OP, coming from the assignment(OP,PAT) output of the board phase, to which a unique ID is added (to discriminate between morning and afternoon shifts). • Instances of session type(ID,OP,TYPE) represent that the session with identifier ID assigned to operator OP is of type TYPE (which can be individual or supervised ). • Instances of session macro location(ID,MLOC) represent that the session with identifier ID has to be held in the macro-location MLOC.
• Instances of session length(ID,MIN,IDEAL) represent that the session ID has a minimum length (MIN) that has to be performed in individual, and an ideal length (IDEAL) that would be beneficial to the patient, but it is not mandatory to perform. • Instances of mandatory session(ID) and optional session(ID) identify sessions that are mandatory and optional, respectively. • Instances of forbidden(PAT,PER,STA,END) represent an unavailability of the patient PAT in the period PER from the time unit from STA to END. • Instances of session preference(ID,PER,START,TYPE) represent the preference of the patient, stating that the session should be held during the period PER and it must start at the time unit START, where TYPE indicates if the preference is high or low.
The output is represented by atoms start(ID,PER,T), length(ID,PER,L), and session location(ID,LOC), which indicate the start, length and location of each session, respectively.
Encoding. In Figure 3 the encoding for the agenda is presented. Rules r 1 and r 2 assign a start time to every session: for the optional session, the start atom can be unassigned. Rule r 3 defines a length for all the sessions: the session length cannot be lower than the minimum time of the session and cannot be greater than the ideal time the session should take. Rule r 4 assigns a location for each session. Rules r 5 and r 6 reserve to each session slots of time before it starts and after it ends, in which the session can be performed in a supervised fashion.
Then, rules r 7 and r 8 define auxiliary atoms extstart(ID,PER,TS) and extlength(ID,PER,TS) using TS slots of times for the session with identifier ID on period PER reserved for the start and length extensions, respectively. Rule r 9 defines an auxiliary atom of the form individual session location(ID,LOC,OP,MIN,IDEAL) which represents that an individual session ID in the location LOC is assigned to the operator OP, and its minimum and ideal lengths are equal to MIN and IDEAL, respectively. Rule r 10 defines session time(ID,OP,PL,PER,T) which states that during time T of period PER the session ID is being performed by operator OP.
Rule r 11 states that two individual assignments shall not overlap. Rule r 12 imposes that each patient is assigned to at most one session per period. Rules r 13 trough r 15 impose that the optional individual time (i.e., the difference between the minimum length of the session and the planned length) is added fairly to all individual sessions, starting with shorter ones. Rule r 16 imposes that for each time slot, the operator is not in two different places. Rule r 17 states that patients must have their minimum time reserved. Rule r 18 imposes a limit on the concurrent use of locations with limited capacity. Rules r 19 through r 21 impose that a session cannot happen during a forbidden time. Rule r 22 avoids that, during a time slot, the distribution of sessions between each pair of locations inside the same macro location is unfair (i.e., a location is at its full capacity while another is empty).
The weak constraint r 23 states that each session duration should be as close as possible to the ideal duration. Rules r 24 and r 25 minimize the distance between the actual and the preferred starting time for the sessions with high priority. Rule r 26 maximizes the number of optional sessions included in the scheduling. Rules r 27 and r 28 are similar to r 24 and r 25 , respectively, but for the sessions having low priority.
Experimental Analysis
In this section, the analyses performed on the two encodings is presented. The first part of our analysis is performed on real data coming from the institutes of Genova Nervi and Castel Goffredo; then, in order to evaluate the scalability of the approach and to analyse how our solution would behave in larger institutes having similar characteristics, an analysis is performed on synthetic instances with increasing dimensions, but considering real parameters. A comparison between the real and synthetic instances vali- dates the approach and demonstrates that synthetic instances can reasonably model the problem at hand. All these three parts are included, in separate paragraphs, in a first subsection, while a second subsection is devoted to a comparison to alternative logicbased formalisms. Encodings and benchmarks used in the experiments can be found at: http://www.star.dist.unige.it/~marco/RuleMLRR2021TPLP/material.zip.
Results of the encoding
Real data. ICS Maugeri utilizes a web-based software called QRehab (Saverino et al. 2021), which is built on top of the specified encoding; thus, analysis can be performed on real data coming from the institutes of Genova Nervi and Castel Goffredo, which tested and used this software since mid 2020 for Genova Nervi and the beginning of 2021 for Castel Goffredo. This allowed us to access 290 instances for Genova Nervi and 100 for Castel Goffredo. Table 1 provides an overview of the dimension of the instances in the two institutes in terms of number of physiotherapists, number of daily patients, density of patients per operator, number of floors (i.e., macro-locations) and number of total gyms (which we recall are locations in which multiple sessions can be performed in parallel). In Table 2, the results obtained by the two encodings are presented in terms of percentage of instances for which an optimal/satisfiable/no solution is computed, which also correspond to the three outcomes of interest for a practical use of our solution. The last two rows report the mean time of instances solved optimally and of the last computed solution for all satisfiable instances, respectively. The scheduling was performed using the ASP solver clingo (Gebser et al. 2012) with a cut-off of 30s using two different optimization methods: The first is the default Branch&Bound (BB) optimization method (Gebser et al. 2015) with the option --restart-on-model enabled; the second leverages instead the Unsatisfiable Core (USC) algorithm (Andres et al. 2012) with the clingo options --opt-strategy=usc,k,0,4 and --opt-usc-shrink=bin enabled (which turn on the algorithm k (Alviano and Dodaro 2020) and the shrinking of the unsatisfiable cores (Alviano and Dodaro 2016), respectively). The cut-off of 30s was chosen in order to be able to analyse a vast amount of experiments in overall reasonable time, and has proven to be a sufficient amount of time to achieve meaningful results; in the software used daily by the ICS Maugeri the cut-off is set to 300s, as a means to solve even the hardest instances, having a limited number of instances to be run daily. As it can be seen in Table 2, results are mixed: the USC algorithm performs better in the agenda encoding while the BB algorithm is better on the board scheduling; moreover, 100% of the board instances are solved, while for approximately one third of the agenda instances from Castel Goffredo a solution cannot be found. Considering these are hard real instances and cut-off time is limited, results are positive and highly appreciated by ICS Maugeri members. Synthetic data. In order to understand how our solution scales to larger institutes having similar characteristics, a simulated approach is needed. For this reason, a generator able to produce random instances with features as close as possible to the ones of real hospitals was developed. Some examples of real data utilized are: the percentage of individual and supervised sessions, the medium length of operator's shifts, the occurrence of forbidden time slots for patients, and the ideal length of sessions. For every new instance created, each feature was extracted from a random distribution which was modelled from the real data coming from the hospitals or from the knowledge of institute administrators and managers. In Figure 4 results of the scheduling of the board encoding, computed from the synthetic data, are presented. The x-axis defines the number of patients and the y-axis the number of operators; white lines represent points in which the density is an integer. Every pixel of the image depicts the mode of the results of 5 simulations executed with the corresponding number of patients and operators with a cut-off of 30s using the BB optimization algorithm (left) and the USC optimization algorithm (right). The colour of a pixel thus signals if the majority of instances with that particular number of operators and patients resulted in: (i) Optimum Found, signalling that the optimal stable model was found, (ii) Satisfiable, when at least one suboptimal stable model was found, but the solution is not guaranteed to be optimal, (iii) Unknown, if no stable model could be found before the cut-off, or (iv) Unsatisfiable, when no stable model exists which can satisfy all the constraints. As it can be seen from the figure, the results of the scheduling are directly proportional to the density (i.e., the average number of patients for every operator), changing from Optimum Found to Satisfiable when reaching a density of approximatively 2.4 patients per operator. Notably, despite the use of random instances, no instance results Unsatisfiable since the fictitious operator can always catch the patients which cannot be assigned to any operator (due to all the operators reaching full capacity). The position of the hospitals of Genova Nervi and Castel Goffredo are highlighted with a circle. In this figure it can be noted how BB gives better results than USC, by being able to find, before the cut-off, at least a suboptimal stable model for instances of higher densities, where, instead, the USC algorithm returns Unknown. In Figure 5 the results of the agenda encoding, scheduled with the BB algorithm (left) and USC algorithm (right), are presented in the same format as for the previous experiment. The instances for this experiment are the same as the previous one, but are augmented with the assignment among patients and operators found by clingo with the board encoding and other needed parameters. As previously stated, each pixel represents 5 instances and its colour represents the mode of the clingo results. Here two things can be noted: (i) unlike the board results, which showed a proportionality with the density, these results show a correlation only with the number of patients, and (ii) some red dots scattered in the image indicate that some instances result Unsatisfiable: this can happen since the random data could create some instances with features that cause an impossibility to schedule. With the BB optimization algorithm, the transition between the Optimum Found and Satisfiability results is located near 40 patients, and near 120 patients for the transition between Satisfiability and Unknown. As it can be seen in Figure 5 (right), the USC algorithm performs instead better and moves the transition between the Optimum found and Satisfiable results from 40 to 60 patients but, on the other hand, the transition between Satisfiable and Unknown slightly decreases from 120 patients to 110. The improvements on the transition betwee Optimum found and Satisfiable is very important in our setting, since Genova Nervi and Castel Goffredo fall into this area, confirming the improvements obtained in Table 2.
Validation of synthetic instances. In order to understand if the simulated instances correctly represent the real data and can be therefore used to predict the behaviour of the system in larger institutes with similar characteristics, a validation is needed to compare the results obtained on real and synthetic instances. Intuitively, we have considered the data presented in Table 2 and compared it to the results of the instances within the circles around Genova Nervi and Castel Goffredo in Figures 4 and 5, to check if they "coincide". For doing so, a decision tree was trained, taking as dataset all the features of the Fig. 6: Visual representation of the decision tree trained on the results found by clingo on real data utilizing the BB+RoM algorithm. The tree nodes represent features of the instance (density and average qualifications) and the leafs represent the result given by clingo (optimal found, satisfiable, unsatisfiable, unknown). simulated instances, some of them listed in the previous paragraph. Then, a test dataset with features extracted from the real instances was produced and given as input to the decision tree, and the predicted result was then compared to the result given by clingo on the real instances. Figure 6 shows a visual representation of the decision tree trained on the results found by clingo on real data utilizing the BB+RoM approach. The tree nodes represent the most important features found by the decision tree approach, which allows a correct classification of the results of clingo, shown in the leaf nodes. The shown features are (i) the density, i.e., the proportion patients/operator ratio, and (ii) the average number of qualifications, i.e., the type of patients (orthopaedic, neurological, covid-positive etc). Synthetic resources were then used as new data and given as input to the decision tree. The output of the decision tree was then compared with the actual result given by clingo. It can be seen, in fact, how the decision tree in Figure 6 is able to explain the results of the synthetic instances depicted in Figure 4 (left): the colour green (representing optimality) is indeed classifiable only by the patients/operator density (i.e., the white diagonal lines) and the transition between the two results happen when the density is near 2.4, as classified, from the real data, by the decision tree. When the density is between 2.42 and 2.77 it can be noted how the average number of qualification (not explicitly shown in Figure 4) makes the difference between an optimal and suboptimal result (the more specialized the operators are, the fewer patients there are available for the planner to choose). This test showed that for the board encoding, all the results on real instances were equal to the predicted ones for both institutes; the agenda encoding produced instead the same results in 93% of the cases for Genova Nervi and in 67% of the cases for Castel Goffredo, thus showing that, overall, the synthetic data behaves similarly to the real one and can be used for predicting the behavior of instances in larger First 58% 41,7% 0,3% ---77,1% 15,6% 7,3% --Second 41,7% 58,3% 1% ---14,7% 54,2% 29,4% --Third 0,3% -10,3% ---6,4% 28,4% 61,5% --Solver TO --9,3% 20,9% 20,9% 100% 1,8% 1,8% 1,8% 100% 100% Pypblip TO --79,1% 79,1% 79,1% ------institutes having similar characteristics. Finally, the computed decision trees also confirm what are the most relevant features outlined above by inspecting the figures. In fact, if the height of the decision tree is increased, the accuracy of prediction does not improve that much, signalling that the features shown in Figure 6 are sufficient to explain the differences in the results of clingo.
Comparison to alternative logic-based formalisms
In the following, an empirical comparison of our ASP-based solution to alternative logicbased approaches is presented, obtained by applying automatic translations of our ASP encoding. In more detail, the ASP solver wasp (Alviano et al. 2019) was used, with the option --pre=wbo, which converts ground ASP instances into pseudo-Boolean instances in the wbo format (Olivier Roussel and Vasco Manquinho 2012). Then, the tool pypblib (Ansótegui et al. 2019) was employed to encode wbo instances as MaxSAT instances. Moreover, given that other formalisms cannot handle multi-level optimizations, in order to provide a fair comparison, the ASP instances were processed using wasp with the option --pre=lparse, which collapses all weak constraints levels into one single level using exponential weights. With this approach, the costs found by the different approaches can be straightforwardly compared. Three state-of-the-art MaxSAT solvers were considered, namely MaxHS (Saikko et al. 2016), open-wbo (Martins et al. 2014), and rc2 (Ignatiev et al. 2019), as well as the industrial tool for solving optimization problems gurobi (Gurobi Optimization, LLC 2021), which is able to process instances in the wbo format. Concerning clingo, the already presented BB+RoM and USC algorithms were used. The latter enables the usage of algorithm oll (Morgado et al. 2014), which is the same algorithm employed by the MaxSAT solver rc2.
These experiments were run using the ASP encoding coming from the already presented real-world instances of the hospitals of Genova Nervi and Castel Goffredo. The experiments were conducted in the following way: firstly, the ASP encoding in which the weak constraints have been collapsed, was transformed in the wbo and MaxSAT formats, then all the solvers were called with a cut-off of 30s (the same used in all the other experiments). Then, for every formalism the following metrics were recorded: if it has found the optimum, the final cost and the time of computation. With these metrics, the formalisms can be ordered from best to worst based on their result: an optimal solution is better than one not declared optimal; if both are suboptimal then the one with the lowest cost is better; if both are optimal then the one which took less to declare optimality is better. In Table 3 the ranking among the formalisms is presented. For each of the different formalisms, the table shows the percentage of how many times it has arrived first, second or third. Solver TO represents cases in which the solver was unable to find a solution before the cut-off (Unknown). Pypblib TO represents cases in which the tool pypblib was unable to encode wbo instances as MaxSAT instances in a cut-off time of 60s. For the board encoding both clingo's algorithms USC and BB+RoM are presented, since they showed comparable result; instead, in the agenda encoding only the USC algorithm is presented since it outperformed the BB+RoM in the previous tests.
The results show that for the board encoding clingo is the most performant algorithm, coming first in almost all the experiments. In particular, clingo with the BB+RoM optimization algorithm resulted more performant than the algorithm relying on the Unsatisfiable Core strategy, which is conformant with the experiments run on the multi-level version of the ASP encoding. For this encoding, it can be seen that for a high number of instances, around 80%, the tool pypblib was unable to encode the MaxSAT instances within the cut-off. Still, in the remaining 20%, clingo remains the most performant algorithm. For the agenda encoding, clingo is still the best solver, but a more precise ranking among solvers can be noticed with MaxHS coming second and OpenWBO third. Notably, RC2 and Gurobi are, with both encoding, always unable to find a solution within the cut-off.
Domain Specific Optimisations
Motivated by the analysis performed in the previous section, in which ASP outperformed other formalisms on translated (MaxSAT and pseudo-Boolean) formulas, we apply domain specific optimizations to our ASP encoding, with the aim of further improving the solving time and move towards solving larger instances. In Section 4, benchmarks for the board and agenda phases were presented, showing different results based on the optimization algorithm chosen (i.e., BB+RoM or USC). The domain specific optimisations are presented to mainly decrease grounding, and consequently planning times, with the aim, as mentioned, of being able to find optimum solutions in larger instances, e.g., possibly corresponding to larger hospitals. The optimizations are presented only on the agenda encoding, which is the more complicated of the two phases and has still a great margin of improvement via changes in the encoding. These changes all rely on the knowledge of the RSP domain and on the possibility to prune impossible solutions already in the grounding process, avoiding wasting time in search. The section is organized in two subsections, in which the first presents the changes and improvements done on the previous agenda encoding, while the second subsection focuses on the results.
Optimized encoding
The next two paragraphs present the specific domain optimizations introduced.
Pruning of session starts. As it can be seen in Figure 3, in rules r 1 and r 2 the start of a session is guessed between all the possible time slots in the shift of an operator, expressed via the atom time(PER,OP,TS). These guess rules can be improved by reducing the number of time slots in which it is possible to start a session with the following constraints: 1. a session cannot start in a time slot near to the operator' shift's end. This is because the minimum specified time of the session would not be satisfied, given the shift's end; 2. if a patient has a forbidden time (i.e., a time interval where the patient cannot be scheduled), the session cannot start during the forbidden time. Moreover, some timeslots before the forbidden times should be excluded beforehand since, if the session started in these timeslots, this would not allow it to end before the forbidden time starts.
In Figure 7 the ASP encoding for pruning the session starts is shown. In r 29 a new atom forbiddenRange is defined for the purpose of extending the start of forbidden times by including the time slots which would not allow the session to end before the start of the forbidden time. Rule r 30 spreads forbiddenRange in all the time slots (forbiddenSlot) within the range. In rule r 31 a new atom allowedTime is defined as a time slot in the shift of the operator which is not a forbiddenSlot, and which allows the session, with its min length MIN, to end before the end of the shift END. In rules r 32 and r 33 , the atom allowedTime replaces the more broad atom time in the guess rule of the start of the session. In the optimized encoding, rules from r 29 to r 33 replace rules r 1 and r 2 of the original encoding (Figure 3).
Pruning of session extension. As stated in Section 2, the agenda encoding relies on two auxiliary atoms (extstart and extlength) as a means to reserve slots of time before it starts and after it ends to each session, in which the session can be performed in a supervised fashion. These before (after) slots of time are decided with a guess rule on the atom before (after) in rule r 5 (r 6 ) of Figure 3. The definition of these atoms can be improved in order to reduce the number of ground instantiations, in the following way:
1. as described in the previous paragraph, the extended part of a session cannot start during a forbidden time; 2. the before and after timeslots cannot be greater than the difference between the ideal length of a session and its minimum length. Since the weak constraints impose a minimization of the distance between the final length of the session and its ideal length, this last acts as an upper bound of the length of the session; 3. if there is already an extension before the session, the extended length of the session cannot be longer than the ideal length of the session.
Rules r 5 , r 6 , r 7 and r 8 of the agenda encoding presented in Figure 3 can be substituted by the encoding presented in Figure 8. Rule r 34 states that a value for the before extension can be computed taking the difference between the start of the session and an allowed time slot distant no more than the difference between the ideal and minimum length of a session. Rule r 35 defines the auxiliary atom extstart via the previously guessed before atom. Rule r 36 finds the amount to reserve after the session in the same way as expressed with the before atom. Rule r 37 defines the auxiliary atom extlength now being limited by the ideal length. Rule r 38 imposes that a session must have an extended length (which can correspond to the individual length if no supervised time is needed); this is to avoid solutions in which the planner increases to the maximum both the before and after extensions of a session as a shortcut to falsify all instantiations of rule r 37 by not having an extended length less than the ideal length.
Results of the optimized encoding
The next two paragraphs present the performance of the optimized encoding on real and synthetic instances, respectively.
Real Data. In Table 4 and 5 the results of the optimized encoding on real instances of the hospitals in Genova Nervi and Castel Goffredo are presented. Table 4 presents a comparison about the grounding between the two encodings, showing the significant reduction in terms of time, number of variables and number of rules that the optimized encoding brings. Table 5 then shows the results for the basic agenda encoding presented in Section 3.2 with the two algorithms BB+RoM and USC (this part of the table is Table 2). The other half of the table shows the results for the optimized encoding presented in the previous subsection, again with the two algorithms. As it can be seen, the optimized encoding boosts the performances, especially when combined with the USC algorithm. Comparing the two encodings, the first thing that can be noticed is that the optimized encoding is able to find a solution for each instance. Moreover, it can be seen that:
• even if in Genova Nervi the percentages remain the same for the BB algorithm, the average time in which the last stable model is outputted decreases. • with the USC algorithm, using the optimized encoding, for most of the instances both in Genova Nervi and Castel Goffredo, an optimal solution can be found.
Branch & Bound + Restart On Model Unsatis able Core Fig. 9: Results of the synthetic benchmarks of the agenda produced by clingo with the optimized encoding Synthetic Data. As previously explained in Section 4, testing an encoding on synthetically generated instances is important to understand how our solution could scale to larger institutes having similar characteristics. Figure 9 shows the results of the scalability test performed with the new optimized encoding, where the meaning of the colours, axes and lines has been already explained in Section 4. On the left, the results of running the optimized encoding using clingo with BB+RoM settings; on the right, the result are computing against the optimized encoding leveraging the USC algorithm. Comparing Figure 9 (optimized encoding, Opt) with Figure 5 (basic encoding, Basic) serious improvements can be noted:
• comparing the best combinations, i.e., Basic+USC and Opt+BB+RoM, it can be noted how the transition between solution with Optimum Found and Satisfiable stays approximatively the same near 50 patients, but Unknown results no longer appear, meaning that in the cut-off of 30s clingo can find a suboptimal solution.
In fact, the aim of the optimization was to reduce to the minimum the grounding time, which has left now the largest part of the 30s cut-off to be spent in actually solving. • focusing on Opt, the results of Opt+BB+RoM and Opt+USC show the supremacy of the USC algorithm.
Focusing on optimization algorithms, as it can be seen from the figures, the results on these benchmarks are comparable with the ones performed on real data: using the BB+RoM algorithm in fact it can be noted how the area of the graph with properties similar to the hospitals of Nervi and Castel Goffredo (the orange circle) have most of its area of a blue colour (representing Satisfiable results) and only for easy examples an optimal solution can be found; using the USC approach, similarly, we can see that in the circle fall most solutions found with optimality, thus confirming the results on the real instances. In fact, comparing the results of Basic+USC and Opt+USC it can be seen a real improvement in the number of instances which can be now solved with optimality: before, with the Basic+USC approach, the transition between solutions Optimum Found and Satisfiable lied near 50 patients, while now has reached almost 90 patients. At last, we also compared our approach with algorithm USC (that the analysis demonstrated to be the best) to the other logic-based formalisms already employed in Table 3, using the same evaluation metric and presentation. As it is clear from Table 6, the ASP approach is the best also when considering the optimized encoding.
Related Work
This paper is an extended and revised version of (Cardellini et al. 2021), having the following main consistent additions: (i) a comparison to alternative logic-based formalisms on real instances (Subsection 4.2), and (ii) the definition and related experimental evaluation of two domain specific optimizations (Section 5).
There have been few attempts in the literature to solve rehabilitation scheduling, since most hospitals are still doing it in a manual way. Among the automated solutions, often they are applied to real world data. However, their results are not directly comparable to ours, since their constraints and objective functions are different from the ones that emerged from our meetings with the physiotherapists and management at ICS Maugeri. In particular, to our knowledge, no other solution takes into account several aspects like the preferred time for the session scheduling and the preferences in the assignment of the patient to the operator. Huang et al. (2012) developed a system, equipped with a Graphical User Interface, which can generate the optimal schedules for rehabilitation patients to minimize waiting time and thus enhance service quality and overall resource effectiveness of rehabilitation facilities. More recently, Huynh et al. (2018) further refined the algorithm in order to develop a hybrid genetic algorithm (GASA) that integrates genetic algorithm (GA) and simulated annealing (SA). Recently, Li and Chen (2021) designed a genetic algorithm based on Waiting Time Priority Algorithm (WTPA), which was tested on a rehabilitation department. Schimmelpfeng et al. (2012) developed a decision support system for the scheduling process based on mixed-integer linear programs (MILPs), to determine appointments for patients of rehabilitation hospitals, subject to numerous constraints that are often found in practice. We already mentioned in the introduction that ASP has been already successfully used for solving application problems in several research areas (see, e.g., Gebser et al. (2016) for routing driverless transport vehicles, Ricca et al. (2012) for team scheduling, and Erdem et al. (2016) for a general overview) including scheduling problems in the Healthcare domain (see, e.g., for an overview focused on them). Differently from previous papers in the Healthcare domain, the current work focuses on the rehabilitation scheduling problem, that was not addressed before using ASP; and it combines a two-phase encoding, rather than the usually employed direct encoding, with the evaluation of the solution on real benchmarks. Concerning the experimental analysis, similarly to Dodaro et al. (2021), in this work we compared our ASP-based solution with alternative logic-based approaches.
Finally, in (Saverino et al. 2021) an analysis of the usage of the tool in the hospital of Genova Nervi for a period of approx. 6 months is reported. As an example, statistics about the sessions planned by our ASP encodings and their actual durations in the hospital usage are recorded. As shown in the paper, reported and planned session lengths are similar, with the ratio between these two quantities has been between 0.95 and 1.05 for the 95% of the considered time span.
Conclusion
In this paper, we have presented a two-phase ASP encoding for solving rehabilitation scheduling. We have started from a general solution, then extended with domain specific optimizations. Our solution has been tested with clingo and both real and synthetic benchmarks, the former provided by ICS Maugeri while the latter created with real parameters and employed to understand a possible behavior of the solution on upcoming institutes where the solution will be employed. Results are satisfying for the institutes employed at the moment and give some indications about the upcoming institutes ICS Maugeri plans to instrument with this solution. Domain specific optimizations further improve the results, by also diminishing the number of instances for which a solution cannot be found in short time. Future research includes a more fine-grained analysis of our solution by, e.g., combining the strengths of the optimization algorithms, analysing further dimensions of our encoding, e.g., number of floors and gyms for synthetic benchmarks, and benchmarking the impact of the introduced domain specific optimizations separately.
Competing interests: The authors declare none.
Fig. 1 :
1Result of the scheduling of the agenda in a real case scenario in the hospital of Genova Nervi. Light blue (yellow) squares represent time units in which the sessions will be performed in an individual (supervised) fashion. The ticks on the left keep track of the period (morning or afternoon) and time slot in which the session will start or end.
Fig. 2 :
2ASP Encoding for the board problem.
ID,PER,NL) : time(PER,OP,L), NL=L-ST, TS+NL <= END, NL>= MIN, NL<= IDEAL} = 1 :start(ID,PER,TS), period(PER,OP,ST,END), session(ID,_,OP), session_length(ID,MIN,IDEAL). 4 {session_location(ID,LOC): macro_location(MAC,LOC)} = 1 :-session_macro_location(ID,MAC). 5 {before(ID,NL): time(PER,OP,L), NL=L-ST, NL<=TS-ST} = 1 :-start(ID,PER,TS), period(PER,OP,ST,_), session(ID,_,OP). 6 {after(ID,NL): time(PER,OP,L), NL=L-ST, NL<=END-TS-LEN} = 1 :-start(ID,PER,TS), period(PER,OP,ST,END), length(ID,PER,LEN), session(ID,_,OP). 7 extstart(ID,PER,TS-LB) :-start(ID,PER,TS), before(ID,LB). 8 extlength(ID,PER,L+LA+LB) :-length(ID,PER,L), after(ID,LA), before(ID,LB). 9 individual_session_location(ID,LOC,OP,MIN,IDEAL) :-session_type(ID,OP,individual), session_location(ID,LOC), session_length(ID,MIN,IDEAL). 10 session_time(ID,OP,PL,PER,TS..TS+L-1) :-session(ID,_,OP), session_location(ID,PL), extstart(ID,PER,TS), extlength(ID,PER,L). 11 :-start(ID,PER,TS), length(ID,PER,L), session_type(ID,OP,individual), start(ID2,PER,TS2), session_type(ID2,OP,individual), ID!=ID2, TS2>=TS, TS2<TS+L.
12 :
12-session(ID1,PAT,_), session(ID2,PAT,_), start(ID1,PER,_), start(ID2,PER,_), ID1!=ID2.
13 :
13-individual_session_location(ID1,LOC,OP,MIN1,OPT1), length(ID1,PER,L1), individual_session_location(ID2,LOC,OP,MIN2,OPT2), length(ID2,PER,L2), OPT1-L1 <= OPT2-MIN2, OPT2-L2 <= OPT1-MIN1 , |OPT1 -L1 -OPT2 + L2| > 1.
14 :
14-individual_session_location(ID1,LOC,OP,MIN1,OPT1), length(ID1,PER,L1), individual_session_location(ID2,LOC,OP,MIN2,OPT2), length(ID2,PER,L2), OPT1-L1 > OPT2-MIN2, L2 > MIN2.15 :-individual_session_location(ID1,LOC,OP,MIN1,OPT1), length(ID1,PER,L1), individual_session_location(ID2,LOC,OP,MIN2,OPT2), length(ID2,PER,L2), OPT1-L1 <= OPT2-MIN2, OPT2-L2 <= OPT1-MIN1, OPT2 < OPT1, OPT1-L1 < OPT2-L2.16 :-session_time(ID,OP,PL,PER,T), session_time(ID2,OP,PL2,PER,T), ID != ID2, PL != PL2. 17 :-patient(PAT,MIN), #sum{LEN, ID: session(ID,PAT,_), extlength(ID,_,LEN)} < MIN. 18 :-location(LOC,LIM,PER,ST,END), LIM>0, time(PER,_,T), T>=ST, T<END, #count{ID: session_time(ID,_,LOC,PER,T)} > LIM.
19 :
19-forbidden(PAT,PER,ST,_), session(ID,PAT,_), extstart(ID,PER,TS), extlength(ID,PER,L), ST>=TS, ST<TS+L.20 :-forbidden(PAT,PER,_,END), session(ID,PAT,_), extstart(ID,PER,TS), extlength(ID,PER,L), END>TS, END<=TS+L. 21 :-forbidden(PAT,PER,ST,END), session(ID,PAT,_), extstart(ID,PER,TS), extlength(ID,PER,L), ST<=TS,END>TS.
22 :
22-time(PER,_,T), macro_location(MAC,LOC1), macro_location(MAC,LOC2), #sum{1,ID1:session_time(ID1,_,LOC1,PER,T); -1,ID2:session_time(ID2,_,LOC2,PER,T)} > 2.23 :∼ length(ID,_, L), session_length(ID,MIN,IDEAL), D=|L-IDEAL|. [D@6, ID] 24 :∼ start(ID,PER,_), session_type(ID,_,individual), session_preference(ID,PER2,_,high), D=|PER-PER2|. [D@5, ID] 25 :∼ start(ID,PER,TS), session_type(ID,_,individual), session_preference(ID,PER,TS2,high), D=|TS-TS2|. [D@4, ID] 26 :∼ optional_session(ID), time(PER,_,TS), not start(ID,PER,TS). [1@3,ID] 27 :∼ start(ID,PER,_), session_preference(ID,PER2,_,low), session_type(ID,_,individual), optional_session(ID), D=|PER-PER2|. [D@2, ID] 28 :∼ start(ID,PER,TS), session_preference(ID,PER,TS2,low), session_type(ID,_,individual), optional_session(ID), D=|TS-TS2|. [D@1, ID] Fig. 3: ASP Encoding for the agenda problem.
Fig. 4 :
4Results of clingo using the BB optimization algorithm and the option --restart-on-model enabled (left) and the USC optimization algorithm (right) on synthetic benchmarks of the board.
Fig. 5 :
5Results of clingo using the BB optimization algorithm and the option --restart-on-model enabled (left) and the USC optimization algorithm (right) on synthetic benchmarks of the agenda.
29 forbiddenRange(ID,PER,XSTA,END) :-forbidden(PAT,PER,STA,END), session(ID,PAT,_), session_length(ID,MIN,_), XSTA = STA -MIN + 1. 30 forbiddenSlot(ID,PER,STA..END-1) :-forbiddenRange(ID,PER,STA,END). 31 allowedTime(ID,PER,T) :-time(PER,OP,T), session(ID,_,OP), session_length(ID,MIN,_), period(PER,OP,_,END), T <= END -MIN, not forbiddenSlot(ID,PER,T). 32 1 {start(ID,PER,TS) : allowedTime(ID,PER,TS)} 1 :-session(ID,_,OP), mandatory_session(ID). 33 0 {start(ID,PER,TS) : allowedTime(ID,PER,TS)} 1 :-session(ID,_,OP), optional_session(ID).
Fig. 7 :
7Optimized encoding for pruning the session starts
Fig. 8 :
8Optimized encoding for pruning of session extension
Table 1 :
1Dimensions of the ICS Maugeri's institutes.Institute
# Operators # Patients Density # Floors # Gyms
Genova Nervi
[9,18]
[37,67]
[2.4,5.2]
1
1
Castel Goffredo
[11,17]
[51,78]
[3.5, 6.4]
2
3
Table 2 :
2Results on ICS Maugeri institutes.Branch & Bound + RoM
Unsatisfiable Core
Genova Nervi Castel Goffredo Genova Nervi Castel Goffredo
Board Agenda Board Agenda Board Agenda Board Agenda
% Optimum
35%
0%
0%
0%
22%
45%
0%
0%
% Satisfiable
65%
100% 100%
67%
78%
55%
100%
70%
% Unknown
0%
0%
0%
33%
0%
0%
0%
30%
Avg Time for opt
1.1s
-
-
-
10s
0.01s
-
-
Avg Time Last SM 1.3s
30s
5.2s
30s
12.1s
21.3s
10.4s
30s
Table 3 :
3Comparison between alternative logic-based formalisms for the board and agenda phase.Board
Agenda
BB+RoM USC MaxHS OpenWBO RC2 Gurobi
USC MaxHS OpenWBO RC2 Gurobi
Table 4 :
4Comparison, in terms of grounding, between the basic and the optimized encoding on real instances coming from the Maugeri's hospitals.Basic
Optimized
Nervi C.G. Nervi C.G.
Avg Grounding Time
8.3s 11.5s 0.7s 0.9s
Avg Number of Variables 323k 587k
51k 59k
Avg Number of Rules
3.8M 11.4M 177k 327k
Table 5 :
5Results of the optimized agenda encoding on real instancesBasic+BB+RoM Basic+USC Opt+BB+RoM Opt+USC
Nervi
C.G.
Nervi C.G Nervi
C.G.
Nervi C.G.
% Optimum
0%
0%
45% 0%
0%
0%
82% 74%
% Satisfiable
100%
67%
55% 70% 100% 100%
18% 26%
% Unknown
0%
33%
0% 30% 0%
0%
0%
0%
Avg Time for opt
-
-
0.01s
-
-
-
0.01s 4.7s
Avg Time Last SM 30s
30s
21.3s 30s 19.6s
20s
20.4s 8.0s
Table 6 :
6Comparison between alternative logic-based formalisms for the optimized agenda phase.Agenda
USC MaxHS OpenWBO RC2 Gurobi
First
94,2% 3,1%
0,0%
-
-
Second
3,1% 72,3%
21,5%
-
-
Third
0,0% 21,5%
75,5%
-
-
Solver TO
2,7%
3,1%
3,0%
100% 100%
Pypblip TO
-
-
-
-
-
https://www.icsmaugeri.it/.
A person who goes to a hospital for a daily treatment, without staying the night.
{assignment(OP, PAT) : operator(OP)} = 1 :-patient(PAT). 2 uniqueLocationLength(OP,PAT,DUR) :-assignment(OP,PAT), patient_session(PAT,_,LOC), patient_data(PAT,_,DUR), #count{ID:patient_session(ID,_,LOC), assignment(OP,ID)} < 2. 3 sameLocationLength(OP,PAT,DUR) :-assignment(OP,PAT), patient_session(PAT,DUR,LOC), #count{ID:patient_session(ID,_,LOC), assignment(OP,ID)} > 1.
This is because, for practical reasons, we always want to have a solution.
. :-Operator_Contract( Op, Time , _ ) #sum{u, Pat Op, :-operator_contract(OP,TIME,_), #sum{U,PAT:uniqueLocationLength(OP,PAT,U);
. S , Pat : , ( Op, Pat , S)} > Time, -operator_contract(OP,_,N), #count{PAT:assignment(OP,PAT)} > N. 6 :-operator_limit(OP,T,N), #count{PAT:assignment(OP,PAT), patient_data(PAT,T,_} > N. 7 :∼ #sum{W, PAT:assignment(OP,PAT), patient_preference(PAT,OP,W)} = N.5N@3] 8 :∼ #count{PAT: assignment(-1, PAT)} = N. [N@2S, PAT:sameLocationLength(OP,PAT,S)} > TIME. 5 :-operator_contract(OP,_,N), #count{PAT:assignment(OP,PAT)} > N. 6 :-operator_limit(OP,T,N), #count{PAT:assignment(OP,PAT), patient_data(PAT,T,_} > N. 7 :∼ #sum{W, PAT:assignment(OP,PAT), patient_preference(PAT,OP,W)} = N. [N@3] 8 :∼ #count{PAT: assignment(-1, PAT)} = N. [N@2]
. :∼ #sum{w, Pat ) Op, Pat, W Op, )} = N, ID,NL): allowedTime(ID,PER,T), T<=TS, NL=TS-T, NL<=IDEAL-MIN} 1 :-start(ID,PER,TS), session(ID,_,OP), session_length(ID,MIN,IDEAL341:∼ #sum{W, PAT:assignment(OP,PAT), history_preference(PAT,OP,W)} = N. [N@1] 34 1 {before(ID,NL): allowedTime(ID,PER,T), T<=TS, NL=TS-T, NL<=IDEAL-MIN} 1 :-start(ID,PER,TS), session(ID,_,OP), session_length(ID,MIN,IDEAL).
Per Id, Ts-Lb) , -start(ID,PER,TS), before(ID,LB). extstart(ID,PER,TS-LB) :-start(ID,PER,TS), before(ID,LB).
. N L Id, ( ): Time, Per, T ) Op, Nl=t-Ts-Len T>=ts+len, Nl<=ideal , -start(ID,PER,TS), length(ID,PER,LEN), session(ID,_,OP), session_length(ID,MIN,IDEAL{after(ID,NL): time(PER,OP,T), T>=TS+LEN, NL=T-TS-LEN, NL<=IDEAL-MIN} 1 :-start(ID,PER,TS), length(ID,PER,LEN), session(ID,_,OP), session_length(ID,MIN,IDEAL).
. Per Id, Len) ; Id, Per , L ) , ( Id, La ) , ( Id, ( Lb), Session, _ Id, _ ) , ( Id, _ Ideal), Len Len=l+la+lb, <=, _ Id, _ ) , 38-length-startnot extlength(ID, _, _)extlength(ID,PER,LEN) :-length(ID,PER,L), after(ID,LA), before(ID,LB), session(ID,_,_), session_length(ID,_,IDEAL), LEN=L+LA+LB, LEN <= IDEAL. 38 :-start(ID, _, _), not extlength(ID, _, _).
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An ASP-based solution to the chemotherapy treatment scheduling problem. Theory and Practice of Logic Programming. C Dodaro, G Galatà, A Grioni, M Maratea, M Mochi, I Porro, 21Dodaro, C., Galatà, G., Grioni, A., Maratea, M., Mochi, M., and Porro, I. 2021. An ASP-based solution to the chemotherapy treatment scheduling problem. Theory and Practice of Logic Programming, 21, 6, 835-851.
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Applications of answer set programming. E Erdem, M Gelfond, Leone , N , AI Magazine37Erdem, E., Gelfond, M., and Leone, N. 2016. Applications of answer set programming. AI Magazine, 37, 3, 53-68.
Theory solving made easy with clingo 5. M Gebser, R Kaminski, B Kaufmann, M Ostrowski, T Schaub, P Wanko, :15. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik. Carro, M., King, A., Saeedloei, N., and Vos, M. D.52Technical CommunicationsProceedings of ICLPGebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., and Wanko, P. Theory solving made easy with clingo 5. In Carro, M., King, A., Saeedloei, N., and Vos, M. D., editors, Proceedings of ICLP (Technical Communications) 2016, volume 52 of OASICS, pp. 2:1-2:15. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik.
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Core-Guided MaxSAT with Soft Cardinality Constraints. A Morgado, C Dodaro, J Silva, Proceedings of Principles and Practice of Constraint Programming -20th International Conference. Principles and Practice of Constraint Programming -20th International ConferenceLyon, FranceSpringerMorgado, A., Dodaro, C., and Marques-Silva, J. Core-Guided MaxSAT with Soft Car- dinality Constraints. In Proceedings of Principles and Practice of Constraint Programming - 20th International Conference, CP 2014 2014, pp. 564-573, Lyon, France. Springer.
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| {'fraction_non_alphanumeric': 0.05650019062142585, 'fraction_numerical': 0.02048544923116025, 'mean_word_length': 4.773367571533382, 'pattern_counts': {'":': 0, '<': 24, '<?xml version=': 0, '>': 19, 'https://': 1, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "A core part of the rehabilitation scheduling process consists of planning rehabilitation physiotherapy sessions for patients, by assigning proper operators to them in a certain time slot of a given day, taking into account several legal, medical and ethical requirements and optimizations, e.g., patient's preferences and operator's work balancing. Being able to efficiently solve such problem is of upmost importance, in particular after the COVID-19 pandemic that significantly increased rehabilitation's needs.In this paper, we present a two-phase solution to rehabilitation scheduling based on Answer Set Programming, which proved to be an effective tool for solving practical scheduling problems. We first present a general encoding, and then add domain specific optimizations. Results of experiments performed on both synthetic and real benchmarks, the latter provided by ICS Maugeri, show the effectiveness of our solution as well as the impact of our domain specific optimizations.Under consideration in Theory and Practice of Logic Programming (TPLP).", 'arxivid': '2303.08709', 'author': ['Matteo Cardellini ', 'Ics Maugeri ', 'Italy Carmine Dodaro ', 'ItalySurgiq Srl ', 'Anna Giardini ', 'Ics Maugeri ', 'Italy Marco Maratea ', '\nPolytecnic of Torino\nTorinoItaly\n', '\nUniversity of Genova\nGenovaItaly\n', '\nUniversity of Calabria\nRendeItaly\n', '\nDIBRIS\nUniversity of Genova\nGenovaItaly\n', '\nUniversity of Calabria\nRendeItaly\n', '\nIVAN PORRO\nSurgiQ srlItaly\n'], 'authoraffiliation': ['Polytecnic of Torino\nTorinoItaly', 'University of Genova\nGenovaItaly', 'University of Calabria\nRendeItaly', 'DIBRIS\nUniversity of Genova\nGenovaItaly', 'University of Calabria\nRendeItaly', 'IVAN PORRO\nSurgiQ srlItaly'], 'corpusid': 257532498, 'doi': '10.1017/s1471068423000030', 'github_urls': [], 'n_tokens_mistral': 21849, 'n_tokens_neox': 19653, 'n_words': 11868, 'pdfsha': '5ff6a51dfc7435b9c2ccda80c5cb6de637575c53', 'pdfurls': ['https://export.arxiv.org/pdf/2303.08709v1.pdf'], 'title': ['Solving Rehabilitation Scheduling problems via a Two-Phase ASP approach * PAOLO DE NARDI GIUSEPPE GALATÀ', 'Solving Rehabilitation Scheduling problems via a Two-Phase ASP approach * PAOLO DE NARDI GIUSEPPE GALATÀ'], 'venue': []} |
arxiv |
In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe 2
6 Jan 2023
Chia-Hao Lee
Huije Ryu
Gillian Nolan
Yichao Zhang
Yangjin Lee
Siwon Oh
Hyeonsik Cheong
Kenji Watanabe
Takashi Taniguchi
Kwanpyo Kim
Gwan-Hyoung Lee
¶
Pinshane Y Huang
§Department of Physics
Seoul National University
08826SeoulKorea
Department of Physics
Yonsei University
03722SeoulKorea
⊥Research Center for Functional Materials
Sogang University
04107SeoulKorea
#International Center for Materials Nanoarchitectonics
National Institute for Materials Science
1-1 Namiki305-0044TsukubaJapan
@Materials Research Laboratory
National Institute for Materials Science
1-1 Namiki305-0044TsukubaJapan
University of Illinois at Urbana-Champaign
61801UrbanaIllinoisUnited States
In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe 2
6 Jan 2023†Department of Materials Science and Engineering, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, United States ‡These authors contributed equally to this work ¶Department of Materials Science and Engineering,
Understanding the phase transition mechanisms in two-dimensional (2D) materials is a key to precisely tailor their properties at the nanoscale. Molybdenum ditelluride (MoTe 2 ) exhibits multiple phases at room temperature, making it a promising candidate for phase-change applications. Here, we fabricate lateral 2H-T d interfaces with laser irradiation and probe their phase transitions from micro-to atomic scales with in situ heating in the transmission electron microscope (TEM). By encapsulating the MoTe 2 with graphene protection layers, we create an in situ reaction cell compatible with atomic resolution imaging. We find that the T d -to-2H phase transition initiates at phase boundaries at low temperatures (200-225 • C) and propagates anisotropically along the b-axis in a layer-by-layer fashion. We also demonstrate a fully reversible 2H-T d -2H phase transition cycle, which generates a coherent 2H lattice containing inversion domain boundaries. Our results provide insights on fabricating 2D hetero-phase devices with atomically sharp and coherent interfaces.KeywordsIn situ heating, anisotropic phase transition, laser irradiation, molybdenum ditelluride, transmission electron microscopy, 2D materials Phase transformations in two-dimensional transition metal dichalcogenides (2D TMDCs) are an emerging research area due to their polymorphism-the ability to host different phases of the same chemical composition with distinct crystal structures. Utilizing phases with diverse electronic properties, multiple functionalities can be compactly packaged into nanoscale devices, such as monolithic 2D electronics 1-3 and phase-change memories. 4,5 Among the group VI 2D TMDCs, molybdenum ditelluride (MoTe 2 ) has been widely studied because of the minimal energy difference (40 meV per formula unit) 6-8 between the trigonal prismatic (2H), monoclinic (1T'), and orthorhombic (T d ) phases shown inFigure 1a. While the honeycomb lattice of the 2H structure is distinct, the 1T' and T d phase share the same
monolayer structure, but have different stacking structure in multilayers. The 1T' phase has a monoclinic structure with β = 93.9 • , while the T d phase is orthorhombic with β = 90 • . 9 This stacking difference results in the broken inversion symmetry of the T d phase and its unique quantum properties, including type-II Weyl fermions, 10,11 quantum spin Hall effect, 12 giant magnetoresistance, 13 and superconductivity. 14 Phase transitions between 2H-and 1T'-MoTe 2 have been demonstrated using electric biasing, 4,15 strain, 16 heat, 17,18 ion intercalation, 19 and laser irradiation. 20-22 However, the phase transitions between 2H-and T d -MoTe 2 remain largely unexplored because the T d phase is less thermodynamically stable than other phases under ambient conditions, which makes it difficult to study its room temperature properties and phase-change behaviors. Conventionally, the T d phase is obtained by cooling 1T'-MoTe 2 crystals down to 250 K 23,24 or through chemically alloying with W substitutions. 25-27 Very recently, the 2H-to-T d transition has been reported with high temperature annealing of hBN-encapsulated MoTe 2 , 28 In situ TEM is a powerful technique to investigate phase transformations in 2D materials. 29,30 However, electron beam irradiation 31,32 and vacuum annealing 33,34 can cause major degradation of MoTe 2 and other 2D TMDCs due to the significant loss of chalcogen atoms, making it particularly challenging to probe their phase transitions without altering the chemical composition.
Here, we combine in situ TEM with graphene encapsulation to study the reversible phase transitions of MoTe 2 from micro-to atomic scales. We first use laser irradiation to locally convert few-layer MoTe 2 flakes from the 2H to a mixture of 1T' and T d phases, which we find is primarily T d in the regions examined by our TEM experiments. Then, we apply in situ pulsed heating to monitor the reverse phase transition from the T d to 2H phase with a combination of aberration-corrected scanning transmission electron microscopy (STEM) and dark-field TEM (DFTEM). We find that the T d -to-2H phase transition initiates at the 2H-T d interface at around 200-225 • C. Between 200-400 • C, we observe a highly anisotropic phase transition: the 2H phase fronts progress along the b-axis of the T d grains, in a layer-by-layer fashion. The ability to visualize each 2H phase front enables measurements of T d -to-2H phase transition kinetics of individual MoTe 2 layers. Lastly, we demonstrate the reversibility of phase transitions between 2H and T d phases with cycles of laser irradiation and vacuum heating. Figure S1). The encapsulation is essential because it creates an enclosed reaction cell that acts as physical and chemical barrier for MoTe 2 , minimizing the sublimation of Te and interactions with the atmosphere during further processing. If the MoTe 2 were not encapsulated, it would be nearly impossible to observe the phase transition without modifying the crystal stoichiometry through the loss of Te atoms, which has been shown to impact the phase transition. Using encapsulated samples, we did not observe any Te vacancy formation via ADF-STEM during heating. Importantly, the graphene contributes minimal background signal to the TEM images, enabling atomic-resolution imaging. 31,37,38 We then irradiate the encapsulated 2H-MoTe 2 with a 532 nm laser to locally initiate the phase transition from the 2H to a primarily T d phase (SI Section 2). Because it is difficult to distinguish 1T' and T d phases, the phases of pristine and laser-irradiated MoTe 2 are characterized by multiple techniques, including aberration-corrected annular dark-field STEM (ADF-STEM) images (Figure 1c-d), TEM diffraction, and polarized Raman spectroscopy (SI Figure S2). TEM diffraction and polarized Raman measurements indicate that the resulting materials contain a mixture of 1T' and T d phases, which is in agreement with previous reports. 39,40 The potential for mixed phases occurs because the calculated energy difference between 1T' and T d phase is less than 3 meV per unit cell. 8,39 In the TEM samples analyzed below, however, atomic resolution STEM imaging (Figure 1d) indicates that the laser-irradiated material is primarily T d (see SI Figure S3 for top-down ADF-STEM image simulation of 1T' and T d phases). Therefore, we refer to the transformed phase as T d .
For in situ heating, we transfer the laser-irradiated, encapsulated MoTe 2 specimens to a microelectromechanical system (MEMS)-based heating TEM chip (SI Section 1). Bright-field TEM (BFTEM) imaging before in situ heating ( Figure 2a) shows very little contrast between the 2H and T d phases, indicating a uniform thickness across the hetero-phase interface. The selected-area electron diffraction (SAED) patterns in Figure 2b-c exhibit the characteristic hexagonal and rectangular lattice of the 2H and T d phases.
We use DFTEM to map the real-space location and orientation of the 2H and T d phases (SI Section 3). DFTEM has been widely used to determine the crystal orientation and stacking order of 2D materials 41-43 and operates by selecting specific Bragg spots in the diffraction pattern with an objective aperture, so that only crystal grains that diffract to a narrow range of k-vectors appear bright in the image. DFTEM images of the 2H and T d phases in Figure 2d-e are obtained by selecting the (1100) 2H and (210) T d Bragg reflections, marked with blue and orange circles. We observe T d grains with three orientation directions, rotated 120 • from each other. The b-axis of each T d orientation is parallel to one of the three zig-zag directions of the three-fold symmetric 2H matrix (SI Figure S4). Figure 2f shows a false-colored DFTEM overlay image mapping the four grains present after laser-irradiation: the 2H phase (red) and the three T d orientations (green, yellow, and blue). The majority of the T d region in Figure 2f is oriented in one of the three orientations (green), with needle-like inclusions of the other two orientations.
Next, we perform in situ heating to investigate the reverse T d -to-2H phase transition.
We use heat pulses 18 instead of continuous heating for three reasons: (1) Pulsing provides flexibility to "halt" the phase transition at any time, rapidly jump to specific temperatures, and even hold at different temperatures for more detailed kinetic studies. (2) The ability to pause between pulses makes it possible to acquire both large field-of-view (FOV) DFTEM and atomic-resolution ADF-STEM images at several positions between pulses, which provides both a large-scale view of the phase transition kinetics and atomic scale snapshots at the interfaces. (3) Heat pulsing minimizes the energy input and potential sublimation during the phase transition. While the graphene encapsulation minimizes damage and sublimation to the MoTe 2 , we find that heating at temperatures above 600 • C for 0.5 s can produce small (5-10 nm) voids (see SI Movie S1).
We use DFTEM imaging to track the propagation of 2H phase between heat pulses, with pulse durations of 0.5 to 60 s, and temperatures from 200 to 275 • C; note that all images are acquired between pulses, when the sample is at room temperature. DFTEM images of the 2H phase after different heat pulse temperatures (Figure 3a-d and SI Movie S1 ) show that the 2H region at the phase boundary propagates anisotropically toward the T d grain during heating, forming a belt-shaped inclusion. Figure 3e shows a large-FOV DFTEM image where newly grown 2H regions, marked in red, inherit the orientation of the 2H matrix. We occasionally observed inversion domains in the 2H phase, which we discuss later in Figure 5.
Contrary to previous reports of high (500-600 • C) 1T'-to-2H phase transition temperatures in bulk samples, 44 we observe the T d -to-2H phase transition initiates at temperatures as low as 200-225 • C, from the existing 2H-T d interfaces. There are two reasons for such low transition temperatures: First, the T d phase is thermodynamically unstable under ambient condition, so only the kinetic barrier needs to be overcome. Second, the existing 2H-T d interfaces act as nucleation sites, which further reduce the kinetic barrier of the T d -to-2H phase transition. Figure 3f shows a contour overlay of the 2H phase fronts captured between heat pulses of 200-275 • C in a 4-layer thick sample. We outline the propagating 2H regions that contain at least a monolayer of 2H phase. This image shows that the 2H phase growth is anisotropic in-plane, progressing along the [010] T d (b-axis direction) of the T d grain. This result is in contrast to previous reports of an isotropic 1T'-to-2H transition, which is an averaged result from large-scale polycrystalline 1T' grains. 44,45 The preferential b-axis growth of the 2H phase is observed for all T d orientations (SI Figure S5). The anisotropy occurs mainly at low-temperatures, and we find the phase transition becomes more isotropic above 400 • C (SI Movie S2).
As shown in the atomic models in Figure 3f, 2H-T d phase boundaries can be classified into two types 1 based on their symmetry: Type 1, where the phase boundary is parallel to the b-axis of T d and Type 2, where the phase boundary is rotated by 120 • from the b-axis of T d .
The anisotropic propagation of the 2H phase suggests that the propagation (growth) rate of the type 2 interface is much faster than for type 1, resulting in the formation of belt-shaped 2H grains with mostly type 1 interfaces. This behavior can be described by a kinetic Wulff construction, [46][47][48][49] where the final crystal shape is predicted using thermodynamic and kinetic factors including the interface energy and relative growth rate of different facets. The type 2 interface energy is estimated to be 70 meV/Å higher than type 1 interface, 50 making type 1 interfaces more thermodynamically stable. This is consistent with our observations that there are more kinks and steps in the atomic-resolution ADF-STEM images of type 2 interfaces than in the type 1 interfaces (Figure 3g,h). The 2H growth preferentially propagates along the [010] T d direction due to the higher kink formation and expansion rate of type 2 interfaces. show that the T d region transformed from 2H phase is uniform, with 2H-T d boundaries that are sharp on the micron scale. After annealing at 800 • C for 3 hours, Raman mapping indicates the structure is fully and uniformly converted to 2H phase, as shown in Figure 5d.
This result shows that a reversible phase transition of 2H-and T d -MoTe 2 can be achieved by laser irradiation and vacuum annealing. When we anneal multiple samples at different temperatures (300-800 • C), all of them exhibit the T d -to-2H phase transition.
Finally, we examine the crystal structure of MoTe 2 after a full conversion cycle from 2H, to T d , and back to 2H phase in Figure 5e-h. The six-fold symmetry of the SAED pattern in Figure 5e indicates that the converted sample has no rotational grain boundaries. However, the DFTEM image (Figure 5f) shows that for a selected (1100) 2H Bragg reflection (marked with a green circle in Figure 5e), one of the two grains appears brighter due to the breaking of Friedel's law. 43,52 By selecting a neighboring Bragg reflection (orange circle in Figure 5f), the contrast of the two grains is reversed (Figure 5g). This indicates that the 2H grains have inversion symmetric orientations separated by an inversion domain boundary (IDB), a twin boundary commonly observed in 2D TMDCs. 34,43,53 Figure 5h shows an atomic resolution STEM image of an IDB in the 2H phase region after a full conversion cycle of 2H-T d -2H.
We also observe an IDB running through only 3 of the layers in a 5-layer MoTe 2 sample (SI Figure S6), indicating the IDBs do not necessarily go all the way through the sample.
IDBs are likely generated during the T d -to-2H phase transition because the T d grain has two equivalent transition pathways (SI Figure S7). As a result, cyclic phase transitions from 2H-T d -2H convert a 2D single crystal to coherent 2H polycrystals stitched with IDBs. Our work shows that cyclic phase transitions are a promising technique to fabricate the IDBs, which act as one-dimensional metallic tunnels 54,55 embedded in 2D semiconductors.
In conclusion, we have demonstrated that encapsulated, few-layer MoTe 2 can be reversibly phase engineered between the semiconducting 2H phase and the T d phase using laser irradiation and thermal annealing. Using in situ pulsed heating and DFTEM, we show that the T d -to-2H phase transition initiates at the 2H-T d interfaces at temperatures as low as 200-225 • C. Moreover, we observe anisotropic growth of the 2H phase front, which preferentially propagates along the b-axis of the nearby T d grains. Our findings can be applied to fabrication of coplanar 2D circuitry, including 2D Josephson junctions, 56 broadband photodetectors, 57 and other hetero-phase devices. Finally, we demonstrate a new approach for in situ studies of 2D materials using graphene encapsulation and pulsed heating, which can be applied to other micro-to atomic scale in situ studies of solid state phase transitions. Lastly, we placed the entire sample in chloroform for 30 min to remove the PC film.
To image the MoTe 2 at atomic resolution and reduce multiple scattering from thick hBN layers, we removed the hBN layers with XeF 2 dry etching. 2 The sample fabrication process was similar with the one used for ex situ experiments but with some modifications, see Figure S1 for the schematic. The bottom hBN layer was etched away by the XeF 2 exposure, and the etching process was self-limited at the graphene layer. We transferred the stack onto a clean SiO 2 /Si substrate and exposed it with chloroform, oxygen plasma, and XeF 2 again to remove the top hBN layers. After these steps, the stack was encapsulated with fluorinated graphene and ready for transferring onto an in situ heating TEM chip (E-FHDC-VO-10, Protochips). We used the conventional polymer transfer technique with poly(methyl methacrylate) (PMMA) and KOH to transfer the stack 3 from SiO 2 /Si substrate. 4 After transferring, the PMMA film was removed by placing the samples in acetone for 12 hours.
Laser irradiation parameters for phase transition
To initiate the 2H-to-T d phase transition of MoTe 2 , we irradiated the encapsulated samples using continuous wave (CW) 532 nm laser and power of 21 mW at ambient conditions. The laser was focused by a 100× objective lens (N.A. = 0.9) and the resulting spot size on the substrate was around 1µm. The laser-irradiated area was patterned by rastering the laser spot with 200 nm point-to-point distance and 0.1 s exposure time per step. to 200 mrad collection semi angles, 20 pm pixel size and a total dwell time of 20 µs/pixel using 10-frame averages. For BFTEM, SAED, and DFTEM, the data were acquired with a Ceta 16M camera at parallel illumination using the three-condenser TEM mode. The electron dose rate was around 10 3 e -/nm 2 /s and the exposure times for SAED and DFTEM were 2 to 5 s. Note that sparse dark pits were observed in DFTEM at the end of the in situ imaging after an accumulated total dose around 4 × 10 5 e -/nm 2 . While the graphene encapsulation was unlikely to form holes under this condition, these dark pits were likely to be crystallographic defects such as voids formed by displacing atoms of the MoTe 2 flakes.
S/TEM measurement
Ex situ T d -to-2H phase transition with annealing
The ex situ T d -to-2H phase transition of MoTe 2 ( Figure 5 of the main text) was performed by an annealing process in a vacuum furnace. We annealed the laser-irradiated sample in a vacuum chamber (10 -4 Torr) and slowly ramped up to targeted temperatures in 3 hours and held for another 3 hours. The targeted temperatures were set from 300 to 800 • C. The furnace was naturally cooled to room temperature.
Raman spectroscopy measurement
The linearly polarized Raman measurements (SI Figure S2) were carried out in the backscattering geometry using 514.5 nm laser excitation. The input laser beam was focused onto the samples by a 50× microscope objective lens (0.8 NA), and the scattered light was collected and collimated by the same objective lens. To access the low-frequency range below 50 cm -1 , volume holographic filters (OptiGrate) were used to clean the laser lines and reject the Rayleigh-scattered light. A laser with a low power of 300 µW was used to avoid local heating. The Raman scattering signals were dispersed by a Jobin-Yvon iHR550 spectrometer with a 2400 grooves/mm grating (400 nm blaze) and detected by a liquid-nitrogen-cooled, back-illuminated CCD detector. An achromatic half-wave plate was used to rotate the polarization of the linearly polarized laser beam to the desired direction. The analyzer angle was set such that photons with polarization parallel to the incident polarization passed through.
Another achromatic half-wave plate was placed in front of the spectrometer to keep the polarization direction of the signal entering the spectrometer constant with respect to the groove direction of the grating. The Raman spectra ( Figure 5 of the main text) were acquired using a HORIBA LabRAM HR Evolution with the laser wavelength at 532 nm. To minimize the irradiation damage, the laser power was set below 5 mW with an acquisition time of 60 s. All measurements were conducted at ambient conditions. Polarized Raman spectra of T d + 1T'-(purple) and 2H-MoTe 2 (orange). The 1T' and T d phase are typically characterized by the peak splitting around 128 cm -1 : those with a split peak were identified as the T d phase, and those with a single peak as the 1T' phase. 5 The blue peak in the inset indicates the presence of T d phase. However, considering the SAED result in (a), our specimen shows a spatial inhomogeneity of mixture of 1T' and T d phases. (e) Overlay of the simulated diffraction patterns of 2H and T d phase. The T d phase can be derived by shifting the chalcogen layers in 2H phase along one of the three arm-chair directions followed by some metal atom dimerization. Therefore, the b-axis of the derived T d variants are parallel to one of the three zig-zag directions of the 2H matrix. The field-of-view is marked by the yellow square. The IDBs are marked by the blue arrows. In this specific region, the IDB is only observed at the 4-layer region, indicating the IDB does not go all-the-way-through this 5-layer sample and suggesting an independent layer-by-layer phase transition mechanism. The layer number is determined by quantitative ADF-STEM intensities. Figure S7: Formation mechanism of the inversion domain boundary. Atomic models of two potential pathways of T d -to-2H phase transition in (a) side-view and (b,c) top-views. By sliding either the top or bottom chalcogen layers along the a-axis direction, 2H grains with opposite orientations can be derived from a single crystalline T d grain. Therefore, shifting opposite layers of the chalcogen atoms in a T d grain will generate an inversion domain boundary. The dark (Pathway 1) and light (Pathway 2) blue arrows indicate the sliding directions of each chalcogen layer.
Movie S2: DFTEM video acquired using the (1100) 2H spot at room temperature after each heat pulse from 200 to 700 • C, showing the anisotropic T d -to-2H phase transition at lower temperatures. The phase transition then become more isotropic at temperatures above 400 • C. We applied two rounds of heating, the first round is 200-400 • C, while the second round is 200-700 • C. The temperature intervals are all 25 • C and the heat pulses are all 0.5 s.
while this work focuses on the reverse T d -to-2H transition and its atomic-scale mechanisms. Understanding the micro-to atomic scale phase transition mechanisms between the semiconducting 2H and the topological T d phase may open up new possibilities towards low-dissipation 2D electronics and spintronics.
Figure
1b shows a schematic of the phase conversion process. To create an encapsulated cell, we fabricate hBN/graphene/MoTe 2 /graphene/hBN heterostructures using a PDMSassisted pick-up technique. 35 The MoTe 2 flakes are mechanically exfoliated with lateral size around tens of microns and 4-5 layers in thickness. The MoTe 2 is encapsulated by both monolayer graphene and 10 nm thick hBN on the top and bottom surfaces. The hBN layers improve adhesion with the polymer film used in the pick-up technique and are removed before (S)TEM analysis using XeF 2 etching 36 (see Supporting Information (SI) Section 1 and
Figure 4a
4ashows that the newly grown 2H region in the 4-layer MoTe 2 has multiple phase fronts (I, II, III, and IV ).Figure 4bschematically shows horizontally staggered 2H phase fronts and the resulting DFTEM intensities. In the kinematic limit, DFTEM intensities scale quadratically with the number of 2H layers, making it possible to individually probe the position of the 2H phase front at each layer. However, we are not able to determine the depth of each phase front because TEM produces images that are averaged in projection (along the direction of the electron beam path). InFigure 4c, we measure the 2H-T d interface positions of each layer as a function of accumulated heating time. The calculated propagation rates range from 0.07 to 0.4 nm/s at 225 • C and exhibit wide variability. For example, both interfaces II (orange) and III (green) exhibit a sudden jump in 2H phase front position at t = 300 sec after the fifteenth 250 • C pulse is applied. The non-uniform propagation rates might be due to strain, defects, and differing surface energies between atomic layers (2H, T d , and graphene encapsulation), which can locally alter the energy barrier of MoTe 2 phase transitions.51 Next, we demonstrate that the laser-induced T d phase can be transformed back to the 2H phase via ex situ vacuum annealing (Figure 5and SI Section 4). We characterize the pristine, laser-irradiated, and annealed MoTe 2 with Raman spectroscopy(Figure 5a-d and SI Section 5). Pristine (as-stacked) MoTe 2(Figure 5a,b) exhibits the three characteristic Raman peaks (E 2g , A 1g , and B 2g ) of the 2H phase, while laser-irradiated regions (red areas inFig 5c)exhibit the A 1 and A 2 modes of the T d phase. The Raman maps(Figure 5b,c)
Figure 1 :
1Characterization and fabrication of different phases of MoTe 2 . (a) Atomic structure models of 2H-, 1T'-, and T d -MoTe 2 with top and side views. Monolayer models are made for top view for clarity. (b) Schematic of the reversible phase transition of MoTe 2 . The 2H-MoTe 2 flakes are encapsulated by graphene and hBN layers. Local laser-irradiation induces 2H-to-T d phase transition of MoTe 2 , while the T d phase reverts back to 2H phase after thermal annealing. (c,d) Aberration-corrected ADF-STEM images for the 2H and T d phases, respectively.
Figure 2 :
2Phase and grain orientation mapping of laser-irradiated MoTe 2 with DFTEM. (a) BFTEM image of the suspended, graphene-encapsulated MoTe 2 containing both 2H and T d grains. The T d phase region is delineated by the laser trajectory and outlined by the black dashed lines. The minimum width of the T d region is determined by the radial laser intensity profile. (b,c) SAED patterns, (d,e) DFTEM images of 2H and T d phase, respectively. The diffraction patterns (b,c) are acquired with zone axis perpendicular to the basal planes. The weaker diffraction spots are generated by the graphene encapsulating layers. The DFTEM images (d-e) are formed by selecting the (1100) 2H and (210) T d Bragg reflections in (b) and (c) with the objective aperture position marked with blue and orange circles. The objective aperture and selected Bragg reflections are centered on the optical axis to reduce image aberrations. (f) Overlay of false-colored DFTEM images of the 2H matrix (red) and three different orientations of T d grains (green, yellow, and blue). The DFTEM images (d-f) are acquired at the region marked by the yellow dashed square in (a).
Figure 3 :
3Anisotropic, low-temperature T d -to-2H phase transition. (a-d) DFTEM images formed from the (1100) 2H spot are acquired at room temperature after heat pulses of 0.5 s from 200 to 275 • C. The T d -to-2H transition initiates at the interface, and the 2H phase front anisotropically propagates into the T d phase region. (e) Overlay of a low magnification DFTEM image with newly grown 2H regions marked in red. The T d -to-2H phase transition occurs primarily at 2H-T d interfaces. (f) Contour plot of the 2H phase front in the same region as (a-d) shows propagation along the b-axis of nearby T d grain. The insets are the atomic models of 2 different types of interface. The anisotropy arises from the different interface energy of type 1 and 2 interfaces. ADF-STEM images of (g) type 1 and (h) type 2 2H-T d interfaces with atomic kinks indicates a step-flow growth model.
Figure 4 :
4Layer-by-layer phase transition and growth kinetics measurement. (a) DFTEM image that shows the layer-by-layer phase transition. The 2H-T d interface of different layers are individually identified by their intensity difference. (b) Schematic of the intensity differences of 4-layer MoTe 2 2H-T d interfaces in DFTEM. The mono-, bi-, tri-, and quad-layer 2H phase fronts are labeled as I, II, III, and IV respectively. Note that the relative positions (in the z-direction) of each phase front are unknown due to the projection nature of TEM. (c) Plot of 2H phase front positions of different layers as a function of accumulated heating time. We perform a series of short heat pulses to capture the phase transition. Each dot corresponds to a heat pulse. The pulsing time ranges from 0.5 s to 1 min and can be read from the horizontal spacing between the dots. The pulsing temperatures (200-275 • C) are color-coded by the background shades. The propagation rates are extracted by the slope of the curves, which have a strong temperature dependence.
Figure 5 :*
5Cyclic phase transition and recovery of MoTe 2 (2H − → T d − → 2H phase) via laser irradiation and annealing. (a) Raman spectra and (b-d) Raman mapping of pristine (asstacked), laser-irradiated, and annealed MoTe 2 . The Raman maps are visualized with the E 2g (2H) and A 1 (T d ) peak, respectively. (e) SAED pattern of the 2H phase region. The orange and green circles denote the objective aperture positions that are used to generate DFTEM images (f,g) from different (1100) 2H Bragg reflections. The brighter region corresponds to the specific 2H orientation that generates the stronger Bragg reflection. The inversion domain boundary is outlined by the white dashed line. (h) ADF-STEM image at the inversion domain boundary of 2H grains with opposite orientations. The atomic models are overlaid with arrows indicating opposing orientations. I ASSOCIATED CONTENT Supporting Information Sample fabrication workflow, 1T' and T d mixture analysis, simulated ADF-STEM images, ADF-STEM images and atomic models of inversion domain boundaries, and in situ movies of MoTe 2 phase transition. Email: [email protected] *Email: [email protected]
by P.Y.H., C.-H.L., G.N. and Y.Z. acquired and analyzed the in situ heating DFTEM and ADF-STEM images. Under supervision by G.-H.L., H.R. fabricated the MoTe 2 samples, performed ex situ annealing experiments and Raman spectroscopy. Under supervision by K.K., Y.L. perform TEM analysis of phase-engineered MoTe 2 . Under supervision by H.C., S.O. conducted polarized Raman measurements. K.W. and T.T. synthesized the hBN flakes. All authors read and contributed to the manuscript. Notes The authors declare no competing financial interest. Acknowledgement This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award number DE-SC0020190, which supported the electron microscopy and related data analysis. This work was carried out in part in the Materials Research Laboratory Central Facilities at the University of Illinois at Urbana-Champaign. G.-H.L. acknowledges support by the Creative-Pioneering Researchers Program through Seoul National University (SNU), the National Research Foundation (NRF) of Korea (NRF-2021R1A2C3014316, SRC program: vdWMRC center 2017R1A5A1014862, NRF-2021M3F3A2A01037858), the Research Institute of Advanced Materials (RIAM), Institute of Engineering Research, and Institute of Applied Physics at SNU, which supported the sample fabrication and ex situ characterization. K.W. and T.T. acknowledge support from the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Numbers 19H05790 and 20H00354) and A3 Foresight by JSPS, which supported the h-BN synthesis. Di, Z. Atomistic Observation of the Local Phase Transition in MoTe 2 for Application in Homojunction Photodetectors. Small 2022, 18, 2200913. (19) Eshete, Y. A.; Ling, N.; Kim, S.; Kim, D.; Hwang, G.; Cho, S.; Yang, H. Vertical Heterophase for Electrical, Electrochemical, and Mechanical Manipulations of Layered MoTe 2 . Advanced Functional Materials 2019, 29, 1904504. (20) Cho, S.; Kim, S.; Kim, J. H.; Zhao, J.; Seok, J.; Keum, D. H.; Baik, J.; Choe, D.-H.; Chang, K. J.; Suenaga, K.; Kim, S. W.; Lee, Y. H.; Yang, H. Phase patterning for ohmic homojunction contact in MoTe 2 . Science 2015, 349, 625-628. (21) Tan, Y.; Luo, F.; Zhu, M.; Xu, X.; Ye, Y.; Li, B.; Wang, G.; Luo, W.; Zheng, X.; Wu, N.; Yu, Y.; Qin, S.; Zhang, X.-A. Controllable 2H-to-1T' phase transition in fewlayer MoTe 2 . Nanoscale 2018, 10, 19964-19971. (22) Bae, G. Y.; Kim, J.; Kim, J.; Lee, S.; Lee, E. MoTe 2 Field-Effect Transistors with Low Contact Resistance through Phase Tuning by Laser Irradiation. Nanomaterials 2021, 11, 2805. (23) Zhang, K.; Bao, C.; Gu, Q.; Ren, X.; Zhang, H.; Deng, K.; Wu, Y.; Li, Y.; Feng, J.; Zhou, S. Raman signatures of inversion symmetry breaking and structural phase transition in type-II Weyl semimetal MoTe 2 . Nature Communications 2016, 7, 13552. (24) Chen, S.-Y.; Goldstein, T.; Venkataraman, D.; Ramasubramaniam, A.; Yan, J. Activation of New Raman Modes by Inversion Symmetry Breaking in Type II Weyl Semimetal Candidate T'-MoTe 2 . Nano Letters 2016, 16, 5852-5860. (25) Rhodes, D. et al. Engineering the Structural and Electronic Phases of MoTe 2 through W Substitution. Nano Letters 2017, 17, 1616-1622. (26) Lv, Y.-Y. et al. Composition and temperature-dependent phase transition in miscible Mo 1−x W x Te 2 single crystals. Scientific Reports 2017, 7, 44587. ard, K. L.; Dean, C. R. One-Dimensional Electrical Contact to a Two-Dimensional Material. Science 2013, 342, 614-617. (36) Son, J.; Kwon, J.; Kim, S.; Lv, Y.; Yu, J.; Lee, J.-Y.; Ryu, H.; Watanabe, K.; Taniguchi, T.; Garrido-Menacho, R.; Mason, N.; Ertekin, E.; Huang, P. Y.; Lee, G.-H.; M. van der Zande, A. Atomically precise graphene etch stops for three dimensional integrated systems from two dimensional material heterostructures. Nature Communications 2018, 9, 3988. (37) Huang, P. Y.; Kurasch, S.; Srivastava, A.; Skakalova, V.; Kotakoski, J.; Krasheninnikov, A. V.; Hovden, R.; Mao, Q.; Meyer, J. C.; Smet, J.; Muller, D. A.; Kaiser, U. Direct Imaging of a Two-Dimensional Silica Glass on Graphene. Nano Letters 2012, 12, 1081-1086. (38) Algara-Siller, G.; Kurasch, S.; Sedighi, M.; Lehtinen, O.; Kaiser, U. The pristine atomic structure of MoS 2 monolayer protected from electron radiation damage by graphene. Applied Physics Letters 2013, 103, 203107. (39) Huang, F.-T.; Joon Lim, S.; Singh, S.; Kim, J.; Zhang, L.; Kim, J.-W.; Chu, M.-W.; Rabe, K. M.; Vanderbilt, D.; Cheong, S.-W. Polar and phase domain walls with conducting interfacial states in a Weyl semimetal MoTe 2 . Nature Communications 2019, 10, 4211. (40) Hart, J. L.; Bhatt, L.; Han, M.-g.; Hynek, D.; Schneeloch, J. A.; Tao, Y.; Louca, D.; Zhu, Y.; Kourkoutis, L. F.; Cha, J. J. Layer Stacking Determination in Topological Semimetal MoTe 2 via STEM Imaging, Liquid He TEM, and Quantitative Electron Diffraction. Microscopy and Microanalysis 2022, 28, 1746-1748. (41) Huang, P. Y.; Ruiz-Vargas, C. S.; van der Zande, A. M.; Whitney, W. S.; Levendorf, M. P.; Kevek, J. W.; Garg, S.; Alden, J. S.; Hustedt, C. J.; Zhu, Y.; Park, J.; Supporting Information In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe 2 Chia-Hao Lee, Huije Ryu, Gillian Nolan, Yichao Zhang, Yangjin Lee, Siwon Oh, Hyeonsik Cheong, Kenji Watanabe, Takashi Taniguchi, Kwanpyo Kim, Gwan-Hyoung Lee, * and Pinshane Y. Huang * * E-mail: [email protected] * E-mail: [email protected] 1. Sample preparation for in and ex situ experiments To fabricate the hBN/Gr/MoTe 2 /Gr/hBN heterostructures, we mechanically exfoliated thin layers of 2D materials (MoTe 2 , hBN, graphene) from bulk crystals (MoTe 2 : HQ graphene, graphene: NGS Naturgraphite GmbH, hBN: NIMS) onto SiO 2 /Si substrate. We then used the pick-up transfer technique 1 with a poly(bisphenol A carbonate, Sigma Aldrich) (PC)coated poly (dimethyl siloxane) (PDMS) lens mounted on a microscope slide to pick-up and released the constituent flakes on the substrate. The PC/PDMS/glass slide was held in a 3-axis micromanipulator to control the position of the contact area with the 2D materials.The substrate was placed on a heating stage. By controlling the temperature of the heating stage (80-130 • C), the 2D flakes were picked up by the PC with minimal cracking or folding, leaving the substrate on the heating stage. The hBN/Gr/MoTe 2 /Gr/hBN heterostructures were fabricated by repeating the above steps, and then transferred onto a clean SiO 2 /Si substrate by releasing the PC film from the PDMS lens at a temperature above 180 • C.
In situ TEM experiments were done in a Thermo Fisher Scientific Themis-Z aberrationcorrected S/TEM operated at 80 kV. For atomic-resolution ADF-STEM imaging, the point resolution was about 1Å with 25 mrad convergence semi angle, 35 pA probe current, 63
Figure S1 :
S1Sample preparation for in situ TEM experiments. (a-1) The schematic illustration of hBN/Gr/MoTe 2 /Gr/hBN structure on SiO 2 (285 nm)/Si++ substrate. (a-2,3) Pick-up the structure by polycarbonate (PC) film on polydimethylsiloxane (PDMS). (b-1,2) Etching the bottom side of the hBN by exposing to XeF 2 gas. (b-3) After exposing to XeF 2 gas, the bottom hBN is completely etched, while the graphene layers and the encapsulated layers remain. In addition, the PC exposed by XeF 2 is also chemically modified, which can not be dissolved by chloroform. (c-1,2) One-side-etched sample is transferred to another SiO 2 (285 nm)/Si++ substrate at about 180 • C and separated from the PDMS lens. The Si substrate was treated with O 2 plasma to increase the adhesion energy of SiO 2 . (c-3,4) Remove the PC film by chloroform bath. Note that the fluorinated PC was not dissolved in chloroform. (c-5,6) Etch off the fluorinated PC layer by O 2 plasma. Since the etch rate of hBN is much slower than fluorinated PC layer using O 2 plasma, the fluorinated PC layer is removed while the Gr/MoTe 2 /fluorinated Gr structure remains. (c-7,8) Remove the top hBN by XeF 2 gas etching. Finally, we transferred the heterostructures on MEMS TEM chips using the PMMA-assisted, wet-transfer method.
Figure S2 :
S2Mixture of 1T' and T d phase characterized by TEM diffraction and polarized Raman spectroscopy. (a) Selected area electron diffraction pattern (SAED) acquired at a region with mixed 1T' and T d phases. The red circles are Bragg peaks of T d phase that would be absent if it were 1T' phase, however, the intensities are too weak for the region to be pure T d phase, indicating a mixture of 1T' and T d phases. The Bragg peaks inside the blue circles are also characteristic peaks of T d phase that are absent in the 1T' phase. (b)
Figure S3 :
S3ADF-STEM image simulation of 1T'-and T d -MoTe 2 . (a-d) Simulated ADF-STEM images of 1T' and T d phase at different orientations using semi-quantitative image simulation package 6 (e-h) Atomic models of 1T' and T d phase at different orientations. The 3.9 • tilt angle is chosen to match the β angle of 1T' phase.
Figure S4 :
S4Crystallographic relation between the T d and the 2H matrix. (a,b) Atomic models of top-view, monolayer 2H and T d phases. (c,d) Simulated diffraction patterns of 2H and T d phase.
Figure S5 :
S5Anisotropic phase transition for all 3 T d orientations. (a) Overlay of falsecolored DFTEM images of the 2H matrix (red) and three different orientations of T d grains. Reproduced from Figure 2f of the main text. (b-d) DFTEM images of 2H-T d interfaces with different T d phase orientations. (e-g) DFTEM images of 2H-T d interfaces after heat pulses. The white arrows indicate the growth directions of the 2H phase front. The growth directions are parallel to the b-axis directions of the nearby T d grains. (h-j) Atomic models of 2H-T d interface with three different orientations. The b-axis directions of the T d phases are marked by the black arrows.
Figure S6 :
S6ADF-STEM images of an inversion domain boundary (IDB) at (a) lower and (b) higher magnifications.
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| {'fraction_non_alphanumeric': 0.06630684894423389, 'fraction_numerical': 0.034329317812669194, 'mean_word_length': 4.097455799913756, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Understanding the phase transition mechanisms in two-dimensional (2D) materials is a key to precisely tailor their properties at the nanoscale. Molybdenum ditelluride (MoTe 2 ) exhibits multiple phases at room temperature, making it a promising candidate for phase-change applications. Here, we fabricate lateral 2H-T d interfaces with laser irradiation and probe their phase transitions from micro-to atomic scales with in situ heating in the transmission electron microscope (TEM). By encapsulating the MoTe 2 with graphene protection layers, we create an in situ reaction cell compatible with atomic resolution imaging. We find that the T d -to-2H phase transition initiates at phase boundaries at low temperatures (200-225 • C) and propagates anisotropically along the b-axis in a layer-by-layer fashion. We also demonstrate a fully reversible 2H-T d -2H phase transition cycle, which generates a coherent 2H lattice containing inversion domain boundaries. Our results provide insights on fabricating 2D hetero-phase devices with atomically sharp and coherent interfaces.KeywordsIn situ heating, anisotropic phase transition, laser irradiation, molybdenum ditelluride, transmission electron microscopy, 2D materials Phase transformations in two-dimensional transition metal dichalcogenides (2D TMDCs) are an emerging research area due to their polymorphism-the ability to host different phases of the same chemical composition with distinct crystal structures. Utilizing phases with diverse electronic properties, multiple functionalities can be compactly packaged into nanoscale devices, such as monolithic 2D electronics 1-3 and phase-change memories. 4,5 Among the group VI 2D TMDCs, molybdenum ditelluride (MoTe 2 ) has been widely studied because of the minimal energy difference (40 meV per formula unit) 6-8 between the trigonal prismatic (2H), monoclinic (1T'), and orthorhombic (T d ) phases shown inFigure 1a. While the honeycomb lattice of the 2H structure is distinct, the 1T' and T d phase share the same", 'arxivid': '2301.02694', 'author': ['Chia-Hao Lee ', 'Huije Ryu ', 'Gillian Nolan ', 'Yichao Zhang ', 'Yangjin Lee ', 'Siwon Oh ', 'Hyeonsik Cheong ', 'Kenji Watanabe ', 'Takashi Taniguchi ', 'Kwanpyo Kim ', 'Gwan-Hyoung Lee ', '¶ ', 'Pinshane Y Huang ', '\n§Department of Physics\nSeoul National University\n08826SeoulKorea\n', '\nDepartment of Physics\nYonsei University\n03722SeoulKorea\n', '\n⊥Research Center for Functional Materials\nSogang University\n04107SeoulKorea\n', '\n#International Center for Materials Nanoarchitectonics\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan\n', '\n@Materials Research Laboratory\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan\n', '\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIllinoisUnited States\n'], 'authoraffiliation': ['§Department of Physics\nSeoul National University\n08826SeoulKorea', 'Department of Physics\nYonsei University\n03722SeoulKorea', '⊥Research Center for Functional Materials\nSogang University\n04107SeoulKorea', '#International Center for Materials Nanoarchitectonics\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan', '@Materials Research Laboratory\nNational Institute for Materials Science\n1-1 Namiki305-0044TsukubaJapan', 'University of Illinois at Urbana-Champaign\n61801UrbanaIllinoisUnited States'], 'corpusid': 250947991, 'doi': '10.1021/acs.nanolett.2c04550', 'github_urls': [], 'n_tokens_mistral': 19438, 'n_tokens_neox': 16147, 'n_words': 9556, 'pdfsha': 'e6c8a1087ed8eae0b6ee23455f5bf2249968bdea', 'pdfurls': ['https://export.arxiv.org/pdf/2301.02694v1.pdf'], 'title': ['In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe 2', 'In situ Imaging of an Anisotropic Layer-by-Layer Phase Transition in Few-Layer MoTe 2'], 'venue': []} |
arxiv |
Secure synchronization of artificial neural networks used to correct errors in quantum cryptography
Marcin Niemiec *[email protected]†[email protected]‡[email protected]
Tymoteusz Widlarz
Miralem Mehic
Department of Telecommunications
Faculty of Electrical Engineering
University of Sarajevo
Zmaja od Bosne bb71000Sarajevo
Bosnia and Herzegovina ‡ VSB -Technical University of Ostrava
17. listopadu 2172/15708 00OstravaCzechia
AGH University of Science and Technology
al. Mickiewicza 3030-059KrakowPoland
Secure synchronization of artificial neural networks used to correct errors in quantum cryptography
Index Terms-quantum cryptographykey reconciliationerror correctionartificial neural networks
Quantum cryptography can provide a very high level of data security. However, a big challenge of this technique is errors in quantum channels. Therefore, error correction methods must be applied in real implementations. An example is error correction based on artificial neural networks. This paper considers the practical aspects of this recently proposed method and analyzes elements which influence security and efficiency. The synchronization process based on mutual learning processes is analyzed in detail. The results allowed us to determine the impact of various parameters. Additionally, the paper describes the recommended number of iterations for different structures of artificial neural networks and various error rates. All this aims to support users in choosing a suitable configuration of neural networks used to correct errors in a secure and efficient way.
I. INTRODUCTION
The emergence and intensive development of the field of quantum computing has put many cryptography algorithms at risk. However, quantum physics also allows to achieve multiple cryptography tasks. One of the most popular is quantum key distribution [1]. Unfortunately, quantum communication is not perfect and additional solutions are required to correct any errors after the key distribution in the quantum channel. Artificial neural networks can be utilized to correct these errors [2]. It is a recently proposed solution which provides high level of security and efficiency comparing to other existing error correction methods. This paper analyzes the impact of different neural networks' parameters on the synchronization process. These parameters influence the number of iterations required as well as the security and efficiency of quantum cryptography. Therefore, it is important to know which neural network scheme should be chosen and which should be avoided. Additionally, the synchronization requires the number of iterations to be specified. Therefore, a recommended number of iterations for a particular multiple neural network's scheme is provided.
The paper is structured as follows. Related work is reviewed in Section 2. Section 3 presents the basics of quantum cryptography, the architecture of the tree parity machine, and error correction using this structure of artificial neural networks. Analysis of synchronization parameters including the recommended number of iterations for typical keys and error rates is described in Section 4. Section 5 concludes the paper.
II. RELATED WORK
The first quantum key distribution (QKD) protocol, introduced in 1984 by Bennet and Brassard, is BB84 [3]. This scheme uses the polarization state of a single photon to transmit information. Since then, several other protocols have been presented. One of them is the E91 protocol introduced in 1991 by Ekerd [4]. It utilizes entangled pairs of photons in the QKD process. However, some errors usually appear during data exchange in the quantum channel. After the initial QKD, there is a specific step: quantum bit error rate (QBER) estimation based on the acquired keys. The QBER value is usually low [5]. It must to be lower than the chosen threshold used to detect the eavesdropper.
Several methods of correcting error incurred in the quantum key distribution process have been developed. The first described method -BBBSS -was proposed in 1992 [6]. However, the most popular is the Cascade key reconciliation protocol [7]. It is based on multiple random permutations. The Winnow protocol, based on the exchange of parity and Hamming codes, is another method of error correction in the raw key [8]. Its main improvement is the reduction of the required communication between both parties. The third most popular error reconciliation scheme is the low density parity check approach. It offers a significant reduction of exchanged information; however, it introduces more computation and memory costs than the Cascade and Winnow protocols [7].
In 2019, another method of error correction in quantum cryptography was proposed by Niemiec in [2]. The solution uses mutual synchronization of two artificial neural networks (ANN) to correct the errors. The tree parity machine (TPM) is proposed as a neural network used in this approach. It is a well-known structure in cryptography -the synchronization of two TPMs can be used as a key exchange protocol. TPMs cannot be used as a general method to correct a selected error because it is not possible to predict the final string of bits after the synchronization process. However, it is a desirable feature for shared keys which should be random strings of bits.
III. QUANTUM CRYPTOGRAPHY SUPPORTED BY
ARTIFICIAL NEURAL NETWORKS Symmetric cryptography uses a single key to encrypt and decrypt secret messages. Let's assume that Alice and Bob, the two characters used in describing cryptography protocols, are using symmetric encryption. The goal is to send information from Alice to Bob in a way that provides confidentiality. To achieve this, Alice and Bob need to agree on a shared secret key. Alice encrypts confidential data using the previously chosen key and Bob decrypts it using the same key. The same key is applied to encrypt and decrypt the information, hence the name: symmetric-key encryption. It is worth mentioning only the one-time-pad symmetric scheme has been proven secure but it requires a key not smaller than the message being sent.
In general, symmetric-key encryption algorithms -for example the Advanced Encryption Standard (AES) [9] -perform better than asymmetric-key algorithms [10]. However, symmetric-key algorithms have an important disadvantage compared to asymmetric-key schemes. In the symmetric key encryption scheme, the key needs to be safely distributed or established between Alice and Bob [11]. The symmetric key can be exchanged in a number of ways, including via a trusted third party or by direct exchange between involved parties. However, both methods introduce some vulnerabilities, including passive scanning of network traffic. A method where the eavesdropper can be easily detected uses quantum mechanics to establish keys between Alice and Bob. It is called the quantum key distribution protocol.
A. Quantum key distribution
Quantum mechanics allows for secure key distribution 1 among network users. Two main principles are the core of the security of QKD: an unknown quantum state cannot be copied [12], and the quantum state cannot be estimated without disturbing it. One of the most popular QKD protocols which uses those principles is the BB84 scheme [3].
The BB84 protocol uses photons with two polarization bases: rectilinear or diagonal. Alice encodes a string of bits using photons on a randomly chosen basis. After that, all the photons are sent through a quantum channel. Bob randomly chooses a basis for each photon to decode the binary 0 or 1. Alice and Bob's bases are compared through a public communication channel. Each bit where both parties chose the same basis should be the same. However, when Bob measures the photon in a different basis than Alice, this bit is rejected. The remaining bits are the same for both parties and can be considered as a symmetric key. Next, the error estimation is performed. Randomly chosen parts of the keys between Alice and Bob are compared to compute the QBER value. If the comparison results in a high error rate, it means that the eavesdropper (Eve) is trying to gain information about the exchanged photons. However, the quantum channel is not perfect, and errors are usually detected due to disturbance, noise in the detectors or other elements. The number of errors introduced by the quantum channel's imperfections must be considered while deciding the maximum acceptable error rate.
The differences between Alice and Bob's keys need to be corrected. Several error correction methods are known. BBBSS is the earliest scheme proposed in [6]. It is mainly based on parity checks. The most popular method is the Cascade protocol [13]. It is an improved version of BBBSS and requires less information to be sent between Alice and Bob through the public channel. The Cascade protocol and its predecessor are based on multiple parity checks. The basic idea is that the keys are divided into blocks of a fixed size. The number of bits in each block depends on the previously calculated QBER value. Alice and Bob compare the parities of each block to allow them to find an odd number of errors. If errors are detected in a given block, it is split into two. The process is repeated recursively for each block until all errors are corrected. It concludes a single iteration after which Alice and Bob have keys with an even number of errors or without any errors. Before performing the following iterations, the keys are scrambled, and the size of the block is increased. The number of iterations is predetermined. As a result of this process, Alice and Bob should have the same keys. However, it is not always the case. A number of iterations or block sizes can be chosen incorrectly and cause failure in error correction. Additionally, the algorithm performs multiple parity checks over the public channel, which can be intercepted by an eavesdropper (Eve). As a result, Eve can construct a partial key. Alice and Bob should discard parts of their keys to increase the lost security. This reduces the performance of this method since the confidential keys must be shortened in the process. Another error reconciliation method is based on mutual synchronization of artificial neural networks.
B. Tree parity machine
An artificial neural network (ANN) is a computing system inspired by biological neural networks [14]. ANNs are used to recognize patterns and in many other solutions in the fields of machine learning. ANNs consist of multiple connected nodes (artificial neurons), with each neuron representing a mathematical function [15]. These nodes are divided into three types of layers: the first (input) layer, at least one hidden layer, and the output layer. The connections between neurons in each layer can be characterized by weights.
In cryptography, the most commonly used neural network is the tree parity machine (TPM) [16]. A scheme of this model is presented in Fig. 1. There are K ×N input neurons, divided into K groups. There is a single hidden layer with K nodes. Each of these nodes has N inputs. The TPM has a single output neuron. The connections between input neurons and hidden layer neurons are described by weights W -integers in the range [−L, L], thus L is the maximum and −L is the minimum weight value. The values of σ characterize the connections between the hidden layer neurons and an output neuron. The output value of the TPM is described by τ .
The value of σ is calculated using the following formulas:
σ k = sgn( N n=1 x kn * w kn )(1)sgn(z) = −1 z ≤ 0 1 z > 0(2)
Due to the usage of the presented signum function, σ can take two values: 1 or −1. The output value of TPM is calculated as:
τ = K k=1 σ k(3)
This neural network has two possible outcomes: 1 or −1.
For the TPM structure, multiple learning algorithms are proposed. Most popular are Hebbian, anti-Hebbian, and random walk. The leading is the Hebbian rule [17]. The Hebbian algorithm updates ANN weights in the following manner:
w * kn = v L (w kn + x kn * σ k * θ(σ k , τ ))(4)
where θ limits the impact of hidden layer neurons whose value was different than τ :
θ(σ k , τ ) = 0 if σ k = τ 1 if σ k = τ(5)
The v L function makes sure that the new weights are kept within the [−L, L] range:
v L (z) = −L if z ≤ −L z if − L < z < L L if z ≥ L(6)
The TPM structure allows for mutual learning of the two neural networks [18], primarily based on updating weights only when the outputs from both neural networks are the same. The input values are random and the same for both Alice and Bob's TPMs. Inputs are updated in each iteration. The security of this process relies on the fact that cooperating TPMs can achieve convergence significantly faster than Eve's machine, which can update weights less frequently. The TPM is most commonly used in cryptography to exchange a secret key. This usage is defined as neural cryptography [19]. Alice and Bob mutually synchronize their TPMs to achieve the same weights. After the synchronization process, these weights provide a secure symmetric key.
C. Error correction based on TPMs
TPMs can be utilized during the error correction process in quantum cryptography [2]. The neural network's task is to correct all errors to achieve the same string of confidential bits at both endpoints. Firstly, Alice and Bob prepare their TPMs. The number of neurons in the hidden layer (K) and the number of input neurons (N ) is determined by Alice and passed on to Bob. The value L must also be agreed between the users. The keys achieved using the QKD protocol are changed into integer values in the range [−L, L]. These values are used in the appropriate TPMs as weights between neurons in the input layer and the hidden layer. Since Alice's string of bits is similar to Bob's (QBER is usually not high), the weights in the created TPMs are almost synchronized. At this point, Alice and Bob have constructed TPMs with the same structure but with a few differences in the weight values.
After establishing the TPM structure and changing bits to weights, the synchronization process starts. It consists of multiple iterations, repeated until common weights are achieved between Alice and Bob. A single iteration starts from Alice choosing the input string and computing the result using the TPM. After that, the generated input string is passed on to Bob, who computes the output of his TPM using the received input. Then, the results are compared. If the outputs of both TPMs match, the weights can be updated. Otherwise, the process is repeated with a different input string.
After an appropriate number of iterations, the TPMs are synchronized and Alice and Bob can change the weights back into a string of bits. The resulting bits are the same. However, the privacy amplification process after error correction is still recommended [20]. The reduction of the key protecting Alice and Bob from information leakage is defined as [2]:
Z = log 2L+1 2 i (7)
where i is the number of TPM iterations. This usage of TPM is safer than the neural cryptography solution, because weights are similar before the synchronization. Therefore, significantly fewer iterations are required to achieve convergence than the randomly initialized weights in key establishing algorithms. It is worth mentioning this method of error correction is characterized by high efficiency, e.g. requires approximately 30% less iterations than Cascade algorithm [2].
IV. ANALYSIS OF THE SYNCHRONIZATION PROCESS
The crucial decision regarding the error detection approach based on TPMs is the number of iterations during the synchronization process. This value should be as low as possible for security reasons. However, it cannot be too low, since neural networks will not be able to correct all errors in the key otherwise. It is the user's responsibility to select the appropriate value for the error correction. The main objective of the analysis is to determine the impact of various neural network parameters on the synchronization process. Another goal is to provide a recommended number of iterations for users.
A. Testbed
The experiments require an application to simulate the error correction process based on artificial neural networks. The application for correcting errors arising in quantum key distribution was written in Python and uses the NumPy package -a library for scientific computing which provides fast operations on arrays required by the TPM. The functions provided by NumPy satisfy all necessary calculations to achieve neural network convergence. Synchronization of TPMs is performed over sockets to allow real-world usage of this tool. The Hebbian learning algorithm for updating weights is used.
The developed application makes it possible to correct errors in the keys using quantum key distribution protocols. The users are also able to correct simulated keys with the chosen error rate. It helps if users do not have strings of bits created by a real QKD system. An important feature of the tool is its ability to select neural network parameters. The user can personalize the synchronization process, starting from the key length and error rate. The least sufficient number of bits was used for translation into a single integer (values of the weights must be in the range [−L, L]). It was demonstrated that the number of hidden neurons and the number of inputs depend on the chosen key length and L value. Therefore, users need to select these parameters taking into account the requirements and needs.
During the experiments the minimum number of returned required iterations for a single TPM configuration was set to 200. The maximum number of iterations was limited to 1000. Additionally, the maximum number of retries in a single iteration was limited to 10 to speed up the simulation process. Finally, 1880 different scenarios were analyzed. All possible TPM configurations for key lengths varying between 100 and 700 with a 100 bit step are available. Moreover, the data is available for other keys with lengths varying between 128 and 352 with an 8 bit step. Between 350 and 500 synchronizations were performed for each TPM. It was assumed that this number of iterations is sufficient to achieve convergence.
B. Recommended number of iterations
To obtain the recommended number of iterations of TPMs for successful error correction, the sum of means and standard deviations of the results was calculated. The median and variance values were calculated as well for comparison. The full results are available online 2 . The selected part -the neural network configurations where the key length equals 256 bits with the recommended number of iterations -is presented in Tab. I. Fig. 2 shows the histogram of data gathered for a single neural network configuration. The distribution is rightskewed. The mean value is greater than the median. It is a common characteristic for other tested TPM configurations. If the distribution is not positively skewed, it is symmetrical. The recommended number of iterations for the presented configuration, according to Tab. I, equals 302. It is based on the sum of the mean and standard deviation values. For all presented TPM configurations, this sum gives an 84% chance of successful synchronization, assuming a normal distribution of results. For the right-skewed distribution, similar to the one presented in Fig. 2, the probability of success is higher. The 85-th percentile for the given set is equal to 276 -less than the proposed value. In this case, after choosing the suggested number of iterations the user has more than an 88% chance of success.
Knowing the lowest required number of iterations is important because it reduces the risk of a successful attack by Eve. The attacker could create independent TPMs and try to synchronize one of them with Alice or Bob's machine. The recommended number of iterations increases the security of this solution because Alice and Bob require far fewer iterations to synchronize, compared to Alice (or Bob) and Eve synchronizing using random weights.
C. Impact of TPM structures
The results of simulations allow us to analyze how TPM structures affect the number of required iterations during the synchronization process. Fig. 3 shows the number of required iterations depending on the K and N parameters. It shows two different TPM configurations: one with a 144 bit key and another with a 216 bit key. These configurations were chosen due to having a similar number of possible K and N pairs. For a given key length, L value and error rate there is a limited number of possible N and K values. The K value changes in inverse proportion to the N value. As presented in Fig. 3 the speed of the TPM synchronization process depends on the neural network structure (N and K values). The number of required iterations increases alongside the higher number of neurons in the hidden layer (K). The trend is similar for both presented TPMs. After achieving a certain threshold, the number of recommended iterations increases slowly. The results fit the logarithmic trend line. It means that after a certain K value, increasing this parameter further does not affect the synchronization speed as much as under a certain threshold. Other configurations of the selected TPMs were studied based on the increasing error rate of the keys. Two configurations with 128 and 256 bit keys were tested. The average of every possible configuration of the recommended number of iterations was calculated for different QBER values. The results are presented in Fig. 4. This confirms that a greater number of errors results in a higher average number of recommended iterations. It confirms the applicability of TPMs to correct errors emerging in quantum key distribution, where the error rate should not be higher than a few percent. Therefore, the eavesdropper needs more iterations to synchronize its TPM.
Additionally, it was verified that value L has an exponential impact on the average recommended number of iterations. The data was gathered using a similar approach to the study with the impact of QBER. The average recommended number of iterations of each configuration for a given L was calculated. Fig. 5 shows the exponential trend line. It is worth mentioning that the impact of L value on the synchronization time is significant. It is the user's responsibility to choose the best possible configuration for a given key length and QBER value. The analysis shows that the L value should be chosen carefully since it exponentially affects the required number of iterations. Additionally, the choice of the K value should be made with caution due to its logarithmic impact on the number of iterations.
V. SUMMARY
The analysis of the TPM synchronization process used for error correction purposes was presented in this paper. It shows that the parameters of the TPM structure have an impact on the synchronization time and security of this error correction method. However, different parameters of artificial neural networks have different effects. Therefore, users should be aware of how to choose the configuration of neural networks used to correct errors in a secure and efficient way. One of the deciding factors which need to be selected is the number of iterations. The paper describes the recommended number of iterations for different TPM structures and QBER values to assist users in this step. The numbers recommended by the authors are as low as possible but with a high probability of successful synchronization to ensure secure and efficient error correction based on artificial neural networks.
Fig. 2 .
2Histogram for number of iterations (TPM with a 256 bit key, N = 16, K = 4, L = 4, QBER = 3%).
2
Recommended numbers of iterations for 1880 different scenarios -TPM structures and QBER values -are available from: http://kt.agh.edu.pl/ ∼ niemiec/ICC-2023 This is mainly based on possible key lengths which vary between 128 and 500 bits with 4 bit steps. Additionally, keys with lengths between 500 and 700 with 100 bit steps are included.
Fig. 3 .
3Number of iterations for TPMs with 144 and 216 bit keys for different K value.
Fig. 4 .
4Number of iterations for TPMs with 128 and 256 bit keys depended on the QBER.
Fig. 5 .
5Number of iterations for TPMs with 128 and 256 bit keys dependent on the L value.
Fig. 1. Model of tree parity machine.X 11
X 12
X 13
X 1N
X 21
X 22
X 23
X 2N
X K1
X K2
X K3
X KN
∑
∏
∑
∑
W 11
W 1N
W 21
W 2N
W KN = {-L, … ,L}
W K1
σ K = {-1, 1}
σ 2
σ 1
τ={-1, 1}
In fact, a key is not distributed but negotiated. However, the term 'distribution' is consistently used in this paper to be consistent with the commonly accepted name of the technique.
ACKNOWLEDGMENT This work was supported by the ECHO project which has received funding from the European Union's Horizon 2020 research and innovation programme under the grant agreement no. 830943.
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| {'fraction_non_alphanumeric': 0.03512963740273159, 'fraction_numerical': 0.016789900229246715, 'mean_word_length': 4.584565452578435, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Quantum cryptography can provide a very high level of data security. However, a big challenge of this technique is errors in quantum channels. Therefore, error correction methods must be applied in real implementations. An example is error correction based on artificial neural networks. This paper considers the practical aspects of this recently proposed method and analyzes elements which influence security and efficiency. The synchronization process based on mutual learning processes is analyzed in detail. The results allowed us to determine the impact of various parameters. Additionally, the paper describes the recommended number of iterations for different structures of artificial neural networks and various error rates. All this aims to support users in choosing a suitable configuration of neural networks used to correct errors in a secure and efficient way.', 'arxivid': '2301.11440', 'author': ['Marcin Niemiec *[email protected]†[email protected]‡[email protected] ', 'Tymoteusz Widlarz ', 'Miralem Mehic \nDepartment of Telecommunications\nFaculty of Electrical Engineering\nUniversity of Sarajevo\nZmaja od Bosne bb71000Sarajevo\n\nBosnia and Herzegovina ‡ VSB -Technical University of Ostrava\n17. listopadu 2172/15708 00OstravaCzechia\n', '\nAGH University of Science and Technology\nal. Mickiewicza 3030-059KrakowPoland\n'], 'authoraffiliation': ['Department of Telecommunications\nFaculty of Electrical Engineering\nUniversity of Sarajevo\nZmaja od Bosne bb71000Sarajevo', 'Bosnia and Herzegovina ‡ VSB -Technical University of Ostrava\n17. listopadu 2172/15708 00OstravaCzechia', 'AGH University of Science and Technology\nal. Mickiewicza 3030-059KrakowPoland'], 'corpusid': 256358720, 'doi': '10.48550/arxiv.2301.11440', 'github_urls': [], 'n_tokens_mistral': 7670, 'n_tokens_neox': 6795, 'n_words': 4886, 'pdfsha': '9329f66c5cb1778c07b6afdc7daa98be199c4897', 'pdfurls': ['https://export.arxiv.org/pdf/2301.11440v1.pdf'], 'title': ['Secure synchronization of artificial neural networks used to correct errors in quantum cryptography', 'Secure synchronization of artificial neural networks used to correct errors in quantum cryptography'], 'venue': []} |
arxiv |
Nonlinear Compton scattering and nonlinear Breit-Wheeler pair production including the damping of particle states
20 Sep 2022
T Podszus *[email protected]†[email protected]‡[email protected]
Max Planck Institute for Nuclear Physics
Saupfercheckweg 1D-69117HeidelbergGermany
V Dinu
Department of Physics
University of Bucharest
P.O. Box MG-11077125MȃgureleRomania
A Di Piazza
Max Planck Institute for Nuclear Physics
Saupfercheckweg 1D-69117HeidelbergGermany
Nonlinear Compton scattering and nonlinear Breit-Wheeler pair production including the damping of particle states
20 Sep 2022(Dated: September 21, 2022) 1
In the presence of an electromagnetic background plane-wave field, electron, positron, and photon states are not stable, because electrons and positrons emit photons and photons decay into electronpositron pairs. This decay of the particle states leads to an exponential damping term in the probabilities of single nonlinear Compton scattering and nonlinear Breit-Wheeler pair production.In this paper we investigate analytically and numerically the probabilities of nonlinear Compton scattering and nonlinear Breit-Wheeler pair production including the particle states' decay. For this we first compute spin-and polarization-resolved expressions of the probabilities, provide some of their asymptotic behaviors and show that the results of the total probabilities are independent of the spin and polarization bases. Then, we present several plots of the total and differential probabilities for different pulse lengths and for different spin and polarization quantum numbers.We observe that it is crucial to take into account the damping of the states in order for the probabilities to stay always below unity and we show that the damping factors also scale with the intensity and pulse duration of the background field. In the case of nonlinear Compton scattering we show numerically that the total probability behaves like a Poissonian distribution in the regime where the photon recoil is negligible. In all considered cases, the kinematic conditions are such that the final particles momenta transverse to the propagation direction of the plane wave are always much smaller than the particles longitudinal momenta and the main spread of the momentum distribution on the transverse plane is along the direction of the plane-wave electric field.
I. INTRODUCTION
With the ongoing progress in laser technology toward higher intensities, lasers represent a promising tool to test QED in a regime where quantum effects induced by the laser field play an important role. In QED the vacuum is pictorially described as being populated by fluctuations of virtual electron-positron pairs which can be polarized by a sufficiently large electromagnetic field. The so called "critical" field of QED F cr = m 2 /|e| = 1.3×10 16 V/cm = 4.4 × 10 13 G (here m and e < 0 are the electron mass and charge, respectively, and we use units where 0 = = c = 1) [1][2][3] determines the typical field scale of QED. In the presence of an electric field of the order of F cr the vacuum becomes unstable under electron-positron pair production and the interaction energy of a Bohr magneton with a magnetic field of strength F cr is of the order of m. The field strength F cr corresponds to a critical laser intensity I cr ∼ 10 29 W/cm 2 , which is far from being reached by available lasers. Indeed, today's record for the laser peak intensity I 0 is about 1.1 × 10 23 W/cm 2 [4], and even upcoming laser facilities are aiming for intensities of the order of I 0 ∼ 10 23 −10 24 W/cm 2 [5][6][7][8][9]. However, due to the Lorentz-invariance of QED, physical observables like transition probabilities depend on the electromagnetic field only via Lorentz-and gauge-invariant parameters, such that the interesting regime where field-induced quantum effects dominate the dynamics can be efficiently entered experimentally already with today's technology. For an electron (photon) of four-momentum p µ = ( , p) (q µ = (ω, q)) moving in a background field, represented by the field tensor F µν 0 = (E 0 , B 0 ) in the laboratory frame, the probability of a physical process depends on the so-called quantum nonlinearity parameter χ 0 = |(F µν 0 p ν ) 2 |/mF cr (κ 0 = |(F µν 0 q ν ) 2 |/mF cr ), with the metric tensor η µν = diag(+1, −1, −1, −1) [10][11][12][13][14][15][16][17]. In case of an electron or positron this parameter corresponds to the field strength that the electron or positron experiences in its rest frame in units of the critical field.
First experiments probing laser-electron interactions in the regime χ 0 1 were performed in the late 90s at SLAC [18][19][20] and recently two experiments have been carried out close to the χ 0 ∼ 1 regime by using an all-optical setup where the electron beam was generated via laser wake-field acceleration [21,22]. Further experiments for testing the strong-field regime of QED with intense lasers are planned at DESY [23] and at SLAC [24].
Already at intensities lower than I cr classical nonlinear effects due to the interaction with the background field become significant and complicate the theoretical description of the electron and positron dynamics. These nonlinear effects are controlled by the so-called classical nonlinearity parameter ξ 0 = |e|E 0 /mω 0 , where E 0 is the electric field amplitude and ω 0 is the central angular frequency of the laser pulse. For ξ 0 1 the energy that an electron or a positron gains by the acceleration in the background field in one laser wavelength is comparable to its rest energy and the interaction of the charge with the background field cannot be treated perturbatively anymore [10][11][12][13][14][15][16][17]. Indeed, for optical lasers the parameter ξ 0 exceeds unity already at intensities of the order of 10 18 W/cm 2 and the interaction with the background field has to be treated exactly in the calculations in this case. This problem is commonly solved by working in the so-called Furry picture [25]. Here the background field is taken into account in the quantization procedure of the fermion field [1,2], such that the corresponding Dirac equation includes the interaction with the background field. An analytical solution of the Dirac equation can be found in the case of a plane-wave background field [26] (see also Ref. [1]) and the corresponding states are known as Volkov states.
Two elementary processes which have been thoroughly investigated by employing Volkov states are the emission of a photon by an electron or positron (nonlinear Compton scattering) and the decay of a photon into an electron-positron pair (nonlinear Breit-Wheeler pair production), which, among others, were studied in Refs. [11, (nonlinear Compton scattering) and Refs. [13,28,52,[54][55][56][57][58][59][60][61][62][63][64][65] (nonlinear Breit-Wheeler pair production).
Considering an electron (photon) moving in a plane wave laser pulse, it turns out that the leading-order in the fine-structure constant α = e 2 /4π ≈ 1/137 total probability of nonlinear Compton scattering (nonlinear Breit-Wheeler pair production) exceeds unity for a sufficiently long total phase duration Φ L of the laser pulse (or for a sufficiently large laser intensity). Probabilities larger than unity are of course unphysical and in contradiction with the unitarity of the S-matrix. Instead, for nonlinear Compton scattering it can be interpreted as the average number of photons emitted by the electron in the classical limit rather than as a probability [66]. In general, from the QED perspective the reason behind this apparent contradiction is that higher-order loop corrections and processes become significant. This is intuitively clear for nonlinear Compton scattering since the emission of several photons by an electron due to nonlinear multiple Compton scattering processes becomes sizable with an increasing pulse length [67][68][69][70][71][72]. A first investigation to taking into account these effects was carried out in Ref. [73] for χ 0 1. The probability for an arbitrary number of consecutive incoherent photon emissions by an electron was calculated, taking into account the recoil at each photon emission. Each probability was then "renormalized" by imposing that the total probability of emitting either no photons or an arbitrary number of photons was unity.
The renormalization ensures that the final probability of nonlinear (multiphoton) Compton scattering stays below unity even for large phase lengths of the laser pulse. These results where confirmed in Refs. [74,75] by means of a kinetic approach, which also included the effects of nonlinear Breit-Wheeler pair production [75]. In Refs. [74,75] also inclusive quantities like average momenta are computed and an approach to obtain the momentum expectation values of an electron including multiple photon emissions and loops has been put forward in Ref. [76].
In Ref. [77] the probability of an electron emitting an arbitrary number of photons was derived via the following considerations. The probability of an electron emitting N photons was calculated by first combining the probability of the electron to emit N − 1 photons until a certain time t, with the probabilities to emit one photon between t and t + dt, with dt being an infinitesimal positive time interval, and with the probability of emitting no photons from time t + dt on, and finally by integrating the result over all possible times t (a similar method was used in Ref. [78] to compute the pair-production yield as an observable to diagnose the intensity of the laser beam producing the pairs). Additionally, the photon recoil was taken into account. The obtained recursive equation for nonlinear multiphoton Compton scattering contains an exponential damping term describing the "decay" of the electron state by emitting a photon. This damping term depends on time and on the energy of the electron, and ensures that the total probability of emitting either no or an arbitrary number of photons is unity. The results in the regime χ 0 1 were in agreement with those in Ref. [73].
In Ref. [79] the probability of nonlinear Compton scattering and of nonlinear Breit-Wheeler pair production was computed from first principles. The derived expressions are equivalent to the resummation of all one-particle reducible diagrams containing an arbitrary number of corrections to the electron and photon states by the one-loop mass and polarization operators, respectively (see Fig. 1 for the case of nonlinear Compton scattering).
This was achieved by calculating the S-matrix with the "exact" electron and photon states, which are the solutions of the Schwinger-Dyson equation. This work and Refs. [73][74][75][76][77] were framed within the so-called locally-constant-field approximation (LCFA). In the LCFA probabilities of QED processes reduce to the corresponding probabilities in a constant crossed field averaged over the phase-dependent plane-wave profile. This is reasonable in the limit of low-frequency plane waves with fixed electric-field amplitude, since here the formation length of QED processes is much smaller than the typical wavelength of the plane wave [11].
The assumption is valid if ξ 0 1 at χ 0 , κ 0 ∼ 1, which was assumed throughout the derivation (we did not consider the problems which occur for the LCFA at low photon energies in the case of nonlinear Compton scattering [50,53] Breit-Wheeler pair production comprise an exponential damping term, describing the "decay" of the electron states by emitting a photon and the decay of the photon state into an electron-positron pair. This decay of the particle states turned out to become significant
if αξ 0 Φ L 1.
It is interesting to notice that the result for the probability of nonlinear Compton scattering is structurally similar to the single photon emission probability derived by the probabilistic approach in Ref. [77]. However, the probability in Ref. [79] additionally includes spin and polarization effects as well as the decay of the photon state into an electron-positron pair.
In Ref. [72] an approach has been developed to investigate higher-order QED processes also beyond the LCFA but for sufficiently long plane-wave pulses that the dynamics is dominated by the so-called cascade channel. In the cascade channel the higher-order process occurs as a sequence of the elementary building blocks represented by nonlinear Compton scattering and nonlinear Breit-Wheeler pair production.
The aim of the present work is to present new analytical insights and numerical examples on the probabilities of nonlinear Compton scattering and nonlinear Breit-Wheeler pair production including the particle states decay derived in Ref. [79]. The paper is organized as follows. First, we give a detailed analytical evaluation of the differential and the total probabilities, together with the asymptotic behavior for one final particle getting all or none of the initial light-cone energy. Further, we prove that the results of the total probabilities are independent of the spin and polarization basis that we use in the calculations. Then, we pass to the numerical evaluation of the results. We present plots of the differential and the total probabilities at two different pulse lengths for both nonlinear Compton scattering and nonlinear Breit-Wheeler pair production. In the case of nonlinear Compton scattering we compare the results with a Poissonian distribution. For the differential probability we show different plots corresponding to different combinations of spin and polarization of the initial and final particles. Finally, the main conclusions of the paper are presented. An appendix contains the explicit computation of the traces of Dirac matrices, which are not essential for the understanding of the main results of the paper.
II. ANALYTICAL CALCULATIONS
A. Notation
As indicated in the introduction, we investigate the probability of nonlinear Compton scattering and nonlinear Breit-Wheeler pair production including the particle states decay, which was presented in Ref. [79] and for better comparison we employ the same notation here.
We consider a plane-wave background field with central photon four-momentum k µ 0 = ω 0 n µ , where we introduced the quantity n µ = (1, n), with the three-dimensional unit vector n pointing along the direction of propagation of the background field. Additionally, we introduce the quantityñ µ = (1, −n)/2 such that (nñ) = 1. Since the background field is a plane wave, its four-potential A µ (φ) = (A 0 (φ), A(φ)) only depends on the light-cone time φ = (nx) = t − n · x. Further, it is a solution of the free wave equation ∂ µ ∂ µ A ν = 0, as we assume that it fulfills the Lorentz gauge condition ∂ µ A µ (φ) = 0. By additionally fixing the gauge such that A 0 (φ) = 0, the vector potential is perpendicular to the direction of propagation of the plane wave, i.e., n · A(φ) = 0, if A(φ) → 0 at φ → ±∞, which we also assume in the following.
By introducing the two four-vectors a µ j = (0, a j ) with j = 1, 2, which obey the relations (na j ) = −2(ña j ) = −n · a j = 0 and (a j a j ) = −a j · a j = −δ jj , with j, j = 1, 2, the vector potential of the plane wave can be expressed as A(φ) = ψ 1 (φ)a 1 + ψ 2 (φ)a 2 . Here, ψ j (φ) denotes the jth pulse shape function and it vanishes for φ → ±∞. In the following we will only consider the case of a linearly polarized plane wave, such that we choose without loss of generality ψ 2 (φ) = 0 and ψ 1 (φ) = A 0 ψ(φ), with A 0 < 0 being related to the amplitude of the electric field of the plane wave. The functions ψ(φ) and ψ (φ) are assumed to be such that |ψ(φ)|, |ψ (φ)| 1. Here and in the following a prime at a function denotes the derivative of the function with respect to its argument. The electromagnetic field tensor is
then given by F µν (φ) = n µ A ν (φ) − n ν A µ (φ) = A µν 0 ψ (φ),
where we have introduced the notation A µν 0 = A 0 (n µ a ν 1 − n ν a µ 1 ). The dual of the field tensor isF µν (φ) =à µν 0 ψ (φ), wherẽ A µν 0 = (1/2)ε µνλρ A 0,λρ and we define the four-dimensional Levi-Civita tensor as ε 0123 = +1 (note that in the chosen reference frame it isà µν 0 = A 0 (n µ a ν 2 − n ν a µ 2 )). Since the fourpotential and the field tensor always occur multiplied by the electron charge, we introduce the notation A µ (φ) = eA µ (φ), A 0 = eA 0 , and F µν (φ) = eF µν (φ).
The quantities n µ ,ñ µ , and a µ j fulfill the relation η µν = n µñν +ñ µ n ν − a µ 1 a ν 1 − a µ 2 a ν 2 . It is useful to employ light-cone coordinates and for an arbitrary four-vector v µ = (v 0 , v) the components in light-cone coordinates are v − = (nv) = v 0 − n · v, v + = (ñv) = (v 0 + n · v)/2, and v ⊥ = (v ⊥,1 , v ⊥,2 ) = −((va 1 ), (va 2 )) = (v · a 1 , v · a 2 ). Further we introduce the notation v = γ µ v µ , where γ µ are the Dirac-matrices, and we define γ 5 = iγ 0 γ 1 γ 2 γ 3 .
B. Nonlinear Compton Scattering
In the case of nonlinear Compton scattering, we assume the incoming (outgoing) electron to have four-momentum p µ = (ε, p) (p µ = (ε , p )), with energy ε = m 2 + p 2 (ε = m 2 + p 2 ), and an asymptotic spin quantum number s = ±1 (s = ±1). The spin quantization axis of the incoming (outgoing) electron is chosen along the four-vector
ζ µ = −Ã µν 0 p ν /(p − A 0 ) (ζ µ = −Ã µν 0 p ν /(p − A 0 ))
, which corresponds to the three-dimensional spin vector ζ (ζ ) pointing in the same direction of the magnetic field in the case of a constant crossed field and in the rest frame of the incoming (outgoing) electron. For the outgoing photon the four-momentum is q µ = (ω, q), with energy ω = |q| and we define its two transverse polarization states, identified by the index j = 1, 2, along the four-vector Λ µ 1 (q) = A µν 0 q ν /(q − A 0 ) and the pseudo-four-vector Λ µ 2 (q) =Ã µν 0 q ν /(q − A 0 ), which fulfill the relation (Λ j (q)Λ j (q)) = −δ jj with j, j = 1, 2.
With these definitions the probability of nonlinear Compton scattering including the damping of particle states within the LCFA, which was derived in Ref. [79], is given by the
expression P (e − →e − γ) j,s,s = d 3 q 16π 2 α p − p − ω dφ + e 2Im m p − φ + −∞ dϕMs(p,ϕ)+ ∞ φ + dϕ m p − M s (p ,ϕ)+ m q − P j (q,ϕ) × dφ − e i m 2 2p − q − p − [1+π 2 ⊥,e (φ + )]φ − + E 2 (φ + ) m 2 φ 3 − 12 T j,s,s ,(1)
where we introduced the trace
T j,s,s = 1 4 tr 1 −n [Â(φ + ) +Â (φ + )φ − /2] 2p − Λ j (q) 1 +n [Â(φ + ) +Â (φ + )φ − /2] 2p − ×(p + m)(1 + sγ 5ζ ) 1 −n [Â(φ + ) −Â (φ + )φ − /2] 2p − Λ j (q) × 1 +n [Â(φ + ) −Â (φ + )φ − /2] 2p − (p + m)(1 + s γ 5ζ ) ,(2)
the transverse momentum
π ⊥,e (φ) = p ⊥ m − p − q − q ⊥ m − A ⊥ (φ) m ,(3)
and the plane-wave electric field (times the electron charge) E(φ) = −A (φ). Due to energymomentum conservation the minus component and the perpendicular component of the outgoing electron momentum are fixed to p − = p − − q − and p ⊥ = p ⊥ − q ⊥ , respectively.
As we explained in the introduction, the probability comprises a damping term due to the particle states decay, which is the first exponential function in Eq. (1). The exponent contains the imaginary parts of the mass and polarization operator, more precisely, the one-loop mass operator within the LCFA, given by the expression [80][81][82]
M s (p, φ) = αm 2π ∞ 0 du ∞ 0 dv (1 + v) 3 e −iu 1+ 1 3 χ 2 p (φ) v 2 u 2 5 + 7v + 5v 2 3 χ 2 p (φ) v 2 u − isχ p (φ) ,(4)
for s = ±1, with the φ-dependent quantum nonlinearity parameter defined as χ p (φ) = −(p − /m)A 0 ψ (φ)/F cr , and the transverse part of the one-loop polarization operator within the LCFA, given by [83][84][85]
P j (q, φ) = α 48π mjκ 2 q (φ) ∞ 0 du u 1 0 dve −iu 1+ (1−v 2 ) 2 48 κ 2 q (φ)u 2 (1 − v 2 )[3 − (−1) j v 2 ],(5)for j = 1, 2, with κ q (φ) = −(q − /m)A 0 ψ (φ)/F cr .
According to the optical theorem, the quantity −(2m/p − )Im[M s (p, φ)] is equal to the total probability per unit φ that an electron of four-momentum p and spin quantum number s emits a photon [80] and the quantity −(2m/q − )Im[P j (q, φ)] is equal to the total probability per unit φ that a photon of four-momentum q and polarization quantum number j decays into an electron-positron pair [84]. Hence, this exponential damping can be understood as the electron and photon states not being stable in the background field but decaying, where the electron "decays" by emitting a photon and the photon decays into an electron-positron pair.
Further, we notice that the damping exponential depends on the spin of the incoming and outgoing electrons and on the polarization of the outgoing photon. This prevents one from employing the commonly used spin and polarization sum rules when solving the trace in Eq. (2), such that the spin-and polarization-resolved traces have to be calculated. A complete analytical derivation can be found in the appendix. Here, we only present the main steps of the calculations. With our choice of the spin and polarization basis, the trace for the two polarization states j = 1, 2 reduces in a linearly-polarized field to
T 1,s,s = (1 + ss ) (pp ) − m 2 − 1 2 q − p − q − p − − q − p 1 − p − q − q 1 2 − 2 + 1 2 q − p − q − p − − q − A 2 0 ψ 2 (φ + ) φ 2 − 4 + 2 + 1 2 q − p − q − p − − q − p 1 − p − q − q 1 + A 0 ψ(φ + ) 2 − i(s + s ) m 2 A 0 ψ (φ + )φ − q − p − 2 + q − p − − q − − ss q − p − q − p − − q − p 2 − p − q − q 2 2(6)
and
T 2,s,s = (1 − ss ) (pp ) − m 2 − 1 2 q − p − q − p − − q − p 1 − p − q − q 1 2 − 1 2 q − p − q − p − − q − A 2 0 ψ 2 (φ + ) φ 2 − 4 + 1 2 q − p − q − p − − q − p 1 − p − q − q 1 + A 0 ψ(φ + ) 2 + (1 + ss )2 p 2 − p − q − q 2 2 + ss q − p − q − p − − q − p 2 − p − q − q 2 2 + i(s − s ) m 2 A 0 ψ (φ + )φ − q − p − q − p − − q − .(7)
We observe that the traces depend on the pulse shape function ψ(φ + ). However, it should be possible to express the dependence on ψ(φ + ) in a manifestly gauge-invariant way. In order to achieve this, we consider the transverse momentum π ⊥,e (φ + ) defined in Eq. (3).
After some simplifications all the dependence on ψ(φ + ) and q 1 turns out to be in the electron quasi-momentum π ⊥,e (φ + ), which can be removed by performing the integral in φ − and using the properties of the Airy functions. Indeed, terms proportional to the derivative in φ − of the second exponential function in Eq. (1), i.e., terms proportional to
1 + π 2 ⊥,e (φ + ) + (A 2 0 ψ 2 (φ + )/m 2 )(φ 2 − /4)
, vanish when performing the integral over φ − . By performing the appropriate substitutions, the traces ultimately depend only on the derivative ψ (φ + ) of the pulse shape function and the probability is therefore manifestly gaugeinvariant. Ignoring the corresponding vanishing terms, the traces can be equivalently written as
T 1,s,s = −2(1 + ss )m 2 − (1 + ss ) 4 + q − p − q − p − − q − A 2 0 ψ 2 (φ + ) φ 2 − 4 − i(s + s ) m 2 A 0 ψ (φ + )φ − q − p − 2 + q − p − − q − − 2 + 2ss + ss q − p − q − p − − q − p 2 − p − q − q 2 2(8)
and
T 2,s,s = −(1 − ss ) q − p − q − p − − q − A 2 0 ψ 2 (φ + ) φ 2 − 4 + i(s − s ) m 2 A 0 ψ (φ + )φ − q − p − q − p − − q − + 2 + 2ss + ss q − p − q − p − − q − p 2 − p − q − q 2 2 .(9)
We transform the integral in the photon momentum into light-cone coordinates using the
relation d 3 q = (ω/q − )dq − d 2 q ⊥ and introduce the notatioñ T j,s,s = − 1 4π 2 m 2 p − q − p − dφ − d 2 q ⊥ e i m 2 q − 2p − p − [1+π 2 ⊥,e (φ + )]φ − + E 2 (φ + ) m 2 φ 3 − 12 T j,s,s .(10)
Now, the integral in the perpendicular photon momentum q ⊥ can be computed analytically by using the two basic integrals [86]
d 2 q ⊥ e i m 2 q − 2p − p − π 2 ⊥,e (φ + )φ − = 2πi q − p − p − (φ − + i0) ,(11)d 2 q ⊥ p 2 − p − q − q 2 2 e i m 2 q − 2p − p − π 2 ⊥,e (φ + )φ − = −2π p 2 − (φ − + i0) 2 .(12)
For the integral in φ − we use the integral representation Ai(z) =
∞ −∞ (dφ/2π) exp(izφ + iφ 3 /3) of the Airy function [86]. With the substitutionsφ = q − E 2 (φ + )/(8p − p − ) 1/3 φ − and z = q − /(p − χ p (φ + )) 2/3 , the integral in φ − can be taken.
Hence, the probability of nonlinear Compton scattering including the damping of particle states is finally given by
P (e − →e − γ) j,s,s = − αm 2 4p 2 − p − 0 dq − dφ +Tj,s,s × e 2Im m p − φ + −∞ dϕMs(p,ϕ)+ ∞ φ + dϕ m p − M s (p ,ϕ)+ m q − P j (q,ϕ) ,(13)whereT 1,s,s = 1 + ss 1 − q 2 − 2p − (p − − q − ) Ai 1 (z) + 3 + q 2 − p − (p − − q − ) + ss 3 + q 2 − 2p − (p − − q − ) Ai (z) z + (s + s ) 2 q − p − + q 2 − p − (p − − q − ) Ai(z) √ z sgn(ψ (φ + ))(14)
andT 2,s,s = 1 + ss 1 +
q 2 − 2p − (p − − q − ) Ai 1 (z) + 1 + q 2 − p − (p − − q − ) + ss 1 − q 2 − 2p − (p − − q − ) Ai (z) z + (s − s) q 2 − p − (p − − q − ) Ai(z) √ z sgn(ψ (φ + )),(15)
with Ai 1 (z) = ∞ z dxAi(x) and with sgn(ψ (φ + )) denoting the sign of ψ (φ + ). Note that without the exponential damping term the results reduce to the expressions of the spin-and polarization-resolved probabilities of nonlinear Compton scattering, which can be found in Ref. [87]. This observation can be used to prove analytically that the probability
P (e − →e − γ) s = j,s P (e − →e − γ) j,s,s
is always smaller than unity. In fact, since the damping exponentials are smaller or equal to unity, it is
P (e − →e − γ) s < − αm 2 4p 2 − j,s p − 0 dq − dφ +Tj,s,s e 2Im m p − φ + −∞ dϕMs(p,ϕ) = dφ + ∂P NC s,p ∂φ + e − φ + −∞ dϕ ∂P NC s,p ∂ϕ = − dφ + ∂ ∂φ + e − φ + −∞ dϕ ∂P NC s,p ∂ϕ = 1 − e − ∞ −∞ dϕ ∂P NC s,p ∂ϕ < 1,(16)
where ∂P NC s,p /∂φ indicated the probability without damping of nonlinear Compton scattering per unit of light-cone time φ.
At this point we also want to investigate the asymptotic behavior of the differential probability for the two cases q − p − and p − − q − p − . In the first case, the photon recoil is negligible, whereas in the second case, almost all light-cone energy of the incoming electron goes into the photon. The differential probability is obtained from Eq. (13) and it is given by
∂P (e − →e − γ) j,s,s ∂q − = − αm 2 4p 2 − dφ + e D NC j,s,s T j,s,s ,(17)
where we have renamed the exponent of the exponential damping function as
D NC j,s,s = 2Im m p − φ + −∞ dϕM s (p, ϕ) + ∞ φ + dϕ m p − M s (p , ϕ) + m q − P j (q, ϕ) .(18)
As already mentioned, according to the optical theorem, the probabilities of nonlinear Compton scattering and nonlinear Breit-Wheeler pair production are related to the imaginary part of the mass and polarization operator, respectively. Hence, the exponent of the damping term, D NC j,s,s , is equal to minus the sum of the probability of nonlinear Compton scattering between −∞ and φ + for the incoming electron with light-cone energy p − and spin quantum number s, and of the probabilities of nonlinear Compton scattering of the outgoing electron with light-cone energy p − and spin quantum number s and of nonlinear Breit-Wheeler pair production of the outgoing photon with light-cone energy q − and polarization quantum number j, both between φ + and +∞ [80,84], i.e.,
D NC j,s,s = − φ + −∞ dϕ ∂P NC s,p ∂ϕ − ∞ φ + dϕ ∂P NC s ,p ∂ϕ + ∂P NBW j,q ∂ϕ .(19)
1. Asymptotic expression for q − p −
We first analyze the asymptotic expression of the differential probability in the asymptotic region q − p − . Also, we assume that the quantum nonlinearity parameter χ p (ϕ) of the electron is fixed, such that the absolute value of the quantum nonlinearity parameter κ q (ϕ) = (q − /p − )χ p (ϕ) of the photon is much smaller than unity (if |χ p (ϕ)| is larger than unity, the ratio q − /p − is assumed to be sufficiently small that |κ q (ϕ)| 1). Thus, we can use in the damping term the corresponding asymptotic expression for the probability of nonlinear Breit-Wheeler pair production: [11] ∂P NBW
j,q ∂ϕ κq(ϕ) 1 ≈ 3 2 αm 2 |κ q (ϕ)|j 8q − e − 8 3|κq (ϕ)| .(20)
Since it is exponentially suppressed in the limit κ q (ϕ) → 0, we neglect it below. Furthermore, due to the conservation of the minus component of the four-momentum, if q − p − then p − ≈ p − , and the damping function reduces to
D NC j,s,s q − p − ≈ 2m p − Im φ + −∞ dϕM s (p, ϕ) + ∞ φ + dϕM s (p, ϕ) .(21)
Now, we will shortly see that for q − p − the spin-dependent terms of the probability of nonlinear Compton scattering can be neglected [29], such that the damping exponent effectively reduces to a constant, which is equal to minus the probability of an electron of momentum p − emitting a photon between phase −∞ and +∞ averaged over the electron spin.
In the functionsT j,s,s we can expand the Airy functions for z =
q − p − χp(φ + ) 2/3 ≈ q − p − χp(φ + ) 2/3
1 (we assume that χ p (φ + ) is fixed and that the ratio q − /p − is much smaller than 1/|χ p (ϕ)| if |χ p (ϕ)| < 1) and we obtaiñ
T 1,s,s z 1 ≈ − 2 × 3 2/3 Γ 1 3 z ,T 1,s,−s z 1 ≈ − q 2 − 2p 2 − 1 3 1/3 Γ 1 3 z ,(22)T 2,s,s z 1 ≈ − 2 3 1/3 Γ 1 3 z ,T 2,s,−s z 1 ≈ − q 2 − 2p 2 − 3 2/3 Γ 1 3 z .(23)
As expected in the classical limit where the photon recoil is small, we see that the probability of spin flip is substantially suppressed as compared to the case s = s . Further, the damping function in Eq. (21) in the dominant case s = s is independent of the integration variables and corresponds in the classical limit to the mean number of photons emitted by an electron.
Hence, within this limit the total probability of emitting a single photon is in agreement with that obtained from the Poissonian distribution.
Asymptotic expression for
p − − q − p −
Now, we evaluate the asymptotic expression of the differential probability in Eqs. (17) and (18) in the region p − = p − − q − p − . In this case, since we again assume χ p (ϕ) to be fixed, the absolute value of the quantum nonlinearity parameter
χ p (ϕ) = (p − /p − )χ p (ϕ)
of the outgoing electron is assumed to be smaller than unity. We use the corresponding asymptotic expression for the probability of nonlinear Compton scattering in Eq. (19), which is independent of p and given by [11] ∂P NC
s ,p ∂ϕ χ p (ϕ) 1 ≈ 5 2 √ 3 αm 2 |χ p (ϕ)| p − .(24)
The damping function is then
D NC j,s,s p − p − ≈ 2 Im m p − φ + −∞ dϕM s (p, ϕ) + m p − ∞ φ + dϕP j (p, ϕ) − ∞ φ + dϕ 5 √ 3 αm 2 |χ p (ϕ)| p − .(25)
In the functionsT j,s,s we assume that the ratio p − /p − is sufficiently small that z =
q − p − χp(φ + ) 2/3 ≈ p − p − χp(φ + ) 2/3
1. We obtain for photon polarization j = 1 and identical spin quantum numbers (s = s )
T 1,s,s z 1 ≈ − 1 √ π z −3/4 e − 2 3 z 3/2 p − p − (1 − s sgn(ψ (φ + ))) + 2 p − p − s sgn(ψ (φ + )) − 1 96 √ π z −9/4 e − 2 3 z 3/2 (124 + 20s sgn(ψ (φ + ))) − 1 9216 √ π z −15/4 e − 2 3 z 3/2 p − p − (3938 − 770s sgn(ψ (φ + ))) .(26)
Note that here in the special case of s = sgn(ψ (φ + )) compensations occur and this is why higher-order terms have been reported in the expression above. However, by keeping in mind that the functionsT j,s,s are ultimately integrated over the light-cone time φ + to compute the emission probability, the function sgn(ψ (φ + )) takes both values +1 and −1 (we implicitly assume here that the plane wave describes an oscillating laser wave). Therefore, the scaling of the probability will be determined by the term inT 1,s,s scaling as z −3/4 /p − and we can approximateT 1,s,s
z 1 ≈ − 2 √ π z −3/4 e − 2 3 z 3/2 p − p − , for s = −sgn(ψ (φ + )).(27)
For opposite spin quantum numbers (s = −s ) the asymptotic expression is
T 1,s,−s z 1 ≈ − 1 4 √ π z −9/4 e − 2 3 z 3/2 p − p − .(28)
With photon polarization j = 2 we have for identical spin quantum numbers (s = s )
T 2,s,s z 1 ≈ − 1 4 √ π z −9/4 e − 2 3 z 3/2 p − p −(29)
and for opposite spin quantum numbers (s = −s )
T 2,s,−s z 1 ≈ − 1 √ π z −3/4 e − 2 3 z 3/2 p − p − [1 + s sgn(ψ (φ + ))] − 1 9216 √ π z −15/4 e − 2 3 z 3/2 p − p − [3938 + 770s sgn(ψ (φ + ))] .(30)
Analogously as above, the scaling of the emission probability will be determined by the approximated expressioñ
T 2,s,−s z 1 ≈ − 2 √ π z −3/4 e − 2 3 z 3/2 p − p − , for s = sgn(ψ (φ + )).(31)
C. Nonlinear Breit-Wheeler pair production Now, we pass to the probability of nonlinear Breit-Wheeler pair production including the particle states decay.
Here, the incoming photon has four-momentum q µ = (ω, q), with polarization quantum number j = 1, 2 and the outgoing positron (electron) has four-momentum p µ = (ε, p) (p µ = (ε , p )) and spin quantum number s = ±1 (s = ±1). The symbols of quantum numbers of the particles have been chosen in order to exploit the crossing symmetry between the amplitudes of nonlinear Compton scattering and of nonlinear Breit-Wheeler pair production.
The expression of the probability was computed in Ref. [79] and it is given by
P (γ→e − e + ) j,s,s = d 3 p 16π 2 α q − p − ε dφ + e 2Im m q − φ + −∞ dϕP j (q,ϕ)+ ∞ φ + dϕ m p − M s (p ,ϕ)+ m p − Ms(−p,ϕ) × dφ − e i m 2 2p − q − p − [1+π 2 ⊥,p (φ + )]φ − + E 2 (φ + ) m 2 φ 3 − 12 G j,s,s ,(32)
with the trace
G j,s,s = 1 4 tr 1 −n [Â(φ + ) +Â (φ + )φ − /2] 2p − Λ j (q) 1 −n [Â(φ + ) +Â (φ + )φ − /2] 2p − ×(p − m)(1 + sγ 5ζ ) 1 +n [Â(φ + ) −Â (φ + )φ − /2] 2p − Λ j (q) × 1 +n [Â(φ + ) −Â (φ + )φ − /2] 2p − (p + m)(1 + s γ 5ζ )(33)
and with
π ⊥,p (φ) = p ⊥ m − p − q − q ⊥ m + A ⊥ (φ) m .(34)
The trace can be simplified and the integrals in the perpendicular positron momentum and φ − can be taken analogously as in the case of nonlinear Compton scattering. An important difference is however that the relations for momentum conservation are instead p ⊥ = q ⊥ −p ⊥ and p − = q − − p − .
By performing the trace over the matrix contractions (see the appendix for the derivation), the expressions for the two polarization quantum numbers j = 1 and j = 2 become G 1,s,s = (1 + ss ) 1 2
q 2 − p − p − m 2 + m 2 π 2 ⊥,p (φ + ) − A 2 0 ψ 2 (φ + ) φ 2 − 4 − 2 m 2 π 2 ⊥,p (φ + ) − A 2 0 ψ 2 (φ + ) φ 2 − 4 + i(s + s ) m 2 A 0 ψ (φ + )φ − q − p − 2 − q − p − + 2 + 2ss − ss q 2 − p − p − p 2 − p − q − q 2 2(35)
and
G 2,s,s = (1 − ss ) 1 2 q 2 − p − p − m 2 + m 2 π 2 ⊥,p (φ + ) − A 2 0 ψ 2 (φ + ) φ 2 − 4 + i(s − s ) m 2 A 0 ψ (φ + )φ − q 2 − p − p − − 2 + 2ss − ss q 2 − p − p − p 2 − p − q − q 2 2 .(36)
Here we have already expressed the pulse shape function in terms of π ⊥,p (φ + ). The dependence on π ⊥,p (φ + ) can in turn be removed by using the fact that the integral over terms proportional to 1 + π 2 ⊥,p (φ + ) + (A 2 0 ψ 2 (φ + )/m 2 )(φ 2 − /4) vanish when performing the integral in φ − , due to the properties of the Airy functions. In this way, by adding and subtracting suitable terms, the traces can be written in a manifestly gauge-invariant way. Furthermore, we transform the integral in the positron momentum into light-cone coordinates by using that d 3 p = (ε/p − )dp − d 2 p ⊥ and we employ the notatioñ
G j,s,s = − 1 4π 2 m 2 q − p − p − dφ − d 2 p ⊥ e i m 2 q − 2p − p − [1+π 2 ⊥,p (φ + )]φ − + E 2 (φ + ) m 2 φ 3 − 12 G j,s,s .(37)
The integral over the transverse positron momentum p ⊥ is taken by employing the two Gaussian integrals [86]
d 2 p ⊥ e i m 2 q − 2p − p − π 2 ⊥,p (φ + )φ − = 2πi p − p − q − (φ − + i0) ,(38)d 2 p ⊥ p 2 − p − q − q 2 2 e i m 2 q − 2p − p − π 2 ⊥,p (φ + )φ − = −2π p − p − q − (φ − + i0) 2 ,(39)
and the integral in φ − results again in Airy functions. In this way, the probability of nonlinear Breit-Wheeler pair production including the damping of particle states finally reads
P (γ→e − e + ) j,s,s = − αm 2 4q 2 − q − 0 dp − dφ +Gj,s,s × e 2Im m q − φ + −∞ dϕP j (q,ϕ)+ ∞ φ + dϕ m p − M s (p ,ϕ)+ m p − Ms(−p,ϕ) ,(40)withG 1,s,s = −(1 + ss ) − ss q 2 − 2p − p − Ai 1 (z) + −3(1 + ss ) + 1 + ss 2 q 2 − p − p − Ai (z) z − (s + s ) q − p − − q − p − Ai(z) √ z sgn(ψ (φ + ))(41)
andG 2,s,s = −(1 + ss ) + ss
q 2 − 2p − p − Ai 1 (z) + −(1 + ss ) + 1 − ss 2 q 2 − p − p − Ai (z) z + (s − s) q 2 − p − p − Ai(z) √ z sgn(ψ (φ + )).(42)
Without the exponential damping term the above probability reduces to the result of the spin-and polarization-resolved probability of nonlinear Breit-Wheeler pair production calculated in Ref. [87]. Analogously as in the case of nonlinear Compton scattering, it can be proved analytically that the probability in Eq. (40) is always smaller than unity [see the discussion below Eq. (15)]. Now, we investigate the asymptotic behavior in the two regions q − − p − q − and p − q − and for the differential probability of nonlinear Breit-Wheeler pair production
∂P (γ→e − e + ) j,s,s ∂p − = − αm 2 4q 2 − dφ + e D NBW j,s,s G j,s,s ,(43)
with D NBW j,s,s = 2Im
m q − φ + −∞ dϕP j (q, ϕ) + ∞ φ + dϕ m p − M s (p , ϕ) + m p − M s (−p, ϕ) ,(44)= − φ + −∞ dϕ ∂P NBW j,q ∂ϕ − ∞ φ + dϕ ∂P NC s ,p ∂ϕ + ∂P NC s,p ∂ϕ .(45)
1. Asymptotic expression for q − − p − q − First, we consider the asymptotic region p − = q − − p − q − . Thus, by assuming that the quantum nonlinearity parameter κ q (ϕ) of the photon is fixed, the absolute value of the quantum nonlinearity parameter of the electron χ p (ϕ) = (p − /q − )κ q (ϕ) is much smaller than unity (if κ q (ϕ) is larger than unity, the ratio p − /q − is assumed to be sufficiently small that |χ p (ϕ)| 1) and we use for the damping term in Eq. (45) the corresponding asymptotic expression for the probability of nonlinear Compton scattering, which is independent of p and given in Eq. (24). The exponent of the damping term becomes (we use the fact that
χ p (ϕ) = (p − /q − )κ q (ϕ)) D NBW j,s,s p − q − ≈ 2m q − Im φ + −∞ dϕP j (q, ϕ) + ∞ φ + dϕM s (−q, ϕ) − ∞ φ + dϕ 5 √ 3 αm 2 |κ q (ϕ)| q − .(46)
Concerning the functionG j,s,s , we can expand the Airy functions for z =
q − p − χp(φ + ) 2/3 ≈ q − p − κq(φ + ) 2/3
1. In the case of photon polarization j = 1 and identical spin quantum numbers (s = s ), we obtain the expressioñ
G 1,s,s z 1 ≈ − 1 √ π z −3/4 e − 2 3 z 3/2 q − p − (1 − s sgn(ψ (φ + ))) + 2 p − q − s sgn(ψ (φ + )) + 1 96 √ π z −9/4 e − 2 3 z 3/2 [124 + 20s sgn(ψ (φ + ))] − 1 9216 √ π z −15/4 e − 2 3 z 3/2 q − p − [3938 − 770s sgn(ψ (φ + ))] .(47)
As in the case of nonlinear Compton scattering, the scaling of the probability is determined by the case s = −sgn(ψ (φ + )), wherẽ
G 1,s,s z 1 ≈ − 2 √ π z −3/4 e − 2 3 z 3/2 q − p − .(48)
For opposite spin quantum numbers (s = −s ) we havẽ
G 1,s,−s z 1 ≈ − 1 4 √ π z −9/4 e − 2 3 z 3/2 q − p − .(49)
With photon polarization j = 2 the asymptotic expansion is for identical spin quantum numbers (s = s )G 2,s,s
z 1 ≈ − 1 4 √ π z −9/4 e − 2 3 z 3/2 q − p −(50)
and for opposite spin quantum numbers (s = −s )
G 2,s,−s z 1 ≈ − 1 √ π z −3/4 e − 2 3 z 3/2 q − p − [1 + s sgn(ψ (φ + ))] − 1 9216 √ π z −15/4 e − 2 3 z 3/2 q − p − [3938 + 770s sgn(ψ (φ + ))] .(51)
The scaling of the probability of nonlinear Breit-Wheeler pair production is determined by the case s = sgn(ψ (φ + )), i.e.,G 2,s,−s In the case of the spin state of an electron the two free, positive-energy spinors u 1 (p) and
z 1 ≈ − 2 √ π z −3/4 e − 2 3 z 3/2 q − p − .(52)
u −1 (p) form a spin basis [1]. They are normalized as u † s (p)u s (p) = 2εδ ss , with s, s = ±1 and fulfill the relation γ 5ζ u s (p) = su s (p) (for the positron the spin basis is formed by the two free negative-energy spinors v 1 (p) and v −1 (p), normalized as v † s (p)v s (p) = 2εδ ss ) [1]. A spinor corresponding to an arbitrary spin direction, indicated here as u + (p), can be expressed by a linear combination of the two basis spinors u 1 (p) and u −1 (p) as
u + (p) = β 1 u 1 (p) + β −1 u −1 (p),(53)
where β 1 , β −1 are two complex numbers such that |β 1 | 2 + |β −1 | 2 = 1. The two coefficients β 1 and β −1 are related to the polar angle θ and the azimuthal angle ϕ between the spin vector ζ and the new spin axis [1]. The spinor u + (p) in Eq. (53) forms a basis together with the
spinor u − (p) = β * −1 u 1 (p) − β * 1 u −1 (p),(54)
which is perpendicular to u + (p). Now, in Ref. [79] the probabilities were calculated by using the exact electron, positron and photon states, which were obtained by solving the corresponding Schwinger-Dyson equations. For the electron out-state Ψ
(out) e (x) the Schwinger-Dyson equation is {γ µ [i∂ µ −A µ (φ)]−m}Ψe,+ (x) =β 1 Ψ (out) e,1 (x) + β −1 Ψ (out) e,−1 (x), Ψ (out) e,− (x) =β * −1 Ψ (out) e,1 (x) − β * 1 Ψ (out) e,−1 (x).(55)
The physical meaning of the states with the above choice of β 1 and β −1 is that at x 0 → ∞ in the rest frame of the electron the spin axis points along the chosen axis and fulfill the abovementioned relations. Considering now the process of nonlinear Compton scattering, the S-matrix element for an incoming electron of spin quantum number s, an outgoing photon of polarization j, and an outgoing electron with spin quantum number b is proportional to
S (e − →e − γ) s,j,+ = β * 1 S (e − →e − γ) s,j,1 + β * −1 S (e − →e − γ) s,j,−1(56)
and
S (e − →e − γ) s,j,− = β −1 S (e − →e − γ) s,j,1 − β 1 S (e − →e − γ) s,j,−1 .(57)
By using these S-matrix elements, we then easily obtain the probability of nonlinear Compton scattering for the two spin quantum numbers + and − of the final electron as
P (e − →e − γ) s,j,+ = d 3 q (2π) 3 d 3 p (2π) 3 |S (e − →e − γ) s,j,+ | 2 = d 3 q (2π) 3 d 3 p (2π) 3 |β 1 | 2 |S (e − →e − γ) s,j,1 | 2 + |β −1 | 2 |S (e − →e − γ) s,j,−1 | 2 + β 1 β * −1 S (e − →e − γ) * s,j,1 S (e − →e − γ) s,j,−1 + β * 1 β −1 S (e − →e − γ) s,j,1 S (e − →e − γ) * s,j,−1(58)
and
P (e − →e − γ) s,j,− = d 3 q (2π) 3 d 3 p (2π) 3 |S (e − →e − γ) s,j,− | 2 = d 3 q (2π) 3 d 3 p (2π) 3 |β −1 | 2 |S (e − →e − γ) s,j,1 | 2 + |β 1 | 2 |S (e − →e − γ) s,j,−1 | 2 − β 1 β * −1 S (e − →e − γ) * s,j,1 S (e − →e − γ) s,j,−1 − β * 1 β −1 S (e − →e − γ) s,j,1 S (e − →e − γ) * s,j,−1 .(59)
Taking now the sum of the probabilities for both spin states b = {+, −}, i.e., taking the sum of Eqs. (58) and (59), we obtain
b={+,−} P (e − →e − γ) s,j,b = d 3 q (2π) 3 d 3 p (2π) 3 |S (e − →e − γ) s,j,+ | 2 + |S (e − →e − γ) s,j,− | 2 = d 3 q (2π) 3 d 3 p (2π) 3 |S (e − →e − γ) s,j,1 | 2 + |S (e − →e − γ) s,j,−1 | 2 = s ={1,−1} P (e − →e − γ) s,j,s ,(60)
which is identical to the analogous result obtained for the original spin axis. Hence, the probability summed over a spin quantum number is independent of the used quantization axis. It can easily be shown that the same holds for electron in-states and for the positron in-and out-states.
In order to describe the photon polarization states, we have chosen the two four-vectors Λ µ 1 (q) and Λ µ 2 (q). An arbitrary polarization basis is given by the two four-vectors
Λ µ + (q) = b 1 Λ µ 1 (q) + b 2 Λ µ 2 (q) Λ µ − (q) = b 2 Λ µ 1 (q) − b 1 Λ µ 2 (q)(61)
with b 1 , b 2 being two real numbers such that b 2 1 + b 2 2 = 1. Analogously to the case of the spin, one can show that also in this case the probability summed over the polarization indexes does not depend on the polarization basis.
III. NUMERICAL CALCULATIONS
A. Methods
The results in the following section have been produced using two types of integration: 1) a purely numerical one in order to obtain probability densities, fully differential with respect to an outgoing particle's momentum and 2) a quadrature which takes advantage of the fact that in lightfront coordinates the transverse momenta yield analytical Gaussian integrals, thus leaving just one longitudinal momentum integral to be done numerically in order to derive the total probability of the process.
We successfully compared the two methods against each other, by integrating the results of method 1 with respect to the transverse momenta and leaving aside the longitudinal integral in method 2. In fact, since the longitudinal momentum is practically proportional to the energy for the ultrarelativistic particles we consider, the latter is a good method of obtaining the energy spectrum. In both methods, before proceeding to perform the φ + integral in Eqs. (1) and (32) we expressed the integrals over φ − through Airy functions and stored them in an interpolation table in logarithmic form. Then, we also produced interpolation tables that store, for different spin and polarization numbers and longitudinal momentum values, the mass and polarization integrals in Eqs. (4) and (5), respectively.
These were then used in computing differential probabilities and total ones. All the numerical quadratures were performed using adaptive Gauss-Konrod rules.
An important step for the production of fully differential spectra was to divide the φ + integration domain into some well-chosen subintervals so that the adaptive integrator did not miss any relevant region. For large values of ξ 0 the integrand has some very narrow peaks apart from which it is almost vanishing and, unless the integrator is led to those peaks, it would stop short of finding them, yielding a negligible result for the whole integral. To find the peaks we first had to identify and store the intervals of monotonicity of the vector potential and then identify the possible peaks in each such interval using a nonlinear equation solver. As the positions of these sharp peaks change with the transverse momenta, a prior analytical (sharp) Gaussian integration with respect to the latter yielded a φ + integral that is easy to handle without any subdivision of intervals. For the Compton case the longitudinal momentum adaptive quadrature has been made faster through a deformation of its variable, to reduce the number of subdivisions needed to get a good precision at very small q − , while keeping in mind, though, that the behavior of the LCFA result is different from the exact one in this limit.
B. Results
In this section we present plots based on numerical implementations of the analytical results obtained above. We use the linearly-polarized (along the x direction) plane-wave laser pulse with Gaussian envelope described by the vector potential We present both probability densities, differential with respect to the momentum of one of the outgoing particles (which determines the momentum of the other outgoing particle through the conservation laws) and total probabilities. Concerning the probabilities integrated over the unconstrained momentum of one of the final particles, we observe the following. The meaning of the spin and polarization quantum numbers implicitly depends on the momenta of the particles, because the spin and polarization four-vectors depend on the particles' momenta. Thus, in general, once one integrates over the unconstrained momentum of one of the final particles, the physical meaning of the discrete quantum numbers of the final particles is unclear. However, we will always consider head-on collisions with the incoming particle having an energy much larger than mξ 0 , such that the angular spread of the produced particles is small. In this case, one can then conclude that the spin and polarization linearly-independent directions approximately correspond to the directions of the electric and magnetic field of the (linearly-polarized) background plane wave in the laboratory frame. Within this approximated framework one can then investigate, for instance, the occurrence of electron spin flip for nonlinear Compton scattering and the distinction between same-spin and different-spin states of the pair produced in nonlinear Breit-Wheeler process. It is interesting to discriminate between such cases, for their probabilities can be very different.
A(φ) = A 0 e −(φ/τ ) 2 sin(ω 0 φ)a 1 .(62)
It is convenient to introduce the notation P . For sufficiently small values of η 0 , such that χ 0 1, the photon recoil is negligible and we expect that the emissions are independent of each other. In Ref. [66] it was shown that in this classical limit the probability of emitting an arbitrary number of photons follows a Poissonian distribution. Indeed, we observe that at such low values of χ 0 the Poissonian distribution well approximates the full QED results [see also the discussion below Eq. (21)]. As χ 0 increases, important differences start to be seen.
Two pulse length parameters τ have been considered: τ = 5 fs (Fig. 2) and τ = 20 fs (Fig. 3). In the first case the probability increases as ξ 0 increases, reaches a maximum at ξ 0 smaller than about 10, and then decreases. In the classical regime of low values of χ 0 , the maximum value is e −1 , as predicted by the Poissonian P NC s,p e −P NC s,p . For the longer pulse used in Fig. 3, the whole plot is in the region where the probability decays with ξ 0 .
Indeed, the decay of the states is now stronger and essentially any increase of ξ 0 reduces the single-photon emission probability, as multiple photon emissions become favored over the single-photon emission. As mentioned in the introduction, the decay of the particle states becomes significant if the quantity αξ 0 Φ L is of the order of unity or larger. This corresponds to values of ξ 0 9.9 and ξ 0 2.5 for a pulse with τ = 5 fs (Fig. 2) and τ = 20 fs (Fig. 3),
respectively. Here, we estimated the total phase duration by the FWHM of the intensity,
i.e., we set Φ L = √ 2 ln 2 ω 0 τ . We see that for larger values of τ the damping effect becomes significant already at smaller values of ξ 0 .
For the probability of nonlinear Breit-Wheeler pair production we show plots similar to the aforementioned ones, in Figs. 4 (for a pulse duration corresponding to τ = 5 fs) and 5
(for a pulse duration corresponding to τ = 20 fs). A logarithmic scale is used for the energy of the incoming particle in both cases, but in the present case we started from higher values than for nonlinear Compton scattering, due to the exponential suppression of the process that occurs at low κ 0 = ρ 0 ξ 0 . In both cases the probability shows a maximum in ξ 0 (see also
Ref. [88]). This feature is observable in the plots only at high incoming photon energies for the shorter pulse, but also at lower incoming photon energies for the longer pulse, indicating that the size of the effect also depends on the damping of the states.
It can be seen from Fig. 6 that, as ρ 0 increases toward unity, the probability for (j = 1, s = s ) reduces to around 80% of the total nonlinear Breit-Wheeler pair production probability (implying that for (j = 1, s = −s ) it grows to around 20%), whereas the probability for (j = 2, s = s ) stays smaller than 3 . Analogous behaviors can be observed in nonlinear Compton scattering, according to the crossing symmetry existing between the two processes. In all cases we fix the energy of the incoming particle to 10 GeV (recall that the incoming particle is counterpropagating with respect to the plane wave) and we consider two laser and (j = 2, s = s ) for q 2 = 0 and p 2 = 0, respectively. Now, since the incoming particle is assumed to counterpropagate with respect to the plane wave for both processes, we have that Finally, for nonlinear Breit-Wheeler pair production the probability density always vanishes for the energy of any final particle approaching the energy of the incoming photon. In addition, for (j = 2, s = s ) it also vanishes when the final particles both have half of the total available energy (see Figs. 10 and 12).
IV. CONCLUSIONS
In conclusion, we presented analytical expressions and numerical evaluations of the probabilities for nonlinear Compton scattering and nonlinear Breit-Wheeler pair production including the particle states decay. The probabilities take into account that in a planewave background field the electron and photon states are not stable because electrons and positrons emit photons and because photons themselves decay into electron-positron pairs.
Within our approach based on the locally-constant field approximation, the decay of the states leads to an exponential damping term in the expressions of the probabilities, which depends on the plane-wave light-cone time as well as on the light-cone energies and discrete quantum numbers of the participating particles.
In the analytical part, we first calculated the spin-and polarization-resolved traces and took the integrals over the transverse momenta and phase differences. The final probabilities depend on the spin and polarization quantum numbers, on the incoming particle fourmomentum, as well as on the quantum nonlinearity parameter. Furthermore, we computed the asymptotic expressions for the probabilities differential in the light-cone momentum of one final particle for arbitrary spin and polarization quantum numbers in the limit of one of the outgoing particles gaining all or none of the light-cone energy of the incoming particle.
All calculations were carried out using particular spin and polarization four-vectors and we proved that the results for the total probabilities are independent of the chosen spin and polarization bases.
In the numerical part we presented plots of the total and differential probabilities for nonlinear Compton scattering and nonlinear Breit-Wheeler pair production by considering two different pulse lengths. Due to the particle states decay, the total probabilities stay below unity in all cases. The damping becomes important for αξ 0 Φ L 1 at values of the quantum nonlinearity parameter of the order of unity, such that it increases more rapidly with ξ 0 for larger values of the pulse phase length Φ L . For nonlinear Compton scattering we saw that the probability behaves like a Poissonian distribution for low values of η 0 such that photon recoil is negligible, whereas important differences arise when η 0 increases. For the differential probability we found it has its maximum at vanishing perpendicular momentum in the case of polarization j = 1 and same spins s = s , and in the case of polarization j = 2 and opposite spins s = −s , otherwise it vanishes at q y = 0. The same behavior is observed for nonlinear Breit-Wheeler pair production for the same spin and polarization Also, for (j = 2, s = s ) the differential probability vanishes if the electron and the positron have the same light-cone energy, i.e., half the light-cone energy of the incoming photon.
Finally, in all the cases we investigated numerically, the kinematic conditions were such that 1) the transverse momenta of produced particles are much smaller than the corresponding longitudinal momentum and 2) the main spread of the transverse momenta is along the direction of the plane-wave electric field.
where the coefficients are obtained by solving the following traces
c 1 = 1 4 Tr [1 4×4 Q p,s (φ + , φ − )] ,(A5)c 5 = 1 4 Tr γ 5 Q p,s (φ + , φ − ) ,(A6)c µ = 1 4 Tr [γ µ Q p,s (φ + , φ − )] ,(A7)c 5µ = 1 4 Tr iγ µ γ 5 Q p,s (φ + , φ − ) ,(A8)c µν = 1 8 Tr [σ µν Q p,s (φ + , φ − )] .(A9)
These traces can be calculated and we present them here for a linear polarized plane wave background field
c 1 = m − i s 4p − αβγδ ζ α F βγ ψ (φ + )φ − p δ ,(A10)c 5 = 0, (A11) c µ =p µ − A µ (φ + ) − 1 p − n µ p ⊥ · A ⊥ (φ + ) + 1 2p − n µ A 2 ⊥ (φ + ) − A 2 ⊥ (φ + ) φ 2 − 4 − i ms 4p − η µν αβδν ζ α F βδ ψ (φ + )φ − ,(A12)c 5µ =imsζ µ + 1 4p − η µν αβδν p α F βδ ψ (φ + )φ − ,(A13)
and
c µν = 1 2 − i m 2p − F µν ψ (φ + )φ − − s µνρσ p ρ ζ σ + s 2p − (η νρ µστ δ − η µρ νστ δ )p ρ ζ σ F τ δ ψ(φ + ) + s 2p − µντ ρ n τ ζ ρ A δ (φ + )A δ (φ + ) − A δ (φ + )A δ (φ + ) φ 2 − 4 .(A14)
The total trace of nonlinear Compton scattering, given in Eq. (A2), contains also the function Q p ,s (φ + , −φ − ) for which we introduce the notation Q p ,s (φ + , −φ − ) = c 1 1 4×4 + c 5 γ 5 + c τ γ τ + c 5τ iγ τ γ 5 + c τ λ σ τ λ ,
where the primed coefficients are obtained analogously to Eqs. (A5)-(A9). With this the trace for nonlinear Compton scattering can be reduced to the form T j,s,s = − c 1 c 1 + c 5 c 5 + (c µ c τ − c 5µ c 5τ )(2Λ µ j (q)Λ τ j (q) + η µτ ) − 2c µν c τ λ η νλ (4Λ µ j (q)Λ τ j (q) + η µτ ).
(A16) and
FIG. 1 .
1The amplitude of nonlinear Compton scattering computed with the exact electron and photon states (thick straight and wiggly lines, respectively) is equal to the resummation of all one-particle reducible diagrams with corrections to the one-loop mass operator (M) on the Volkovelectron states (thin double lines) and to the one-loop polarization operator (P) on the photon states (thin wiggly lines) (see Ref.[79]).
which we obtained from Eq.(40). According to the optical theorem, the exponent of the damping term is here equal to minus the sum of the total probability of nonlinear Breit-Wheeler pair production between −∞ and φ + for the incoming photon with light-cone energy q − and polarization quantum number j and the total probabilities of nonlinear Compton scattering between φ + and +∞ for the outgoing electron with light-cone energy p − and spin quantum number s and for the outgoing positron with light-cone energy p − and spin quantum number s[80,84], i.e.D NBWj,s,s
2 .
2Asymptotic expression for p − q − Due to the symmetry of the probability of pair production under the exchanges p − ↔ p − , s ↔ s , and ψ (φ + ) ↔ −ψ (φ + ) [see Eqs. (40)-(42)], the asymptotic expressions of the differential probability of nonlinear Breit-Wheeler pair production in the region p − q − can be easily obtained from those derived in the region p − q − via the corresponding substitution rules. D. Arbitrary spin and polarization basis So far, both here and in Ref. [79], the calculations were performed by employing a special direction for the electron or positron spin and for the photon polarization. This choice of the spin and polarization four-vectors has the advantage that the mass and polarization operator become diagonal. Consequently, the equations including the damping of the states can be solved in a relatively straightforward way. However, if the final probabilities are summed over a spin and/or a polarization quantum number, the results should be in the end independent of the choice of the corresponding spin and polarization basis, which we would like to prove explicitly below.
M
L (y, x) = γ 0 M † L (y, x)γ 0 and M L (y, x) being the mass operator in a plane wave. This equation is linear in the spin basis u 1 (p) and u −1 (p), such that we can decompose an arbitrary electron state in terms of the two states Ψ are solutions of the Schwinger-Dyson equation constructed via the spinors u 1 (p) and u −1 (p), respectively. The electron out-state Ψ(out) e,b (x) of an arbitrary spin direction b = {+, −}, given by the solution of the Schwinger-Dyson equation for the spinor u b (p), can be expressed then by the
All plots have been made for a carrier angular frequency ω 0 corresponding to 1.55 eV in our units, which also corresponds to a wavelength of 0.8 µm. The parameter τ describes the length of the pulse and we have chosen the two values τ = 5 fs and τ = 20 fs. Within the full width at half maximum (FWHM) of the intensity the pulse contains about 2.2 cycles for τ = 5 fs and 8.8 cycles for τ = 20 fs. Since our analytical results are valid within the LCFA, we consider below amplitudes of the vector potential corresponding to ξ 0 = 5 and we also restrict the parameters η 0 = χ 0 /ξ 0 = (k 0 p)/m 2 for nonlinear Compton scattering and ρ 0 = κ 0 /ξ 0 = (k 0 q)/m 2 for nonlinear Breit-Wheeler pair production, with k µ 0 = ω 0 n µ to values not exceeding (approximately) unity, by setting an upper bound of 100 GeV for the incoming particle's energy. For larger values of ξ 0 one can further relax the condition on η 0 and ρ 0 , allowing them to take even larger values.
− e + ) j,s,s for nonlinear Compton scattering and nonlinear Breit-Wheeler pair production, respectively. In Figs. 2 and 3 we show the total probability P (e − →e − γ) s of nonlinear Compton scattering (top panels) and the one-event value of the Poisson distribution corresponding to the average P NC s,p given by the first-order probability of nonlinear Compton scattering, computed by ignoring the decay of particles states, i.e. [see also the discussion below Eq. (15)], − dφ +Tj,s,s (bottom panels)
FIG. 2 .
2Nonlinear Compton scattering total probability P (e − →e − γ) s including the decay of the wave functions (top panel), as compared to the result obtained from a Poissonian distribution whose average photon number is the total "undamped emission probability" P NC s,p (bottom panel). The pulse length corresponds to τ = 5 fs and the initial electron spin corresponds to s = 1.
FIG. 3 .
3Same as inFig. 2, but for τ = 20 fs. Now, we present sections of the fully differential distribution of the photon momentum for the nonlinear Compton scattering effect(Figs. 7 and 8)and of the momentum of the positron produced in the nonlinear Breit-Wheeler pair production process(Figs. 9 -12).
intensities corresponding to ξ 0 = 10 and 50, which are equivalent to 2.1 × 10 20 W/cm 2 and 5.4 × 10 21 W/cm 2 , respectively. The sections are plotted as functions of q x = −q 1 and q y = −q 2 for nonlinear Compton scattering and of p x = −p 1 and p y = −p 2 for nonlinearFIG. 4. The probability P (γ→e − e + ) j of nonlinear Breit-Wheeler pair production in a short, τ = 5 fs pulse, by a photon with polarization quantum number j = 1 (left panel) and j = 2 (right panel). FIG. 5. Same as in Fig. 4, but for τ = 20 fs.Breit-Wheeler pair production, by keeping the light-cone energy of the emitted photon and of the produced positron fixed (which also means, with very good approximation, to keep fixed q 3 or q − and p 3 or p − , respectively). Indeed, the plots show that the outgoing particles are most probably produced with transverse momenta much smaller than |q 3 | and |p 3 |, respectively (see that the scale of the sections is in MeV units, instead of the GeV unitsFIG. 6. Fraction of the total probability of nonlinear Breit-Wheeler pair production corresponding to same-spin pairs, ( s P (γ→e − e + ) j,s,s )/P (γ→e − e + ) j for j = 1 (left panel) and for j = 2 (right panel). In both cases, the pulse duration corresponds to τ = 20 fs. used for the longitudinal momentum). As expected, the sections are asymmetric, spreading more along the direction of the electric field of the plane wave, i.e., the x direction. Both in the case of nonlinear Compton scattering and nonlinear Breit-Wheeler pair production, for some quantum number combinations the spectra vanish if the component of the momentum of the particle along the magnetic field of the wave, i.e., along the y direction, vanishes. This can be explained analytically looking at Eqs. (8) and (9) for nonlinear Compton scattering and at Eqs. (35) and (36) for nonlinear Breit-Wheeler pair production. These equations show that for both processes the probabilities vanish in the cases (j = 1, s = −s )
p 2
2= 0 in nonlinear Compton scattering and q 2 = 0 in nonlinear Breit-Wheeler pair production. Thus, if in addition q 2 = 0 in nonlinear Compton scattering and p 2 = 0 in nonlinear Breit-Wheeler pair production, the corresponding probability vanishes for the mentioned spin and polarization combinations. Furthermore, we see in Figs. 7 -12 that the maximum of each section is found at the origin for the complementary combinations (j = 1, s = s ) and (j = 2, s = −s ). For the examples of nonlinear Breit-Wheeler pair production processes of Figs. 9 -12 the aforementioned cases (j = 1, s = s ) and (j = 2, s = −s ) also have an alto-for τ = 20 fs, ε = 10 GeV, ξ 0 = 10, and incoming electron spin quantum number s = 1. The color levels indicate percentages of the maximum reached by the distribution in that section. The approximate value of that maximum is shown at the upper right corner of each subplot. gether larger probability than the complementary ones (j = 1, s = −s ) and (j = 2, s = s ), as it can be recognized also noticing the different scales of the panels. For nonlinear Compton scattering we only show results for the initial spin state defined FIG. 8. Sections through the nonlinear Compton scattering probability distribution in GeV −3 , for τ = 20 fs, ε = 10 GeV, ξ 0 = 50, and incoming electron spin quantum number s = 1. The color levels indicate percentages of the maximum reached by the distribution in that section. The approximate value of that maximum is shown at the upper right corner of each subplot.by s = 1 (see Figs. 7 and 8) as the s = −1 case gives very similar results, provided one changes the sign of s , too. The different scales in the transverse photon momenta for each individual plot show a general tendency of broadening of the probability distribution on the transverse plane for increasing energies of the emitted photon.
FIG. 9 .
9Sections through the nonlinear Breit-Wheeler pair production probability distribution in GeV −3 , for τ = 20 fs, ω = 10 GeV, ξ 0 = 10, and incoming photon polarization state j = 1. The horizontal axes correspond to p x [MeV] and the vertical ones correspond to p y [MeV].
FIG. 11 .
11Sections through the nonlinear Breit-Wheeler pair production probability distribution in GeV −3 , for τ = 20 fs, ω = 10 GeV, ξ 0 = 50, and incoming photon polarization state j = 1. The horizontal axes correspond to p x [MeV] and the vertical ones correspond to p y [MeV]. combination and for p y = 0, i.e., for the component of the positron momentum along the magnetic field of the wave. For nonlinear Breit-Wheeler pair production we have also seen that the differential probability vanishes for all spin and polarization combinations if the light-cone energy of the incoming photon all goes to either the electron or the positron.
). It can be shown that the solution of the Schwinger-Dyson equation intrinsically includes the resummation of all corrections by the one-loop mass and polarization operator for the exact electron or positron and photon state, respectively. The final probabilities for nonlinear Compton scattering and nonlinear
(A1)With this notation the trace for nonlinear Compton scattering is given byand for nonlinear Breit-Wheeler pair production the trace readsNow the function Q p,s (φ + , φ − ) can be decomposed into a linear combination of the matricesThe trace turns out to only depend on contractions of the primed and the corresponding not primed coefficients. These contractions can be calculated and they are given by− 2ss (pζ )(p Λ j (q))(ζΛ j (q)) + (ζp )(ζ Λ j (q))(pΛ j (q)) − (pp )(ζΛ j (q))(ζ Λ j (q)) − (ζζ )(pΛ j (q))(p Λ j (q))
A16) and simplifying the expression for the two polarization states j = 1 and j = 2 leads finally to Eqs. (6) and (7), respectively. The trace for nonlinear Breit-Wheeler pair production given in Eq. Inserting Eqs. (A17)-(A21) into Eq.. A3) can be derived directly from the result of nonlinear Compton scattering in Eqs. (A16)-(A21) by multiplyingInserting Eqs. (A17)-(A21) into Eq. (A16) and simplifying the expression for the two polarization states j = 1 and j = 2 leads finally to Eqs. (6) and (7), respectively. The trace for nonlinear Breit-Wheeler pair production given in Eq. (A3) can be derived directly from the result of nonlinear Compton scattering in Eqs. (A16)-(A21) by multiplying
A16) by an overall minus sign and by changing the sign of the four-momentum p µ in the. Eq, Eq. (A16) by an overall minus sign and by changing the sign of the four-momentum p µ in the
With that one finally obtains Eqs. (35) and (36) for the two polarization states j = 1 and j = 2, respectively. Eqs, A17)-(A21)Eqs. (A17)-(A21). With that one finally obtains Eqs. (35) and (36) for the two polarization states j = 1 and j = 2, respectively.
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arxiv |
Article Mechanical transistors for logic-with- memory computing
Huyue Chen
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
School of Mechanical Engineering
Shanghai Jiao Tong University
ShanghaiP. R. China
Chao Song
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
Jiahao Wu
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
Bihui Zou
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
Zhihan Zhang
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
An Zou
Yuljae Cho
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
Zhaoguang Wang
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
Wenming Zhang
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
School of Mechanical Engineering
Shanghai Jiao Tong University
ShanghaiP. R. China
Lei Shao
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
School of Mechanical Engineering
Shanghai Jiao Tong University
ShanghaiP. R. China
Jaehyung Ju
University of Michigan -Shanghai Jiao Tong University Joint Institute
Shanghai Jiao Tong University
ShanghaiP. R. China
Article Mechanical transistors for logic-with- memory computing
As a potential revolutionary topic in future information processing, mechanical computing 1,2 has gained tremendous attention for replacing or supplementing conventional electronics vulnerable to power outages, security attacks, and harsh environments 3,4 . Despite its potential for constructing intelligent matter towards nonclassical computing systems beyond the von Neumann architecture, most works on mechanical computing demonstrated that the ad hoc design of simple logic gates cannot fully realize a universal mechanical processing framework involving interconnected arithmetic logic components and memory 5-16 . However, such a logic-with-memory computing architecture is critical for complex and persistent state-dependent computations such as sequential logic. Here we propose a mechanical transistor (M-Transistor), abstracting omnipresent temperatures as the input-output mechanical bits, which consists of a metamaterial thermal channel as the gate terminal driving a nonlinear bistable soft actuator to selectively connect the output terminal to two other variable thermal sources. This M-Transistor is an elementary unit to modularly form various combinational and sequential circuits, such as complex logic gates, registers (volatile memory), and long-term memories (non-volatile memory) with much fewer units than the electronic counterparts. Moreover, they can establish a universal processing core comprising an arithmetic circuit and a register in a compact, reprogrammable network involving periodic read, write, memory, and logic operations of the mechanical bits. Additionally, we demonstrate the self-unfolding of aerospace solar sails deployed by sequential logic functions with environmental thermal inputs. Our work contributes to realizing a non-electric universal mechanical computing architecture that combines Article multidisciplinary engineering with structural mechanics, materials science, thermal engineering, physical intelligence, and computational science.
Introduction
Electrical computing has been indispensable since the invention of transistors 17,18 .
However, electronics fail in harsh environments, such as extreme temperatures and radiation 3,4 , and they cannot be engineered to directly interact with external stimuli such as heat, force, pressure, etc. Recently, mechanical computing has been gaining attention as a new computational paradigm to supplement or replace conventional computing 1,2 .
Unlike the earlier obsolete analog mechanical computing engines 19,20 , modern mechanical logic gates integrate abstracted digital processing [21][22][23] , which can increase the computational density and respond to environmental stimuli. For instance, several mechanical devices were explored to implement binary information processes with mechanical metamaterials [5][6][7][8]21 , specific solvents 9,10 , pneumatic 11,12 , magnetic 13,14 , and thermal control 15,16 . However, most previous studies only covered basic logic computing or memory bits [24][25][26][27] separately, without forming integrated architectures to demonstrate a full processing core with key features, such as computing, transferring, and registering mechanical bit signals. This restriction is due to the ad hoc design of the above building blocks, causing an intractable barrier between logic and memory, limiting their development towards higher levels of physical intelligence and distributed information processing power [28][29][30] . Additionally, some metamaterial-based computing systems heavily rely on manual resetting or electrical signals for either inputs or outputs [31][32][33][34][35] . This results in a heterogeneous signal flow and inconvenience in logic-memory integration.
Therefore, a more fundamental approach is required, wherein an elementary mechanical transistor is proposed to construct interconnected logic and memory components that perform persistent computation and complex decision-making.
Inspired by the omnipresence of natural heat as a mechanical signal, we created a mechanical transistor with four thermal terminals driven by and output heat. This transistor, which abstracts high or low temperatures as mechanical 1 or 0, respectively, comprises a mechanical metamaterial temperature-sensing channel that actuates a bistable soft element to connect the output terminal from two other variable thermal sources. This unconventional transistor design, with one more source terminal, Article significantly promotes reprogrammability and reduces the number of components required to construct various logic and memory units, which is especially desirable for resource-constrained scenarios. Thus, we used far fewer transistors to construct a mechanical logic-with-memory processing core. This shows the multicycle computing of adding two 4-bit numbers in an arithmetic logic unit while continuously memorizing and reading from an on-site register. This fully mechanical logic-with-memory computing framework demonstrates the potential towards and beyond the von Neumann architecture and could be used for high-level autonomous matter and physical intelligent systems. Fig.1|Mechanical transistor and logic-with-memory computing architecture. a, A 3D schematic of a mechanical transistor (M-Transistor) comprising (i)-(iii) three input terminals and one output terminal for carrying temperature signals, (iv) a bistable soft actuator, and (v) a mechanical metamaterial sensor. The value of l and w are 250 mm and 85 mm, respectively. b, Reversible switching mechanism between the two thermal conditions as the mechanical sensor expands when sensing a high temperature and shrinks when exposed to a low temperature. c, Snap-through and snap-back actuation in experiments. The blue lines are the coefficient of thermal expansion (CTE) of polycarbonate (PC) and invar alloy; while the red curves represent reversible bistability within a preset displacement (5mm). d, Thermal response of an M-Transistor as its terminals set up in b. High (abstracted as 1, >60℃) and low (abstracted as 0, <40℃) output temperature states exhibit a quantitative and significant distinction, thus facilitating the binary abstraction. e, The mechanical logic-with-memory processing unit network, with an integrated arithmetic logic unit and an on-site register memory in one core. f, Five-level hierarchy of mechanical computing, showing that M-Transistors eliminate the barriers between logic and memory, promoting higher-level architectural integrations.
Article
Mechanical transistors
As the building block of a non-electric computer, we create an elementary mechanical transistor (Fig. 1a), instead of directly starting with logic-level designs, to modularly construct fully functional logic circuits and with-memory processing networks. The M-Transistor consists of three thermal inputs (i)-(iii) and one thermal output, a bistable soft Article actuator (iv), and a flexible metamaterial sensor (v) (details in "Materials and Methods").
In contrast to previous manual operations of mechanical logic devices [5][6][7][8]21,[31][32][33] , we encode natural heat into the M-Transistor, abstracting "1" for temperatures above 60 °C (red) and "0" for below 40 °C (blue), to self-switch between states due to thermomechanical actuation. Depending on the adaptive expansion or shrinkage of the metamaterial sensor with respect to environmental temperatures (input i), inputs (ii) or (iii) are selected to connect with the output channel through a triangular copper block mounted at the end of a soft bistable element (Fig. 1b). This copper maintains conductivity in one of the thermal channels, while disconnecting the other channel.
Article
Notably, due to the distinction between logic and memory, scientists have failed to create information interactions between mechanical arithmetic 19-22,31-34 and storage [24][25][26][27] . Herein, we demonstrate a mechanical computing core integrated with an arithmetic logic unit (ALU) and a register (Reg) based on M-Transistors (Fig. 1e). Future mechanical computing expects a neuromorphic morphology, instead of the von Neumann architecture which separates logic and memory (Fig. 1f). A significant advantage of logicwith-memory computing is that we can save the required information after one computation cycle, and subsequently replenish the recorded data. M-Transistors, as a universal unit, could be used to demonstrate all logic gates, rewritable registers, nonvolatile memories, full adders, and other physical intelligence architectures (details in Supplementary Video 1 "Mechanical logic-with-memory architecture"). an output opposite to input (i), with constant heat (ii) and cold (iii) sources. Infrared images and heat path diagrams describe two reversible states. Red represents a high temperature as a bit value of 1, and blue denotes a low temperature as a bit value of 0. c, A Reprogrammable strategy utilizing the identical physical configuration. An OR gate calculates inputs X and Y, and an AND gate for inputs X and Z. d,e, An XOR gate built with only two M-Transistors, with an experimental demonstration of the thermal pathway in four states. The symbol ↮ represents "exclusive disjunction."
Article
Mechanical logic gates
Article
For a 1-bit input, the M-Transistor could be configured to function as a NOT gate, as the output thermal signal is the opposite of the input's signal, shortly $%& = ¬ *+ (Fig. 2a,b). Similarly, we can build a buffer (a.k.a. YES gate) by exchanging the two constant sources (ii) and (iii) (details in Supplementary Materials "Other logic gates"). The 2-bit logic gates (OR and AND) can also be exhibited in the same M-Transistor without requiring cascaded multiple units. For an OR gate, we select inputs X and Y as the binary inputs ( $%& = -∪ / ) while keeping the input Z as 1 (1). Its "disjunction" logic defines that one or both of the alternatives is a high temperature (1); if neither of them is satisfied, we get a low temperature (0) as a result. Constructing an AND gate requires keeping input Y as 2 (0) and selecting inputs X and Z as the inputs ( $%& = -∩ 4 ).
Notably, a single M-Transistor switches functions among NOT, YES, OR, and AND logics by simply reconfiguring the sources, leading to much more compact logic circuits than their electronic counterparts. Also, this reprogrammable strategy is distinct to existing designs that cannot be altered once fabricated [6][7][8][9][10][14][15][16] . In addition, all the above logic functions are persistently reversible without requiring manual resets, which is another key factor for constructing complex networks and has not been successfully achieved in many previous devices [6][7][8][19][20][21][32][33][34] .
In conventional computing, complex gates inevitably require the layering of multiple transistors, thus necessitating substantial efforts and dissipation in circuit construction and tuning. Electrical XOR operation cascades ten electrical transistors (five PMOS and five NMOS) or four basic logic gates (one NOT, one OR, and two AND gates). However, we can build NOR, NAND, XOR, and XNOR logic gates by combining only two M-Transistors c, Experimental results of the mechanical register showing a continuous and periodic operation, with erasable and rewritable bit information. d, Mechanism of a type of non-volatile memory in which the polymer of the bistable element is replaced with a shape memory polymer (SMP). The self-locking status of the SMP at low temperatures limits the snap-through between (i) and (iv). e, Energy portrait of the bistable actuator with SMP, showing that heating softens the SMP; thus, the actuator provides enough energy to overcome the energy barrier between (ii) and (iii). f, Experimental results of the non-volatile mechanical memory, which has the same configuration as the inverter (NOT gate) but maintains its previous state after removing energy (compared with Fig. 1d).
Data storage is equally important as logic operations in computing architectures, but strict barriers exist to differentiate their functions. While various in-memory computing electronics are proposed to eliminate this barrier 39,40 , mechanical memory and logic are Article currently two separate systems with distinct and incompatible designs. Owing to the reprogrammable properties of M-Transistors, we achieve both mechanical logic and memory units using the same building blocks. Technically, a set-reset latch as a universal register furnishes erasable and rewritable operations of a binary bit 10,11 . The sequential outputs depend not only on its current input but also on its current operation, such as holding previous values or redefining values (Fig. 3a). Interestingly, two mirrored M-Transistors compose a locally closed loop to circulate current and future outputs +$6 & +89& (Fig. 3b). Periodic set-reset experiments verified the repeated registration of mechanical bits as shown in Fig. 3c. After the "set" signal (orange line) reaches logic "1", the output heats up and is considered as "write". Only when the "reset" signal (blue line) kicks in to "1", the output rapidly cools down, considered as "erase" (details in Supplementary Video 4 "Erasable, rewritable mechanical register"). We note that transmission delays and losses can be reduced by optimizing the contact (Rc) and assembly (R0) thermal resistances. Besides, continuous operation over 1000 min also further verifies the stability and durability of the mechanical latch (see Supplementary Video 5, "Durability tests of mechanical transistors").
Neuromorphic computing typically requires non-volatile memories 41,42 , which could be achieved by prominent and sustained mechanical bistability, even after completely losing energy sources. To transform the M-Transistor into a non-volatile device, we harness the self-locking mechanism by replacing the materials of the soft bistable beams with a shape memory polymer (SMP), namely polylactic acid (Fig. 3d). Specifically, the stiffness and energy barrier of SMP beams are closely related to temperatures, and thus the metamaterial sensor could only trigger the bistable actuator when the SMP exceeds its glass transition temperature (Tg~60°C); Otherwise, the self-locking of the SMP bistable element prevents snap-through, creating a non-volatile operation. We use energy portrait to further illustrate the energy evolution of the SMP element (Fig. 3e). While softening the SMP with heat, the two stable states of the bistability could be switched in the low-energy channel, resulting in an inverter similar to the dynamic behavior shown in Fig. 1b-d. On the other hand, the energy barrier between the two stable states significantly increases when the SMP actuator is cooled to room temperature, causing the actuator to lock to its position as the sensor's thermomechanical force does not suffice to overcome the energy barrier, thus maintaining long-term memory. This effect is Article experimentally shown as the output stays at "0" while the input switched from "1" to "0", which is clearly opposite to the inverter behavior in Fig. 1b-d (Fig. 3f, and details in Supplementary Video 6 "Non-volatile mechanical memory"). We note that a power outage does not affect the non-volatile state; only if we reheat the SMP, the bistable actuator can return to its original shape as a bit rewriting operation. These dynamic and non-volatile memories promote a general logic-with-memory mechanical computing architecture.
Article
Computing architecture Fig.4|Computing architectures with M-Transistors for aerospace applications and logic-withmemory computing. a, Unfolding deployment of aerospace solar sails due to sequential logic functions utilizing environmental heat inputs. b, Implementation of a logic-with-memory mechanical calculator. The ALU (full adder) comprises seven M-Transistors for arithmetic logic and the Reg (setreset latch) for periodically registering the "carry" bit. c, Experimental results of the mechanical full adder. M-Transistors are compatible with sequential and combinatorial logic, firstly caching data, then operating "1+0" with carry "1"; finally inverting all bytes. d, Thermal simulation of a four-bit Article processing network. We take X: 0011 + Y: 0111 = Sum: 1010 as an example, accompanied by iterative carry. The "carry out" of the previous period is transferred to the "carry in" of the next period. Notably, the disturbance at 6τ is self-corrected in the end. e, Benchmarking cutting-edge mechanical computing.
We present a comprehensive summary of existing mechanical computing, highlighting our intelligent architecture with M-Transistors and the logic-with-memory architecture.
The main goal of modern mechanical computing is to compensate for electronic malfunctions in harsh conditions, such as aerospace, nuclear, and polar environments.
Here, we demonstrate that natural heat sequentially deploys folded solar panels with embedded mechanical logic gates (Fig. 4a). In one scenario, a Mars rover experiences power loss and goes to hibernate due to neither external solar power nor internally More importantly, we create a modular physical architecture with very few M-Transistors for logic-with-memory computing. Taking an electronic full adder as an example, 38 electronic transistors (20 MOSFETs for two XOR gates, 12 MOSFETs for two AND gates, 6 MOSFETs for one OR gate) barely meet the demand for 1-bit "sum" and "carry" as outputs. In contrast, by leveraging the reprogrammability of our platform, only seven M-Transistors are sufficient to perform the equivalent arithmetic function, with two additional M-Transistors to periodically store the "carry" bit for the next computing period (Fig. 4b). With such a configuration, we have experimentally revealed the timedomain signals of a mechanical full adder (Fig. 4c). We define a dimensionless time constant (τ=160 min) as the time scale based on thermal modeling for the heat transfer between two neighboring M-Transistors approaching a steady state (details in Supplementary Materials "Heat transfer modeling"). The M-Transistors within the Article network execute logic in a sequential manner from the periphery to the center because of heat transfer delays. First, the input signals transmit around the periphery (X="carry in"=1, Y=0), and the M-Transistors for the final states "sum" and "carry out" remain at their initial conditions, similar to "caching data" in the central processing unit (CPU).
After 0.5τ, all the M-Transistors are actuated and implementing designed logic operations, with outputs "sum"=0 and "carry out"=1. Finally, with the updated inputs, the M-Transistor network reboots itself to implement a new instruction with X="carry in"=0 and Y=1 (details in Supplementary Materials "Full adder verification", and
Supplementary Videos 8,9).
Further, the logic-with-memory processing core (Reg + ALU) could persistently calculate the final "sum" using the outputs from previous periods. The spatiotemporal heat transfer of the transistor network was simulated to verify its feasibility (Fig. 4d). As one of the most complex four-digit additions, X: 0011 + Y: 0111, iterative "carry" bit poses a grand challenge to the mechanical computing, which integrating combinational logic and sequential circuits. The blue and orange curves represent the time-domain signals of X and Y, respectively, with the least significant bit being the first value entering the mechanical processing core (entry sequence for X being 1, 1, 0, 0 and Y being 1, 1, 1, 0).
Because the first-bit operation does not produce a "carry" bit, the initial instruction that we give to the register is "reset". As a result, all iterative "carry out" can be stored as the signal "1" in the register, and precisely be held for one period. The fourth bit, "carry out, " accumulates all the previous hysteresis and delays, and it takes 3τ to reach a steady state. Notably, due to the extra time required for thermal signal propagation and bistable actuation in M-Transistors, a sudden nonideal disruption at the beginning of the third cycle shows up, but it does not affect subsequent calculations as the mechanical network has self-correcting stability. The purple and green areas convey "carry" and "sum" in the adder architecture, and a correct sum of 1010 is reached in the end. We note that, on the one hand, the significant thermal inertia in M-Transistors is the leading cause of delay, and on the other hand, the insensitivity to short-term disturbance ensures excellent robustness. Finally, owing to a lack of metrics for assessing different designs of mechanical computing, we benchmark all recent studies against the functional completeness and the required number of units (Fig. 4e). This work not only shows complete functions of logic and memory but also achieves an integration of them for a Article logic-with-memory computing architecture, with significantly fewer number of units as well.
Article Outlook
According to the equivalent electronic circuits, we have verified all eight logic gates and designed essential computing circuits, such as a full adder, an SR register, and a nonvolatile memory, with a tremendous decrease in the number of transistors. Second, the inevitable heat dissipation and conduction loss are the main limitations of a further larger scale of integration, which can be improved by miniaturization to the mesoscale or performing all the computing in a vacuum environment, as in real space. On the other hand, this delay also allows an insensitivity to unexpected disturbances, as the computation will not be interrupted even if the heating energy is missing for a short period. Third, although the miniaturization of the M-Transistor is possible 43,44 for a faster speed and a higher integration, real-world applications 45,46 may not easily provide a small-scale alternative hot and cold temperature distribution and could also accommodate macroscopic thermal devices, such as nuclear facilities, space exploration, and intelligent buildings.
In summary, as a new paradigm of mechanical computing, our concept of a fundamental mechanical transistor exhibits its versatility and indispensability. Compared with existing mechanical logic gates, we highlight the emergence of "mechanical logic-withmemory computing" and corresponding combinational (logic) and sequential (memory) circuits with full functional completeness. We anticipate that our design would be a milestone for more sophisticated physical intelligence and may inspire engineering in broader fields such as integrated logic and memory in molecular computing or living organisms 47,48 .
Kinematic models of the CPS
We define input and output mechanical signals as *+ and $%& , respectively. We provided an analytical expression of the thermomechanical response of a compliant porous structure (CPS) with compliant hinges whose main loading mode is pure bending (see Supplementary Materials "Thermomechanical deformation of Compliant Porous Structure (CPS)").
The input displacement *+ is calculated as
*+ = < + > + 2 @ − B C (1)
The output displacement $%& is determined as
$%& = 2 D E B2 F > + < + > G > + E + J − & F > G > sin N O >(2)
, where Li and th represent the length and thickness of hinges, respectively, and * (i=1-3, h) is the angle of the curve, as shown in Figure S1. The amplification ratio = $%& *+ ⁄ increased with a decrease in both < and *+ . The maximum application ratio ST9 (= 47.4) can be obtained with < = 10 and *+ = 0.1 , as shown in Figure S2-S4.
Spring modeling of the bistable element
To better understand the physics behind the snap-through behavior, we develop an analytical model of a bistable structure based on a soft spring model 49,50 . Finally, we can obtain the force-displacement relationship of the system as The bistable soft actuator comprises a photosensitive resin element and two triangular copper prisms. The soft matter is fabricated using stereolithography (Formlabs Inc., USA) or direct ink writing 6,51 . Cu (type T2 with 99.91% Cu, Guanye Co., Ltd., China). was manufactured and polished using computerized numerical control (CNC). We replaced the soft beam with a 3D-printed polylactic acid (PLA) element for the non-volatile memory unit. To avoid the need for additional energy, the main body of the bistable actuator adopted a hollow copper block, connecting heat or cold sources to the PLA element.
Assembly and heat transfer optimizations
To reduce the heat loss and achieve a higher temperature output, we cored M-Transistors including attaching a heated ceramic, water-cooling head, copper, and aerogel.
Thermal resistance models of M-Transistors
To verify the logic operation of the M-Transistor, we use a heat transfer model and analyze an inverter (NOT gate) as an example. Based on our previous work 38 , the contact thermal resistance is described through an elastic model: (K·m 2 /W), as the contact becomes increasingly tight with the increasing input temperature.
Dimensionless time constant
We define dimensionless time to guide possible miniaturization and material selection for computing systems. The relaxing time τ (systematic time constant) represents the time it takes to reduce the temperature difference to f< of the initial state. Considering the time required to heat the copper via a single M-Transistor:
(²f² ³ ) (² ŠŽŠ•Š‰´f ² ³ ) = exp`−1 š º »ˆa = f<(7)
, where ρ = 8978 (kg/m 3 ) is copper's density, cp = 381 (J /kg·K) indicates the heat capacity of copper, and V indicates the volume of the copper block. By fitting the experimental results and thermal simulations, we obtain τ = 160 min.
Article
Data availability
The codes and source data supporting the findings of this study are available from the corresponding authors upon reasonable request.
Author Contributions
Depending on the various setup of sources (ii) and (iii), we could implement diverse logic and memory functions in a single M-Transistor.The mechanical sensor yields a large displacement triggered by a thermal expansion mismatch between a polycarbonate Kirigami-inspired displacement amplifier and an Ishape invar bar 35-38 . We further maximize the vertical displacement by breaking the mirror symmetry about the vertical centerline of the kirigami structure (details in Supplementary Materials "Thermomechanical deformation of Compliant Porous Structure (CPS)"). Therefore, as the sensor temperature increases from room temperature (25 °C) to 80 °C, the bistable actuator snaps through and converts the continuous signal of the metamaterial sensor into a distinct binary, while the 'trigger point' is between 50 and 60℃(Fig. 1c). To avoid ambiguity in the transistor operation, the pre-designed asymmetric bending curves in the soft beams prescribes the bistable elements to reside at one side in the initial cold state (details in Supplementary Materials "Bistable modeling"). Because the sensor's thermomechanical forces at high temperatures (TH) and low temperatures (TL) are designed to overcome the bistable energy barriers, the snap-through/back features regulate the reversibility of M-Transistors. We verify the M-Transistors by recording the thermal signals at the T-shaped copper (i) and the output with constant heating (>130 °C) and cooling (< 30 °C) sources at inputs (ii) and (iii), respectively. The output thermal signal responds in the opposite manner to the input, as shown inFig. 1d(details in Supplementary Video 2 "Thermomechanical actuating of the transistor").
Fig
.2|Reprogrammable mechanical logic gates with complete functions. a,b, NOT gate yielding
in series or parallel (Fig. 2d, with more details in Supplementary Materials "Other logic gates"). Because thermal signals are used for both inputs and outputs, M-Transistors can connect and share data with each other, permitting the previous input/output as the subsequent input. An XOR gate (Fig. 2e) is experimentally verified only if two mutually exclusive inputs would result in a high temperature (1), which could enable local summation in arithmetic logic units. Article Mechanical memory Fig.3|Rewritable volatile and non-volatile mechanical memories. a, Principle and transition table of a mechanical register (volatile), consisting of two loop-connected M-Transistors. b, Schematic of the mechanical register with "Set" and "Reset" commanding heating and cooling inflows, respectively.
reserved energy (0, 0). Later, because of heating by reappeared solar power, the thermally responsive clamps first released a small solar panel as a result of the OR logic (1, 0). However, the main solar panel is still folded because catastrophes like Martian sandstorms can pollute and damage large-area panels, and thus it waits for mechanical logic gates to make further decisions. The main solar panel is only allowed to fully deploy only when the small solar panel harvests sufficient energy under a calm weather and provides a heated second input due to the result of an AND logic (1, 1) (details in Supplementary Video 7 "Thermomechanical computing for aerospace"). This is the first demonstration of self-implementation of mechanical logic driven by environmental heat.
Articlewhere k1,2 indicate linear spring coefficients.Figures S5-S7describe the initial geometry of the spring system, including the spring's length , and the inclined angle . An external vertical force F results in a vertical displacement d (see Supplementary Materials "Bistable modeling").Design and fabrication of M-TransistorsThe mechanical metamaterial sensor comprises a Kirigami-inspired CPS and an I-shapeinvar bar. The two parts are designed in CAD software SolidWorks 2019 (Dassault Systemes SE, France). We used polycarbonate (PC) as the main material for CPS, which is sliced at 100% infill density by Ultimaker Cura (Ultimaker B.V., USA) to generate a Gcode file for 3D printing. Besides, the supports of the CPS and the bistable structure are also made from Ultimaker polycarbonate through 3D printing. CPS can function as a displacement amplifier by embedding a waterjet-cut invar bar (type 4J36 with 60% iron, 32%-36% nickel, Mingshang Metal, China) under a certain temperature difference. The alloy's density and coefficient of thermal expansion are 8.1 B ⁄ and 1 × 10 f€ ⁄ , respectively. The details of the parameters are shown in Supplementary Materials "Material properties."
… and † represent the temperatures of the heat and cold sources, respectively. * ( = 1, 2, 3, 4) denotes the thermal resistance of copper (see Supplementary Materials "Heat transfer modeling of M-Transistors"); š$+&Tš& represents the contact thermal resistance, and T*› indicates the thermal resistance of air (relatively large, can be regarded as an open circuit).
H.C., C.S., and J.J. conceived the research contents. H.C. and J.W. fabricated the devices and conducted experiments. H.C. analyzed the data and produced the figures and videos. C.S. and B.Z. performed mechanical models and tested material properties. H.C., Z.Z., and Z.W. built thermal models. A.Z., and Y.C. contributed to the background and new concepts of electrical computing. H.C. and J.J. wrote the manuscript. W.Z., L.S., and J.J. reviewed the manuscript. L.S. and J.J. supervised the project. All authors contributed to the discussion and editing. H.C. and C.S. equally contributed to this work.
with aerogels (type JN650, Hegao Materials, China). The density and coefficient of heat aerogel was the base of M-Transistors; 2mm-thick strips are covered on the surface of Article the copper at the output. A liquid metal 52 with good thermal conductivity was smeared on the interfaces to fill micro air gaps, thus reducing the thermal contact resistance. The liquid metal mainly consists of gallium and indium (Yelengxing, China), which can keep its properties stable at -50−150℃. The density and the coefficient of heat conductivity Three kinds of glue were used during the assembly, including the cyanoacrylate adhesive, Teflon tape, and Loctite EA E-120HP adhesive. A cyanoacrylate adhesive was applied to connect the bistable element, its support, and the aerogel base. Teflon tape was used to adhere the thermocouple and heated ceramic to the copper. The Loctite EA E-120HP adhesive with good heat resistance can maintain stickiness under high-temperature conditions,conductivity are 0.18 − 0.20
B
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and 0.018
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, respectively. The 8mm-thick
are approximately 5.8
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and 140
·
⁄
, respectively.
roughness, †%; 2¬ and †%; 2¬ are thermal conductivities and Poisson's ratios of copper block and liquid metal, respectively. E is the effective Young's modulus, and the mechanical sensor with an asymmetric Kirigami structure generates a contact pressure P with high-temperature input signals. We adopt numerical fitting with experimental data to quantify contact thermal resistance, Rcontact changes from 0.8 to 1.5 × 10 -3š$+&Tš& = 1.25
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AcknowledgementsCompeting interestsThe authors declare no competing interests.
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| {'fraction_non_alphanumeric': 0.055957288639589105, 'fraction_numerical': 0.025050122772634093, 'mean_word_length': 4.662244897959184, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 14, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'As a potential revolutionary topic in future information processing, mechanical computing 1,2 has gained tremendous attention for replacing or supplementing conventional electronics vulnerable to power outages, security attacks, and harsh environments 3,4 . Despite its potential for constructing intelligent matter towards nonclassical computing systems beyond the von Neumann architecture, most works on mechanical computing demonstrated that the ad hoc design of simple logic gates cannot fully realize a universal mechanical processing framework involving interconnected arithmetic logic components and memory 5-16 . However, such a logic-with-memory computing architecture is critical for complex and persistent state-dependent computations such as sequential logic. Here we propose a mechanical transistor (M-Transistor), abstracting omnipresent temperatures as the input-output mechanical bits, which consists of a metamaterial thermal channel as the gate terminal driving a nonlinear bistable soft actuator to selectively connect the output terminal to two other variable thermal sources. This M-Transistor is an elementary unit to modularly form various combinational and sequential circuits, such as complex logic gates, registers (volatile memory), and long-term memories (non-volatile memory) with much fewer units than the electronic counterparts. Moreover, they can establish a universal processing core comprising an arithmetic circuit and a register in a compact, reprogrammable network involving periodic read, write, memory, and logic operations of the mechanical bits. Additionally, we demonstrate the self-unfolding of aerospace solar sails deployed by sequential logic functions with environmental thermal inputs. Our work contributes to realizing a non-electric universal mechanical computing architecture that combines Article multidisciplinary engineering with structural mechanics, materials science, thermal engineering, physical intelligence, and computational science.', 'arxivid': '2306.02352', 'author': ['Huyue Chen \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n\nSchool of Mechanical Engineering\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Chao Song \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Jiahao Wu \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Bihui Zou \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Zhihan Zhang \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'An Zou ', 'Yuljae Cho \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Zhaoguang Wang \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Wenming Zhang \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n\nSchool of Mechanical Engineering\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Lei Shao \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n\nSchool of Mechanical Engineering\nShanghai Jiao Tong University\nShanghaiP. R. China\n', 'Jaehyung Ju \nUniversity of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China\n'], 'authoraffiliation': ['University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'School of Mechanical Engineering\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'School of Mechanical Engineering\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China', 'School of Mechanical Engineering\nShanghai Jiao Tong University\nShanghaiP. R. China', 'University of Michigan -Shanghai Jiao Tong University Joint Institute\nShanghai Jiao Tong University\nShanghaiP. R. China'], 'corpusid': 259076432, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12514, 'n_tokens_neox': 10683, 'n_words': 6461, 'pdfsha': 'af7c3f1aa2cc438e6bbd84383f5c514657a90413', 'pdfurls': ['https://export.arxiv.org/pdf/2306.02352v1.pdf'], 'title': ['Article Mechanical transistors for logic-with- memory computing', 'Article Mechanical transistors for logic-with- memory computing'], 'venue': []} |
arxiv |
Privacy in Practice: Private COVID-19 Detection in X-Ray Images Extended Version
Lucas Lange
Leipzig University & ScaDS.AI
Dresden/Leipzig, LeipzigGermany
Maja Schneider
Leipzig University & ScaDS.AI
Dresden/Leipzig, LeipzigGermany
Peter Christen [email protected]
The Australian National University
CanberraAustralia
Erhard Rahm
Leipzig University & ScaDS.AI
Dresden/Leipzig, LeipzigGermany
Privacy in Practice: Private COVID-19 Detection in X-Ray Images Extended Version
Privacy-Preserving Machine LearningDifferential PrivacyMembership Inference AttackPractical PrivacyCOVID-19 DetectionDifferentially-Private Stochastic Gradient Descent
Machine learning (ML) can help fight pandemics like COVID-19 by enabling rapid screening of large volumes of images. To perform data analysis while maintaining patient privacy, we create ML models that satisfy Differential Privacy (DP). Previous works exploring private COVID-19 models are in part based on small datasets, provide weaker or unclear privacy guarantees, and do not investigate practical privacy. We suggest improvements to address these open gaps. We account for inherent class imbalances and evaluate the utility-privacy trade-off more extensively and over stricter privacy budgets. Our evaluation is supported by empirically estimating practical privacy through black-box Membership Inference Attacks (MIAs). The introduced DP should help limit leakage threats posed by MIAs, and our practical analysis is the first to test this hypothesis on the COVID-19 classification task. Our results indicate that needed privacy levels might differ based on the task-dependent practical threat from MIAs. The results further suggest that with increasing DP guarantees, empirical privacy leakage only improves marginally, and DP therefore appears to have a limited impact on practical MIA defense. Our findings identify possibilities for better utility-privacy trade-offs, and we believe that empirical attack-specific privacy estimation can play a vital role in tuning for practical privacy.
INTRODUCTION
The COVID-19 pandemic pushed health systems worldwide to their limits, showing that rapid detection of infections is vital to prevent uncontrollable spreading of the virus. Detecting COVID-19 in patients can be achieved using a RT-PCR test 1 . Although they are more reliable in terms of sensitivity than rapid antigen tests, results can take hours to arrive, and even if displaying negative, the virus could have already left the throat and manifested itself in the lungs, rendering it undetectable for either test (Albert et al., 2021).
In hospitals, chest X-rays can mitigate these drawbacks by enabling a fast and reliable diagnosis. (Chowdhury et al., 2020;Rahman et al., 2021). COVID-19 positive scans are characterized by patchy consolidations of the lungs.
in the COVID-19 scans, such X-rays remain challenging to interpret. Specialists, however, are able to identify the severity of a case early on and can take measures without waiting for lab results. Table 1: Existing solutions from related work next to our private model at ε = 1 (Müftüoglu et al., 2020;Zhang et al., 2021;Ho et al., 2022). The methods refer to training private models. For differentiating the tasks, we assign the classes as COVID-19 (C), Normal (N), or Pneumonia (P). A performance comparison is difficult due to the different characteristics. Baseline shows the best non-private models. Our best private result is based on accuracy for comparison to related work. We further include two proposed additions for filling open gaps: F1-score and MIA. Machine Learning (ML) techniques can effectively assist medical professionals in an initial screening by quickly classifying large numbers of images. However, the amount of data needed for training such classifiers poses problems due to clinical data privacy regulations, which present strict limitations to data sharing between hospitals. All sensitive patient information must be treated confidentially before, during, and after processing. (Balthazar, 2018) To complicate matters, not only the dataset itself but also the models resulting from ML can compromise privacy. Published models are vulnerable to attacks, including leaking details about their training data (Shokri et al., 2017). Such leaks allow adversaries to potentially deduce sensitive medical facts about individuals in the dataset, for instance by exposing a patient's genetic markers (Homer et al., 2008).
In the case of COVID-19 detection, an attacker could be able to reveal if a person was infected, which would already violate privacy. While the specific risk of X-ray-based attacks might be low, such data should be handled with caution, especially since even with anonymization, results can still be linked to other information like related medications. Furthermore, we cannot rule out an attacker with internal access to images, e.g. doctors utilizing the model in a hospital.
Privacy-Preserving ML (PPML) is a collection of methods for creating trustworthy ML models, enabling, for example, the development of medical applications while maintaining patient privacy. In this work, we apply PPML that satisfies Differential Privacy (DP) (Dwork, 2008) in training a COVID-19 detection model, thus limiting attacks on the resulting classifier from incurring information leakage.
Our investigation is divided into three successive Our contributions are:
• We fill open gaps from previous work (Müftüoglu et al., 2020;Zhang et al., 2021;Ho et al., 2022), where Table 1 shows their characteristics in comparison to our approach. We address the class imbalances and analyze the utility-privacy trade-off more extensively by evaluating multiple and stricter privacy budgets. We further investigate practical privacy by empirically estimating privacy leakage through black-box MIAs. These gaps and our improvements are addressed throughout the following sections. • We are the first to evaluate if DP helps narrow down MIAs on the COVID-19 detection task. We additionally re-examine this hypothesis on a common benchmarking dataset to reveal connections between the two datasets. Our results point towards identifying the benefits from DP in defending against MIAs as task-dependent and plateauing. We are able to gain better utility-privacy trade-offs at no practical cost. These results thus strengthen the belief that empirical privacy analysis can be a vital tool in supporting attack-and task-specific tuning for privacy.
The following Section 2 provides an overview of essential concepts. We then contextualize our work by examining the existing literature in Section 3, and we present our selected solutions to address open research gaps in Section 4. Section 5 lays out our experimental setup, with the results presented in Section 6 and their discussion in Section 7. In closing, Section 8 provides conclusive thoughts and adds an outlook to possible future work.
BACKGROUND
This section establishes a basic understanding of the relevant concepts and algorithms used in this work.
Differential Privacy
DP is the quasi gold standard in private data analysis, which offers a guarantee that the removal or addition of a single dataset record does not (substantially) affect the outcome of any analysis (Dwork, 2008). Thus, an attacker is incapable of differentiating from which of two neighboring datasets a given result originates and has to resolve to a random guess-i.e., a coin flip. DP's provided guarantee is measured by giving a theoretical upper bound of privacy loss, represented as the privacy budget ε. The metric is accompanied by the probability of privacy being broken by accidental information leakage, which is denoted as δ and depends on the dataset size. Formally, an algorithm A training on a set S is called (ε,δ)-differentially-private, if for all datasets D and D' that differ by exactly one record:
Pr[A(D) ∈ S] ≤ e ε Pr[A(D ) ∈ S] + δ(1)
Meaningful privacy guarantees in ML should fulfill ε ≤ 1 and δ 1/n, where n is the number of training samples (Nasr et al., 2021;Carlini et al., 2019). The notation ε = ∞ indicates that no DP criteria are met. The design of DP algorithms is based on one of DP's fundamental properties: composability (Dwork and Roth, 2014). It states that if all the components of a mechanism are differentially-private, then so is their composition. Another essential attribute of DP is its post-processing immunity, implying DP is preserved by all further processing. Therefore, in terms of achieved privacy, it does not matter whether a ML model uses an already DP conform dataset or applies DP while training (Dwork and Roth, 2014).
Differentially-Private Stochastic Gradient Descent
The Differentially-Private Stochastic Gradient Descent (DP-SGD) algorithm introduced by Abadi et al.
(2016) takes widely used SGD and applies a gradient perturbation strategy. Gradient perturbation adds enough noise to the intermediate gradients to obfuscate the largest value, since that original sample inhibits the highest risk of exposure. To generally bound the possible influence of individual samples while training, DP-SGD clips gradient values to a predefined maximum Euclidean norm before adding noise. The noisy gradients are then used to update the parameters as usual. The total noise added through the algorithm is composed over all training iterations using an accounting mechanism and determines the resulting privacy budget.
Membership Inference Attacks
In black-box MIAs, an attacker feeds data samples to a target model and thereby tries to figure out each sample's membership or absence in the model's training set based solely on the returned confidence values. This technique takes advantage of the differences in predictions made on data used for training versus unseen data, where the former is expected to output higher confidence values due to memorization (Carlini et al., 2019). As proposed by Shokri et al. (2017), such attacks can utilize multiple shadow models specifically mimicking a target model's predictions, to train an attack model able to elicit the desired membership information. Salem et al. (2019) relaxed the need for shadow models, by finding that simply using the original model's predictions on given samples can be sufficient to deduce their membership. By revealing the membership of an individual's record in the dataset, an adversary might in turn disclose sensitive information on them. However, a decisive prerequisite is some form of existing prior knowledge of the target data, because the MIA only tests samples available to the attacker (Shokri et al., 2017). Thus, the attacker needs to possess a set of individual samples to uncover their membership.
Private COVID-19 X-Ray Detection
In Table 1, existing works on private COVID-19 detection from X-rays are summarized and compared to our approach. There are multiple factors that impede a fair comparison, which mainly lie in the differences in datasets, tasks, and privacy guarantees (ε). In this section, we show open gaps and then give elaborations in Section 4 on how we address them.
Datasets. A problem regarding (Müftüoglu et al., 2020) is that their results are based on only a small dataset of 139 COVID-19 scans. The COVID-19 Radiography Database used by (Ho et al., 2022) and us, provides a better basis in terms of dataset size. However, the class imbalances result in a rather skewed data basis, which is left unaddressed but could influence MIA threat (Jayaraman et al., 2021). With the FedDPGAN approach, Zhang et al. (2021) try to enlarge and balance their small dataset using synthetic images, but the quality of the generated distribution is left unanswered. This is particularly problematic because GANs trained on imbalanced input data tend to produce data with similarly disparate impacts (Ganev et al., 2022). As a general problem with skewness, the mentioned works solely assess performance using accuracy, although this metric is known to undervalue false negatives for minority classes and could favor classifiers that are actually worse in detecting the COVID-19 minority class (Bekkar et al., 2013).
Privacy budgets. The used ε-values of 5.98 and 39.4 by Müftüoglu et al. (2020) and Ho et al. (2022) respectively, are significantly weaker than the privacy budget of ε ≤ 1, which is commonly assumed to provide strong privacy (Nasr et al., 2021;Carlini et al., 2019). Furthermore, the results by Zhang et al. (2021) lack comparability, since they do not provide their privacy budget. Using their parameters and noise in a standard DP-SGD analysis results in ε > 5 * 10 13 for a client after 500 rounds 2 of federated training. Even with their most private setting they still accumulate ε = 19.6. Thus, no model adheres to ε ≤ 1 and they instead only offer weaker or unclear guarantees.
Practical privacy. Regarding practical privacy, prior work does not include actual attack scenarios. It is therefore left open to what extent the provided models and ε-guarantees retain patient privacy against real adversaries. Such analysis helps in assessing the defense capabilities provided by the achieved privacy budgets and could reveal room for tuning them.
Repelling MIAs
Related work suggests multiple strategies for reducing MIA threats. Shokri et al. (2017) show that limiting the model outputs to only class labels instead of explicit confidence values can be an effective remedy. However, in medical tasks such as COVID-19 detection, where the use case is to help medical professionals in diagnosing a disease, the confidence value is an integral part that indicates how likely a patient is affected. Shokri et al. (2017) also find that model architecture can contribute to MIA defense and Salem et al. (2019) demonstrate that even the training process can hinder MIAs through e.g. model stacking.
DP should limit and oppose the success of MIAs by design, with Jayaraman and Evans (2019) supplying the corresponding reasoning: "[DP], by definition, aims to obfuscate the presence or absence of a record in the data set. On the other hand, [MIAs] aim to identify the presence or absence of a record [. . .]." Rahman et al. (2018) test this hypothesis by evaluating MIAs on different privacy levels. They find their model's MIA resistance to gradually increase when lowering the allowed privacy budget and explain it with less overfitting when adding more noise. Yeom et al. (2018) prove that overfitting in ML models is sufficient to enable MIAs, but at the same time show that overfitting is not a necessary criterion, and stable models can still be vulnerable.
Practical Privacy Analysis
Multiple works examined the possibilities of estimating the practical privacy for ML models by performing an empirical study through attacks, e.g. MIAs. Jagielski et al. (2020) and Nasr et al. (2021) conclude that the assumed theoretical upper bound privacy loss for DP, given in the privacy budget ε, gives a tight worst-case analysis on attack proneness and thereby limits MIA success. However, in many cases actual attacks extract significantly less information than assumed by the theoretical bound, which is also supported by Malek et al. (2021) and Jayaraman and Evans (2019). This discrepancy could possibly enable better utility-privacy trade-offs, but Jayaraman and Evans (2019) warn that privacy always comes at a cost and reducing privacy could ultimately promote information leakage. Malek et al. (2021) propose that a realistic lower bound on the amount of revealed information by a model can be determined by "[considering] the most powerful attacker pursuing the least challenging goal" and that in the case of standard DP, such would be an attacker powerful enough to successfully perform membership inference. Therefore, by attacking our models with MIAs, we can empirically estimate practical privacy leakage which might differ substantially from the upper leakage bound derived from DP theory.
METHODS
As seen in Section 3.1 and Table 1, related work on COVID-19 detection lacks comparability and leaves open research gaps. We therefore do not solely focus on enhancing the performance of former solutions but rather suggest improvements by filling existing gaps, ultimately proposing the following improvements.
Datasets. Since the dataset used by us and Ho et al. (2022) provides a good amount of COVID-19 samples, we instead aim for better handling of the problems arising from the skewed nature of the class distribution. In a first effort, we employ random undersampling and class weights to elevate the underrepresented COVID-19 class, both in database construction and in training, respectively. Furthermore, since accuracy is not representative in cases of skewed data, we improve the evaluation by using the more balanced F1-score metric (Bekkar et al., 2013).
Privacy budgets. We investigate the utilityprivacy trade-off by evaluating multiple and stricter privacy budgets of ε = [∞, 10, 1, 0.1]. To find the best private model and extend the pool of evaluated methods, we propose untested architectural experiments relevant to private DP-SGD training in Section 5.4.
Practical privacy. As seen in the works discussed in Section 3.3, we investigate the practical implications of DP regarding the defense against black-box MIAs by undertaking an empirical analysis through actual attacks, and therefore give a more realistic lower bound to the resulting privacy leakage (Jagielski et al., 2020;Malek et al., 2021). We thereby provide the first attack results in the field of private COVID-19 detection and evaluate possible room for tuning the utility-privacy trade-off. An additional evaluation regarding the privacy leakage of our models on the MNIST database enables us to formulate takeways regarding similarities and disproportions regarding the attack-specific privacy on both datasets. Evaluating another dataset is a first step towards generalization and MNIST is particularly interesting because related works (Rahman et al., 2018;Nasr et al., 2021) previously investigated the connection between DP and MIA on this task.
EXPERIMENTAL SETUP
In this section we provide details on the setups used in our experiments, which are summarized in Table 2.
Reference code is available from our repository 3 .
Environment
The implementation uses Python with Keras 4 and Tensorflow 5 . Further, we employ the modules for DP-SGD training and MIAs from the Tensorflow Privacy library 6 . To enable consistent and reproducible results, random seeds are set to a fixed value, which is 42 in our case. Hardware-wise our machines are equipped with 64GB of RAM and an NVIDIA Tesla V100-PCIE-32GB GPU.
Datasets
For a comprehensible dataset creation, we provide details on the different public datasets we used.
• The COVID-19 Radiography Database 7 (Chowdhury et al., 2020; Rahman et al., 2021) is the most comprehensive collection of COVID-19 chest Xray images, stemming from different databases around the web. In total, this image collection offers chest X-rays of 3,616 COVID-19 positive, 10,192 Normal, and 1,345 Pneumonia cases. For our binary task, we omit the pneumonia samples and employ undersampling to directly reduce class imbalances. Dataset construction takes all COVID-19 scans but only 1.5× the amount for Normal (5,424), resulting in 9,040 images total. When testing hyperparameters, this ratio showed to elevate performance (F1) and reduce privacy risk due to less overfitting (Rahman et al., 2018).
• The Chest X-Ray Images (Pneumonia) 8 (Kermany et al., 2018) offers X-ray images divided into two classes with 1,583 Normal and 4,273 Pneumonia samples. Here, we again apply undersampling to achieve similar class ratios and take all Normal scans but just 1.5× the amount for Pneumonia (2,374). This pneumonia dataset is also part of the COVID-19 Radiography Database, constituting 13% (1,341) of its Normal class images. To fix this issue and enable its use as a public dataset for our private transfer learning approach without compromising privacy, we exclude duplicates when sampling images for the COVID-19 task.
• The ImageNet (Deng et al., 2009) is a vast collection counting 14 million images and covering 20,000 categories from general (mammal) to specific (husky). Non-private models benefit from using this massive dataset for pre-training, introducing many differentiating concepts to a neural network before training on the target data. Our experiments are mainly focusing on the taskrelevant COVID-19 Radiography Database as the evaluation basis. Additionally, models are then evaluated on the MNIST database. Pre-training uses the respective public ImageNet and Pneumonia datasets.
Pre-processing
To build our final splits for model training, we employ necessary sampling and pre-processing steps. Both X-ray datasets, for COVID-19 and pneumonia, use a train-validation-test split of 80% training, 5% validation, and 15% test set. All datasets are handled with three color channels. We therefore convert the MNIST grey-scale images into the RGB space, as this is vital to allow the models pre-trained on color images to still work with the input data. X-ray images are downscaled to 224x224 pixels, while MNIST images keep their size of 28x28. Furthermore, each data point undergoes an image normalization step using the factor of x=1/255.
To combat overfitting, training sets are shuffled and training images from the X-ray datasets are subjected to data augmentation (Shorten and Khoshgoftaar, 2019). An problem already improved from the X-ray datasets' undersampling is their imbalance re-garding Sickness (COVID-19, Pneumonia) and Normal class frequencies. To further improve on this issue, we apply class weights during training to help artificially balance each sample's impact. This process yields class weights of 0.83 for the Normal and 1.25 for the Sickness classes.
Architectural Experiments
In the following, we describe our different architectural choices we used for experiments.
Model Size in DP-SGD
Non-private and private classification perform differently depending on the underlying model architecture . Performance is greatly dependent on model size, i.e., the number of layers or parameters in a model, with non-private training typically benefiting from using bigger models. In contrast, using DP-SGD, the same models suffer from accuracy loss when increasing in size. Taking these findings into account, in our experiments we used two differently-sized architectures.
ResNet Models
The model family of Residual Networks (He et al., 2016), or ResNets, provides well-scaling deep CNN architectures thanks to its residual connections. We utilize two different versions as our basic model architectures for the experiments. For one, the ResNet50 version boasting 50 layers, which is a stretched compromise between size and computing needs, as well as the ResNet18, which is mostly identical except for the lower layer count of 18 layers. When introducing architectural experiments, we change each basic model accordingly, creating different variants.
tanh Activation
A disruptive discovery in DP-SGD research was made by Papernot et al. (2021). In their work, they determined that replacing the de facto standard ReLU activation function with the tanh function in model layers improves performance in DP-SGD. To achieve this boost, they utilize the fact that the tanh activation generally results in smaller gradients than the ReLU function, which in turn reduces the information loss from gradient clipping.
Pre-training
An important consideration in private training is the use of public pre-training datasets. This is due to the advantage that public datasets do not require the same noisy training mechanisms as private datasets. On the basis of the resulting pre-tuned weights, the pretrained model is then fine-tuned to the private target data for the actual task.
A commonly applied strategy to improve performance for non-private classification relies on pretraining using the extensive ImageNet collection. As another method, Abadi et al. (2016) state that DP-SGD models can further profit from pre-training in a domain closely related to the target task. While Im-ageNet resembles a general choice for image-based tasks, pre-training for pneumonia detection is closer to our COVID-19 task due to the similarity in symptoms (Speranskaya, 2020;Lange, 2022).
The pre-training on the Pneumonia dataset is performed using the same settings as on the COVID-19 set, while the ImageNet variants are provided by a library for Keras models 9 . For our tanh variants we take the take the pre-trained ReLU models and change the activation function in each trainable layer before training on our target datasets.
Privacy Experiments
In this section, we elaborate on the used settings and hyperparameters for evaluating privacy.
Private Training Settings
All model variants are first trained non-privately using Adam (Kingma and Ba, 2015) optimization at ε = ∞ to form a baseline. We employ batch sizes of 32 and train for 20 epochs using a learning rate of α = 1e−3, which decays down to a minimum of α = 1e−6 on plateaus. Afterwards, we apply DP-SGD (or here DP-Adam) training to all models, training a private candidate for each ε-guarantee. The DP-SGD algorithm is applied by changing the optimizer and handing in the necessary privacy parameters, like our employed clipping norm of 1.0. DP-SGD training for COVID-19 uses ResNet50 and ResNet18 variants with batch sizes of 8 and 16, instead of 32 respectively. We aim at privacy budgets of ε ≤ 1, since such values present strong privacy guarantees (Nasr et al., 2021;Carlini et al., 2019). We also evaluate budgets neighboring this setting by an order of magnitude, to gain further insights into the performance and estimated privacy on different DP levels. Due to the dataset size, the DP analysis uses δ = 1e−4 for COVID-19 (n = 9, 040) and δ = 1e−5 for MNIST (n = 60, 000). 9 https://github.com/qubvel/classification_models
MIA Settings
For selecting the most potent MIA each run, we try four different attack types based on logistic regression, multi-layered perceptron, k-nearest neighbors, and threshold. These attacks found in the Tensorflow Privacy library are an implementation of the single shadow model black-box attack proposed by Salem et al. (2019), that directly relies on target model predictions instead of training several shadow models. Given a target model, MIAs utilize two types of data for training: (1) the original training data to be inferred and (2) unseen but similar data to differentiate non-training data. In our case, we want to fully empower the attacker for estimating the practical worstcase in an optimal black-box setting (Malek et al., 2021). We satisfy this condition by giving access to the full training and test sets with their corresponding labels, thus, handing the attacker the largest input regarding (1) and the most similar input regarding (2).
Measuring Privacy Leakage
Like Jayaraman and Evans (2019), our used metric for measuring privacy leakage through MIAs is the attacker's membership advantage as introduced by Yeom et al. (2018). The adversarial game is based on an attacker's capabilities in differentiating the membership of a sample that is chosen uniformly at random to originate from the training set or not. The resulting difference in True Positive Rate (TPR) and False Positive Rate (FPR) is then given as the attacker's membership advantage: Yeom et al. (2018) show that if a learning algorithm satisfies ε-DP, then the adversary's membership advantage is bounded by Adv M ≤ e ε −1 in their attack scenario. Transferring the theorem to (ε,δ)-DP given by Equation (1), the upper bound can be derived as:
Adv M = T PR − FPR (2)Adv M ≤ e ε − 1 + δ(3)
Because the theoretical assumption relies on Gaussian distributed training errors and a balanced prior data distribution probability, it might not provide reliable bounds given our differing practical scenario. Since individual MIA results are subject to variability, they need to be experimentally stabilized. Like Malek et al. (2021), we achieve this by running 100 entire MIAs and calculating the corresponding 95% Confidence Interval (CI) for the obtained results. Table 3: Experimental results on the COVID-19 dataset. The Standard, ImageNet and Pneunomia models rely on the ReLU activation function, which is then changed to tanh in the respective counterparts. Model variants are evaluated across multiple DP budgets ε, where ε = ∞ translates to non-private training. They are matched by accuracy and F1-score in %, as well as empirical privacy leakage from MIAs, measured by the membership advantage (Adv M ) and given as a 95% CI over 100 attacks. If training resulted in an F1-score of 0.0, no feasible model was derived, making accuracy and attacks obsolete (NA). Table 3 caption for details. F1-score is given as the macro average over the 10 classes. Pneumonia pre-trained models are omitted from the evaluation, since the tasks are not closely related.
ε = ∞ ε = 10 ε = 1 ε = 0.1 Variant %-Acc. %-F1 Adv M %-Acc. %-F1 Adv M %-Acc. %-F1 Adv M %-Acc. %-F1 Adv MResNet18ε = ∞ ε = 10 ε = 1 ε = 0.1 Variant %-Acc. %-F1 Adv M %-Acc. %-F1 Adv M %-Acc. %-F1 Adv M %-Acc. %-F1 Adv MResNet18
RESULTS
In this section, we present the outcomes of our experiments on the COVID-19 and MNIST tasks. Table 3 summarizes our results in performance (accuracy, F1-score) and practical privacy (empirical privacy leakage as Adv M ) over all privacy levels. The non-private baseline at ε = ∞ is dominated by the ImageNet variants, which reaches 95.9% F1 for ResNet18. The non-private tanh models generally show to perform less than their ReLU counterparts, making tanh less suited for non-private training. Pneumonia pre-training does present an upgrade but smaller than ImageNet. For the weakest privacy setting of ε = 10, we already find a steep utilityprivacy trade-off of 16.2%, when comparing the best F1-scores. The ResNet18 ImageNet keeps the best performance. However, the other tanh models now surpass their ReLU siblings. The ResNet50 ReLU models struggle so much, that they do not provide a working model (0.0% F1). For both ResNet18 and ResNet50, the tanh-ImageNet now falls far behind.
Results on the COVID-19 Database
Our results at ε = 1 present a substantial paradigm shift. The ResNet18 ImageNet, our former winner, stops working at this level, with the only working ReLU model left being the ResNet18 Penumonia Thus, at this stronger privacy setting, the tanh activation is essential to achieve feasible models for all variants. We now see the tanh-Pneumonia delivering the best performing models with 72.5% F1 on ResNet50 and 75.2% accuracy on ResNet18. From ε = 10 to ε = 1, we lose 6.9% F1. At ε = 0.1, the ResNet50 tanh-Pneumonia performs best with 69.4% F1 and 73.0% accuracy. The ResNet18 sibling is still close in accuracy but falls further behind in F1. On the ResNet50, the resulting trade-off of 3.1% F1 compared to ε = 1 is less significant than before.
Regarding privacy leakage given as Adv M , we only see minor improvements when comparing nonprivate and private settings. The initial observation is underlined by mean advantages across privacy levels: 0.23-0.25 (ε = ∞), 0.22-0.23 (ε = 10), 0.20-0.22 (ε = 1), and 0.20-0.22 (ε = 0.1). The differences are especially small in the better models. The lower performance models, however, offer less confidence in their predictions, making it harder for an attacker to correctly classify membership (Rahman et al., 2018).
Results on the MNIST Database
We now refer to Table 4 and omit the Pneumonia variants because there is no relation between the tasks.
The non-private performance trends show that MNIST is an easier task than COVID-19 detection because only one model falls short of reaching 99% F1. Furthermore, the tanh models are on par with the ReLu models even in the non-private setting. Familiarly to the COVID-19 task, the ResNet18 Im-ageNet performs best at 99.5% F1, but shares the crown with both Standard variants. In private training, the ResNet50 ReLU models again start to fail at ε = 10. They still achieve some F1 but scores of 14.5% and 10.1% can also be seen as unusable.
The other models, however, only see a smaller reduction in their performance than on COVID-19. The tanh-ImageNet shines on MNIST, with the ResNet18 model achieving the best results at 97.8% F1 and accuracy, which is just a 1.7% reduction from ε = ∞. The ResNet50 counterpart closely follows at 97.7%. The tanh-ImageNet superiority carries over to ε = 1, where the ResNet18 tanh-ImageNet reigns at 96.7% F1 and accuracy. This again results in a marginal trade-off of 1.1% from ε = 10 to ε = 1. However, other than at ε = 10, the ImageNet on ReLU now also fails to deliver good F1 at 35.2% on ResNet18. The marginal trade-offs cannot be kept for ε = 0.1, where the ResNet50 tanh-ImageNet beats the ResNet18 sibling but now loses 3.4% F1 and lands at 93.2%. The total F1 trade-off from ε = ∞ to ε = 0.1 is still low at 6.3% and considerably better than on the COVID-19 task. F1 for ResNet18 Standard and ImageNet on ReLU decreases steeply at ε = 0.1, delivering rather unusable models. Even the ResNet50 tanh-Standard now falls to a low of 25.4% F1.
Shifting the view to our privacy analysis, nonprivate models on MNIST show a slightly lower proneness to MIAs than on COVID-19. For models who only achieve low F1-scores, we again see lower Adv M values stemming from their corresponding low memorization (Rahman et al., 2018). The better performing variants, on the other hand, again show just minor changes in leakage measurements. When we exclude the misleading low-performing results with under 50% F1, the following mean advantages confirm our point: 0.19-0.21 (ε = ∞), 0.18-0.20 (ε = 10), 0.19-0.21 (ε = 1), and 0.17-0.19 (ε = 0.1). We thus only find a small leakage difference between COVID-19 and MNIST, with COVID-19 models being slightly more prone in general.
DISCUSSION
We now revisit the open gaps from the related work discussed in Section 3 and review the outcomes of our proposed solutions from Section 4. We again refer to Tables 3 and 4 for our evaluation results and to Table 1 for a short and organized showcase of the compared methods from related work (Müftüoglu et al., 2020;Zhang et al., 2021;Ho et al., 2022). In the following we evaluate our proposed improvements:
Datasets. We achieve a more balanced data basis than before by utilizing undersampling and class weights. To better evaluate on the still skewed data, we add the F1-score metric. The advantage regarding accuracy is visible in the COVID-19 results, where both metrics differ regularly and F1 thus reveals models that perform better on the minority class COVID-19. That F1 better represents the underlying class distribution is further demonstrated by comparison to the more balanced MNIST dataset, where accuracy and F1-score are almost identical.
Privacy budgets. In contrast to related work, we are able to achieve working COVID-19 detection models, while adhering to strong privacy budgets of ε ≤ 1. By additionally evaluating different architectures over multiple privacy levels, we deduce favorable architectural decisions for keeping good utilityprivacy trade-offs in DP-SGD. Our findings for training private models are summarized in Section 7.1.
Practical privacy. By including an empirical study on practical privacy though MIAs, we gain insights into the relationship between DP and privacy leakage. In Section 7.2 we derive the implications stemming from our empirical analysis. The results allow us to improve the utility-privacy trade-off while keeping the same practical privacy.
Building Better DP-SGD Models
With our experiments, we transfer the results from Papernot et al. (2021) to deeper and pre-trained networks and are able to confirm the tanh advantage over ReLU in low ε DP-SGD training. Our private models with strong ε-guarantees of ε = 1 and ε = 0.1 rely on this change, while the non-private and less private models still prefer the ReLU activation function.
A commonality between our best performers is that they were subjected to pre-training. While all best non-private models are pre-trained on ImageNet, this trend only continues in all private models on the MNIST database. The same ImageNet-based models underperform on the COVID-19 task, when looking at settings of ε = 1 and ε = 0.1, which might be related to the different contents in both tasks. On the COVID-19 task, we introduce task-specific pretraining on pneumonia images, that leads to superior performance in our most private settings.
We could not fully confirm that larger models perform worse in DP-SGD . The ResNet50 especially wins at the most private setting of ε = 0.1. Model size, however, seems to play a role, when examining the earlier failure of the private ReLU models in the bigger ResNet50.
In summary, our results support the existing belief that model architectures should be specifically adjusted for private DP-SGD training, where established standards from non-private training do not necessarily provide the same advantages Abadi et al., 2016). Examples are the switch from ReLU to tanh activation and the superiority of Pneumonia pre-training to ImageNet pre-training in the private COVID-19 models.
Insights Regarding Practical DP
For this section, we refer to Figure 2 that visualizes the results for our estimated privacy leakage from MIAs at the different ε-budgets.
In both Figures 2a and 2b, we include the (ε,δ)-DP bound on Adv M from Equation (3), which is based on Yeom et al. (2018). The bound already surpasses our plotted maximum of 0.5 Adv M long before ε = 1, which shows the large discrepancy between the theoretically assumed worst-case and practice. Simultaneously, no model actually trained for ε = 0.1 is able to conform to the calculated bound. Such inconsistencies can also be found in related work (Yeom et al., 2018;Jayaraman and Evans, 2019). As an explanation, Yeom et al. (2018) unveil that, in practice, the training set error distributions are not exactly Gaussian, sometimes leading to better attack performance than predicted by theory. Even though COVID-19 and MNIST have rather opposing priors, where the former's classes are skewed and the latter's roughly balanced, we see the same inconsistencies in both evaluations. Thus, the given theoretical bound does not seem reliable for deriving a limit on the real world threat in our case. For both COVID-19 and MNIST, the leakage almost describes a flat line with just negligible changes over all privacy settings. We spot a few outliers 10 , that see a bigger drop in leakage risk, which however, is mainly attributed to their gravely lowered performance (>20% F1 loss) and accordingly reduced memorization (Rahman et al., 2018). Even the non-private models exhibit almost the same leakage as the private models and thus, including DP-guarantees does not imply the expected improvement to practical MIA proneness. The plateau in privacy leakage can en-able the use of lesser DP-guarantees, while still providing the same practical privacy The MNIST models show to generally leak slightly less than on COVID-19, leading to stronger privacy needs for COVID-19. The existing difference in MIA risk between COVID-19 and MNIST suggests, that privacy estimation can be an important tool for assessing task-and data-dependent threats from attacks. Thus, such estimates can in turn support tuning trade-offs according to task-specific privacy needs.
The findings suggest room for utilizing weaker DP-guarantees on both tasks when defending against our MIA-specific setting. Practical privacy is already strong in our less private and even non-private models. We are thus able to improve the utility-privacy trade-off on both datasets at no practical privacy cost. We could introduce even stronger guarantees to possibly further improve MIA defense, however, this would lead to an even bigger utility loss and in turn result in impractical performance.
We want to emphasize that there is still a need for strong theoretical privacy guarantees (Nasr et al., 2021). As stated in Section 3.2, ε-guarantees from DP limit the maximum amount of possible information leakage. In actual attacks, however, the theoretical ceiling might differ notably from the practical threat as shown in this and other works presented in Section 3.3. From this, we should not conclude that DP is unnecessary, since future adversaries could find better attacks that make an earlier empirical evaluation invalid. To cover for such cases it is therefore advised to keep a reasonably strong privacy guarantee even when tuning for better trade-offs. Thus, we would rather choose a COVID-19 model at ε = 10 than at ε = ∞, even though both exhibit almost the same practical privacy levels. The model at ε = 10 performs better than the one at ε = 1 and, in contrast to ε = ∞, still provides a provable DP guarantee to limit future adversaries.
CONCLUSION
Within this piece of work, we close several open gaps in the field of private COVID-19 detection from X-ray images. In comparison to related work on the topic, we improve data handling regarding imbalances, deliver a more robust privacy evaluation, and are the first to investigate the implications concerning practical privacy (Müftüoglu et al., 2020;Zhang et al., 2021;Ho et al., 2022).
We introduce a selection of yet untested architectural ML model choices to the COVID-19 task. Through our evaluation, we are able to compare the setups in a common environment. Since well-known practices from non-private training are not always transferable to DP-SGD training, it is important to gather a wide range of results for finding the best models. We are therefore making a noticeable contribution by exploring a range of different architectures on the COVID-19 and MNIST tasks.
Our practical privacy analysis reveals that assessing attack-specific threats from black-box MIAs in a practical scenario helps finding appropriate privacy attributes and can thus improve the utility-privacy trade-off at no practical cost. On both the COVID-19 and MNIST datasets, we found just minor improvements from the provided theoretical DP-guarantee regarding practical defense against our MIAs. Instead, our tested models almost showed the same strong repelling properties across all privacy levels-even for non-private models. By confirming this plateau for both datasets, we are able to reduce the required DP guarantees for both tasks without sacrificing attackspecific practical privacy. Our attacks are slightly more successful on the COVID-19 task, showing that it needs stricter privacy than MNIST and that practical privacy analysis is important for identifying the task-specific initial MIA threat.
We still advocate the use of DP and would not recommend to risk publishing non-private COVID-19 detection models. Instead, if justified by a practical privacy analysis, the ε-guarantee can be tuned to a more favorable utility-privacy trade-off that through the inclusion of a reasonable DP-guarantee still limits the worst-case privacy leakage from future attacks. As a brief outlook into possible future work, it would be beneficial to extend our evaluation by applying practical privacy analysis to more datasets, especially with different underlying tasks. Another venture could be to derive best practices and ultimately a taxonomy regarding advantageous architectural decisions when training DP-SGD models.
Figure 1 shows chest X-ray scans of healthy (top) and COVID-19 (bottom) patients in direct comparison. Even though patchy consolidations are Transcription Polymerase Chain Reaction (RT-PCR) testing is broadly used for COVID-19 diagnosis.
Figure 1 :
1Chest X-ray images of different patients extracted from the COVID-19 Radiography Database
Figure 2 :
2Empirical privacy leakage results from MIAs are given as our 95% CI membership advantage (Adv M ) and plotted across the different privacy budgets. Model variants can be distinguished with the legend. We exclude data points with <50% F1 because low performance disproportionately reduces leakage. A dotted line shows the DP bound fromYeom et al. (2018).
Table 2 :
2Summary of the experiment parameters. Each combination from left to right constitutes a possible setup (resulting in 2 * 2 * 2 * 3 * 4 = 96 setups).Dataset
Architecture Activation
Pre-training
ε
COVID-19
ResNet18
ReLU
None/Standard
∞
MNIST
ResNet50
tanh
ImageNet
10
Pneumonia
1
0.1
Table 4 :
4Experimental results on the MNIST database. See
RELATED WORKIn the following, we first describe gaps left open by related work in Section 3.1. We then show mitigation strategies for MIAs and methods of practical privacy analysis in Sections 3.2 and 3.3, respectively.
They do not state their exact number of rounds but their graphs show 500 rounds.
https://github.com/luckyos-code/mia-covid 4 https://keras.io/ 5 https://www.tensorflow.org/ 6 https://github.com/tensorflow/privacy 7 https://www.kaggle.com/tawsifurrahman/covid19radiography-database 8 https://www.kaggle.com/paultimothymooney/chestxray-pneumonia
On COVID-19 the outliers are both tanh-ImageNet models, which reduce their leakage from non-private to ε = 10, and the ResNet50 tanh-Standard doing the same from ε = 10 to ε = 1. There is also one outlier on MNIST, where the Resnet18-tanh-Standard improves privacy at ε = 0.1.
ACKNOWLEDGMENTSWe thank our colleagues for insights on earlier drafts. The authors acknowledge the financial support by the Federal Ministry of Education and Research of Germany and by the Sächsische Staatsministerium für Wissenschaft Kultur und Tourismus in the program Center of Excellence for AI-research "Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig", project identification: ScaDS.AI. Computations were done (in part) using resources of the Leipzig University Computing Centre.ETHICAL PRINCIPLESAll patient data originated from public sources provided for research purposes and was solely used within the limited scope of this work.APPENDIX A PROOF OF EQUATION 3In the following Theorem 1, we formally proveEquation (3). With the prerequisites of Experiment 1 and Definition 4 fromYeom et al. (2018), we adapt their Theorem 1 and corresponding proof from ε-DP to (ε,δ)-DP. Theorem 1. Let A be an (ε,δ)-differentially private learning algorithm, A be a membership adversary, Adv M the membership advantage of A, n be a positive integer, and D be a distribution over data points (x, y). Then we have:Proof. According to Definition 4 inYeom et al. (2018), Adv M (A, A, n, D) can be expressed as the difference between A's true and false positive rates(4)where Adv M is a shortcut for Adv M (A, A, n, D).Given S = (z 1 , . . . , z n ) ∼ D n and an additional point z ∼ D, define S (i) = (z 1 , . . . , z i−1 , z , z i+1 , . . . , z n ). Then, A(z , A S , n, D) and A(z i , A S (i) , n, D) have identical distributions for all i ∈ [n], so we can write:The above two equalities, combined with Equation 4, gives:Without loss of generality for the case where models reside in an infinite domain, assume that the models produced by A come from the set {A 1 , . . . , A k }. (ε,δ)-DP guarantees that for all j ∈ [k],Using this inequality, we can rewrite and bound the right-hand side of Equation 5 aswhich is at most e ε −1 +δ since A(z,A j , n, D) ≤ 1 for any z, A j , n, and D.
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| {'fraction_non_alphanumeric': 0.04993062781824489, 'fraction_numerical': 0.023812001387443636, 'mean_word_length': 4.4889100428367446, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 7, 'lorem ipsum': 0, 'www.': 3, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Machine learning (ML) can help fight pandemics like COVID-19 by enabling rapid screening of large volumes of images. To perform data analysis while maintaining patient privacy, we create ML models that satisfy Differential Privacy (DP). Previous works exploring private COVID-19 models are in part based on small datasets, provide weaker or unclear privacy guarantees, and do not investigate practical privacy. We suggest improvements to address these open gaps. We account for inherent class imbalances and evaluate the utility-privacy trade-off more extensively and over stricter privacy budgets. Our evaluation is supported by empirically estimating practical privacy through black-box Membership Inference Attacks (MIAs). The introduced DP should help limit leakage threats posed by MIAs, and our practical analysis is the first to test this hypothesis on the COVID-19 classification task. Our results indicate that needed privacy levels might differ based on the task-dependent practical threat from MIAs. The results further suggest that with increasing DP guarantees, empirical privacy leakage only improves marginally, and DP therefore appears to have a limited impact on practical MIA defense. Our findings identify possibilities for better utility-privacy trade-offs, and we believe that empirical attack-specific privacy estimation can play a vital role in tuning for practical privacy.', 'arxivid': '2211.11434', 'author': ['Lucas Lange \nLeipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany\n', 'Maja Schneider \nLeipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany\n', 'Peter Christen [email protected] \nThe Australian National University\nCanberraAustralia\n', 'Erhard Rahm \nLeipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany\n', 'Lucas Lange \nLeipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany\n', 'Maja Schneider \nLeipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany\n', 'Peter Christen [email protected] \nThe Australian National University\nCanberraAustralia\n', 'Erhard Rahm \nLeipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany\n'], 'authoraffiliation': ['Leipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany', 'Leipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany', 'The Australian National University\nCanberraAustralia', 'Leipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany', 'Leipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany', 'Leipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany', 'The Australian National University\nCanberraAustralia', 'Leipzig University & ScaDS.AI\nDresden/Leipzig, LeipzigGermany'], 'corpusid': 253734943, 'doi': '10.48550/arxiv.2211.11434', 'github_urls': ['https://github.com/qubvel/classification_models', 'https://github.com/luckyos-code/mia-covid', 'https://github.com/tensorflow/privacy'], 'n_tokens_mistral': 15804, 'n_tokens_neox': 13873, 'n_words': 8859, 'pdfsha': '64313a5536acadc4a48d2c6acd621f0db8cdb71a', 'pdfurls': ['https://export.arxiv.org/pdf/2211.11434v4.pdf'], 'title': ['Privacy in Practice: Private COVID-19 Detection in X-Ray Images Extended Version', 'Privacy in Practice: Private COVID-19 Detection in X-Ray Images Extended Version', 'Privacy in Practice: Private COVID-19 Detection in X-Ray Images Extended Version', 'Privacy in Practice: Private COVID-19 Detection in X-Ray Images Extended Version'], 'venue': []} |
arxiv |
A new helioseismic constraint on a cosmic-time variation of G
Alfio Bonanno
INAF
Osservatorio Astrofisico di Catania
via S. Sofia, 7895123CataniaItaly
Hans-Erich Fröhlich
Leibniz Institute for Astrophysics Potsdam (AIP)
An der Sternwarte 1614482PotsdamGermany
A new helioseismic constraint on a cosmic-time variation of G
10.1140/epjh/e2016-70034-0
Helioseismology can provide strong constraints on the evolution of Newton's constant over cosmic time. We make use of the best possible estimate of 8640 days of low-BiSON data, corrected for the solar cycle variation, to obtain a new constraint on an evolving gravitational constant. In particular, by means of a Bayesian analysis we conclude thatĠ/G today = (1.25±0.30)×10 −13 yr −1 . Our result, a 4-σ effect, is more than one order of magnitude stronger than previous constraints obtained with helioseismology. We also take into account possible systematic effects by considering the theoretical uncertainties on the efficiency of the proton-proton (pp) fusion cross-section. We show that models with variable G significantly outclass models with no secular variation of G, viz by a Bayes factor exceeding 30.
INTRODUCTION
The idea that the Sun can be considered a laboratory for fundamental physics traces back to the early developments in nuclear physics by contributing to the understanding of the basic nuclear processes involved in stellar nucleosynthesis. In recent times, accurate measurements of acoustic p-mode spectrum combined with inversion techniques have further stressed this role [1]. Important examples are the investigation of the equation of state [2], the discovery of neutrino flavour oscillations [3,4], the properties of Dark Matter [5][6][7][8], the constraints on axions emission [9,10] the properties of the screening of nuclear reaction rates [11,12] and constraints on physical constants [13].
A fundamental problem that can be tackled by means of helioseismology is the possibility of limiting secular variations of G, a possibility argued long ago by Dirac [14] and Milne [15]. This initial intuition has been further elaborated in [16,17] and is nowadays an important ingredient of various scalar-tensor theories [18], quantumgravity inspired models of modified gravity [19,20], and string theory low-energy models [21].
In this context a widely used approach to promote the gravitational constant to a dynamical variable is to extend the general relativistic framework in which gravity is mediated by a massless spin-2 graviton, to include a spin-0 scalar field which couples universally to matter fields. As the universality of free-fall is maintained theories that predict that the locally measured gravitational constant vary with time often violate the equivalence principle in its strong form. For this reason empirical constraints onĠ/G today , where the dot indicates a derivative with respect to the cosmic time t, have been obtained in several contexts [22,23]. Current limits oṅ G/G today span fromĠ/G today = (4 ± 9) × 10 −13 yr −1 obtained from the Lunar Laser Ranging (LLR) experiment [24], to −3 × 10 −13 <Ġ/G today < 4 × 10 −13 yr −1 from BBN [25], orĠ/G today ∼ 10 −12 yr −1 from white dwarfs [26].
Helioseismology is able to provide independent constraints on possible time evolution of the gravitational constant G over cosmic time because the stellar luminosity L varies as ∼ G 7 [27]. For example, a monotonically increasing Newton's constant must be compensated for a systematic decrease of core temperature and a corresponding change in the hydrogen abundance in order to match L , the solar radius R and the metal to hydrogen abundance ratio (Z/X) . In [28] a direct comparison of low-degree p-modes to GONG data has allowed us to ob-tainĠ/G today ≤ 1.6 × 10 −12 yr −1 , assuming a power-law of the type G(t) ∝ t −α . In this paper we shall present a new limit onĠ/G today based on a bayesian approach which makes use of the definitive "best possible estimate" of 8640 days of low-frequency BiSON data, corrected for the solar cycle modulation [29].
SOLAR MODELS AND MODEL UNCERTAINTIES
In this context it is important to reduce as much as possible any source of systematic uncertainties in the input physics of the calibrated solar models in order to obtain a significant constraint onĠ/G today . From this point of view the main problem is clearly our ignorance of the efficiency of the proton-proton (pp) fusion cross-section for which only theoretical estimates are available. An uncertainty of ±3% on the value of S pp (0), the astrophysical S-factor at zero energy, is quoted in [30], in particular. Therefore bothĠ/G today as well as S pp (0), have been estimated from the data in a Bayesian manner.
Our solar models are built using the Catania version of the GARSTEC code [31,32], a fully-implicit 1D code including heavy-elements diffusion and updated input physics. We prescribed the time evolution of the gravitational constant as a power-law [28,33]
G(t) = G 0 t 0 t α(1)
where G 0 is the cosmologically recent value of Newton's constant according to 2010 CODATA so that G 0 = 6.67384 × 10 −8 cm 3 g −1 s −2 and t 0 = 13. [37] are employed and the nuclear reaction rates are taken from the compilation in [30]. Our starting models are chemically homogeneous PMS models with log L/L = 0.21 and log T e = 3.638 K, thus close to the birth line of a 1M object. Initial Helium fraction, (Z/X) and mixing-length parameter are adjusted to match the solar radius R = 6.95613 × 10 10 cm (based on an average of the two values and quoted error bar in Table 3 of [38]), the solar luminosity L = 3.846 × 10 33 erg s −1 [34] and the chemical composition of [39] with (Z/X) = 0.0245 at the surface. We also employed the new accurate meteoritic estimate of the solar age of [40], t = 4.567 Gyr, a value consistent with the helioseismic solar age [41]. We further noticed that models with the so-called "new abundances" for which (Z/X) = 0.0178 [42] would lead to much smaller Bayes factors and we decided not to discuss these models in this work.
In order to define a proper seismic diagnostic we adopted a widely used approach: if ν n,l is the frequency of the mode of radial order n and angular degree , the frequency separation ratios r l,l+2 (n) = ν n,l − ν n−1,l+2 ν n,l+1 − ν n−1,l+1 (2) can be shown to be localized near the core and weakly dependent on the complex physics of the outer layers [43,44]. In particular in the limit n 1
r , +2 (n) ≈ −(4 + 6) 1 4π 2 ν n, R 0 dc s dR dR R (3)
so that a change in temperature (T ) and mean molecular weight (μ) directly impacts on the r , +2 (n) terms as δc s /c s ≈ 1 2 δT /T − 1 2 δμ/μ.
BAYESIAN APPROACH
We consider the following two-dimensional parameter space: −0.1 ≤ α ≤ 0.1 and 0.97 ≤ S/S pp (0) ≤ 1.03. The proposed α range generously covers all previouṡ G/G today limits obtained by independent methods [33]. Moreover, the S interval 0.97-1.03 allows for an up to ±3% deviation from the recommended value S pp (0) = (4.01 ± 0.04) × 10 −22 keV b in [30].
Central to the Bayesian hypothesis testing is the likelihood. In the following, a Gaussian has been assumed,
L(α, S) = N =17 i=1 1 √ 2πσ i exp − (d i − m i (α, S)) 2 2σ 2 i ,(4)
where d i = r 02 (n) are the observed data (n = i + 8, i = 1 . . . N, N = 17), m i the theoretical model values, and σ i the errors (see also [41] for an application of this likelihood to the helioseismic determination of the solar age). All 17 contributions enter the likelihood with the same weight.
The posterior probability distribution is the likelihood (4) weighted with a prior distribution. Obviously, this prior distribution should be a flat one compared to α. Concerning S we decided to take a conservative point of view, i. e. that nothing is known about S pp (0). In that case we are on the safe side and the only eligible prior distribution is a flat one over the logarithm, log(S).
In the end two hypotheses have to been compared: H 1 = H 1 (−0.1 ≤ α ≤ 0.1, 0.97 ≤ S/S pp (0) ≤ 1.03) vs. our zero hypothesis H 0 (α = 0, 0.97 ≤ S/S pp (0) ≤ 1.03).
RESULTS
The posterior probability distribution is indistinguishable from a two-dimensional Gaussian (Fig. 1). The reason is that the theoretical models m i (α, log(S)) are linearly dependent on both α and log(S) (cf. [45]) as we checked in all our models. From α's marginal distribution one reads its mean value and standard deviation: α = −0.0017 ± 0.0004. Formally, this is a 4-σ effect. With t 0 = 13.7 Gyr this translates tȯ G/G today = (1.25 ± 0.30) × 10 −13 yr −1 . As a by-product one gets log(S/S pp (0)) = 0.011±0.008 and a correlation coefficient of -0.62. An enhanced S goes with a reduced α. However, the indicated slight enhancement of Adelberger et al. [30] pp cross-section by 1% proves insignificant. Our result is one order of magnitude stronger than the limit obtained in [28] and comparable in precision to those obtained with LLR [24] or BBN [25].
Integrating the posterior over the whole parameter space or subsections of it, respectively, one gets the required evidences. The evidence in favour of a hypothesis is the prior-weighted mean of the likelihood over parameter space. The ratio of the evidences, E(H 1 )/E(H 0 ), the so-called Bayes factor amounts to 34.0. (If one trusts the relative error in the recommended S pp (0) and applies the appropriate Gaussian prior, this Bayes factor would increase to 51.1.) Despite one parameter more, the α = 0 hypothesis significantly outclasses the zero hypothesis, i. e. no secular variation of G -provided the S factor is the sole and decisive unknown.
FIG. 1
1. 2-dimensional posterior probability distribution. The closed contour with a probability density of 10.3 per cent of the peak density comprises 90 per cent of the total probability. The hatched strip marks the 68.3-per-cent (± 1-σ) interval of α's marginal distribution.
7 Gyr is a reference age of the Universe according to most of ΛCDM estimates. As G 0 M ≡ 1.32712440 × 10 26 cm 3 s −2 [34] is fixed, M = 1.98855×10 33 g is assumed. Irwin's equation of state [35] with OPAL opacities for high temperatures [36] and Ferguson's opacities for low temperatures
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arxiv |
Collapse-in and Collapse-out in Partial Measurement in Quantum Mechanics and its WISE Interpretation
23 May 2021
Gui-Lu Long
Department of Physics
State Key Laboratory of Low-dimensional Quantum Physics
Tsinghua University
100084BeijingChina
Beijing Academy of Quantum Information Sciences
100193BeijingChina
Collaborative Innovation Center of Quantum Matter
100084BeijingChina
Beijing National Research Center for Information Science and Technology
100084BeijingChina
Collapse-in and Collapse-out in Partial Measurement in Quantum Mechanics and its WISE Interpretation
23 May 2021(Dated: June 2, 2021)arXiv:2106.00466v1 [physics.gen-ph]
One central issue in quantum mechanics is the relation between the wavefunction and the quantum system it describes. As quantum mechanics is understood in different ways, the wavefunction is given various explanations. Some regard the wavefunction as "epistemic", that is, something reflected in the human mind, and some regard it as "ontological", i.e., something realistic. The orthodox interpretation of quantum mechanics of Copenhagen [1, 2] is epistemic and treats the wavefunction merely as a mathematical quantity. Recent examples of the ontological interpretation include the random discontinuous motion[3], "Wavefunction Is the System Entity"(WISE) interpretation[4], and information complete interpretation[5]. WISE treats the wavefunction equivalently as the quantum system itself, that is, the quantum system is just the wavefunction, and the wavefunction is just the quantum system. These two are exactly the same. The quantum system, which is also the wavefunction, can exist in disjoint regions of space, travel at a finite speed, and collapse upon measurements. An encounter-delayedchoice experiment has been proposed and experimentally demonstrated recently[6].In this short communication, we will concentrate on the partial measurement issue and give an explanation concerning the WISE interpretation. The essential idea of WISE is given in Ref.[4], together with the linear combination of unitaries (LCU) formalism of quantum computing. LCU has now become one of the major techniques in quantum algorithm design. The quantum circuit implementation of LCU is given in Refs.[7,8], and a review of the subject is given in Ref.[9].Partial measurement postulate. We recall first the measurement postulate in standard quantum mechanics. If a particle is in state |Ψ , a measurement of the variable (corresponding to) Ω will yield one of the eigenvalues ω with probability P (ω) ∝ | ω|Ψ | 2 . The state of the system will change from |Ψ to |ω as a result of the measurement[10].What will happen if the measurement is on part of the wave function (partial measurement) rather than on a full wave function (full measurement)? For instance, in a three-slits system, if one places a detector immediately after the first slit and places no detectors in the remaining * [email protected] two slits, then a partial measurement is established. Here we give the details of the partial measurement.Using the quantum circuit realization in Refs.[7,8], the wavefunction of an electron passing through a d-slit is represented as |Φ = d i=1 c i |ψ i |i , where |i represents the i-th slit, and |ψ i is the sub-wavefunction in the ith slit. For simplicity, we assume that |ψ i = |ψ . The wavefunction is normalized, namely i |c i | 2 = 1.In the WISE interpretation, these sub-waves as a whole form an "electron". Thus, it is easy to comprehend that an "electron" passes through the d-slits simultaneously. The "electron" is no longer a rigid sphere. It is distributed in space, even disjointedly. It changes its shape as the wavefunction changes.If one places a detector just after slit-1, then there is a probability |c 1 | 2 that the detector will measure the electron, and the whole wavefunction will collapse into slit-1. What would happen if the detector at slit-1 does not get a result? To this end, we rewrite |Ψ asEq. (1) is like a double-slit experiment with slit-1 and slit-S2. If we imaginarily place two detectors, one at slit-1 and one at slit-S2, then there is a probability |c 1 | 2 that the detector will measure the electron at slit-1, and the whole wavefunction will collapse into slit-1, and a probability 1-|c 1 | 2 that the detector at nominal slit-S2 will measure the electron. By this heuristic reasoning, we give our partial measurement postulate, which is a novel development of the measurement postulate of standard quantum mechanics: Suppose a quantum system is in state, ω i is one of the eigenvalues of observable Ω, and |ω i is the corresponding eigenvector. If part of the wavefunction, M i=1 c i |ω i , is measured in variable Ω , then the result of the measurement will be one of the following:(1) Collapse-in: One eigenvalue ω i will be obtained with probability | ω i |Ψ | 2 = |c i | 2 , where 1 ≤ i ≤ M . After the measurement, the state of the system will change instantly from |Ψ into |ω i .
One central issue in quantum mechanics is the relation between the wavefunction and the quantum system it describes. As quantum mechanics is understood in different ways, the wavefunction is given various explanations. Some regard the wavefunction as "epistemic", that is, something reflected in the human mind, and some regard it as "ontological", i.e., something realistic. The orthodox interpretation of quantum mechanics of Copenhagen [1,2] is epistemic and treats the wavefunction merely as a mathematical quantity. Recent examples of the ontological interpretation include the random discontinuous motion [3], "Wavefunction Is the System Entity"(WISE) interpretation [4], and information complete interpretation [5]. WISE treats the wavefunction equivalently as the quantum system itself, that is, the quantum system is just the wavefunction, and the wavefunction is just the quantum system. These two are exactly the same. The quantum system, which is also the wavefunction, can exist in disjoint regions of space, travel at a finite speed, and collapse upon measurements. An encounter-delayedchoice experiment has been proposed and experimentally demonstrated recently [6].
In this short communication, we will concentrate on the partial measurement issue and give an explanation concerning the WISE interpretation. The essential idea of WISE is given in Ref. [4], together with the linear combination of unitaries (LCU) formalism of quantum computing. LCU has now become one of the major techniques in quantum algorithm design. The quantum circuit implementation of LCU is given in Refs. [7,8], and a review of the subject is given in Ref. [9].
Partial measurement postulate. We recall first the measurement postulate in standard quantum mechanics. If a particle is in state |Ψ , a measurement of the variable (corresponding to) Ω will yield one of the eigenvalues ω with probability P (ω) ∝ | ω|Ψ | 2 . The state of the system will change from |Ψ to |ω as a result of the measurement [10].
What will happen if the measurement is on part of the wave function (partial measurement) rather than on a full wave function (full measurement)? For instance, in a three-slits system, if one places a detector immediately after the first slit and places no detectors in the remaining * [email protected] two slits, then a partial measurement is established. Here we give the details of the partial measurement.
Using the quantum circuit realization in Refs. [7,8], the wavefunction of an electron passing through a d-slit is represented as |Φ = d i=1 c i |ψ i |i , where |i represents the i-th slit, and |ψ i is the sub-wavefunction in the ith slit. For simplicity, we assume that |ψ i = |ψ . The wavefunction is normalized, namely i |c i | 2 = 1.
In the WISE interpretation, these sub-waves as a whole form an "electron". Thus, it is easy to comprehend that an "electron" passes through the d-slits simultaneously. The "electron" is no longer a rigid sphere. It is distributed in space, even disjointedly. It changes its shape as the wavefunction changes.
If one places a detector just after slit-1, then there is a probability |c 1 | 2 that the detector will measure the electron, and the whole wavefunction will collapse into slit-1. What would happen if the detector at slit-1 does not get a result? To this end, we rewrite |Ψ as
|Φ = c 1 |ψ |1 + 1 − |c 1 | 2 |ψ |S2 ,(1)where |S2 = d i=2 c i 1 − |c 1 | 2 |i .(2)
Eq. (1) is like a double-slit experiment with slit-1 and slit-S2. If we imaginarily place two detectors, one at slit-1 and one at slit-S2, then there is a probability |c 1 | 2 that the detector will measure the electron at slit-1, and the whole wavefunction will collapse into slit-1, and a probability 1-|c 1 | 2 that the detector at nominal slit-S2 will measure the electron. By this heuristic reasoning, we give our partial measurement postulate, which is a novel development of the measurement postulate of standard quantum mechanics: Suppose a quantum system is in state
M i=1 c i |ω i + d j=M+1 u j |ω j , where i |c i | 2 + j |u j | 2 = 1
, ω i is one of the eigenvalues of observable Ω, and |ω i is the corresponding eigenvector. If part of the wavefunction, M i=1 c i |ω i , is measured in variable Ω , then the result of the measurement will be one of the following:
(1) Collapse-in: One eigenvalue ω i will be obtained with probability | ω i |Ψ | 2 = |c i | 2 , where 1 ≤ i ≤ M . After the measurement, the state of the system will change instantly from |Ψ into |ω i .
(2) Collapse-out: The part of the wavefunction being measured will disappear, and the state of the system will change instantly to the unmeasured part, namely, with
probability 1 − M i=1 |c i | 2 , |Ψ → 1 1 − M i=1 |c i | 2 d j=M+1 u j |ω j .(3)
Collapse-in and collapse-out of partial measurement happen randomly not only in space, but also over time.
Though it is seldom discussed, partial measurement appears very often in reality. For instance, the detection of a photon by a detector can be naturally understood in terms of this partial measurement postulate. When the wavefunction of a photon goes to a detector, it is not measured as a whole at the same time. Its front part arrives at the detector first, hitting some area of the detector. It can collapse-in at any point of the intersecting area, with respective probabilities (randomly in space). If the collapse-in does not happen, then collapseout happens instead. The front part of the wavefunction will disappear, and the corresponding probability will be shifted to another part of the wavefunction. At the next instant, collapse-in or collapse-out happens again. This process continues until the photon is detected. If the photon has not been detected until the last part of the wavefunction reaches the detector, then the amplitude of this remaining wavefunction increases to the full so as to give a probability 1, so that the photon will be surely detected at the final step.
Measurement is reaction. The essential process of a measurement is an enhanced and concentrated reaction. A photon may react with a single atom in free space with a tiny probability. In a detector, a huge number of such atoms are concentrated in a small area, thus the probability of such reaction is increased tremendously. Of course, in addition to such concentration effect, there exist other effects in a detecting process, such as the avalanche effect in a photon detector.
When reacting with other quantum systems, a quan-tum system takes part in as a whole according to the measurement postulate, rather than just a part of the quantum system. Schrödinger once wanted to treat the wavefunction of an electron as an electron cloud, but finally abandoned it because no partial electric charge could be found. If we make an assumption that a quantum system takes part in a reaction as a whole, then this difficulty can be overcome easily. The WISE interpretation goes further to treat the wavefunction of a quantum system as the quantum system itself, thus answering similar questions like why no fractional electric charge is found. This wholeness nature of a quantum system is also important in understanding why a photon can preserve its properties after being generated for many years and traveling a long distance of many light years: if it reacts with another quantum system in an interstellar matter on its way to the Earth, it will react wholly (leading to a collapsein measurement); otherwise, it will retain its properties (collapse-out, with part of its wavefunction "bitten" by the encountered interstellar matter, and with respective probabilities shifted to other parts of the wavefunction) and continue its way to the Earth.
The measurement process takes place randomly in space over time. It is NOT governed by Schrödinger equation. Actually, it is governed by something beyond quantum mechanics. Thus, in the view of the WISE interpretation, the world is a mixture of determinism and randomness. All quantum systems are evolving according to Schrödinger equations with respective interactions. Because of the inter-interactions between different systems, reactions occur with varied probabilities, randomly in space over time. These reactions are objective.
This work was supported by the National Key Research and Development Program of China under Grant No.2017YFA0303700, National Natural Science Foundation of China under Grant No.11974205, and Beijing Advanced Innovation Center for Future Chip (ICFC) and Tsinghua University Initiative Scientific Research Program.
Uber den anschaulichen inhalt der quanten theoretischen kinematik und mechanik. W Heisenberg, Zeit. f. Phyzik. 43172W. Heisenberg, Uber den anschaulichen inhalt der quan- ten theoretischen kinematik und mechanik, Zeit. f. Phyzik 43, 172 (1927).
Das quantenpostulat und die neuere entwicklung der atomistik. N Bohr, Naturwissenschaften. 16245N. Bohr, Das quantenpostulat und die neuere entwick- lung der atomistik, Naturwissenschaften 16, 245 (1928).
The interpretation of quantum mechanics (i) and (ii. S Gao, arXiv preprint physics/9907001S. Gao, The interpretation of quantum mechanics (i) and (ii), arXiv preprint physics/9907001 (1999).
G.-L Long, General quantum interference principle and duality computer. 45825G.-L. Long, General quantum interference principle and duality computer, Communications in Theoretical Physics 45, 825 (2006).
Z.-B Chen, arXiv:1412.1079The information-complete quantum theory. arXiv preprintZ.-B. Chen, The information-complete quantum theory, arXiv preprint arXiv:1412.1079 (2014).
Realistic interpretation of quantum mechanics and encounter-delayedchoice experiment. G Long, W Qin, Z Yang, J.-L Li, SCIENCE CHINA Physics, Mechanics & Astronomy. 6130311G. Long, W. Qin, Z. Yang, and J.-L. Li, Realistic inter- pretation of quantum mechanics and encounter-delayed- choice experiment, SCIENCE CHINA Physics, Mechan- ics & Astronomy 61, 030311 (2018).
Duality computing in quantum computers. G.-L Long, Y Liu, Communications in Theoretical Physics. 501303G.-L. Long and Y. Liu, Duality computing in quantum computers, Communications in Theoretical Physics 50, 1303 (2008).
Allowable generalized quantum gates. G.-L Long, Y Liu, C Wang, Communications in Theoretical Physics. 5165G.-L. Long, Y. Liu, and C. Wang, Allowable generalized quantum gates, Communications in Theoretical Physics 51, 65 (2009).
Duality quantum computing and duality quantum information processing. G.-L Long, International Journal of Theoretical Physics. 501305G.-L. Long, Duality quantum computing and duality quantum information processing, International Journal of Theoretical Physics 50, 1305 (2011).
R Shankar, Principles of Quantum Mechanics. New YorkPlenum Press2nd editionR. Shankar, Principles of Quantum Mechanics, 2nd edi- tion (Plenum Press New York, 1994).
| {'fraction_non_alphanumeric': 0.04664981036662453, 'fraction_numerical': 0.020859671302149177, 'mean_word_length': 4.594413012729844, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'One central issue in quantum mechanics is the relation between the wavefunction and the quantum system it describes. As quantum mechanics is understood in different ways, the wavefunction is given various explanations. Some regard the wavefunction as "epistemic", that is, something reflected in the human mind, and some regard it as "ontological", i.e., something realistic. The orthodox interpretation of quantum mechanics of Copenhagen [1, 2] is epistemic and treats the wavefunction merely as a mathematical quantity. Recent examples of the ontological interpretation include the random discontinuous motion[3], "Wavefunction Is the System Entity"(WISE) interpretation[4], and information complete interpretation[5]. WISE treats the wavefunction equivalently as the quantum system itself, that is, the quantum system is just the wavefunction, and the wavefunction is just the quantum system. These two are exactly the same. The quantum system, which is also the wavefunction, can exist in disjoint regions of space, travel at a finite speed, and collapse upon measurements. An encounter-delayedchoice experiment has been proposed and experimentally demonstrated recently[6].In this short communication, we will concentrate on the partial measurement issue and give an explanation concerning the WISE interpretation. The essential idea of WISE is given in Ref.[4], together with the linear combination of unitaries (LCU) formalism of quantum computing. LCU has now become one of the major techniques in quantum algorithm design. The quantum circuit implementation of LCU is given in Refs.[7,8], and a review of the subject is given in Ref.[9].Partial measurement postulate. We recall first the measurement postulate in standard quantum mechanics. If a particle is in state |Ψ , a measurement of the variable (corresponding to) Ω will yield one of the eigenvalues ω with probability P (ω) ∝ | ω|Ψ | 2 . The state of the system will change from |Ψ to |ω as a result of the measurement[10].What will happen if the measurement is on part of the wave function (partial measurement) rather than on a full wave function (full measurement)? For instance, in a three-slits system, if one places a detector immediately after the first slit and places no detectors in the remaining * [email protected] two slits, then a partial measurement is established. Here we give the details of the partial measurement.Using the quantum circuit realization in Refs.[7,8], the wavefunction of an electron passing through a d-slit is represented as |Φ = d i=1 c i |ψ i |i , where |i represents the i-th slit, and |ψ i is the sub-wavefunction in the ith slit. For simplicity, we assume that |ψ i = |ψ . The wavefunction is normalized, namely i |c i | 2 = 1.In the WISE interpretation, these sub-waves as a whole form an "electron". Thus, it is easy to comprehend that an "electron" passes through the d-slits simultaneously. The "electron" is no longer a rigid sphere. It is distributed in space, even disjointedly. It changes its shape as the wavefunction changes.If one places a detector just after slit-1, then there is a probability |c 1 | 2 that the detector will measure the electron, and the whole wavefunction will collapse into slit-1. What would happen if the detector at slit-1 does not get a result? To this end, we rewrite |Ψ asEq. (1) is like a double-slit experiment with slit-1 and slit-S2. If we imaginarily place two detectors, one at slit-1 and one at slit-S2, then there is a probability |c 1 | 2 that the detector will measure the electron at slit-1, and the whole wavefunction will collapse into slit-1, and a probability 1-|c 1 | 2 that the detector at nominal slit-S2 will measure the electron. By this heuristic reasoning, we give our partial measurement postulate, which is a novel development of the measurement postulate of standard quantum mechanics: Suppose a quantum system is in state, ω i is one of the eigenvalues of observable Ω, and |ω i is the corresponding eigenvector. If part of the wavefunction, M i=1 c i |ω i , is measured in variable Ω , then the result of the measurement will be one of the following:(1) Collapse-in: One eigenvalue ω i will be obtained with probability | ω i |Ψ | 2 = |c i | 2 , where 1 ≤ i ≤ M . After the measurement, the state of the system will change instantly from |Ψ into |ω i .', 'arxivid': '2106.00466', 'author': ['Gui-Lu Long \nDepartment of Physics\nState Key Laboratory of Low-dimensional Quantum Physics\nTsinghua University\n100084BeijingChina\n\nBeijing Academy of Quantum Information Sciences\n100193BeijingChina\n\nCollaborative Innovation Center of Quantum Matter\n100084BeijingChina\n\nBeijing National Research Center for Information Science and Technology\n100084BeijingChina\n'], 'authoraffiliation': ['Department of Physics\nState Key Laboratory of Low-dimensional Quantum Physics\nTsinghua University\n100084BeijingChina', 'Beijing Academy of Quantum Information Sciences\n100193BeijingChina', 'Collaborative Innovation Center of Quantum Matter\n100084BeijingChina', 'Beijing National Research Center for Information Science and Technology\n100084BeijingChina'], 'corpusid': 235266075, 'doi': '10.1007/s11433-021-1716-y', 'github_urls': [], 'n_tokens_mistral': 4147, 'n_tokens_neox': 3690, 'n_words': 2471, 'pdfsha': '88b0b4cb08f45962c9ffd43c5b37df3fe32d76d6', 'pdfurls': ['https://arxiv.org/pdf/2106.00466v1.pdf'], 'title': ['Collapse-in and Collapse-out in Partial Measurement in Quantum Mechanics and its WISE Interpretation', 'Collapse-in and Collapse-out in Partial Measurement in Quantum Mechanics and its WISE Interpretation'], 'venue': []} |
arxiv |
SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES
18 Aug 2012
Mohammad Eslamian
SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES
18 Aug 2012Shrinking projection methodequilibrium problemcommon fixed pointrelatively quasi-nonexpansive multivalued mappings 2000 Mathematics Subject Classification: 47H1047H09
Strong convergence of a new iterative process based on the Shrinking projection method to a common element of the set of common fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings and the solution set of an equilibrium problem in a Banach space is established. Our results improved and extend the corresponding results announced by many others.
Introduction
A nonempty subset C of a Banach space E is called proximinal if for each x ∈ E, there exists an element y ∈ C such that
x − y = dist(x, C) = inf { x − z : z ∈ C}.
We denote by N(C), CB(C) and P (C) the collection of all nonempty subsets, nonempty Let T : E −→ N(E) be a multivalued mapping. An element x ∈ E is said to be a fixed point of T , if x ∈ T x. The set of fixed points of T will be denoted by F (T ). The theory of multivalued mappings has applications in control theory, convex optimization, differential equations and economics. Theory of nonexpansive multivalued mappings is harder than the corresponding theory of nonexpansive single valued mappings. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings (see [1][2][3][4][5][6][7]).
Let E be a real Banach space and let E * be the dual space of E. Let C be a closed convex subset of E. Let F be a bifunction from C × C into R, where R is the set of real numbers. The equilibrium problem for F : C × C −→ R is to find x ∈ C such that F ( x, y) ≥ 0, ∀y ∈ C.
The set of solutions is denoted by EP (F ). Equilibrium problems, have had a great impact and influence in the development of several branches of pure and applied sciences.
Numerous problems in physics, optimization and economics reduce to finding a solution of the equilibrium problem . Some methods have been proposed to solve the equilibrium problem in a Hilbert space. See [8][9][10].
Let E be a real Banach space with norm . and let J be the normalized duality mapping from E into 2 E * given by
Jx = {x * ∈ E * : x, x * = x x * , x = x * }
for all x ∈ E, where E * denotes the dual space of E and ., . the generalized duality pairing between E and E * . As we all know that if C is a nonempty closed convex subset of a Hilbert space H and P C : H −→ C is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [11] recently introduced a generalized projection operator E * in a smooth Banach space E which is an analogue of the metric projection in Hilbert spaces. Consider the functional defined by φ(x, y) = x 2 − 2 x, Jy + y 2 , x, y ∈ E.
Observe that, in a Hilbert space H, φ(x, y) reduces to x − y 2 . The generalized projection Π C : E −→ C is a map that assigns to an arbitrary point x ∈ E the minimum point of the functional φ(x, y) that is,
Π C x = x, where x is the solution to the minimization problem φ(x, x) = inf y∈C φ(y, x)
CONVERGENCE THEOREM FOR MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM 3
The existence and uniqueness of the operator Π C follows from the properties of the functional φ(x, y) and strict monotonicity of the mapping J (see, for example, [11,12,13]). In Hilbert spaces, Π C = P C . It is obvious from the definition of function φ that
( y − x ) 2 ≤ φ(x, y) ≤ ( y + x ) 2 ∀x, y ∈ E. (1.1)
Remark 1: If E is a reflexive, strictly convex and smooth Banach space, then for
x, y ∈ E, φ(x, y) = 0 if and only if x = y (see [13,14]).
Let C be a nonempty closed convex subset of a smooth, strictly convex and reflexive
Banach space E, and let T be a mapping from C into itself. We denote by F (T ) the set of fixed points of T . A point p ∈ C is said to be an asymptotic [15] fixed point of T , if C contains a sequence {x n } which converges weakly to p such that lim n−→∞ x n −T x n = 0.
The set of asymptotic fixed points of T will be denoted by F (T ). A mapping T is said to be relatively nonexpansive [16,17], if F (T ) = F (T ) and φ(p, T x) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). T is said to be relatively quasi-nonexpansive ( [18,19]) if F (T ) = ∅ and φ(p, T x) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). The class of relatively quasinonexpansive mappings is bigger than the class of relatively nonexpansive mappings which requires the strong restriction: F (T ) = F (T ).
In the recent years, approximation of fixed points of relatively quasi-nonexpansive mappings by iteration has been studied by many authors, see [16][17][18][19][20][21][22][23].
Very recently, Eslamian and Abkar [7] introduce the relatively quasi-nonexpansive multivalued mapping as follows:
Definition 1.2. Let C be a closed convex subset of a smooth Banach space E, and T : C −→ N(C) be a multivalued mapping. We set
Φ(T x, T p) = max{ sup q∈T p inf y∈T x φ(y, q), sup y∈T x inf q∈T p φ(y, q)}.
We call T is relatively quasi-nonexpansive multivalued mapping if F (T ) = ∅ and
Φ(T x, T p) ≤ φ(x, p), ∀p ∈ F (T ), ∀x ∈ C.
Remark : In a Hilbert space, Φ(T x, T y) = H(T x, T y) 2 , and hence relatively quasinonexpansivness is equivalent to quasi-nonexpansivness.
In [7] the author presented some example of relatively quasi-nonexpansive multivalued mapping. Another generalization of relatively nonexpansive mapping to multivalued mappings presented by Simin Homaeipour [24] as follows.
(i) F (T ) = ∅, (ii) φ(p, z) ≤ φ(p, x) for all z ∈ T x, x ∈ C and p ∈ F (T ), (iii) F (T ) = F (T ).
In a Hilbert space H, condition (ii) is equivalent to
p − z ≤ p − x , ∀z ∈ T x, ∀x ∈ C, ∀p ∈ F (T ). Now if put H = R and T x = [0,
x], we observe that T is a nonexpansive multivalued mapping but T is not relatively nonexpansive mapping. Hence in spite of single valued case, relatively nonexpansive multivalued mapping is not equivalent to quasinonexpansive multivalued mapping.
Definition 1.4. A multivalued mapping T is called closed if x n −→ w and lim n−→∞ dist(x n , T x n ) = 0, then w ∈ T (w).
In [7], Eslamian and Abkar proved the following theorem.
Theorem 1.5. Let E be a uniformly smooth and uniformly convex Banach space, and
let C be a nonempty closed convex subset of E. Let F be a bifunction from C × C into R satisfying (A1) − (A4). Let T i : C −→ P (C), i = 1, 2, .
.., m, be a finite family of multivalued mappings such that P T i is closed and relatively quasi-nonexpansive. Assume
that F = m i=1 F (T i ) EP (f ) = ∅.
For x 0 ∈ C and C 0 = C, let {x n } be a sequences generated by the following algorithm:
y n,1 = J −1 ((1 − a n,1 )Jx n + a n,1 Jz n,1 ), y n,2 = J −1 ((1 − a n,2 )Jx n + a n,2 Jz n,2 ), ... y n,m = J −1 ((1 − a n,m )Jx n + a n,m Jz n,m ),
u n ∈ C such that f (u n , y) + 1 rn y − u n , Ju n − Jy n,m ≥ 0, ∀y ∈ C, C n+1 = {z ∈ C n : φ(z, u n ) ≤ φ(z, x n )}, x n+1 = C n+1 x 0 , ∀n ≥ 0
where z n,1 ∈ P T 1 x n and z n,i ∈ P T i y n,i−1 for i = 2, ..., m and J is the duality mapping on E. Assume that m i=1 a n,i = 1, {a n,i } ∈ [a, b] ⊂ (0, 1) and {r n } ⊂ [c, ∞) for some c > 0. Suppose that P T i is uniformly continuous with respect to the Hausdorff metric for i = 2, 3, ..., m. Then {x n } converges strongly to Π F x 0 .
In this paper, we introduce a new shirking projection algorithm for finding a common element of the set of common fixed points of an infinite family of relatively quasinonexpansive multivalued mappings and the set of solutions of an equilibrium problem in CONVERGENCE THEOREM FOR MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM 5 uniformly smooth and uniformly convex Banach spaces. Strong convergence to common elements of two set is established. Our results improved and extend the corresponding results announced by many others.
Preliminaries
A Banach space E is said to be strictly convex if x+y 2 < 1 for all x, y ∈ E with x = y = 1 and x = y . It said to be uniformly convex if lim n−→∞ x n −y n = 0 for any two sequences {x n } and {y n } in E such that x n = y n = 1 and lim n−→∞ xn+yn 2 = 1.
Let U = {x ∈ E : x = 1} be the unit sphere of E. Then the Banach space E is said to be smooth provided lim t−→0
x + ty − x t exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ E. It is well known that if E * is uniformly convex, then J is uniformly continuous on bounded subsets of E. E : x ≤ r}, for r > 0. Then, for any given sequence {x n } ∞ n=1 ⊂ B r (0) and for any given sequence {a n } ∞ n=1 of positive numbers with ∞ n=1 a n = 1 there exists a continuous, strictly increasing and convex function g : [0, ∞) −→ [0, ∞) with g(0) = 0 such that that for any positive integers i, j with i < j, ∞ n=1 a n x n 2 ≤ ∞ n=1 a n x n 2 − a i a j g( x i − x j .
φ(y, Π C x) + φ(Π C x, x) ≤ φ(y, x), ∀y ∈ C.
For solving the equilibrium problem, we assume that the bifunction F satisfies the following conditions:
(A1) F (x, x) = 0 for all x ∈ C,
(A2) F is monotone, i.e. F (x, y) + F (y, x) ≤ 0 for any x, y ∈ C, (A3) F is upper-hemicontinuous, i.e. for each x, y, z ∈ C,
lim sup t−→0 + F (tz + (1 − t)x, y) ≤ F (x, y) (A4) F (x, .
) is convex and lower semicontinuous for each x ∈ C.
The following lemma was proved in [8].
Lemma 2.5. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E , let F be a bifunction of C × C into R satisfying (A1) − (A4).
Let r > 0 and x ∈ E. Then, there exists z ∈ C such that
F (z, y) + 1 r y − z, Jz − Jx ≥ 0 ∀y ∈ C.
The following lemma was given in [22].
Lemma 2.6. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E , let F be a bifunction of C × C into R satisfying (A1) − (A4).
Let r > 0 and x ∈ E. define a mapping T r : E −→ C as follows:
S r x = {z ∈ C : F (z, y) + 1 r y − z, Jz − Jx ≥ 0, ∀y ∈ C}.
Then, the following hold:
(i) S r is single valued;
(ii) S r is firmly nonexpansive-type mapping, i.e., for any x, y ∈ E, S r x − S r y, JS r x − JS r y ≤ S r x − S r y, Jx − Jy ;
(iii) F (S r ) = EP (F );
(iv) EP (F ) is closed and convex. If F (T ) = ∅, then F (T ) is closed and convex.
In this section, we prove strong convergence theorems for finding a common element of the set of solutions for an equilibrium problem and the set of fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings in a Banach space.
Theorem 3.1. Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let F be a bifunction from C ×C into R satisfying (A1)−(A4). Let T i : C −→ N(C), be a sequence of multivalued mappings such that for each i ∈ N, P T i is closed relatively quasi-nonexpansive multivalued mappings
and F = ∞ i=1 F (T i ) EP (F ) = ∅.
For x 0 ∈ C and C 0 = C, let {x n } be a sequence generated by the following algorithm:
y n = J −1 (a n,0 Jx n + ∞ i=1 a n,i Jz n,i ), u n ∈ C : F (u n , y) + 1 rn y − u n , Ju n − Jy n ≥ 0, ∀y ∈ C,
C n+1 = {z ∈ C n : φ(z, u n ) ≤ φ(z, x n )}, x n+1 = C n+1 x 0 , ∀n ≥ 0,
where ∞ i=0 a n,i = 1 and z n,i ∈ P T i x n . Assume further that lim inf n a n,0 a n,i > 0, {r n } ⊂ (0, ∞) and lim inf n r n > 0. Then {x n } converges strongly to Π F x 0 , where Π F is the projection of E onto F .
Proof. First, we show by induction that F = ( ∞ i=1 F (T i )) EP (F ) ⊂ C n for all n ≥ 0. From C 0 = C, we have F ⊂ C 0 . We suppose that F ⊂ C n for some n ≥ 0. Let u ∈ F , then we have P T i u = {u}, (i ∈ N). Since S rn and T i are relatively quasi-nonexpansive, we have φ(u, u n ) = φ(u, S rn y n ) ≤ φ(u, y n ) = φ(u, J −1 (a n,0 Jx n + ∞ i=1 a n,i Jz n,i )) = u 2 − 2 u, a n,0 Jx n + ∞ i=1 a n,i Jz n,i + a n,0 Jx n + ∞ i=1 a n,i Jz n,i 2 ≤ u 2 − 2a n,0 u, Jx n − 2 ∞ i=1 a n,i u, Jz n,i + a n,0 x n 2 + ∞ i=1 a n,i z n,i 2 = a n,0 φ(u, x n ) + ∞ i=1 a n,i φ(u, z n,i ) = a n,0 φ(u, x n ) + ∞ i=1 a n,i inf
u∈P T i u φ(u, z n,i ) ≤ a n,0 φ(u, x n ) + ∞ i=1 a n,i Φ(P T i u, P T i x n ) ≤ a n,0 φ(u, x n ) + ∞ i=1 a n,i φ(u, x n ) = φ(u, x n ), which implies that u ∈ C n+1 . Hence F = ∞ i=1 F (T i ) EP (F ) ⊂ C n , ∀n ≥ 0.
We observe that C n is closed and convex (see [21,22]). From x n = Π Cn x 0 , we have
x n − z, Jx 0 − Jx n ≥ 0, ∀z ∈ C n . (3.1)
Since F ⊂ C n for all n ≥ 0, we obtain that
x n − u, Jx 0 − Jx n ≥ 0 ∀u ∈ F . From Lemma 2.3 we have φ(x n , x 0 ) = φ(Π Cn x 0 , x 0 ) ≤ φ(u, x 0 ) − φ(u, Π Cn x 0 ) ≤ φ(u, x 0 )
for all u ∈ F ⊂ C n . Then the sequence φ(x n , x 0 ) is bounded. Thus {x n } is bounded.
From x n = Π Cn x 0 and x n+1 ∈ C n+1 ⊂ C n we have φ(x n , x 0 ) ≤ φ(x n+1 , x 0 ), ∀n ≥ 0.
Therefore {φ(x n , x 0 )} is nondecreasing. So the limit of {φ(x n , x 0 )} exists. By the construction of C n for any positive integer m ≥ n we have
x m = Π Cm x 0 ∈ C m ⊂ C n .
It follows that
φ(x m , x n ) = φ(x m , Π Cn x 0 ) ≤ φ(x m , x 0 ) − φ(Π Cn x 0 , x 0 ) = φ(x m , x 0 ) − φ(x n , x 0 ). Letting m, n −→ ∞ we have lim n−→∞ φ(x m , x n ) = 0. (3.2)
It follows from Lemma 2.1 that x m − x n −→ 0 as m, n −→ ∞. Hence {x n } is a Cauchy sequence. Since C is closed and convex subset of Banach space E, we can assume that
x n −→ z as n −→ ∞. Next we show z ∈ ∞ i=1 F (T i ). By taking m = n + 1 in (3.2) we get
lim n−→∞ φ(x n+1 , x n ) = 0. (3.3)
It follows from Lemma 2.1 that We show that {z n,i } is bounded for i ∈ N. Indeed, for u ∈ F we have
lim n−→∞ x n+1 − x n = 0. (3.4) From x n+1 = Π C n+1 x ∈ C n+1 , we have φ(x n+1 , u n ) ≤ φ(x n+1 , x n ), n ≥ 0, it( z n,i − u ) 2 ≤ φ(z n,i , u) ≤ φ(x n , u) ≤ ( x n + u ) 2 .
Since {x n } is bounded, we obtain {z n,i } is bounded for i ∈ N. Let
r = sup n≥0 { x n , z n,i : i ∈ N}.
Since E is a uniformly smooth Banach space, we know that E * is a uniformly convex Banach space. Therefore from Lemma 2.4 there exists a continuous strictly increasing, and convex function g with g(0) = 0 such that φ(u, u n ) = φ(u, T rn y n ) ≤ φ(u, y n ) = φ(u, J −1 (a n,0 Jx n + ∞ i=1 a n,i Jz n,i )) = u 2 − 2 u, a n,0 Jx n + ∞ i=1 a n,i Jz n,i + a n,0 Jx n + ∞ i=1 a n,i Jz n,i 2 ≤ u 2 −2a n,0 u, Jx n −2 ∞ i=1 a n,i u, Jz n,i +a n,0 x n 2 + ∞ i=1 a n,i z n,i 2 −a n,0 a n,i g( Jx n −Jz n,i ) = a n,0 φ(u, x n ) + ∞ i=1 a n,i φ(u, z n,i ) − a n,0 a n,i g( Jx n − Jz n,i ) ≤ a n,0 φ(u, x n ) + ∞ i=1 a n,i Φ(P T i u, P T i x n ) − a n,0 a n,i g( Jx n − Jz n,i ) ≤ a n,0 φ(u, x n ) + ∞ i=1 a n,i φ(u, x n ) − a n,0 a n,i g( Jx n − Jz n,i ) ≤ φ(u, x n ) − a n,0 a n,i g( Jx n − Jz n,i ). (3.8) It follow that a n,0 a n,i g( Jx n − Jz n,
i ) ≤ φ(u, x n ) − φ(u, u n ) n ≥ 0. (3.9)
On the other hand
φ(u, x n ) − φ(u, u n ) = x n 2 − u n 2 − 2 u, Jx n − Ju n ≤ | x n 2 − u n 2 | + 2| u, Jx n − Ju n | ≤ | x n − u n |( x n + u n ) + 2 u Jx n − Ju n ≤ x n − u n ( x n + u n ) + 2 u Jx n − Ju n .
It follows from (3.6) and (3.7) that lim n−→∞ (φ(u, x n ) − φ(u, u n )) = 0. (3.10)
Using (3.9) and by assumption that lim inf a n,0 a n,i > 0 we have that Now by closedness of P T i we obtain that z ∈ ∞ i=1 F (T i ). By similar argument as in [21] (see also [22]) we obtain that z ∈ EP (F ). Therefore z ∈ F . Finally we prove z = Π F x 0 .
By taking limit in (3.1) we have
z − u, Jx 0 − Jz ≥ 0, ∀u ∈ F .
Hence by Lemma 2.2 we have z = Π F x 0 , which complete the proof.
By similar argument as in the proof of Theorem 3.1, we can prove the following theorem.
Theorem 3.2. Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let F be a bifunction from C × C into R satisfying (A1) − (A4). Let T i : C −→ N(C), be a sequence of closed relatively quasi-nonexpansive multivalued mappings such that F = ∞ i=1 F (T i ) EP (F ) = ∅ and for all p ∈ F , T i (p) = {p}. For x 0 ∈ C and C 0 = C, let {x n } be a sequence generated by the following algorithm:
y n = J −1 (a n,0 Jx n + ∞ i=1 a n,i Jz n,i ), u n ∈ C : F (u n , y) + 1 rn y − u n , Ju n − Jy n ≥ 0, ∀y ∈ C,
C n+1 = {z ∈ C n : φ(z, u n ) ≤ φ(z, x n )}, x n+1 = C n+1 x, ∀n ≥ 0,
where ∞ i=0 a n,i = 1 and z n,i ∈ T i x n . Assume further that lim inf n a n,0 a n,i > 0, {r n } ⊂ (0, ∞) and lim inf n r n > 0. Then {x n } converges strongly to Π F x 0 , where Π F is the projection of E onto F .
As a result for single valued mappings we obtain the following theorem. Theorem 3.3. Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty closed convex subset of E. Let F be a bifunction from C × C into R satisfying (A1) − (A4). Let Let T i : C −→ C, be a sequence of closed relatively quasi-nonexpansive mappings such that F = ∞ i=1 F (T i ) EP (F ) = ∅ . For x 0 ∈ C and C 0 = C, let {x n } be a sequence generated by the following algorithm: y n = J −1 (a n,0 Jx n + ∞ i=1 a n,i JT i x n ), u n ∈ C : F (u n , y) + 1 rn y − u n , Ju n − Jy n ≥ 0, ∀y ∈ C, C n+1 = {z ∈ C n : φ(z, u n ) ≤ φ(z, x n )},
x n+1 = C n+1 x, ∀n ≥ 0,
where ∞ i=0 a n,i = 1. Assume further that lim inf n a n,0 a n,i > 0, {r n } ⊂ (0, ∞) and lim inf n r n > 0. Then {x n } converges strongly to Π F x 0 , where Π F is the projection of E onto F .
Remark : Our main result generalize the result of Eslamian and Abkar [7] of a finite family of multivalued mappings to an infinite family of multivalued mappings. We also remove the uniformly continuity of the mappings.
Some Applications to Hilbert Spaces
In the Hilbert space setting, we have φ(x, y) = x − y 2 , Φ(T x, T y) = H(T x, T y) 2 ∀x, y ∈ H.
Therefore Φ(T x, T p) ≤ φ(x, p) ⇔ H(T x, T p) ≤ x − p for every x ∈ C and p ∈ F (T ). We note that in a Hilbert space H, J is the identity operator.
Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C × C into R satisfying (A1) − (A4). Let T i : C −→ P (C),
i ∈ N be a sequence of multivalued mappings such that P T i is closed quasi-nonexpansive.
Assume that F = ∞ i=1 F (T i ) EP (F ) = ∅. For x 0 ∈ C and C 0 = C, let {x n } be a sequences generated by the following algorithm: y n = a n,0 x n + ∞ i=1 a n,i z n,i , u n ∈ C such that F (u n , y) + 1 rn y − u n , u n − y n ≥ 0; ∀y ∈ C, C n+1 = {z ∈ C n : z − u n ≤ z − x n },
x n+1 = P C n+1 x, ∀n ≥ 0 ,
where ∞ i=0 a n,i = 1 and z n,i ∈ P T i x n . Assume further that lim inf n a n,0 a n,i > 0 and {r n } ⊂ [a, ∞) for some a > 0. Then {x n } converges strongly to P F x 0 .
Remark : Theorem 4.1 holds if we assume that T i is closed quasi-nonexpansive multivalued mapping and T i (p) = {p} for all p ∈ F .
closed bounded subsets and nonempty proximinal bounded subsets of C, respectively. The Hausdorff metric H on CB(C) is defined by H(A, B) := max{sup x∈A dist(x, B), sup y∈B dist(y, A)}, for all A, B ∈ CB(C).
Definition 1.1. A multivalued mapping T : E −→ CB(E) is called (i) nonexpansive if H(T x, T y) ≤ x − y , x, y ∈ E.Young Researchers Club, Babol Branch, Islamic Azad University, Babol, Iran.Email: [email protected].
≤ x − p , x ∈ E, p ∈ F (T ).
Definition 1. 3 .
3Let C be a closed convex subset of a smooth Banach space E, and T : C −→ N(C) be a multivalued mapping, T is called relatively nonexpansive mapping if the following conditions are satisfied:
Lemma 2.1. ([12]) Let E be a uniformly convex and smooth Banach space and let {x n } and {y n } be two sequences in E. If φ(x n , y n ) −→ 0 and either {x n } or {y n } is bounded, then x n − y n −→ 0. Lemma 2.2. ([11]) Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E. Then x 0 = Π C x if and only if x 0 − y, Jx − Jx 0 ≥ 0, ∀y ∈ C. Lemma 2.3. ([11]) Let E be a reflexive, strictly convex and smooth Banach space. Let C be a nonempty closed convex subset of E and let x ∈ E. Then
Lemma 2. 4 .
4( [25]) Let E be a uniformly convex Banach space and let B r (0) = {x ∈
Lemma 2. 7 .
7([22]) Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let F be a bifunction of C × C into R satisfying (A1) − (A4), and let r > 0. Then for all x ∈ E and q ∈ F (S r ), φ(q, S r x) + φ(S r x, x) ≤ φ(q, x).
Lemma 2. 8 . [ 7 ]
87Let C be a nonempty closed convex subset of a uniformly convex and smooth Banach space E. Suppose T : C −→ P (C) is a multivalued mapping such that P T is a relatively quasi-nonexpansive multivalued mapping where P T (x) = {y ∈ T x : x − y = dist(x, T x)}.
( 0
0x n+1 − x n + x n+1 − u n ) = 0. (3.6)It follows from x n −→ z as n −→ ∞ that u n −→ z as n −→ ∞. Since J is uniformly norm-to-norm continuous on bounded sets and lim n−→∞ x n − u n =
Jx n − Jz n,i ) = 0, (i ∈ N). Therefore from the property of g , we have lim n−→∞ Jx n − Jz n,i = 0, (i ∈ N).Since J −1 is uniformly norm-to-norm continuous on bounded set, x n , P T i x n ) ≤ lim n−→∞ x n − z n,i = 0, (i ∈ N).
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arxiv |
Kondo Resonance in Transport Properties of Double Barrier Structure
Jul 1995
V V Ponomarenko
Frontier Research Program
RIKEN
Wako-Shi351-01SaitamaJAPAN
Ioffe Physical Technical Institute
194021, St. PetersburgRussia
Kondo Resonance in Transport Properties of Double Barrier Structure
Jul 1995arXiv:cond-mat/9507090v2 31 Permanent address: A.F.
I consider the effect of the finite width of the resonant level on its thermodynamics and tunneling transport properties in the single electron charging regime. The finite width of the levels results from their delocalization with formation of a narrow band due to their mutual overlap. Making use of 1/N technique one can show that there is a fixed energy level position at which the system undergoes the transition from the Kondo regime (above this energy) to the paramagnetic regime (below it). The linear bias conductance calculated as a function of the chemical potential turns out to be suppressed all around except for the vicinity of the transition energy where it has an asymmetrical resonant behavior.
There is much interest in the study of low temperature transport in mesoscopic systems where new phenomena associated with quantum coherence are invoked by the large Coulomb charging energy of a localized constriction. These have been mostly discussed for the case of tunneling through a single spin-degenerate level specified with Anderson model [4]. Another way of manifesting a system with such a quantum correlation was recently suggested for a conducting granule in the Coulomb blockade condition [2], [5]. In these systems, in particular, the quasiclassical size of the granule results in the delocalization of the states and the appearance of a continuum band instead of the separate levels.
In this paper I will consider the effect of the delocalization of the resonant states on the Kondo resonance phenomenon dealing with the problem of tunneling through a double barrier (2D) structure (DBS) located between two conductors. This structure consists of the levels located at the impurity sites between the barriers. The sites are ordered into a lattice, and the levels of the neighbor cites have a small overlap, which leads to the appearance of a narrow band of width E G . First, one could discuss this system as a vertical quantum dot structure where the tunneling layer is inside a constriction of radius R. This band should be considered continuous if its level splitting ∆ G ≈ hv/R ( v is the speed of the band electron with energy near the bottom ) is less than the inverse time Γ =h/τ of escape of the band electron into the electrodes. To make it more clear one could assume that this layer is infinite and the effective tunneling area through the layer is only of R 2 size. Then the probability for the tunneling electron to escape from this area along the layer could be estimated as relation between the squares of appropriate surfaces 2πRvτ and πR 2 . This probability becomes negligible if vτ ≪ R.
The charging energy for accumulation of the charge Q inside the barriers could be evaluated in a standard way [6] as Q 2 /2C and decreases as R 2 due to dependence of the capacitance C(R) on the area of the contact. Since the charging energy of a non-equilibrium distributed charge should be more than the one in equilibrium the criterion of a singleelectron charging of the layer could be formulated as R < l C where e 2 /C(l C ) > ǫ b , ǫ b is the bottom of the impurity band. To evaluate possible parameters of such a system one could use as an example the typical Metal-Oxide-Metal junction, where closely located conductors with the tunneling area of 100nm 2 form a condensator with ≈ 100K charging energy. This value of the charging energy is substantially greater than the band width which, due to the previous restriction according to Γ, should be less than 0.1K for 10 2 − 10 3 lattice sites. The dependence on R in this discussion could become irrelevant if the phase coherence length of the electrodes is less than R. I could generalize the consideration if I will suppose that this length satisfies the above two criteria instead of R.
Tunneling through the DBS is locally described by the Hamiltonian
H = k ǫ a,k c + ks,a c ks,a + q ǫ q d + qs d qs + k,q,a,s (V a,q,k d + qs c ks,a + V * a,q,k c + ks,a d qs )(1)
with a restriction on the Hilbert space that only states of zero |0 > and one-electron |qs > occupation of the DBS band are permissible. Here c ks,a annihilates the electron of a lead with spin s in the state of wave vector k = (k ⊥ , k ) and of energy ǫ a,k . The components k ⊥ , k are perpendicular and parallel to the layer, respectively. The operator d qs annihilates an electron in the state localized inside the layer with spin s, and wave vector q, and energy ǫ q . An expression for the tunneling matrix elements V a,q,k could be generally written making use of the tight binding model for the DBS band as
V a,q,k = 1 ML 1/2 j T aj (k ⊥ )exp(iR j q − iφ aj (k ))(2)
Here M is the effective number of the sites of the layer restricted either by the geometry or by the coherence length. L is the normalization in the direction perpendicular to the layer, and T aj is the modulus of the corresponding tunneling amplitude. The vertex function may be introduced as Γ a,ω (q 1 , q 2 ) = π k δ(ω − ǫ k )V * a,q 1 ,k V a,q 2 ,k which is completely ample for our future needs. Moreover, in the low temperature limit only the transitions between the holes of the Fermi surface (ω ≈ 0) and the electrons of the bottom of the layer band are important.
( I will suppose that the bottom is reached at ǫ q=0 = ǫ b .) To calculate the function in this limit one could exploit either the jelly model for the leads or the tight binding model if all structure are commensurate. Virtually, the result is the same in the above limit. Indeed,
taking φ ja = R j k one finds Γ a,ω (q 1 , q 2 ) = π Q,Q ′ δ k ,q 1 +Q δ k ,q 2 +Q ′ dǫ k ⊥ ρ 1D (ǫ k ⊥ )δ(ω − ǫ k ⊥ − ǫ k )T 2 a (ǫ k ⊥ ) ≃ πT 2 a ρ a,1D (0)δ q 1 ,q 2 (3)
where Q, Q ′ are the vectors of the reciprocal lattice either of the DBS or of the leads. T aj is taken as a constant if the position of the j DBS site corresponds to the position of the lead site ( and in the case of the jelly model). Otherwise, it was put zero. The second equation
in (3) is correct if the DBS lattice is less close than the lattice of the leads. If this condition is not met, the transitions between nonequal q 1 , q 2 become permissible. I will not discuss it specially, though, the consideration could be easy generalized to account for them.
Conservation of the q components in (3) I will restrict my consideration to leading order, giving summation of all terms of (ΓM) n .
Starting from the thermodynamics I will include the DBS contribution into the partition function in terms of the self energy of the slave-boson Σ 0 (z) [1]. To leading order it takes the form
Z (1) Z band = q,s e −βǫq + C dz 2πi e −βz 1 − ∂ z Σ (1) 0 (z) z − Σ (1) 0 (z)(4)
where C encircles all the singularities of the integrand, β = 1/T is the inverse temperature, and Σ (1) 0 (z) is given by
Σ (1) 0 (z) = a,s,q ∞ −E F dωΓ a f (ω) z − ǫ q + ω(5)
In the low temperature limit the Fermi-Dirac function f (ω) can be approximate by the step function. Then Σ
0 for 2D band, which has a constant density of states ρ 2D = m * /(2πh 2 ) near the edge, is
Σ (1) 0 (z) = ∆ǫ b + P [E G ln E G + ǫ b − z E G + (ǫ b − z)ln E G + ǫ b − z ǫ b − z ](6)
where P = 2(Γ/π)ρ 2D S ≃ (Γ/π)M/E G , Γ = a Γ a , S is the square of the tunneling surface,
∆ǫ b = P E G (ln(E G /E F ) − 1)
. P is a number of the states lying inside the energy interval of width Γ/π. In the above condition P ≫ 1. This function (6) coincides with the one of a single level Kondo model for |ǫ b − z| ≫ E G . However, for small |ǫ b − z| its behavior is quite different. In particular, the self-energy Σ (1) 0 (z) converges to the finite negative value ∆ǫ b at ǫ b and acquires an imaginary part on the real axis above ǫ b as ImΣ
(1) 0 (x − i0) = πP (x − ǫ b ).
This means that unless ǫ b becomes lower then ∆ǫ b , the lowest singularity of the integrand in (4) is a simple pole z = ǫ b − T K , which corresponds to the bound state of the slave-boson
with energy T K ≡ E G x ǫ b P E G = ln(1 + x) + xln(1 + x −1 ) + x P ,ǭ b = ǫ b − P E G (ln(E G /E F ) − 1)(7)
In the limit ǫ b /|∆ǫ b | ≪ 1 the solution of (7) loses its dependence on ǫ b and shows that
T K ≃ |∆ǫ b | when E F ≫ E G .
In the more general caseǭ b could reach the zero, where the transition into a paramagnetic state of the DBS occurs. In that regime the low temperature thermodynamics of the DBS is ruled out by the lowest states of its band.
The standard way of calculating [3] the localized charge n G , the charge χ C and the spin χ s susceptibilities results, for the Kondo regime behavior in
F DBS = −T ln Z (1) Z band = ǫ b − T K , n G = ∂F DBS ∂ǫ b = 1 − Z 0 χ C = − P E G Z 3 0 T K (E G + T K ) , χ s = P E G Z 0 8T K (E G + T K ) (8) Z 0 = 1 1 − ∂ z Σ 0 (z)| z=ǫ b −T K = 1 1 + P ln[(E G + T K )/T K ]
The temperature dependence is dropped in Eq. (8). It remains unimportant while P T /Γ ≪ exp(T K /T ). These solutions (7,8) predict the singular behavior of all thermodynamic characteristics atǭ b = 0. The function n G goes to 1 as 1/ln(ǭ b /E G ).
In the paramagnetic regime the low temperature asymptotic for (4) is
Z (1) Z band = MT E G e −ǫ b /T a(ǫ b , T ) (9) a = 1 − β ∞ 0 dx Γ πǭ b + P x (ǭ b + P x(1 + ln(E G /x))) 2 + (πP x) 2
If −ǭ b > T P ln(E G /T ) the parameter a is evaluated as a = 1 − Γ/(πǭ b ). From Eq. (9) one concludes that n G = 1 and χ s diverges as 1/T . Discussing the transport through this structure I will restrict consideration to linear conductance. Then, the conductance can be expressed in terms of the imaginary part of the retarded Green function [7] of the DBS electrons G q (ω) labelled with the quasi momentum q and energy ω indices. Up to T 2 corrections it is given by
σ 0 = − 2e 2 h Γ R Γ L Γ 2 P s dǫ q ImG q (ω)| ω=0(10)
In this technique the full Green function of electron is calculated as a convolution [1] of the slave-boson and the slave-fermion Green functions along the appropriate contour in the complex energy plane. For the imaginary part it is simplified to (11) Here Σ q (x + ) is the self energy of the slave fermion which is taken at the real point x + = x+i0
ImG q (ω + ) = − 1 Z/Z band ∞ ∞ dx π ImΣ 0 (x + )ImΣ q (x + ω + ) |x − Σ 0 (x ) | 2 |x + ω − ǫ q − Σ q (x + ω)| 2 (1 + e −βω )e −βx
with the infinitely small imaginary shift.
In the Kondo regime the calculations straightforwardly follow the ones of the single level case. To leading order in the 1/M expansion, the imaginary parts of both the slave boson and the slave fermion Green functions become strongly localized as Z 0 δ(ω − T K ) and δ(ω − ǫ q ), respectively, at low temperatures by the factor exp(−T /T K ). To obtain a non-zero ImG q (0) I should use the next order expression for ImΣ q (x) standing in the numerator of (11). This is written as
ImΣ q (y + ) = Im Γ π dx y − x − Σ(1)0 (y + − x) = −ΓZ 0 (1 − f (y − ǫ b + T K ))(12)
for y < ǫ b . Finally, I obtain
ImG q (ω + ) = − ΓZ 2 0 (ω − ǫ q + ǫ b − T K ) 2 , σ 0 = 4e 2 h Γ R Γ L Γ 2 P Z 2 0 E G T K (T K + E G )(13)
Taking T K from Eq.(7) and Z 0 from Eq.(9) one finds
σ 0 = 2e 2 h Γ R Γ L Γ 2 Γ πǭ b ln(E G /ǭ b )(14)
for small T K . This shows that the resonance in the σ 0 dependence on the bottom energy is reached at ǫ b = ∆ǫ b , and it decreases with increasingǭ b more slowly than the 1/ǭ b 2 tail of the one electron resonance. The width on this side of the resonance is less than Γ in the factor 1/ln(E G /Γ). Eq. (13) is applicable only far from the resonance. Near the resonance it could be corrected with the summation of the infinite series of the 1/M expansion as the NCA does.
In the paramagnetic phase, when ǫ b lies below ∆ǫ b , the imaginary parts of the self energies give a contribution into the conductance already in the leading order. However,
this contribution is ∝ T since Σ (1) 0 (y) ∝ (y − ǫ b ).
The zero temperature part of σ 0 appears if the imaginary part of Σ 0 standing in the numerator of (11) is taken in the next order. Then it is equal to
Σ (2) 0 (z) = P Γ π 0 −E F dǫ[ 1 z + ǫ − ǫ b − E G − 1 z + ǫ − ǫ b ] E F 0 dǫ ′ z + ǫ − ǫ ′ − Σ (1) 0 (z + ǫ − ǫ ′ )(15)
Extracting from here the imaginary part and collecting the leading terms one could find
σ 0 = 8e 2 h Γ R Γ L Γaǭ b 2 P (Γ φ(ǭ b ) π + T )(16)
for
T < |ǭ b |/(P ln E G |ǭ b | ). The coefficient φ is given by φ(ǭ b ) = − E F 0 dǫ ǫ b − ǫ − Σ (1) 0 (ǫ b − ǫ) ≃ 1 P lnln( E G P |ǭ b | ) + ln( P E G E F + P E G )(17)
Equation (16) describes the decay of the resonance below ǫ b = ∆ǫ b . It reveals that the square of the halfwidth is larger than Γ 2 by a factor P φ.
Finally, one could conclude that the delocalization of the resonant levels with formation of a narrow band which is located inside the barrier in between two electrodes, leads to the suppression of the Kondo conductance all around except for a special position of the level energies at ǫ b = P E G (ln(E G /E F ) − 1). At this point the tunneling structure contributions to the thermodynamic functions become singular, and the linear conductance has a resonant behavior. These results are easy related to the known properties of the Coulomb blockade transport through the granule [6]. The above consideration describes the first steps of the Coulomb staircase. Due to the lack of the electron-hole symmetry in the excitation spectrum of the granule the position of the resonance is shifted from n G = 1/2 [2] to n G = 1, and the form of the resonance undergoes asymmetrical deformation [5].
I am grateful to D. Averin
makes the tunneling resonant and justifies the application of the 1/N expansion technique [1]. In this case it is a 1/M expansion and it means the summing up of the most divergent terms in M of each order of expansion in Γ.
and K. K. Likharev for elucidative discussions during my staying in Stony Brook, and M. Stopa for his interest and kind help, and RIKEN ICO for the hospitality. This work was supported by the STA Fellowship (Japan) and by ONR grant 00014-93-1-0880.
. N E Bickers, Rev. Mod. Phys. 59845N.E. Bickers, Rev. Mod. Phys. 59,845 (1987).
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A C Hewson, The Kondon Problem to Heavy Fermions. Cambridge, EnglandCambridge Univ. PressA. C. Hewson, The Kondon Problem to Heavy Fermions (Cambridge Univ. Press, Cam- bridge, England, 1993).
. L I Glazman, M E Raikh, Pis'ma Zh. Eksp. Teor. Phiz. 47452JETP Lett.L.I. Glazman and M. E. Raikh, Pis'ma Zh. Eksp. Teor. Phiz. 47, 378 (1988) [ JETP Lett. 47, 452 (1988) ];
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. T K Ng, Phys. Rev. Lett. 701768T.K. Ng , Phys. Rev. Lett. 70, 1768 (1993).
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D Averin, K K Likharev, Mesoscopic Phenomena in Solids. B. L. Altshuller et alAmsterdamElsevier173D. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids , edited by B. L. Altshuller et al (Elsevier, Amsterdam, 1991), p.173.
. L I Chen, C S Ting, Phys. Rev. B. 418533L.I. Chen and C.S. Ting, Phys. Rev. B 41, 8533 (1990);
. Y Meir, N S Wingreen, Phys. Rev. Lett. 682512Y. Meir and N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).
| {'fraction_non_alphanumeric': 0.06450990481090815, 'fraction_numerical': 0.027527656290198096, 'mean_word_length': 3.367696629213483, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 26, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'I consider the effect of the finite width of the resonant level on its thermodynamics and tunneling transport properties in the single electron charging regime. The finite width of the levels results from their delocalization with formation of a narrow band due to their mutual overlap. Making use of 1/N technique one can show that there is a fixed energy level position at which the system undergoes the transition from the Kondo regime (above this energy) to the paramagnetic regime (below it). The linear bias conductance calculated as a function of the chemical potential turns out to be suppressed all around except for the vicinity of the transition energy where it has an asymmetrical resonant behavior.', 'arxivid': 'cond-mat/9507090', 'author': ['V V Ponomarenko \nFrontier Research Program\nRIKEN\nWako-Shi351-01SaitamaJAPAN\n\nIoffe Physical Technical Institute\n194021, St. PetersburgRussia\n'], 'authoraffiliation': ['Frontier Research Program\nRIKEN\nWako-Shi351-01SaitamaJAPAN', 'Ioffe Physical Technical Institute\n194021, St. PetersburgRussia'], 'corpusid': 16039361, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 5263, 'n_tokens_neox': 4740, 'n_words': 3065, 'pdfsha': '9efc9b943548d764b235d5c25087a723910ff740', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/9507090v2.pdf'], 'title': ['Kondo Resonance in Transport Properties of Double Barrier Structure', 'Kondo Resonance in Transport Properties of Double Barrier Structure'], 'venue': []} |
arxiv |
On the projective height zero conjecture
12 Jan 2018 October 5, 2018
Benjamin Sambale
Fachbereich Mathematik
67653Kaiserslautern, KaiserslauternTUGermany
On the projective height zero conjecture
12 Jan 2018 October 5, 2018projective height zero conjecture AMS classification: 20C15
Recently, Malle and Navarro put forward a projective version of Brauer's celebrated height zero conjecture on blocks of finite groups. In this short note we show that Brauer's original conjecture implies the projective version.The following is a long-standing conjecture in representation theory of finite groups: Conjecture 1 (Brauer's height zero conjecture [2, Problem 23]). Let B be a block of a finite group with defect D. Then every irreducible character in B has height 0 if and only if D is abelian.Recently, Malle-Navarro [6] proposed the following generalization of Conjecture 1 (the case Z = 1 yields the original conjecture). An equivalent statement in terms of θ-blocks was given by Rizo [9].Conjecture 2 (Malle-Navarro's projective height zero conjecture). Let B be a p-block of a finite group G with defect group D. Let Z be a central p-subgroup of G and let λ ∈ Irr(Z). Then every irreducible character in B lying over λ has height 0 if and only if D/Z is abelian and λ extends to D.
In their paper, Malle and Navarro already proved the "if direction" of Conjecture 2 by making use of the solution [5] of the "if direction" of Conjecture 1. Moreover, they showed that Conjecture 1 implies Conjecture 2 for blocks of maximal defect. Generalizing their argument, we prove that Conjecture 1 always implies Conjecture 2.
Theorem 3. Suppose that Conjecture 1 holds for all blocks of finite groups. Then Conjecture 2 holds for all blocks of finite groups. Our proof uses the notation from [6] and the language of fusion systems. Recall that every block B with defect group D induces a (saturated) fusion system F on D (see [1,Theorem IV.3.2] for instance). The focal subgroup and the center of F are given by Let K G be the kernel of λ. Suppose first that K = 1. Then B dominates a unique block B of G/K with defect group D/K (see [8,Theorem 9.10]). Since the kernel of every χ ∈ Irr(B|λ) contains K, we have Irr(B|λ) = Irr(B|λ) = Irr 0 (B|λ) = Irr 0 (B|λ). By induction, it follows that (D/K)/(Z/K) ∼ = D/Z is abelian.
foc(F) := x −1 x f : x ∈ Q ≤ D, f ∈ Aut F (Q) D, Z(F) := {x ∈ D : x is
Therefore, we may assume that λ is faithful. This implies D ′ ∩ Z = 1, since λ extends to D. Let F be the fusion system of B. Then Z ≤ Z(F) and it follows from [4,Lemma 4.3] that Z ∩ foc(F) = 1.
Let χ ∈ Irr(B) and µ ∈ Irr(Z|χ). Since foc(F) ∩ Z = 1, there exists an extensionμ ∈ Irr(D) of µ with foc(F) ≤ Ker(μ). Similarly, letλ ∈ Irr(D) be an extension of λ with foc(F) ≤ Ker(λ). By Broué-Puig [3,Corollary], we obtain a character ψ := (λμ −1 ) * χ ∈ Irr(B|λ) (cf. [10]). By hypothesis, ψ has height 0 and the same holds for χ, since χ(1) = ψ(1). Consequently, Irr(B) = Irr 0 (B) and Conjecture 1 shows that D is abelian and so is D/Z.
fixed by every morphism in F} ≤ Z(D)respectively. * Fachbereich Mathematik, TU Kaiserslautern, 67653 Kaiserslautern, Germany, [email protected] ] implies that λ extends to D. We show by induction on |G| that D/Z is abelian.Proof of Theorem 3. Let B be as in Conjecture 2. Since the "if direction" of Conjecture 2 holds, we
may assume that Irr(B|λ) = Irr 0 (B|λ). By [8, Theorem 9.4], the set Irr(B|λ) is not empty and a result
of Murai [7, Theorem 4.4
AcknowledgmentThe author is supported by the German Research Foundation (SA 2864/1-1 and SA 2864/3-1).
M Aschbacher, R Kessar, B Oliver, Fusion systems in algebra and topology. CambridgeCambridge University Press391M. Aschbacher, R. Kessar and B. Oliver, Fusion systems in algebra and topology, London Mathe- matical Society Lecture Note Series, Vol. 391, Cambridge University Press, Cambridge, 2011.
R Brauer, Representations of finite groups. New YorkWileyILectures on Modern MathematicsR. Brauer, Representations of finite groups, in: Lectures on Modern Mathematics, Vol. I, 133-175, Wiley, New York, 1963.
Characters and local structure in G-algebras. M Broué, L Puig, J. Algebra. 63M. Broué and L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306-317.
Control of transfer and weak closure in fusion systems. A Díaz, A Glesser, N Mazza, S Park, J. Algebra. 323A. Díaz, A. Glesser, N. Mazza and S. Park, Control of transfer and weak closure in fusion systems, J. Algebra 323 (2010), 382-392.
Quasi-isolated blocks and Brauer's height zero conjecture. R Kessar, G Malle, Ann. of Math. 2R. Kessar and G. Malle, Quasi-isolated blocks and Brauer's height zero conjecture, Ann. of Math. (2) 178 (2013), 321-384.
G Malle, G Navarro, arXiv:1712.08331v1The projective height zero conjecture. G. Malle and G. Navarro, The projective height zero conjecture, arXiv:1712.08331v1.
Block induction, normal subgroups and characters of height zero. M Murai, Osaka J. Math. 31M. Murai, Block induction, normal subgroups and characters of height zero, Osaka J. Math. 31 (1994), 9-25.
Characters and blocks of finite groups. G Navarro, London Mathematical Society Lecture Note Series. 250Cambridge University PressG. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, Vol. 250, Cambridge University Press, Cambridge, 1998.
. N Rizo, preprintN. Rizo, p-blocks relative to a character of a normal subgroup, preprint.
On the focal defect group of a block, characters of height zero, and lower defect group multiplicities. G R Robinson, J. Algebra. 320G. R. Robinson, On the focal defect group of a block, characters of height zero, and lower defect group multiplicities, J. Algebra 320 (2008), 2624-2628.
| {'fraction_non_alphanumeric': 0.07545533391153512, 'fraction_numerical': 0.0398959236773634, 'mean_word_length': 4.080176211453744, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Recently, Malle and Navarro put forward a projective version of Brauer's celebrated height zero conjecture on blocks of finite groups. In this short note we show that Brauer's original conjecture implies the projective version.The following is a long-standing conjecture in representation theory of finite groups: Conjecture 1 (Brauer's height zero conjecture [2, Problem 23]). Let B be a block of a finite group with defect D. Then every irreducible character in B has height 0 if and only if D is abelian.Recently, Malle-Navarro [6] proposed the following generalization of Conjecture 1 (the case Z = 1 yields the original conjecture). An equivalent statement in terms of θ-blocks was given by Rizo [9].Conjecture 2 (Malle-Navarro's projective height zero conjecture). Let B be a p-block of a finite group G with defect group D. Let Z be a central p-subgroup of G and let λ ∈ Irr(Z). Then every irreducible character in B lying over λ has height 0 if and only if D/Z is abelian and λ extends to D.", 'arxivid': '1801.04272', 'author': ['Benjamin Sambale \nFachbereich Mathematik\n67653Kaiserslautern, KaiserslauternTUGermany\n'], 'authoraffiliation': ['Fachbereich Mathematik\n67653Kaiserslautern, KaiserslauternTUGermany'], 'corpusid': 55472278, 'doi': '10.1090/proc/14134', 'github_urls': [], 'n_tokens_mistral': 2035, 'n_tokens_neox': 1695, 'n_words': 947, 'pdfsha': '3788ca7ddc82d1d5b57400e6053706dc128ee6b0', 'pdfurls': ['https://arxiv.org/pdf/1801.04272v1.pdf'], 'title': ['On the projective height zero conjecture', 'On the projective height zero conjecture'], 'venue': []} |
arxiv |
Full-Life Cycle Intent-Driven Network Verification: Challenges and Approaches
Yanbo Song
Chungang Yang
Jiaming Zhang
Xinru Mi
Dusit Niyato
Full-Life Cycle Intent-Driven Network Verification: Challenges and Approaches
1
With the human friendly declarative intent policy expression, intent-driven network can make network management and configuration autonomous without human intervention. However, the availability and dependability of these refined policies from the expressed intents should be well ensured by full-life cycle verification. Moreover, intent-driven network verification is still in its initial stage, and there is a lack of full-life cycle end-toend verification framework. As a result, in this article, we present and review existing verification techniques, and classify them according to objective, purpose, and feedback. Furthermore, we describe intent verification as a technology that provides assurance during the intent form conversion process and propose a novel full-life cycle verification framework that expands on the concept of traditional network verification. Finally, we verify the feasibility and validity of the presented verification framework in the case of an access control policy for different network functions with multi conflict intents.Index Terms-Intent-driven network, network policy, network verification, network management.
I. INTRODUCTION
Software-defined networking (SDN) is distinguished by programmability, flexibility, and decoupling of control and data planes. However, due to the growing network scale and business diversity, network management becomes more complex and challenging. Millions of forwarding rules dictate how vital devices behave in the network. Incorrect policies or configurations can result in network vulnerabilities that lead network outages, routing oscillations, and forwarding black holes, further impairing network availability and dependability [1]. Most network operators use low-level interfaces to program the network, which is practically inefficient and error-prone. It has been proven that the functionality of programmable networks cannot be implemented unless a highlevel abstraction policy is provided to the users [2]. Intentdriven network (IDN) promises to fill this gap by providing a simple, yet expressive high-level abstraction policy over the network controller. This abstraction policy hides unnecessary details of the underlying infrastructure from users and allows them to customize network configuration using human readable intents.
IDN is a novel network paradigm that has gained significant interest from industry and academia. An intent is an abstraction declaration of what applications require from the † Y. Song, C. Yang, J. Zhang network. It is composed of a set of primitive "verb", each describing a specific but high-level operation [3]. High-level abstraction policies decrease the need for specialized expertise. However, network devices cannot directly comprehend and enforce them. There are certain differences between the semantics contained in the intent and the parameters of the physical network. For example, an intent "to establish a highspeed link from node A to node B" is with unclear semantics. Because there exist "distortion" and mismatch between service provider-defined and user-defined high-speed network. IDN must thus be able to translate intents into more detailed lowerlevel rules, and this process is known as intent translation. Each translation adds some "distortion" due to lower-level constraints and hence the intent must be checked at each step to ensure that it stays accurate throughout the continuous translation. Intent is diverse and random, which can lead to uncontrollable network issues like intent conflicts among applications. As a result, intent verification is required. However, there is a lack of evaluation of intent execution effect and the verification of the intent's feasibility and validity is separately implemented.
IDN eliminates the inefficiencies of conventional network management and decreases the risk of misconfiguration by automatically converting abstraction intents to detailed network configurations. IDN contains multiple key functions: Intent Representation, Intent Verification, Policy Mapping, and Situational Awareness. On the one hand, verification does exist in traditional networks, it is a point-to-point verification and focuses on checking compliance with network policies and properties such as paths among network nodes. On the other hand, intent verification has to be structured with a full-life cycle that automatically achieves the goal of an intent and policy correctness. In particular, an intent can be characterized as natural language or graphic language, and there exist different kinds of verification from intents to policies [4].
The verification process incorporates formal methods, mathematical reasoning, computer languages, and networks. The verification approaches fall into two types. The first is conventional network verification techniques, which are often based on solvers and customized network tools. The other is a breakdown of conflicts based on policy consistency. However, there is no standardized definition of verification procedures and a full-life cycle view of verification. As a result, this article first provides a brief survey and classification of verification techniques. We present a framework for full-life cycle intent-driven network verification and develop a use case that employs the policy graph abstraction to resolve disputes and then configures policy in a simulation network. This article framework, we therefore present an intent-driven network full-life cycle verification framework, which is with both intents and network status in the policy graph abstraction. • To evaluate the full-life cycle verification framework, we implement a full-life cycle to black the virtual network function orchestration and the packet arrival rate verification. The remainder of the paper is organized as follows. We first overview the concept and classification of intent verification in Section II and Section III. Then, we propose the fulllife cycle intent-driven network verification framework and conduct simulations and evaluations to verify the effectiveness of the proposed framework in Section IV, followed by future research and concluding remarks in Section V and VI.
II. AN OVERVIEW OF INTENT-DRIVEN NETWORK
VERIFICATION IDN aims to provide a more natural and intuitive network administration technique than conventional network management paradigms. An example of a general configuration and an example intent is presented as follows:
Conf iguration :
If _(match(srcip = ZoneB, dstport = 80, dstip = ZoneA)) Intent :
T he traf f ic f rom ZoneB to ZoneA is allowed.
The Conf iguration contains specific information about what the switch must do (i.e., match the destination IP address and port, and then forward the packet). Meanwhile, the Intent only describes a desire (i.e., traffic from Zoom B to Zoom A is allowed). Since the intent only describes the abstract desire and lacks many configuration information, it is necessary to complete the details and verify the correctness of the process.
We first classify, discuss, and compare the existing network verification techniques. In general, IDN expands the scope of verification, because only high-level abstraction policies have consistent verification problem. Therefore, feasibility verification often occurs in IDN. The verification techniques in conventional networks, such as IP and SDN, focus on the effectiveness of the feedback from underlying devices. Next, we elaborate some details of the related works. The characteristics of each schemes related to intent verification are shown in Table I. We review the state-of-the-art in feasibility verification, validity verification and joint verification. Feasibility indicates whether the policy can run in the network, validity is an attribute that determines whether a policy meets certain requirements, and joint verification combines the above two attributes.
A. Feasibility Verification
Feasibility verification ensures that the policies are executable in the network, conflict-free between policies (internal), and conflict-free between policies and underlying constraints (external).
1) Verification with graph : The northbound interface (NBI) of the IDN allows users express their intentions, and avoids conflicts between intents [3]. The NBI enables intent conflict resolution before it is issued to the SDN controller. The intent conflict handling capability is more challenging when a vast number of policies are issued. For instance, the policy graph abstraction (PGA) provides a simple and intuitive graphical interface that is similar to how network managers typically visualize their policies on a whiteboard. Through the graph editor and graph composer, the output is a conflict-free policy graph. On the basis of PGA, Janus system expands the graph combination to dynamic policy, maximizing the number of configured policies and minimizing the number of path changes caused by either intrinsic dynamics in policies or due to policy churn [5].
2) Verification with natural language: In addition to graphs, an intent can be expressed in natural language. Therefore, several works have focused on natural language processing (NLP). In the LUMI scheme, information extraction by named entity recognition allows collecting feedback from operators and incorporating it into the information extraction process, which is continuously learned or trained to improve the accuracy of information tagging [4]. The LUMI scheme analyzes the conflicts arising from Nile after confirming a successful extraction of entities [6]. Nile is a scheme for learning network behavior expressed by the operator while providing a user-friendly interface to assist in the verification process of intent concretization. The translation process consists of three phases: entity extraction, intent translation, and intent deployment. For identifying the intent, the Evian client uses NLP and machine learning to build an intelligent bot that can have similar human conversations with users [8]. The bot will use multiple conversations with users in English to gather all requirements about their network use cases.
B. Validity Verification
The advantage of validity verification is to ensure the policy is able to achieve the requirements for the network, including the efficient translation of policy to profile (offline) and the effective execution of policy to forwarding behavior (online).
1) Verification using probe: The southbound intent verification is achieved by capturing the configuration of the network data plane or by collecting real traffic as feedback. The control plane adjusts and modifies the policy formulation based on the feedback to achieve consistency before and after policy devolution. For complex network debugging problems, dynamic policy verification methods are gradually replacing static verification. ATPG is an automated and systematic approach to network testing and debugging as a transparent agent deployed in the middle of the control plane and data plane [9]. At the same time, ATPG reads the switch configuration and generates device-independent models, sends test packets at regular intervals to detect faults, and designs fault location mechanisms. The type of verification is validity verification, and the verification taxonomy refers to R = R static verification and R = F dynamic verification. SERVE is an SDN rule verification framework that can automatically identify data plane network problems [9]. By modeling the network device as a stateful multi-root tree of pipeline processing, the number of probes used is reduced.
In data center networks (DCN), TCP packets sent and received by edge servers can continuously detect network conditions and performance issues, such as end-to-end delay. Moreover, anomalies in key performance data in the network can directly react and determine whether there are network problems. In the Pingmesh scenario, the above approach also provides data support for the definition and tracking of service level agreements [9].
2) Verification using models: Network model is an effective method to evaluate network policies by modeling the network state in the data plane; such as firewalls, load balancing, and other network functions, and then considering whether the network violates network policies based on the constructed model.
VeriDP is a proxy deployed between the control plane and the data plane [1]. VeriDP abstracts all rule configurations on the control plane into a path table. It tags data packets and checks the tag information of the data packets to see if the forwarding is correct. The practical deployment verifies that the VeriDP server is on the control plane and the data collection pipeline is on the data plane. However, the switches require hardware and software modifications and are not easily applied directly to the existing network.
Existing tools require fine-grained time scales for checking profiles and data plane states. Static analysis of the network data plane is performed offline, leading to problems such as not detecting or blocking errors when they arise during network operation. The VeriFlow layer is designed between SDN controllers and forwarding devices to obtain a snapshot of the network as it evolves [10]. Furthermore, by dynamically checking the validity of network invariants as each rule is inserted, modified, or deleted. To ensure a real-time response, VeriFlow introduces incremental algorithms to search for possible errors. The key technologies are mathematical modeling, fast rule checking, and analysis. VeriFlow is deployed with a proxy between the control plane and the data plane without feedback from real traffic on the data plane. This type of verification is called validity verification.
C. Joint Verification
The purpose of feasibility verification and validity verification are complementary; feasibility verification ensures that the policies are executable, while validity verification ensures that the policy can satisfy the network's requirements; the combination of the two can produce better results.
Monocle addresses the inconsistency issue in policy as a result of complex network configuration and data plane [11].
The key technique of monocle is mathematical modeling, where the switch forwarding table logic is constructed by representing it as a boolean satisfiability problem, and probe packets check the practical switch behavior. Monocle is deployed as a transparent proxy between the control plane and data plane, again without real traffic feedback and in the form of the packets. Monocle is positioned as a layer between the OpenFlow controller and the switch. This design allows Monocle to intercept all rule modifications issued to the switch and maintain the expected flow x NLP
x PGA&JANUS x NILE
Intent Translation
x NLP
x PGA&JANUS
x NILE
x Merlin
x Pyretic
x Frenetic
x OpenFlow
x FlowVisor
x OpenVirtex on that switch. Its validation types include feasibility and validity verification. To ensure that the network configuration and status derived from network automation match the administrator's specified intent, Epinoia has designed a network intent checker for stateful networks [13]. Epinoia expands the PGA-based on a unified network function model and gradually checks for intent violations within the network to reduce the impacts and costs of network changes. Minesweeper is a tool that verifies whether a network meets a variety of expected properties, including reachability or isolation between nodes, waypoints, black holes, bounded path length, load balancing, functional equivalence between two routers, and fault tolerance [12]. Minesweeper converts the network configuration file into a logical formula that captures the stable states to which network forwarding will converge, and these states are the result of interactions between routing protocols such as open shortest path first (OSPF), border gateway protocol (BGP), and static routing. Then minesweeper combines the constraints that describe the expected properties. If the combination formula is satisfiable, the network has a stable state. Otherwise, there is no steady state that violates the properties. The joint verification approach of feasibility and validity will be more comprehensive in terms of verification effectiveness than verification techniques that verify a certain property separately. It can be concluded that IDN lacks a clear verification definition and classification. Therefore, the joint use of multiple verification techniques is required to guarantee the full-life cycle performance of network configurations. As a result, we define a full-life cycle verification definition to present full-life cycle of intent and the translation of intent language.
III. DEFINITION AND CLASSIFICATION OF INTENT-DRIVEN NETWORK VERIFICATION
Although the standard organization has a preliminary definition and verification classification, it is still insufficient to cover the entire IDN [3]. We define and classify verification based on the objective, purpose, and feedback, as well as characterize in different perspectives.
A. Definition of Intent-Driven Network Verification
IDN is a revolutionary network paradigm in which intent is viewed as a collection of high-level abstraction policies. High-level abstraction policies reduce the need for specialized knowledge. However, they cannot be directly understood and enforced by network devices. Thus, IDN must be capable of translating intents into more precise lower-level policies, a process referred as intent translation. As illustrated in Fig. 1, the form of intent is continually changing and translating. Each translation introduces some "distortion" due to lowerlevel limitations, therefore the intent should be verified at each stage to verify that it remains correct throughout the continuous translation. In IDN, the intent flow is expressed as follows:
• User Intent I in Natural Language: Natural language is the media via which users convey their intents about network functionalities and performance. The users who are unfamiliar with network can also communicate their intents in natural language. • Network Intent I in Domain-Specific Language: The domain-specific language is used to standardize the unstandardized natural language and hence improving the clarity of the network intents. Typically, the domainspecific language is constructed of tuples with specified names, such as {domain, attribute, object, action, and result}. • Logical Rule R in Northbound Programming Language: Northbound language refines intents by encapsulating them in practical network functions or algorithms. Because these network functions and algorithms act on the controller's logical view of the network and have not yet been constructed on the switch, they are referred to as logical rules. • Physical Rule R in Southbound Programming Language: Southbound languages like OpenFlow, sFLow, NetFlow, and simple network management protocol (SNMP) can be used for translating logical rules into physical rules that can be applied to network devices. • Forwarding Behavior F in Flow Rule: Eventually, the intent will be translated into the network's forwarding behavior, which is determined by the flow table rules. The network device processes and forwards data packets based on the rules. The goal of IDN is to ensure that "user intent" is translated into "packet forwarding behavior." The user intent mentioned above can be user intent, network intent, logical rule, physical rule, and forwarding behavior. The "rules" represent the "intents" semantics. The intent translation process begins by adopting natural language and then converting high-level natural language into various levels of rules. As a result, the verification ensures that the semantics carried by the intent are preserved as much as possible during the translation process. The content to be verified at each stage differs. After the intent is converted into rules, the primary goal of the verification is to determine whether the policy implementation meets the expectations of the intent. Thus, full-life cycle verification ensures that the original semantics can be maintained between intents in any form, and the full-life cycle refers to the time between the generation of the intent and the end of the intent. Therefore, Full-Life Cycle Intent Verification can be defined as:
F = R = R = I = I.
Due to the limitation of verification technology, in practical networks, the current researches on verification don't distinguish all the conversions. Researchers tend to focus on one or several parts of the formula. NLP technology can be used to standardize natural language. PGA and Janus platforms resolve internal conflicts of intents through policy graph abstraction [5]. The traditional SDN programming languages, such as Pyretic, Frenetic, and Merlin can be installed in the controller after being compiled [7]. FlowVisor and OpenVirtex can check the correctness of the policy before and after the rule is issued, respectively [14]. Researchers have achieved varying degrees of verification depending on their own technologies. They made a distinction between the control plane and the data plane. However, they lack the complete classify of verification process. As a result, we rearrange the classification of verification techniques according to the definition of verification.
B. Classification of Intent-Driven Network Verification
We summarize and classify the existing verification technologies according to the location in F = R = R = I = I. As shown in Fig. 1, we categorize verification technologies by where it occurs, whether there is feedback, and what purpose is.
• The verification techniques can be classified as "Internal" and "External" depending on the location of a verification object. Internal verification verifies multiple Xs(X = I, R) in a layer, which is simpler to resolve at higher levels. External verification verifies the correctness of translation between layers, which relies more on underlying device feedback. The dedicated network model is insufficient to cover the full-life cycle intent verification. Since an intent may be expressed at several levels of abstraction, e.g., natural language, programming language, and graph, there may be overlap between the representations of I and I. The outcome of high-level abstraction translation needs to be checked only when the representation of I is at a very high degree of abstraction, like in a domain-specific language. Therefore, the top layer verification R = I should focus on internal consistency or conflict resolution to avoid a more severe impact on the network. • The verification techniques can be further classified into "Static" and "Dynamic" verification, or "Offline" and "Online" verification, based on the feedback from the data plan during the verification process. Offline verification verifies the device profile of the data plane, focusing on R = R . Online verification verifies the network status of the data plane by real-time collection, focusing on R = F and whether the forwarding behavior in the network meets the policy requirements. The distinction is primarily in whether the data plane's real-time network state is collected. • According to the verification purpose, "Feasibility" and "Validity" are more general and comprehensive than the other two classifications. Feasibility verification ensures that policies are conflict-free with each other and the device constraints; Validity verification ensures the validity of configurations and forwarding behaviors. In addition, the conflict of high-level intents and policies needs to be considered; for example, the intents may come from different application requests. The objects targeted by the intent are shared resources, i.e., bandwidth, which will cause a conflict in the policy after intent translation. The conflict should be checked and dissipated before intents are finally executed. Verification should focus on the full-life cycle of intent, including the entire verification process. A sound verification scheme should have a full-life cycle with feedback. Way III in Fig. 1 is a more generic and inclusive classification scheme used to classify verification techniques.
IV. A FULL-LIFE CYCLE INTENT-DRIVEN NETWORK VERIFICATION FRAMEWORK AND IMPLEMENTATION
The intent verification is related to the stability and reliability of the whole network. Current verification techniques form independent modules for different aspects. As a result, we integrate these various aspects of verification techniques in this article. We propose a framework for intent verification in IDN, which contains the full-life cycle of F = R = R = I = I, and includes an application layer, an intent enable controller, an infrastructure layer, and two interface, as shown in Fig. 2.
The application layer requests a service, which could be presented as nature language U ser Intent T able I or graph. User's intent expressed by the graph is already a standardized language I , there is no need for I = I verification. However, possible conflicts among different intents still exist. The formalized intents by intent translation module are sent to the controller through NBI 1 . Additionally, the intent Fig. 2: Full-life cycle intent-driven network verification framework. The verification module is deployed in the control layer, which is connected with the application layer and switches through northbound and southbound interfaces.
verification module collects the intents through NBI and forms the N etwork Intent T able I composed of historical intents. When the controller receives an intent and implements a policy, which is generated according to both the intents and the network status. Status Monitor 3 collects the network state. Policies should be combined with the feedback given by the intent verification module. If the policy fails to pass the verification module, the policy will be remapped 6 . The conflict-free policies 4 are sent down to the switches through the southbound interface (SBI) 2 and forms Logic Rule T able R. After the policy is issued, the practical flow behavior of the network traffic 5 is collected and reported to the intent verification module in real-time and form the Behavior T able F 7 . Then, according to the fulllife cycle verification definition:
• The feasibility verification of I and I guarantee the correctness of the intent translation; the verification of R and I guarantees that the issued policy complies with its underlying network constraints, and the verification of R guarantees that the policy is conflict-free. • The validity verification of R and F , R and F guarantee the practical network traffic forwarding behavior conforms to the specified policy and guarantees the validity of the intent. In the verification process, if the verification fails, the policies are corrected and reissued through policy feedback. The intent verification framework proposed in this article realizes the full-life cycle verification of intents, which can be used as a basic framework in developing IDN.
A. Use Case on Full-Life Cycle Intent-Driven Network Verification
Network function virtualization (NFV) has become an important tool to satisfy the needs of heterogeneous services. A series of virtual network functions connected by virtual links can be used to complete the user's end-to-end service request. Therefore, we describe implementation details for realizing the proposed framework of full-life cycle intent verification outlined in the NFV simulation scenario. We represent the intent in a graph and issue the configuration into Mininet by ONOS as shown in Fig. 3 [3]. We adopt a policy graph abstraction to verify policies and convert conflict-free policies to Pyretic [15]. Neo4j (a graph database) stores physical network information so that the intent conflict resolution module can query it. The conflict resolution algorithm is implemented in Python 2.7.0 as a single-threaded program. The network node groups in policy graph abstraction are separated based on various characteristics. All network nodes in our use case are classified based on their geographic location and function, such as Zone A and Zone B representing two buildings, A1 and B1 meaning two academies in the buildings, and world wide web (Web) and domain name system (DNS) describing different services. Finally, the policy mapper in the network controller sends the verified policy to the network. This use case can be extended to multiple network scenarios and can be integrated into other systems. It is also a beneficial research basis for implementing intent verification in future studies.
We create 100 to 500 intent application requests and the intent verification module collects the intent into Intent T able I. Then we realize the intent feasibility verification through graph combination. The policy graph abstraction after verification can be expressed as shown in Fig. 3 (a). Endpoints represent a group of users or network services. The text represents the port number and other attributes, such as bandwidth or network functions: load balancer (LB), intrusion detection systems (IDS), world wide web(W eb), and distributed denial of service (DDoS). Since the intent application requests come from different endpoint groups, they may have conflicts. At this time, the primary purpose of verification is to ensure conflict-free merging between multiple intents. The policy graph abstraction is translated into domain language as shown in Fig. 3 (b), which is conflict-free and executable, indicating the original address, port number, and network function between them. The service function chain planning issue is reduced to a route planning problem, with each network device representing a service function, and we select the path calculation policy from P olicy T able R to implement the service function.
From the curves in Fig. 4 (a), when there are 100-500 intents, the verification time is between 300 and 1160 ms, and the average verification time is about 2-3 ms. The cumulative distribution function (CDF) of verification time is relatively concentrated, with a 90% of the verification time less than 281 ms, 485 ms, 675 ms, 885 ms, and 1070 ms, respectively, which means a relatively stable effect. For the validity verification of the intent verification engine, we take the intents entered in Fig. 3 as an example, and compare the packet arrival rate with different approach. According to the simulation results in Fig. 4 (b), the worst packet arrival rate result, i.e., 10.7% and 21.3%, are from the approaches 1 and 2 that work by directly issuing the packets without processing, by randomly selecting one of the two conflicting intents to execution. NIC approach, i.e., based on network intent composition (NIC), yields the packet arrival rate of 36.7%. The packet arrival rate of our verification engine is 42.6% (approach 4). According to the results in Fig. 4, it can be seen that our verification engine can better realize the conflict detection and decomposition of multi-user input intents.
V. CHALLENGE AND FUTURE WORK
Various aspects of intent verification technology have been investigated. For example, the research has focused on formal verification, which includes formal modeling, fuzzy mathematics, and other techniques. The other research considers network device hardware and software, including controller verification servers, database development, and traffic collection techniques. However, the current technology needs to be improved in order to achieve the full-life cycle system and ensure its autonomy.
• The simplicity of network collection tools. Continuous dynamic verification is required to combine network state. Existing network diagnostic tools are inefficient because they can assess only the forwarding behavior of a single data packet at a time. The data is obtained directly from the underlying equipment (low-level) rather than through the use of a predefined data model. Network administrators lack a global perspective on the impact of individual device modifications on the network. • The lack of collaboration between the network models.
Whether adopting a formal verification or a verification method based on multiple solvers, the network should be modeled, which results in limited scalability and makes it difficult to adapt to a stateful network. Validity verification includes mathematical model and traffic acquisition techniques, which require to characterize both control plane policies and data plane network states. Moreover, designing fast validation algorithms can ensure realtime validation where the mathematical model should be consistent with the feasibility validation, i.e., extend the control plane policy model to data plane modeling. There should also be modules and pipeline designs in place for the SDN controller and switches so that they can obtain and analyze data between the control plane and the data plane, as well as feedback each other. • The safety of intent. Intents represent user preferences for networks and applications, such as demand for services, content, and network traffic. Disclosing intents may lead to leakage of user privacy. The modification of intent can affect the network more than command-line. Current verification techniques are based on the belief whether the user's intent is correct. In addition to the existing security mechanisms, the data level should be untampered with and be able to verify malicious intents. It should not only check grammar rules but semantics. For example, the intents that don't conform to network operation rules should be considered incorrect. Therefore, we should also include the security of the intent in the scope of verification. It is worth noting that current IDN also faces these challenges. Therefore, more network verification tools and mathematical models should be the main focus in the future work, and verification techniques should be paid more attention on intent security, such as preventing intent from tampering.
VI. CONCLUSIONS
This article began by clearly defining and classifying intentdriven network verification from different perspectives, which was presented based on the location of verification, the availability of feedback, and the purpose of verification. Then we presented a brief survey of the existing verification technology. After that, we presented a full-life cycle verification framework with feedback and verified the access control policy of the network function. Our verification engine can better ensure multi-user intents conflict-free and executable. Finally, we summarized the future work and challenges of intent verification in terms of verification tools, verification models, and intent safety.
VII. ACKNOWLEDGMENTS
Fig. 3 :
3The feasibility verification result: conflict-free policy in Pyretic and graph.
The CDF of verification time for 100-500 intents.
Packet arrival rate with proposed verification engine and NIC apporach.
Fig. 4 :
4The validity verification results: the verification time performance and packet arrival rate. Random approaches 1 and 2 are directly executed without processing and randomly selecting the conflicting intents to execute. NIC approach is based on network intent composition and proposed approach is our verification engine.
and X. Mi are with the State Key Laboratory on Integrated Services Networks, Xidian University, Xi'an, 710071 China. (emails: [email protected]; [email protected]; [email protected]; [email protected]). ‡ D. Niyato is with School of Computer Science and Engineering, Nanyang Technological University, 639798 Singapore. (emails: [email protected]).
TABLE I :
IA brief survey of validity, feasibility, and joint verification techniques. Due to the lack of clear definition and classification of IDN verification, we define the IDN verification as a full-life cycle verification from a language translation standpoint, and present a brief survey and a classification of the current verification technology. • As there is no unified design of the IDN verificationType
Solutions
Theory and
Methodology
Verifi Object
Characteristic
Network
Feasibility
Verification
PGA, Janus [5]
Policy Graph
Abstraction
Network Function
Provide an intuitive graph abstraction to express
and compose policies.
IDN
LUMI [4]
Nile [6] and Merlin
[7]
Natural Language
Continuous feedback to improve the accuracy of
information extraction but fixed Priority Policy.
IDN
Evian [8]
Resource Description
Framework
Natural Language
Intentional presentation platform with natural
language interaction.
IDN
Validity
Verification
ATPG [9]
Probes and Header
Space Analysis
Path
Point out the dynamic verification, and generates
fewer test packages.
SDN
SERVE [9]
Probes
Rules
Verification of data plane, less use of probes.
SDN
Pingmesh [9]
Ping
Path
Single Packet .
DCN
VeriDP [1]
Packet Tag
Path
Forwarding behavior verification, but need to
modify the switch hardware and software .
IP
VeriFlow [10]
Mathematical
modeling EC
Rules
Real-time network verification.
SDN
Joint
Verification
Monocle [11]
Agent
Configuration
Express the logic of the switch forwarding table
as a boolean satisfiability problem.
IP
Mineseweeper
[12]
BatFish and SMT
Formulate
Configuration
Encode all possible packet behavior within the
network using first-order logic, but solve all
constraints as a whole.
IP
Epinoia [13]
Graph Abstraction
Network Function
Extends the intent specification in PGA and
supports incremental checks.
IP
makes the following contributions:
•
table contents in each switch. After determining the expected state of a switch, Monocle can calculate the packet headers for rules to be enforcedUser Intent I
Network Intent I
Pyhsical Rule: R
Logical Rule:R
Forwarding
Behavior: F
Intent Compilation
Rule Installation
Rule Execution
Natural Language
Domain-Specific
Language
Northbound
Programming
Language
Flow Rule
Southbound
Programming
Language
Technical
Perspective
Network
Perspective
Languages
Perspective
R = R
F = R
I = I
I = R
Intent Translation
(b) Conflict-free policy in graph.If_(match(srcip=ZoneB, dstport=80, dstip=ZoneA)) if_(match(srcip=DNS, dstport = 53, dstip = A1)), FW > > IDS if_(match(srcip=A1, dstport=22 23 53, dstip=B1)), FW>>DDoS if_(match(srcip=A1, dstport=80, dstip=Web)), FW>>LB if_(match(srcip=DNS, dstport=53, dstip=B1), FW>>IDS if_(match(srcip=B1, dstport=80, dstip=Web)), FW>>LB (c) Conflict-free policy in Pyretic.ZoneA
ZoneB
(a) Network topology.
ZoneA
A1
DNS
DB
Web
port:80;FW,LB;
permit:80
B1
ZoneB
ZoneA
A1
DNS
DB
Web
port:80;FW,LB;
permit:80
B1
ZoneB
This work was supported in part by the National Key Research and Development Program of China (2020YFB1807700).
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He is currently pursuing his doctoral degree in GUIDE family at Xidian University, which is led by Dr. Chungang Yang. His research interests are intent-driven network security and network management. Yanbo Song received his bachelor degree from Xidian UniversityYanbo Song received his bachelor degree from Xidian University. He is currently pursuing his doctoral degree in GUIDE family at Xidian University, which is led by Dr. Chungang Yang. His research interests are intent-driven network security and network management.
His research interests are artificial intelligent 6G wireless mobile networks, intent-driven networks (IDN), space-terrestrial networks (STN), and game theory for emerging communication networks. Chungang Yang is a full professor at Xidian University, where he leads the research team of "GUIDE, Game, Utility, artificial Intelligent Design for Emerging communicationsChungang Yang is a full professor at Xidian University, where he leads the research team of "GUIDE, Game, Utility, artificial Intelligent Design for Emerging communications ". His research interests are artificial intelligent 6G wireless mobile networks, intent-driven networks (IDN), space-terrestrial networks (STN), and game theory for emerging communication networks.
Jiaming Zhang received her master degree from Xidian University in GUIDE family, which is led by Dr. Chungang Yang. Her research interests are intentdriven network. Jiaming Zhang received her master degree from Xidian University in GUIDE family, which is led by Dr. Chungang Yang. Her research interests are intent- driven network.
She is currently pursuing her Ph.D. degree in communication and information system. Her research interests include intent-driven network and space information network. Lanzhou, ChinaXinru Mi received the B.E. degree in Electronic and information engineering from Northwest Normal UniversityXinru Mi received the B.E. degree in Electronic and information engineering from Northwest Normal University, Lanzhou, China. She is currently pursuing her Ph.D. degree in communication and information system. Her research interests include intent-driven network and space information network.
He received B.Eng. from King Mongkuts Institute of Technology Ladkrabang (KMITL), Thailand in 1999 and Ph. SingaporeDusit Niyato is a professor in the School of Computer Science and Engineering ; at Nanyang Technological University ; D. in Electrical and Computer Engineering from the University of ManitobaDusit Niyato is a professor in the School of Computer Science and Engi- neering, at Nanyang Technological University, Singapore. He received B.Eng. from King Mongkuts Institute of Technology Ladkrabang (KMITL), Thailand in 1999 and Ph.D. in Electrical and Computer Engineering from the University of Manitoba, Canada in 2008.
| {'fraction_non_alphanumeric': 0.035174430577865644, 'fraction_numerical': 0.009699740516495737, 'mean_word_length': 4.965479115479115, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 5, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'With the human friendly declarative intent policy expression, intent-driven network can make network management and configuration autonomous without human intervention. However, the availability and dependability of these refined policies from the expressed intents should be well ensured by full-life cycle verification. Moreover, intent-driven network verification is still in its initial stage, and there is a lack of full-life cycle end-toend verification framework. As a result, in this article, we present and review existing verification techniques, and classify them according to objective, purpose, and feedback. Furthermore, we describe intent verification as a technology that provides assurance during the intent form conversion process and propose a novel full-life cycle verification framework that expands on the concept of traditional network verification. Finally, we verify the feasibility and validity of the presented verification framework in the case of an access control policy for different network functions with multi conflict intents.Index Terms-Intent-driven network, network policy, network verification, network management.', 'arxivid': '2212.09944', 'author': ['Yanbo Song ', 'Chungang Yang ', 'Jiaming Zhang ', 'Xinru Mi ', 'Dusit Niyato '], 'authoraffiliation': [], 'corpusid': 254317623, 'doi': '10.1109/mnet.124.2200127', 'github_urls': [], 'n_tokens_mistral': 11330, 'n_tokens_neox': 10346, 'n_words': 7060, 'pdfsha': 'bae2aaf944132de3aedbf384cd294077d042f637', 'pdfurls': ['https://export.arxiv.org/pdf/2212.09944v1.pdf'], 'title': ['Full-Life Cycle Intent-Driven Network Verification: Challenges and Approaches', 'Full-Life Cycle Intent-Driven Network Verification: Challenges and Approaches'], 'venue': []} |
arxiv |
Inverse problem of recovering the time-dependent damping and nonlinear terms for wave equations
Song-Ren Fu
Inverse problem of recovering the time-dependent damping and nonlinear terms for wave equations
semilinear wave equationdamping termCarleman estimatehigher order linearizationGaussian beams
In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients.
Introduction
Let (Ω, g) be a Riemannian manifold of dimension n ≥ 2 with smooth boundary ∂Ω = Γ. Let M = Ω × (0, T ) and Σ = Γ × (0, T ). Assume that (x, t) = (x 1 , · · · , x n , x 0 = t) are local coordinates on M. The nonlinear wave equation considered in this paper is given by u tt − ∆ g u + b(x, t)u t + f (x, t, u) = 0, (x, t) ∈ M, u(x, t) = h(x, t), (x, t) ∈ Σ, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ Ω, (1.1) where f : Ω × R 1 × C → C is a smooth function. In the local coordinates,
∆ g u = div g Du = (det g) − 1 2 n ij=1 ∂ x j [(det g) 1 2 g ij ∂ x i u],
where div g and D are the divergence operator and Levi-Civita connection in the metric g, respectively.
The main goal in this paper is to study the inverse problem of recovering the timedependent coefficient b(x, t) (damping term), and the nonlinear term f (x, t, z) by some suitable boundary measurements. There are lots of literature concerning the inverse problem of recovering time-dependent or time-independent coefficients in PDEs. We will firstly consider the inverse problem of recovering the particular case, where the nonlinear term f is of time-independent. More precisely, we will recovery f (x, z) in (1.1) by means of the Carleman estimates. It is pointed out that, after the methodology created by [6], the Carleman estimates are well used in inverse problems of uniquely determining the time-independent coefficients.
Secondly, for the time-dependent case, we will recovery b(x, t) and f (x, t, z) simultaneously by the higher order linearization method together with constructing some Gaussian beams. We mention that [18] devoted to an inverse boundary problem for a nonlinear parabolic equation, in which the first order linearization of the DN map was proposed. This approach has been developed and applied in different other contexts.
For simplicity, we write the inverse problem of recovering f (x, z) as inverse problem (I), recovering b(x, t) and f (x, t, z) as inverse problem (II).
Let m be a positive integer. We introduce the following energy space
E m = m ∩ k=0 C k ([0, T ]; H m−k (Ω))
with the norm
||u|| 2 E m = sup 0≤t≤T m k=0 ||∂ k t u(t)|| 2 H m−k for u ∈ E m .
The well-posedness of system (1.1) is discussed in the appendix of this present paper.
Recovery of the nonlinear term f (x, z)
We consider the following equation.
u tt − ∆ g u + b(x, t)u t + f (x, u) = 0, (x, t) ∈ Ω × (0, T ), u(x, t) = h(x, t), (x, t) ∈ Γ × (0, T ), u(x, 0) = µ(x), u t (x, 0) = 0, x ∈ Ω.
(1.2)
Let u = u(f ; µ) be a solution to (1.2) with respect to f and µ(x). Define the input-tooutput map as Λ f T (µ(x)) = ∂ ν u(f ; µ)| Γ×(0,T ) , where ν denotes the unit normal field pointing outside Ω along Γ. For simplicity, we assume that µ(x) ∈ C ∞ (Ω). Based on the initial data µ(x), the boundary data is given by
u(x, t)| Σ = h(x, t) = µ(x)+ t 2 2 [∆ g µ(x)−f (x, µ(x))]+ m k=3 t k k! ∂ k t u(x, 0), x ∈ Γ, m ≥ 2, where ∂ k+2 t u(x, 0) = ∆ g (∂ k t u(x, 0)) − ∂ k t (bu t )| t=0 − ∂ k t f (x, u)| t=0 , k = 0, 1, · · · .
It is easy to see that the compatibility conditions hold for system (1.2) up to order m. Main assumptions.
The following are the main assumptions for the inverse problem (I).
(A.1) f (x, z) is analytic on C with values in C ∞ (M). Moreover, we assume that f (x, 0) = 0, and f (k) z (x, 0)| Ω 0 = f 0k (x)| Ω 0 , where f 0k (x) is known for each k = 1, 2, · · · , and Ω 0 is an arbitrary small neighborhood of Γ inside Ω.
(A.2) There exists a non-negative strictly convex function ψ : Ω → R, of class C 3 in the metric g. There exists a positive constant ρ on Ω such that ψ(x) satisfies (i) D 2 ψ(x)(X, X) ≥ 2ρ|X| 2 g , x ∈ Ω, X ∈ Ω x . (ii) ψ(x) has no critical point for x ∈ Ω. Namely, inf x∈Ω |Dψ(x)| > 0.
(A.3) b(x, t) ∈ C ∞ (M) with b(x, 0) = 0. Let δ > 0 be a small constant. Assume that |µ(x)| ≥ µ 0 > 0 for x ∈ Ω, such that ||h|| H m+1 (Σ) + ||µ(x)|| H m+1 (Ω) ≤ δ 2 , m > n 2 ,
We give the admissible sets of functions b(x, t) and f (x, z) as follows. Let M 0 be a positive constant. We define
U 1 = {b(x, t) ∈ C ∞ (M) : b(x, 0) = 0, ||b|| C ∞ (M) ≤ M 0 }, (1.3) U 2 = {f (x, z) ∈ C ∞ (Ω × C), f satisfies assumption (A.1), ||f (k) z (x, 0)|| L ∞ (Ω) ≤ M 0 , k = 1, 2, · · · }.
(1.4) Remark 1.1. Assume that f has the following Taylor expansion
f (x, u) = ∞ k=0 f (k) z (x, 0) u k k! . (1.5)
Let f 1 , f 2 ∈ U 2 and let u j = u j (x, t; µ, f j ) be the solution to (1.2) with respect to f j for j = 1, 2. We see that assumption (A.1) implies that h(x, t; f 1 ) = h(x, t; f 2 ), where h(x, t; f j ) is the given boundary data corresponding to the equation
u jtt − ∆ g u j + bu jt + f j (x, u j ) = 0, for j = 1, 2.
Assumption (A.2) are usually used in the Carleman estimates, see for example [5,30]. The existence of convex functions depends on the curvature of (Ω, g). Particularly, for the Euclidean case, we can take ψ(x) = |x − x 0 | 2 with x 0 ∈ R n \Ω. For general Riemannian manifolds, such ψ exists locally. There are a number of non-trivial examples to give such ψ, see [48,Chapter 2.3].
We are now in a position to state the main theorem of recovering f (x, z).
Theorem 1.1. Let assumptions (A.1)-(A.3) hold. Assume that b(x, t) ∈ U 1 and f 1 , f 2 ∈ U 2 . Let T > T * ,
where T * is given by (2.2). Then
Λ f 1 T (µ(x)) = Λ f 2 T (µ(x)) implies f 1 (x, z) = f 2 (x, z), (x, z) ∈ Ω × C. (1.6) Remark 1.2. Assume that Γ 0 ⊂ Γ such that {x ∈ Γ : Dψ, ν ≥ 0} ⊂ Γ 0 .
Then the measurement can be replaced by ∂ ν u| Γ 0 ×(0,T ) .
Λ b,f T (h(x, t)) = ∂u(x, t; h) ∂ν Σ .
Before giving the main assumptions for inverse problem (II), we give two definitions as follows.
Definition 1.1. A Riemannian manifold (Ω, g) is called a simple manifold if it is simply connected, any geodesic in Ω has no conjugate points, and the boundary Γ is strictly convex with respect to the metric g (the second fundamental form is positive for every point on the boundary). Definition 1.2. A compact Riemannian manifold (Ω, g) satisfies the foliation condition if there is a smooth strictly convex function, and the boundary Γ is strictly convex with respect to the metric g.
The above conditions are crucial in the inverse problems. They are always used as a sufficient condition in the geodesic ray transform. The inversion of the geodesic ray transform on Ω with depends on the geometric properties of Ω. See more details in section 4. Similar to [9], let
D = {(x, t) ∈ M : dist(x, Γ) < t < T − dist(x, Γ)}
be the domain of influence. It is clear that no information can be obtained about the coefficients on M\D, due to the finite speed of propagation of waves. Let T > 2Diam(Ω) be given, where Diam(Ω) = sup{lengths of all geodesics in (Ω, g)} < ∞.
We define a subset of D by
E = {(x, t) ∈ M : D g (x) < t < T − D g (x)},
where D g (x) denotes the length of the longest geodesic passing through the point x ∈ Ω.
The following assumptions are the main assumptions for the inverse problem (II). (B.1) f (x, t, z) is analytic on C with values in C ∞ (M), and f (x, t, 0) = 0. We define an admissible set for f as
U 3 = {f ∈ C ∞ (Ω × R × C) : f satisfies assumption (B.1), suppf (k) z (x, t, 0) ⊂ E, for k = 1, 2}. (1.7) (B.2) b(x, t) ∈ C ∞ (M)
. Similar to f, let b 0 (x, t) be a known smooth function. We define an admissible set for b as
U 4 = {b(x, t) ∈ C ∞ (M) : supp b(x, t) ⊂ E}. (1.8) (B.3) Assume that the initial data u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), and (u 0 (x), u 1 (x), h(x, t)) ∈ H m+1 (Ω) × H m (Ω) × H m+1 (Σ)
satisfying the compatibility conditions (3.30) up to order m > n 2 , such that
||u 0 || H m+1 (Ω) + ||u 1 || H m (Ω) + ||h(x, t)|| H m+1 (Σ) ≤ δ 2 ,
where δ > 0 is a given small constant. Suppose that (Ω, g) is either simple or satisfies the foliation condition, based on the Gaussian beams, which can allow the existence of conjugate points, we have Theorem 1.2. Assume that u j = u(x, t; b j , f j ) solves (1.1) with respect to b j and f j for each j = 1, 2. Let assumptions (B.1), (B.2) and (B.3) hold, and let T > 2Diam(Ω).
Suppose that b 1 , b 2 ∈ U 4 , f 1 , f 2 ∈ U 3 and f (2) z is known. Then Λ b 1 ,f 1 T (h(x, t)) = Λ b 2 ,f 2 T (h(x, t)) implies b 1 = b 2 in E, and f 1 = f 2 in D × C.
Literature review
Inverse problems of PDEs have attracted much attention with lots of literature on inverse elliptic equations, parabolic equations, hyperbolic equations, and plate equations, etc. For example, for the inverse elliptic problems, the well known Calderón type inverse problems were investigated in [13,19,31,42] and many subsequent papers. The linear inverse problems of determining either the time-independent coefficients or the time-dependent coefficients were widely studied. It is known to us that the Carleman estimates method created by [6] have well used in inverse problems to derive the strong Lipschit stability results for the time-independent coefficients. In this present paper, the Carleman estimate will be used for the recovery of the time-independent nonlinearity f (x, z). However such method deals with the time-independent case only. Also, there are some other methods for the recovery of the time-independent case (or for the real analytic time-dependent coefficients), such as the boundary control (BC) method steaming from [2,3] together with Tataru's sharp unique continuation theorem [43]. We are not going to list more literature concerning the linear inverse problems.
The inverse problems of nonlinear PDEs are much less. Among them, the early works [18,20,21] used the linearization procedure to study the recovery of nonlinear terms appearing in elliptic or parabolic equations. It turns out that the nonlinear interaction of waves can generate new waves, which are essential for the nonlinear inverse problems. For instance, [11,17,18,22,27,41] devoted to the unique recovery of nonlinear terms or coefficients appearing in nonlinear elliptic equations. [8] dealt with the stable recovery of a semilinear term appearing in a parabolic equation, and [26] studied the fractional semilinear Schrödinger equations.
For the inverse problems of hyperbolic equations, we refer to [34,35], which respectively concerned the the recovery of a conductivity and quadratic coefficients. We mention that [29] devoted to the recovery of nonlinear terms from source-to-solution map, where method of the nonlinear interactions of distorted plane waves, originated from [25], was used. Such method has been successfully used in inverse problems of nonlinear hyperbolic equations, see for example [15,25,45,47]. Similarly, for some semilinear wave equations, instead of the distorted plane waves, Gaussian beams together with the higher order linearization and the stationary phase method are used to recover the coefficients. For this method, for instance, we refer to [12,14,46]. Among those, [14] studied an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifolds. Both the distorted plane waves and the Gaussian beams were used to derive the uniqueness. In their paper, due to the unsolved inverse problems of recovering the zeroth order term on general manifolds, they assumed the nonlinearity has the following form
H(x, z) = ∞ k=2 h k (z)z k , h k ∈ C ∞ .
In their paper, the recovery of the quadric term f uu is much complex and interesting. The distorted plane waves were used to construct four future light-like vectors to recovery the quadric term, but without recovering the zeroth term f u (x, t, 0). However, the quadric term is unable to recovery by using the Gaussian beams.
It is worth noting that [24] considered the inverse problem of determining a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold (M, g) with boundary of dimension n = 2, 3. They determined the expres-sion F (t, x, u) both on the boundary x ∈ M and inside the manifold x ∈ M from some partial knowledge of the solutions u on the boundary of the time-space cylindrical manifold (0, T ) × M or on the lateral boundary (0, T ) × ∂M. Let us also point out that [32] investigated inverse boundary problems associated with a time-dependent semilinear hyperbolic equation with variable coefficients. They developed a new method, which combined the observability inequality and a Runge approximation with higher order linearization, to derive the uniqueness of both the initial velocity and the nonlinearity. The measurements they used were either active or passive. For the stability of recovering some coefficients, [28] devoted to the uniqueness and stability of an inverse problem for a semi-linear wave equation: u tt − ∆u + a(x, t)u m = 0, where (x, t) ∈ R n × R and m ≥ 2. They used the higher order linearization together with the Radon transform to prove the stability results of recovering a(x, t) by the Dirichlet-to-Neumann map. It is worth mentioning that, instead of differentiating the nonlinear equation by ∂ m ∂ 1 ···∂ m , they used the finite differences operator D m . We note that, all the above literature concerning the nonlinear wave equations do not consider the recovery of the first order coefficients, such as the damping term. Motivated by these previous works, we study the recovery of the time-dependent damping term and the nonlinearity simultaneously, which can be seen as an extended studying of the existing literature.
The rest of this present paper is organized as follows: In Section 2, we prove Theorem 1.1 by the Carleman setimates. We also add some additional contents in this section. Section 4 focuses on the Gaussian beams and proofs of Theorem 1.4. Finally, in the appendix, the well-posedness of the semilinear wave equation (1.1) is discussed.
Proofs of Theorem 1.1 and an additional result
In this section, we focus on proving Theorem 1.1 and give an additional result of recovering a leading coefficient.
Proofs of Theorem 1.1
Let N ≥ 1 be a positive inter, and = ( 1 , · · · , N ). Let µ k (x) = 0 for x ∈ Ω, and
µ(x) = n k=1 k µ k (x) ∈ H m+1 (Ω) with | | = N k=1 | k | sufficiently small, such that ||h(x, t)|| H m+1 (Σ) + || 1 µ 1 + · · · + N µ N || H m+1 (Ω) ≤ δ 2 .
By the same output, we have
Λ f 1 ( 1 µ 1 + · · · N µ N ) = Λ f 2 ( 1 µ 1 + · · · N µ N ),
which gives
∂ | | ∂ 1 · · · ∂ N =0 Λ f 1 ( 1 µ 1 + · · · N µ N ) = ∂ | | ∂ 1 · · · ∂ N =0 Λ f 2 ( 1 µ 1 + · · · N µ N ). (2.1) Clearly, Λ f ( N k=1 k µ k ) contains more information than {Λ f (µ k )} k=1,··· ,N . Indeed, useful information can be extracted from ∂ N ∂ 1 · · · ∂ N =0 Λ f ( N k=1 k µ k ). Let Q = Ω × (−T, T ), Σ = Γ × (−T, T ).
Extend the domains of u j and b(x, t) to the region Q evenly as usual. Notice that
u 1 (x, t) = u 1t = b(x, 0) = 0, then u 1ttt (x, 0) = 0, which implies that the extension of u 1 (x, t) (also u j 1 (x, t)) is smooth. Similar to [5, Chapter 5.2], we set φ(x, t) = ψ(x) − βt 2 + β 0 , ϕ(x, t) = e λφ(x,t) , (x, t) ∈ M,
where λ is a positive constant, β ∈ (0, ρ) and ρ is given by
(i) in assumption (A.2). Moreover, β 0 > 0 is chose such that φ(x) > 0. Let T * = 1 √ ρ max x∈Ω ψ(x) 1 2 . (2.2)
We assume that T > T * . Then we can choose δ > 0 and β > 0 such that
ρT 2 > max x∈Ω ψ(x) + 4δ, βT 2 > max x∈Ω ψ(x) + 4δ.
Thus, φ(x, t) has the following properties:
(1) φ(x, T ) ≤ β 0 − 4δ uniformly for x ∈ Ω.
(2) There exists a small constant ε > 0 such that
φ(x, t) ≤ β 0 − 2δ for (x, t) ∈ Ω × [T − 2ε, T ] ∪ [−T, −T + 2ε]. Therefore ϕ(x, t) ≤ e λ(β 0 −2δ) =: d 1 < d 0 =: e λβ 0 uniformly in Ω × [T − 2ε, T ].
We next give a key Carleman estimate for the linear wave operator
Lv = v tt − ∆ g v + b(x, t)v t + q(x, t)v, where q, b ∈ L ∞ (M). Let H 1 0 ( Q) = {u ∈ H 1 0 (Ω × [−T, T ]) : u| Γ = 0, ∂ l t u(x, ±T ) = 0, l = 0, 1}.
We have the following from [4].
= s(λ) such that Q s[(|Dv| + v 2 t ) + s 2 v 2 ]e 2sϕ dgdt ≤ C Q |P v| 2 e 2sϕ dgdt + C Σ |∂ ν v| 2 e 2sϕ dΣ (2.3)
holds for all v ∈ H 1 0 ( Q) and s > s 0 > 1.
Proof of Theorem 1.1. Based on the above linearization procedure and the Carleman estimate, we divide the proofs into two steps.
Step 1. First order linearization. Let
u j = u j (x, t; f j , ) ∈ E m+1 be a solution to (1.2) with respect to f j and µ(x) = 1 µ 1 (x) + · · · + N µ N (x) for j = 1, 2. Then u j = u j (x, t; f j , 0) solves u jtt − ∆ g u j + b(x, t) u jt + f j (x, u j ) = 0, (x, t) ∈ Ω × (0, T ), u j (x, t) = 0, (x, t) ∈ Γ × (0, T ), u j = u jt (x, 0) = 0, x ∈ Ω, (2.4)
which admits a zero solution u j = 0 since f j (x, 0) = 0 for each j = 1, 2. We next linearize the system (1.2) around u j = 0.
Let N = 1 and let u j
1 = ∂ ∂ 1 =0 u j . Then u j 1 satisfies u j 1tt − ∆ g u j 1 + b(x, t)u j 1t + f ju (x, 0)u j 1 = 0, (x, t) ∈ Q, u j 1 (x, t) = ∂ ∂ 1 =0 h := h 1 , (x, t) ∈ Σ, u j 1 (x, 0) = µ 1 (x), u j 1t (x, 0) = 0, x ∈ Ω.
(2.5)
Set u 1 = u 1 1 − u 2 1 and q(x) = f 2u (x, 0) − f 1u (x, 0), then u 1tt − ∆ g u 1 + b(x, t)u 1t + f 1u (x, 0)u 1 = q(x)u 2 1 , (x, t) ∈ Q, u 1 (x, t) = 0, (x, t) ∈ Σ, u 1 (x, 0) = u 1t (x, 0) = 0, x ∈ Ω. (2.6) Let y 1 = u 1t . Then y 1tt − ∆ g y 1 + by 1t + (b t + f 1u (x, 0))y 1 = q(x)u 2 1t , (x, t) ∈ Q, y 1 (x, t) = 0, (x, t) ∈ Σ, y 1 (x, 0) = 0, y 1t (x, 0) = q(x)µ 1 (x), x ∈ Ω. (2.7) Next, we chose a cut-off function χ(t) ∈ C ∞ 0 ([−T, T ]) satisfying 0 ≤ χ(t) ≤ 1, χ(t) = 0, t ∈ [−T, −T + ε) ∪ (T − ε, T ], 1, t ∈ [−T + 2ε, T − 2ε].
(2.8)
Letŷ 1 = χ(t)y 1 .
Then
y 1tt − ∆ gŷ1 + bŷ 1t + (b t + f 1u (x, 0))ŷ 1 = χq(x)u 2 1t + χ t (2y 1t + by 1 ) + χ tt y 1 .
Since the assumption that |µ 1 (x)| ≥ c 1 > 0, then
Ω |q(x)| 2 e 2sϕ(x,0) dg ≤ C Ω |q(x)µ 1 (x)| 2 e 2sϕ(x,0) dg = − T 0 ∂ ∂t Ω |y 1t (x, t)χ(t)| 2 e 2sϕ dtdg = −2 Q (χ 2 y 1t y 1tt + χχ t y 2 1t + sϕ t χ 2 y 2 1t )e 2sϕ dgdt ≤ Cs Q [|(χy 1 ) t | 2 + χ 2 t (y 2 1 + y 2 1t )]e 2sϕ dgdt + 2 Q χ 2 y 1t y 1tt e 2sϕ dgdt. (2.9)
We compute the term χ 2 y 1t y 1tt e 2sϕ as follows.
χ 2 y 1t y 1tt e 2sϕ = χ 2 y 1t [∆ g y 1 − by 1t − (b t + f 1u (x, 0))y 1 + q(x)u 2 1t ]e 2sϕ = χ 2 div (y 1t e 2sϕ Dy 1 ) − 1 2 (χ 2 e 2sϕ |Dy 1 | 2 ) t + (χχ t + sχ 2 ϕ t )|Dy 1 | 2 e 2sϕ −2sχ 2 y 1t Dy 1 , Dϕ e 2sϕ − χ 2 [(b t + f 1u (x, 0))y 1 − q(x)u 2 1t ]e 2sϕ ≤ χ 2 div (y 1t e 2sϕ Dy 1 ) − 1 2 (χ 2 e 2sϕ |Dy 1 | 2 ) t +Cs[|D(χy 1 )| 2 + |(χy 1 ) t | 2 + χ 2 t (y 2 1t + |Dy 1 | 2 )]e 2sϕ + χ 2 q(x)u 2 1t e 2sϕ .
Moreover, for system (3.24), the hyperbolic regularity (see,
Lemma A.1 in the ap- pendix) implies that u 2 1t ∈ C([−T, T ]; H m (Ω)) for m > n 2 .
Then the Sobolev embedding theorem shows that u 2 1t ∈ L ∞ ( Q). Thus, there exists a positive constant C = C(M 0 ) such that
− Q {χ 2 y 1t [by 1t + (b t + f 1u (x, 0))y 1 − q(x)u 2 1t ]}e 2sϕ dgdt ≤ C Q [χ 2 t y 2 1 + |(χy 1 ) t | 2 + q(x)]e 2sϕ dgdt.
Notice that 1 2 (χ 2 |Dy 1 | 2 e 2sϕ )| T 0 = 0. Therefore, by (2.9) and (2.10), with v =ŷ, it follows from the Carleman estimate (2.3) that
Ω q 2 e 2sϕ(x,0) dg ≤ Cs Q (ŷ 2 + |Dŷ| 2 +ŷ 2 t )e 2sϕ dgdt +Cs Q χ 2 t (y 2 1 + y 2 1t + |Dy 1 | 2 )e 2sϕ dgdt + C Q q 2 e 2sϕ dgdt ≤ C Q q 2 e 2sϕ dgdt + Q (χ 2 t + χ 2 tt )(y 2 1 + y 2 1t + |Dy 1 | 2 )e 2sϕ dgdt + Ce Cs ||∂ ν y 1 || 2 L 2 (Σ) .
By the standard energy estimate of system (2.7), there exists a positive constant C = C(T, M 0 ) such that
Ω (y 2 1 + y 2 1t + |Dy 1 | 2 )dg ≤ C Ω q 2 dg + C Σ |∂ ν y 1 | 2 dΣ. (2.10)
Since χ t , χ tt = 0 in the case where ϕ(x, t) ≤ d 1 , with (2.10), we have
Ω q 2 e 2sϕ(x,0) dg ≤ C Q q 2 e 2sϕ dgdt + Ce Cs ||∂ ν y 1 || 2 L 2 (Σ) . (2.11)
Moreover, by the Lebesgue's theorem, we have
Q q 2 e 2sϕ dgdt = Ω q 2 e 2sϕ(x,0) T 0 e −2s(ϕ(x,0−ϕ(x,t))) dt = o(1) Ω q 2 e 2sϕ(x,0) dg.
Thus,
||q|| 2 L 2 (Ω) ≤ Ce Cs ||∂ ν y 1 || 2 L 2 (Σ) = Ce Cs ||∂ ν (u 1 1 − u 2 1 ) t || 2 L 2 (Σ) . Hence ∂ ν u 1 1 = ∂ ν u 2 1 on Σ implies that f 1u (x, 0) = f 2u (x, 0) = f u (x, 0) for x ∈ Ω.
Step 2. Higher order linearization. Based on Step 1, we set
f 1u (x, 0) = f 2u (x, 0) = f u (x, 0) for simplicity. Let u j 2 = ∂ 2 ∂ 1 ∂ 2 =0 u j for j = 1, 2. Then u j 2tt − ∆ g u j 2 + b(x, t)u j 2t + f u (x, 0)u j 2 + f juu (x, 0)u 1 1 u 2 1 = 0, (x, t) ∈ Ω × (0, T ), u j 2 (x, t) = 0, (x, t) ∈ Γ × (0, T ), u j 2 (x, 0) = 0, u j 2t (x, 0) = 0, x ∈ Ω. (2.12) Here u 1 1 and u 2 1 satisfy u j 1tt − ∆ g u j 1 + b(x, t)u j 1t + f u (x, 0)u j 1 = 0, (x, t) ∈ Ω × (0, T ), u j 1 (x, t) = ∂ ∂ j =0 := h j (x, t), (x, t) ∈ Γ × (0, T ), u j 1 (x, 0) = µ j (x), u j 1t (x, 0) = 0, x ∈ Ω.
(2.13)
Set y 2 = (u 1 2 − u 2 2 ) t , then y 2tt − ∆ g y 2 + b(x, t)y 2t + (b t + f u (x, 0))y 2 = F (x, t), (x, t) ∈ Ω × (0, T ), y 2 (x, t) = 0, (x, t) ∈ Γ × (0, T ), y 2 (x, 0) = 0, y 2t (x, 0) = −(f 1uu − f 2uu )(x, 0)µ 1 (x)µ 2 (x), x ∈ Ω, (2.14) where F (x, t) = −(f 1uu − f 2uu )(x, 0)(u 1 1t u 2 1 + u 1 1 u 2 1t )
. By a similar argument as that in Step 1, we have
∂ ν u 1 2 = ∂ ν u 2 2 on Σ ⇒ f 1uu (x, 0) = f 2uu (x, 0) ⇒ u 1 2 = u 2 2 := u 2 . Suppose that f (N −1) 1u (x, 0) = f (N −1) 2u (x, 0), x ∈ Ω. Let w N j = ∂ N ∂ 1 ··· N =0
u j . By the recursive assumption, for the expression
f j (x, u) = ∞ k=0 f (k) u (x, 0) u k k! , and v k = ∂ ∂ k =0
u, we know that
∂ N ∂ 1 · · · N =0 f j (x, u) − f u (x, 0)w N j − f (N ) ju (x, 0)v 1 · · · v N
is already known. The above procedure can be proceeded for N -th linearization to obtain
f (N ) 1u (x, 0) = f (N ) 2u (x, 0), x ∈ Ω.
Up to now, by the analyticity of f, we have
f 1 (x, z) = ∞ k=1 f (k) 1u (x, 0) z k k! = ∞ k=1 f (k) 2u (x, 0) z k k! = f 2 (x, z).
Thus, the uniqueness result in Theorem 1.1 holds and the proof of Theorem 1.1 is complete.
An additional result of recovering a leading coefficient
Based on the Carleman estimate, we introduce here a stability result of recovering a leading coefficient. Suppose that there is a leading coefficient appearing in the wave equation
(x)u tt − ∆ g u + bu t + f (x, t, u) = 0, (x, t) ∈ M. (2.15)
The unique recovery of the mass density (x) for a wave equation is essential in inverse problems. There are literature concerning this topic, see for example [1,33,40]. We give a brief discussion of recovering (x). Due to the presence of (x), we consider a new metricĝ = g, then ∆ĝu = 1 ∆ g u + n − 2 2 2 D , Du .
LetD be the Levi-Civita connection in the metricĝ. It is well known that, for any vectors X, Ŷ
D X Y = D X Y + 1 2 g(D ln , Y )X + 1 2 g(D ln , X)Y + 1 2 g(X, Y )D ln .
If ψ(x) is strictly convex in the metricĝ, then one needŝ Dψ(ln ))|X| 2
g = 1 (2 + 1 2 Dψ(ln ))|X| 2 g ≥ ϑ 0 |X| 2 g ,
where ϑ 0 > 0 is a constant. Therefore, we need further assumption on (x):
g(Dψ, D ) ≥ 2 (ϑ 0 − 2).
Let u j (x, t) = u(x, t; ρ j ) be a solution to (2.15) with respect to j for each j = 1, 2. Let
w = u 1 − u 2 ,ˆ = ρ 1 − ρ 2 . Then 2 w tt − ∆ g w + bw t + cw = −ˆ u 1tt ,(2.16)
where c(x, t) = 1 0 f u (x, t, ru 1 + (1 − r)u 2 )dr ∈ L ∞ (M) for sufficiently smooth u 1 and u 2 . Suppose that |∆ g µ(x) − f (x, 0, µ(x))| > 0 for x ∈ Ω, where (u j (x, 0), u jt (x, 0)) = (µ(x), 0) for j = 1, 2. Then a similar argument with the proof of Theorem 1.1 yields the following stability of recovering (x):
|| 1 (x) − 2 (x)|| L 2 (Ω) ≤ C||∂ ν (u 1 − u 2 ) t || L 2 (Σ) .
Remark 2.1. Clearly, as we have discussed above, the non-degeneracy of initial data u(x, 0), which is viewed as the input is needed. Moreover, the existence of some strictly convex functions, which seems not sharp, should be assumed. Such functions guarantee that interior information of solutions to the system arrives at boundary in a finite time.
Gaussian beams and proofs of Theorem 1.4
In this section, as usual, the boundary data h is the input. A geodesic β(t) ⊂ M is called a null geodesic in the metric g = −dt 2 + g, if D gββ = g(β,β) = 0, where D g is the connection in the metric g. We intend to construct some Gaussian beams around a null geodesic. For completeness, we give the details of constructing the Gaussian beams.
Gaussian beams
Let β(t) be a null geodesic. We firstly introduce the Fermi coordinates in a neighborhood of the null geodesic β. We follow the constructions in [10], see also [12]. Recall that Ω ⊂⊂ Ω 1 and the functions are extended smoothly to Ω 1 . Let β(t) = (t, γ(t)) ⊂ R×Ω 1 , where γ(t) is a unit-speed geodesic in the Riemannian manifold (Ω 1 , g). Assume that β(t) passes through a point (t 0 , x 0 ), where t 0 ∈ (0, T ) and γ(t 0 ) = x 0 ∈ Ω. Let β join two points (t − , γ(t − )) and (t + , γ(t + )) with t ± ∈ (0, T ) and γ(t ± ) ∈ Γ. We extend β to a larger manifold M 1 = (0, T )×Ω 1 such that γ(t) is well defined on [t − − , t + + ] ⊂ (0, T ) with > 0 sufficiently small.
Since the geodesic γ is parallel along itself, we can choose {e 2 , · · · , e n } such that {γ(t 0 ), e 2 · · · , e n } forms an orthonormal basis of Ω x 0 . Let s denote the arc length along γ from x 0 . Let E k (s) ∈ Ω γ(s) be the parallel transport of e k along γ to the point γ(s).
We now define a map F 1 : R n+1 → M 1 such that F 1 (y 0 = t; s = y 1 , y 2 · · · , y n ) = (t, exp γ(s) (y 2 E 2 (s) + · · · + y n E n (s))), where exp p (·) denotes the exponential map on Ω 1 at the point p. In the new coordinates, we have g| γ = n k=1 dy 2 k and ∂g ij ∂y k γ = 0 for 1 ≤ i, j, k ≤ n.
On the Lorentzian manifold (M 1 , −dt 2 + g), we introduce the well known Fermi coordinates near the null geodesic β : (t − − , t + + ) → M 1 as follows. Let
a = √ 2(t − − ), b = √ 2(t + + ), a 0 = √ 2(t − − √ 2 ), b 0 = √ 2(t + + √ 2 ), and z 0 = 1 √ 2 (t + s) + a 2 , z 1 = 1 √ 2 (−t + s) + a 2 , z j = y j for 2 ≤ j ≤ n.
Then, we have g| β = 2dz 0 dz 1 + n k=2 dz 2 k and ∂g ij ∂z k β = 0 for 0 ≤ i, j, k ≤ n. For simplicity, we use the notation z = (z 0 , z ) = (z 0 , z 1 , z ) to denote the so called Fermi coordinates. The following lemma from [10, Lemma 3.1] (see also [12,Lemma 1]) is essential for the construction of Gaussian beams.
Lemma 3.1. Let β : (t − − , t + + ) → M 1 be a null geodesic as above. Then there exists a coordinate neighborhood (U, Φ) of β(t − − 2 , t + + 2 ), with the coordinates denoted by (z 0 , z ) such that
V = Φ(U ) = (a, b) × B(0, δ),
where B(0, δ) denotes a ball in R n with a small radius δ.
Based on the above coordinates, we will construct some approximate Gauss beams in a neighborhood of β by defining V = {(z 0 , z ) ∈ M 1 : z 0 ∈ (a 0 , b 0 ), |z | ≤ δ } with 0 < δ < δ sufficiently small such that the set V does not intersect the sets {0}×Ω and {T } × Ω. We use the shorthand notation L b,q u = u tt − ∆ g u + bu t + qu. We consider the WKB ansatz u σ = e i σϕ a + r σ ,
where σ > 0 is a constant, r σ is the reminder term. ϕ and a are called the amplitude and phase respectively. In particular, we will construct ϕ ∈ C ∞ (V) and a ∈ C ∞ 0 (V). Directly calculation yields
L b,q (e i σϕ a) = e i σϕ L b,q a + 2 i σ(ϕ t a t − Da, Dϕ g )e i σϕ + i σa(ϕ tt − ∆ g ϕ + bϕ t )e i σϕ + aσ 2 (|Dϕ| 2 g − ϕ 2 t )e i σϕ . (3.2)
Based on the above equation, we respectively solve The eikonal equation
Sϕ = |Dϕ| 2 g − ϕ 2 t = dϕ, dϕ g = n k,l=0
g kl ∂ k ϕ∂ l ϕ = 0, and the transport equation
T b (a, ϕ) = −2 dϕ, da g − (∆ g ϕ − bϕ t )a = 2(ϕ t a t − Da, Dϕ g ) + (ϕ tt − ∆ g ϕ + bϕ t )a = 0. (3.3)
To achieve this, we make the following ansatz for ϕ, a, namely
ϕ(z 0 , z ) = N k=0 ϕ k (z 0 , z ), a = N k=0 σ −k χ( |z | δ )a k (z 0 , z ), a k = N j=0 a k,j (z 0 , z ).
Here ϕ k is a homogeneous polynomial of degree k with respect to the variables z i for k = 0, 1, · · · , N. In terms of the Fermi coordinates z = (z 0 , z 1 , · · · , z n ) for g = −dt 2 + g, we need
∂ |Θ| ∂z Θ (Sϕ)(z 0 , 0) = ∂ |Θ| ∂z Θ dϕ, dϕ g | β = 0 for Θ = (0, θ 1 , · · · , θ n ), z 0 ∈ (a 0 , b 0 ),
where θ j ≥ 0 are integers for 1 ≤ j ≤ n, and |Θ| = n j=1 θ j ≤ N, Moreover, for k = 1, · · · , N, we need
∂ |Θ| ∂z Θ T (a 0 , ϕ)(z 0 , 0) = 0, ∂ |Θ| ∂z Θ ( i T (a k , ϕ) + L b,q a k−1 )(z 0 , 0) = 0, z 0 ∈ (a 0 , b 0 ).
Construction of the phase. We firstly solve equation (3.4) with |Θ| = 0. That is, n k,l=0 g kl ∂ k ϕ∂ l ϕ = 0 on β.
By (3.1), this reduces to
2∂ 0 ϕ∂ 1 ϕ + n k=2 (∂ k ϕ) 2 = 0. (3.4)
Similar, for |Θ| = 1, we have n kl=0 ∂ 2 jk ϕ∂ l ϕ = 0 for 1 ≤ j ≤ n.
ϕ 0 = 0, ϕ 1 = z 1 = 1 √ 2 (−t + s) + a 2 .
For the case where |Θ| = 2, we set
ϕ 2 (z 0 , z ) = n i,j=1 H ij (z 0 )z i z j ,(3.(∂ 2 ij g kl ∂ k ϕ∂ l ϕ + 2g kl ∂ 3 kij ϕ∂ l ϕ + 2g kl ∂ 2 ki ϕ∂ 2 lj ϕ + 4∂ i g kl ∂ 2 jk ϕ∂ l ϕ)| β = 0. (3.7)
By the choices of ϕ 0 , ϕ 1 and ϕ 2 , (3.7) implies that (∂ 2 ij g 11 + 2g 10 ∂ 3 0ij ϕ + 2 n k=2 ∂ 2 ki ϕ∂ 2 kj ϕ)| β = 0.
We finally obtain the following Riccati equation for H(z 0 ), namely, Moreover, the matrix Y (z 0 ) is non-degenerate on (a 0 , b 0 ), and there holds det(Im H(z 0 )) · |det(Y (z 0 ))| 2 = det(Im H 0 ).
For the case where |Θ| = 3, 4 · · · , the polynomials ϕ j of higher degree are constructed analogously. We omit the details.
Construction of the amplitude. Let us consider the transport equation (3.3). Let |Θ| = 0. It follows from (3.1) that bϕ t = −b √ 2 and ∆ g ϕ = n k,l=0
g kl D 2 g ϕ(∂ k , ∂ l ) = n k,l=0 g kl ∂ 2 kl ϕ = Tr(AH) on β.
Therefore, the transport equation (3.3) reduces to
2∂ z 0 a 0,0 + [Tr(AH) − b √ 2 ]a 0,0 = 0, z 0 ∈ (a 0 , b 0 ). (3.9)
Notice that
Tr(AH) = Tr(A(z 0 )Z(z 0 )Y −1 (z 0 )) = Tr( dY dz 0 Y −1 (z 0 )) = d dz 0 log(det Y (z 0 )), then a 0,0 (z 0 ) = (det (Y (z 0 ))) − 1 2 e 1 2 √ 2 z 0 s b(τ,0)dτ , z 0 ∈ (a 0 , b 0 ) (3.10)
is a solution to (3.9). The subsequent terms a k,0 can be constructed by solving some linear ODEs of first order. We refer to [12] for more details.
Construction of the remainder terms. By a similar proof with [12, Lemma 2], the Gaussian beam has the following property.
Lemma 3.3. Let u σ = e i σϕ a be an approximate Gaussian beam of order N along the null geodesic β. Then for all σ > 0
||L b,q u σ || H k (M) ≤ Cσ −K , where K = N +1−k 2 − 1.
Based on the above lemma and the Sobolev embedding theorem, for sufficient large N, the remainder term r σ satisfies the estimate (cf. [14], [36, Proposition 2.2])
||r σ || H k+1 (M) ≤ Cσ −K ⇒ ||r σ || C(M) ≤ Cσ − n+1 2 −2 .
(3.11)
Proofs of Theorem 1.4
Let us introduce some basic notations on the geodesic ray transform, which are explicitly discussed in, e.g., [9,10,38]. The following contents are mainly from [9], we present here for completeness.
Let SΩ ∈ T Ω be the unit sphere bundle of (Ω, g), and by γ(·; x, v) the geodesic with initial data (x, v) ∈ SΩ. For all (x, v) ∈ SΩ int , we define the exist times as
τ ± (x, v) = inf{r > 0 : γ(±r; x, v) ∈ Γ}.
Assume that (Ω, g) is simple, then τ ± < Diam(Ω). Define
∂ ± SΩ = {(x, v) ∈ SΩ : x ∈ Γ, ± v, ν(x) g > 0}.
All geodesics in Ω int can be parametrized by γ(·; x, v) ⊂ Ω for (x, v) ∈ ∂ − SΩ. The geodesic ray transform on (Ω, g) is defined for f ∈ C ∞ (Ω) by
If (x, v) = τ + (x,v) 0 f (γ(r; x, v))dr for (x, v) ∈ ∂ − SΩ.
Let β be a null geodesic (also called light ray). By the product structure of the Lorentzian manifold R × Ω, we can parametrize the null geodesic β as
β(r; s, x, v) = (r + s, γ(r; x, v)), ∀(s, x, v) ∈ R × ∂ − SΩ.
Then, all the null geodesics β through β(·; s, x, v) with (s, x, v) ∈ R × ∂ − SΩ over their maximal intervals [0, τ + (x, v)] can be identified. The so called light ray transform on R × Ω can be defined as Remark 3.1. For the non-simple case, such ray transform has been also discussed e.g., in [39] and references therein.
Gf (s, x, v) = τ + (x,v) 0 f (r + s, γ(r; x, v))dr ∀(s, x, v) ∈ R × ∂ − SΩ.
We begin with the first order linearized equation (3.12) where q j = f ju (x, t, 0). By the definition of L b,q , we have
L b j ,q j u j = u jtt − ∆ g u j + b j u jt + q j u j = 0, (t, x) ∈ M, u j (t, x) = h 1 (t, x), (t, x) ∈ Σ, u j (0, x) = u jt (0, x) = 0, x ∈ Ω,L * b,q u = u tt − ∆ g u − bu t + (q + b t )u,
where L * b,q denotes the formal adjoint of L b,q with respect to the L 2 (M) inner product. The formal adjoint system of (3.12) with j = 1 is given by
L * b 1 ,q 1 v = v tt − ∆ g v − b 1 v t + (q 1 + b 1t )v = 0, (t, x) ∈ M, v(T, x) = v t (T, x) = 0,
x ∈ Ω, (3.13)
Based on the above constructions of Gaussian beams, we seek such solutions for systems (3.12) and (3.13), respectively. More precisely, we let
u 2 = e i σϕ a 1 + r 1σ = e i σϕ σ n 4 χ( |z | δ )a 10,0 + r 1σ , v = e − i σϕ a 2 + r 2σ = e i σϕ σ n 4 χ( |z | δ )a 20,0 + r 2σ ,
where · means the conjugate of ·. Then r 1σ and r 2σ respectively solve (3.14) and
L b 2 ,q 2 r 1σ = −L b 2 ,q 2 (e i σϕ a 1 ) in M, r 1σ = 0 on Σ, r 1σ (0) = r 1σt (0) = 0 in Ω, L * b 1 ,q 1 r 2σ = −L * b 1 ,q 1 (e − i σϕ a 2 ) in M, r 2σ = 0
on Σ, r 2σ (T ) = r 2σt (T ) = 0 in Ω, (3.15) According to (3.2), we have L b 2 ,q 2 (e i σϕ a 1 ) = e i σϕ [L b 2 ,q 2 a 1 + σ 2 (Sϕ)a 1 + i σT b 2 (a 1 , ϕ)],
and L * b 1 ,q 1 (e − i σϕ a 2 ) = e − i σϕ [L * b 1 ,q 1 a 2 + σ 2 (Sϕ)a 2 − i σT −b 1 (a 2 , ϕ)].
As we have discussed in Section 4.1, we choose
a 10,0 (z 0 ) = (det Y (z 0 )) − 1 2 e 1 2 √ 2 z 0 s b 2 (τ,0)dτ , z 0 ∈ (a 0 , b 0 ), and a 20,0 (z 0 ) = (det Y (z 0 )) − 1 2 e − 1 2 √ 2 z 0 s b 1 (τ,0)dτ , z 0 ∈ (a 0 , b 0 ).
Clearly, L * b 1 ,q 1 (e − i σϕ a 2 ) and L * b 1 ,q 1 (e − i σϕ a 2 ) are compactly supported in a small tubular region around the null geodesic where the Fermi coordinates are well defined.
The following lemma shows that the remainder terms r 1σ and r 2σ vanish as σ → +∞. We are now in a position to prove Theorem 1.4. We only prove Theorem 1.4 for the case where (M, g) is simple. If (Ω, g) satisfies the foliation condition, as in [44] (see also [14]), for any point q ∈ Γ, there exists a wedge-shaped neighborhood O q ⊂ Ω of q such that any geodesic in (O q , g) has no conjugate points. Therefore, we can now recover f k z (x, t, 0) for k ≥ 3. Then the foliation condition allows a layer stripping scheme to recover the coefficients in the whole domain. However, the recovery of f z (x, t, 0) and f (2) z (x, t, 0) is quite different, which needs the inversion of some new ray transforms.
Proof of Theorem 1.4. We divide the proof into three steps.
Step 1. Let h 1 = e i σϕ a 1 | Σ in (3.12). Let
w = u 1 − u 2 , b = b 1 − b 2 , q = q 1 − q 2 . Then w tt − ∆ g w + b 1 w t + q 1 w = −(bu 2t + qu 2 ), (t, x) ∈ M, w(t, x) = 0, (t, x) ∈ Σ, w(0, x) = w t (0, x) = 0, x ∈ Ω. (3.16)
Notice that v solves system (3.13) and Λ b 1 ,q 1 (h 1 ) = Λ b 2 ,q 2 (h 1 ) on Σ implies ∂ ν w| Σ = 0, we multiply the first equation of (3.16) by v and integrate over M to obtain
(L b 1 ,q 1 w, v) L 2 (M) = (w, L * b 1 ,q 1 v) L 2 (M) = 0 ⇒ M (bu 2t + qu 2 )vdV g = 0.
(3.17)
Here dV g = |g| 1 2 dt ∧ dx denotes the volume form of the metric g = −dt 2 + g. Recalling that u 2 = e i σϕ a 1 + r 1σ and v = e − i σϕ a 2 + r 2σ in a neighborhood of the null geodesic β, we have
0 = i σ M bϕ t a 1 a 2 e −2σImϕ dV g + M qa 1 a 2 e −2σImϕ dV g + M b
[a 1t a 2 e −2σImϕ + ( i σϕ t a 1 + a 1t )r 2σ e i σϕ + e − i σϕ a 2 r 1σt + r 1ht r 2σ ]dV g + M q(e i σϕ a 1 r 2σ + e − i σϕ a 2 r 1σ + r 1σ r 2σ )dV g Since the functions a 1 , a 2 are supported in a small tubular neighborhood of the null geodesic β, the integrand in (3.19) is supported near β. Therefore we can use the Fermi coordinates z = (z 0 , z 1 , z ) to compute the limit. Recall that a 1 a 2 = σ n 2 a 10,0 a 20,0
χ 2 ( |z | δ ) + O(σ −1 ) = σ n 2 |det Y (z 0 )| −1 e − 1 2 √ 2 z 0 s b(τ,0)dτ + O(σ −1 ).χ 2 ( |z | δ )e −2σImϕ |det Y (z 0 )| −1 b(z 0 , z )e − 1 2 √ 2 z 0 s b(τ,0)dτ dz 0 ∧ z = 0.
We proceed to calculate the term σ n 2 |z |<δ
χ 2 ( |z | δ )b(z 0 , z )e −2σImϕ dz = R n e −2σx T P x η(x)dx, where η(x) = χ 2 ( |x| δ )b(z 0 , x)
is a smooth function with compact support B(0, δ ), P = ImH(z 0 ) is a positive-definite matrix. By the following well-known formula
F(e − 1 2 x T P x )(ξ) = (2π) n 2 (det P ) 1 2 e − 1 2 ξ T P −1 ξ ,
where F denotes the Fourier transform, we have
F(e −2σx T P x )(ξ) = (2π) n 2 (det P ) 1 2 (4σ) n 2 e − 1 8σ ξ T P −1 ξ .
Equivalently, we have
F[e − 1 2 x T ( 1 4σ P −1 )x ] = (2π) n 2 det ( 1 4σ P −1 ) 1 2 e −2σx T P x . Since R n F[f ](x)g(x)dx = R n f (x)F[g](x)dx, for f, g ∈ L p (R n ), p ∈ (1, +∞), we have R n e − 1 8σ ξ T P −1 ξ Fη(ξ)dξ = R n F[e − 1 8σ x T P −1 x ]η(x)dx = (2π) n 2 (4σ) n 2 (det P ) 1 2 R n e −2σx T P x η(x)dx.
Let P(ξ) : R n → C be a measurable function. The well known theory of pseudodifferential operators tells us
F[P(D)u](ξ) = P(ξ)û(ξ) = P(ξ)F[a](ξ), P(ξ) ∈ S m ,
where D is the differential operator, S m is the symbol class of order m, and P(D)u = 1 (2π) n R n e i x·ξ P(ξ)û(ξ)dξ.
Therefore R n e −2hx T P x η(x)dx = 1 (2π) n 2 (4h) n 2 (det P ) 1 2 R n e − 1 8h ξ T P −1 ξ Fη(ξ)dξ = 1 (2π) n 2 (4σ) n 2 (det P ) 1 2 +∞ k=0 1 k! (− 1 8σ ) k R n (ξ T P −1 ξ) k Fη(ξ)dξ = 1 (4σ) n 2 (det P ) 1 2 +∞ k=0 1 k! (− 1 8σ ) k (P P −1 (D)) k η(0) = 1 (4σ) n 2 (det P ) 1 2 (η(0) + O(σ −1 )),
where P P −1 (D)η(ξ) is defined by
(ξ T P −1 ξ) k Fη(ξ) = F[(P P −1 (D)) k η(ξ)].
By Lemma 3.2, we have
|det Y (z 0 )| −1 = (det Im H(z 0 )) 1 2 (det Im H 0 ) − 1 2
Then lim σ→∞ σ n 2 |z |<δ holds for all maximal null geodesics β in R × Ω. Together with Proposition 3.1, we conclude that q = 0.
χ 2 ( |z | δ )e −2σImϕ |det Y (z 0 )| −1 e − 1 2 √ 2 z 0 s b(τ,0)dτ dz = 1 4 n (det Im H 0 ) − 1 2 b(z 0 , 0)e − 1 2 √ 2 z 0 s b(τ,0)dτ .
Step 3. Since f (2) z is known, we proceed with the third and higher order linearization. Let
W (123) = ∂ 3 ∂ 1 ∂ 2 ∂ 3 =0 u(f ), W (ij) = ∂ 2 ∂ i ∂ j =0 u(f ), 1 ≤ i, j ≤ 3.
Set m(x, t) = f uuu (x, t, 0). Let Σ(3) be the permutation group of {1, 2, 3}. Then (123) ,
L b,q W (123) + fuu(x,t,0) 2 ζ∈Σ(3) W (ζ(1)ζ(2)) v ζ(3) + mv 1 v 2 v 3 = 0 in M, W (123) = 0 on Σ, W (123) (x, 0) = W (123) t (x, 0) = 0 in Ω, (3.23) where L b,q W (123) = W (123) tt − ∆ g W (123) + bW (123) t + qWand for each k = 1, 2, 3, v k = ∂ ∂ k =0 u(f ) solves v ktt − ∆ g v k + bv kt + qv k = 0 in M, v k = h k on Σ, v k (x, 0) = v kt (x, 0) = 0 in Ω.
(3.24)
Let v 0 solve the following adjoint system of (3.24)
v 0tt − ∆ g v 0 − bv 0t + (q + b t )v 0 = 0 in M, v 0 = h 0 on Σ, v 0 (x, T ) = v 0t (x, T ) = 0 in Ω. (3.25)
Integrating by parts over M yields We will use special solutions v 1 , v 2 , v 3 , v 0 in the above identity. Concretely, we shall use the following Gaussian beam solutions e i σϕ a + r σ constructed in section 4.1.
Σ v 0 ∂ 3 ∂ 1 ∂ 2 ∂ 3 =0 Λ( h 1 + 2 h 2 + 3 h 2 )dΣ = M mv 1 v 2 v 3 v 0 + M f uu (x, t, 0) 2 ζ∈Σ(3) W (ζ(1)ζ(2)) v ζ(3) v 0 dgdt.
where ||F || 2 Xm = m k=0 ||∂ k t F || 2 L 1 ([0,T ];H m−k ) .
Assume further that F ∈ E m and b, q ∈ C m (M) with m > n 2 . Assume that the compatibility conditions hold for (3.32) up to order m. Applying lemma 3.6, and by the fact that E m is a Banach algebra when m > n 2 , we know that
v tt − ∆ g v + bv t + qv = F in M, v = h on Σ, v(x, 0) = v 0 , v t (x, 0) = v 1 (x) in Ω.
(3.32) admits a unique solution v ∈ E m+1 with ∂ ν v ∈ H m (Σ). Moreover, the following energy estimate holds
||v|| E m+1 + ||∂ ν v|| H m (Σ)
≤ C(T )(||v 0 || H m+1 (Ω) + ||v 1 || H m (Ω) + ||h|| H m+1 (Σ) + ||F || Xm ).
(3.33)
We now in a position to prove the well-posedness of the nonlinear system (1.1) with small initial data (u 0 , u 1 ) and small boundary data h.
Let v solve the following non-homogeneous linear wave equation Let u 0 , u 1 ∈ B m+1 (ε 0 /3), and h ∈ N m+1 (ε 0 /3) for ε 0 sufficiently small. Then ||v|| E m+1 + ||∂ ν v|| H m (Σ) ≤ C(T )ε 0 .
v tt − ∆ g v + bv t + f z (x, t, 0)v = 0 in M, v = h on Σ, v(x, 0) = u 0 , v t (x, 0) = u 1 (x) in Ω.
(3.35)
For given δ 0 > 0 sufficiently small, let
Z m+1 (δ 0 ) = {ŵ ∈ E m+1 : ||ŵ|| E m+1 ≤ δ 0 } ⊂ E m+1 .
Let w solve the following homogeneous equation
w tt − ∆ g w + bw t + f z (x, t, 0)w + ∞ k=2 f (k) (x, t, 0) (v+ŵ) k k! = 0 in M,
w = 0 on Σ, w(x, 0) = w t (x, 0) = 0 in Ω, (3.36) where v ∈ E m+1 is the solution of (3.34) with estimate (3.35). We define a map A : Z m+1 (δ 0 ) → E m+1 , which sends the givenŵ ∈ Z m+1 (δ 0 ) to the solution of (3.36). For any positive integers k and R, one has ||f (k) (x, t, 0)|| E m+1 ≤ k! R k sup |z|=R ||f (x, t, z)|| E m+1 .
By the a priori estimate of (3.36) and the property of Banach algebra, we have
||Aŵ|| E m+1 ≤ C(T ) ∞ k=2 1 k! ||f (k) (x, t, 0)|| E m+1 ||v +ŵ|| k E m+1 ≤ C(T ) ∞ k=2 1 R k sup |z|=R ||f (x, t, z)|| E m+1 2 k−1 (||v|| k E m+1 + ||ŵ|| k E m+1 ) ≤ C(T ) R sup |z|=R ||f (x, t, z)|| E m+1 ∞ k=1 2 k R k (δ 0 δ k 0 + ε 0 ε k 0 ) = 2C(T ) R sup |z|=R ||f (x, t, z)|| E m+1 (δ 0 δ 0 R − 2δ 0 + ε 0 ε 0 R − 2ε 0 ). (3.37)
Let R = 2 and ε 0 = δ 0 ≤ 1 2 small enough such that
C(T ) sup |z|=R ||f (x, t, z)|| E m+1 δ 0 ≤ 1 2 .
Then we have ||Aŵ|| E m+1 ≤ δ 0 , which implies that A : Z m+1 (δ 0 ) → Z m+1 (δ 0 ) is well defined. Let Therefore, taking δ 0 small enough, we see that
||Aŵ 1 − Aŵ 2 || E m+1 = ||w 1 − w 2 || E m+1 ≤ C(T ) ∞ k=2 k R k sup |z|=R ||f (x, t, z)|| E m+1 (3δ 0 ) k−1 ||ŵ 1 −ŵ 2 || E m+1 ≤ 1 2 ||ŵ 1 −ŵ 2 || E m+1 . (3.38)
Thus, A : Z m+1 (δ 0 ) → Z m+1 (δ 0 ) is a contraction. The Banach's fixed point theorem implies that u = v + w ∈ E m+1 is a solution to system (1.1). Moreover, we have ||u|| E m+1 + ||u|| C(M) + ||∂ ν u|| H m (Σ) ≤ C(||u 0 || H m+1 (Ω) + ||u 1 || H m (Ω) + ||h|| H m+1 (Σ) ),
where C > 0 is independent of u 0 , u 1 and h.
Theorem 2. 1 .
1Under assumptions (A.1) and (A.2), there exist constants C = C(M 0 ) > 0 and λ * > 0, such that for any λ > λ * , there exists s 0
D
2 ψ(X, X) = g(D XD ψ, X) = [g(D X ( −1 Dψ), X)]
Dψ(ln )X + X(ln )Dϑ + X(ψ)D(ln ), X) = D 2 ψ(X, X) + 1 2 Dϑ(ln )|X| 2 g ≥ (2 + 1 2
+ HAH + B = 0, H(s) = H 0 , with Im H 0 > 0 and z 0 ∈ (a 0 , b 0 ),(3.8) where B = 1 4 ∂ 2 ij g 11 , s = √ 2t − and the components of A = (A ij ) satisfy ii = 2, i = 2, · · · , n, A ij = 0, otherwise.For the above Riccati equation, we haveLemma 3.2. [23, Section 8] The Riccati equation (3.8) admits a unique solution. The solution H is symmetric and Im (H(z 0 )) > 0 for all z 0 ∈ (a 0 , b 0 ). We have H(z 0 ) = Z(z 0 )Y −1 (z 0 ), where the matrix valued functions Z(z 0 ), Y (z 0 ) solve the first order linear system dZ dz 0 = −BY, dY dz 0 = AZ, subject to Y (s) = I, Z(s) = H 0 .
By [ 9 ,
9Proposition 1.3] and the discussions in [10, Section 2.2], together with [37, Theorem 1.6], we have Proposition 3.1. Suppose that either (Ω, g) is simple or (Ω, g) satisfies the foliation condition. Let f ∈ C 1 (M)) vanish on the set M\E. Then Gf = 0 for all maximal null geodesic β ⊂ D implies f = 0.
Lemma 3.4. [10] Let the remainder terms r 1σ and r 2σ respectively solve (3.14) and (3.15). Then r jσ ∈ C([0, T ]; H 1 0 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)), and lim σ→∞ (||r jσ || L 2 (M) + σ −1 ||r jσ || H 1 (M) ) = 0, for j = 1, 2.
t a 1 a 2 e −2σImϕ dV g = 0.(3.19)
any maximal null geodesic in R×Ω. Then Proposition 3.1 is applied to obtain b = 0, which implies that b 1 = b 2 in E.We return to the equation(3.18) with b = 0 to have lim σ→∞ M qa 1 a 2 e −2σImϕ dV g qχ 2 ( |z | δ )|det Y (z 0 )| −1 e −2σImϕ dz 0 ∧ dz = 0.
2uu (x, t, 0) is known. Then the Dirichlet to Neumann map Λ determinesM mv 1 v 2 v 3 v 0 dV g .(3.27)
B
m+1 (L) = {(y 1 , y 2 ) ∈ H m+1 (Ω) × H m (Ω) : ||y 1 || H m+1 + ||y 2 || H m ≤ L}, N m+1 (L) = {h ∈ H m+1 (Σ) : ||h|| H m+1 (Σ) ≤ L} ⊂ H m+1 (Σ).
FF
((x, t,ŵ 1 ) − F (x, 1 ) + (1 − τ )(v +ŵ 2 )] k−1 dτ.
Moreover, by[10, Lemma 3.6], we know that ϕ t does not vanish in V. Notice that b, q = 0 on the set M 1 \M. Then(3.19) yieldslim
σ→∞
σ
n
2
b 0
a 0
|z |<δ
For given p = (t 0 , x 0 ) ∈ E, we choose local coordinates such that g coincides with the standard Minkowski metric at p. Define the light cone at p as C(p) = {(t, X) ∈ T p M : t 2 = |X| 2 g }.Similar to[7,Lemma 1](see also,[14, section 3.2],[12,Section 5]), we can assume without loss of generality that ζ 0 , ζ 1 ∈ C(p) satisfyingfor some θ ∈ [0, 1]. Taking θ > 0 small and introduceBy[7,Lemma 1], ζ 0 , ζ 1 are linear-independent, and there are constants k 0 , k 1 , k 2 , k 3 such thatDenote β k to be the null geodesic with cotangent vector ζ k and p. Taking v k = e i σk k ϕ k a k + r kσ for k = 0, 1, 2, 3 as the Gaussian beams concentrating near the null geodesic β k . Notice that the manifold (Ω, g) is simple, the null geodesic β k (k = 0, 1, 2, 3) can intersect only at p.Inserting v k = e i σk k ϕ k a k + r kσ into (3.27), with estimate (3.11), the Dirichlet-to-Neumann map determinesClearly, the product a 0 a 1 a 2 a 3 is supported in a neighborhood of p. We introduce a lemma to deal with the above integral.Based on the above lemma, applying the stationary phase (e.g., see[16,Theorem 7.75]) to (3.28), wewhere c denotes some explicit constant. Thus, the Dirichlet-to-Neumann map determines m(p). For the recovery the higher order coefficients f (k)u (x, t, 0) for k ≥ 4, we can achieve this by induction. We refer to[14,Section 4]for such an operation and omit the details. Therefore, the proof of Theorem 1.4 is complete.Remark 3.2. We notice that the recovery of q and b is much different from that of higher order terms f It seems that we can not recover f uu by the same method as that for terms f (k) u for k = 1 or k ≥ 3. One of the reason is that we can not choose three time-like vectors ζ 0 , ζ 1 , ζ 2 such that ζ 0 , ζ 1 are linear-dependent but ζ 0 , ζ 1 , ζ 2 are linear-dependent. We mention that, in[46], due to the presence of two different matrics g P and g S , the authors have chosen three different vectors satisfying the above Lemma 3.5. Therefore, they proved the unique recovery of coefficients by the Gaussian beams in place of the distorted plane waves method used in e.g.,[14].Appendix Well-posednessWe prove the well-posendess result to (1.1). We begin with the following linear wave equation; H m−k (Ω)) for k = 0, 1, · · · m. Moreover, we assume that the compatibility conditions hold up to order m, which are given byAccording to [23, Theorem 2.45], for system 3.29 with the above conditions, we have Lemma 3.6. Let m be a positive integer and T > 0. Then system (3.29) admits a unique solution u ∈ E m+1 and ∂ ν u ∈ H m (Σ). Moreover, the dependence of u and ∂ ν u on u 0 , u 1 , h, F is continuous in the corresponding spaces, i.e., ||u|| E m+1 + ||∂ ν v|| H m (Σ) ≤ C(T )(||u 0 || H m+1 (Ω) + ||u 1 || H m (Ω) + ||h|| H m+1 (Σ) + ||F || Xm ),(3.31)
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arxiv |
Entropic Compressibility of Lévy Processes
15 May 2022
Julien Fageot
Alireza Fallah
Thibaut Horel
Entropic Compressibility of Lévy Processes
15 May 2022Lévy processesdiscrete entropydifferential entropylocal limit theoremsBlumenthal-Getoor index
In contrast to their seemingly simple and shared structure of independence and stationarity, Lévy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari [33], we characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretization steps. For a Lévy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times, for which we obtain a new local central limit theorem. We generalize known results for stable processes to the non-stable case, with a special focus on Lévy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal-Getoor index.
Introduction
Lévy processes generalize the Wiener process by relaxing the Gaussianity of increments, while retaining their stationarity and independence. They have proven to be a fruitful and flexible stochastic continuous-domain model in many applications, including financial mathematics [67], movement patterns of animals [41], turbulence [5], and sparse signal processing [76], among others.
In this paper, we follow an entropy-based approach initiated in [33] to quantify the compressibility of a Lévy process, assuming that the process is approximated by uniform sampling in time and uniform quantization in amplitude. Specifically, let L = (L t ) t≥0 be a Lévy process and for n ≥ 1, consider the vector (L 1/n , L 2/n , . . . , L (n−1)/n , L 1 ) ∈ R n obtained by sampling L over the interval [0, 1] with a period of 1/n. Since the increments of L are independent and stationary, it is more convenient to work with the vector of independent and identically distributed increments (∆ 1 , . . . , ∆ n ) where ∆ k = L k/n − L (k−1)/n for k = 1, . . . , n (and L 0 = 0 by definition). For an integer m ≥ 1 and x ∈ R, denote by
[x] m = 1 m ⌊mx⌋,
the quantization of x with step size 1/m. Then, we define the entropy of the Lévy process L for sampling period 1/n and quantization step 1/m as
where H(X) denotes the entropy of the discrete random variable X.
The main goal of this paper is to understand the asymptotic behavior of (1) when n, m → ∞. Due to the characterization of entropy in terms of compressibility, a slower asymptotic growth of H n,m (L) implies that fewer bits are required to encode an approximation of L of a given quality. Therefore, the asymptotic behavior of H n,m (L) is a measure of the entropic compressibility of the Lévy process L.
Contributions: the Asymptotic Entropy of Lévy Processes
Our work is directly inspired by the contribution of Hamid Ghourchian, Arash Amini, and Amin Gohari in [33], who introduced the quantity H n,m (L)-with a slightly different but equivalent perspective-and characterized its asymptotic behavior for important classes of Lévy processes, mainly the compound Poisson and stable processes.
We provide a general analysis of the entropy H n,m (L) of a Lévy process L, with a particular focus on Lévy processes whose marginals are absolutely continuous, thus allowing us to reduce this question to the study of the differential entropy h(L t ) at small times t → 0 + . This reduction mostly benefits from the seminal contributions of [33], to which we bring some useful complements. Our main results, detailed below, extend the ones of [33] to all (possibly non-stable) Lévy processes whose marginals are absolutely continuous. • The entropy of locally self-similar Lévy processes. A Lévy process is locally symmetric and self-similar if its rescaled versions converge locally to a symmetric non-trivial random process (see Definition 4.1). The rescaling depends on the local self-similarity order of the Lévy process, that is, the self-similarity order of its local limit, which is itself related to its Blumenthal-Getoor index β (see Section 2.3). We show in Theorem 4.3 that the differential entropy h(L t ) of a locally symmetric and self-similar Lévy process has the same asymptotic behavior as the symmetric-α-stable process with α = β; that is,
h(L t ) ∼ t→0 + 1 β log t.(2)
• An Upper-Bound for the Entropy of Lévy Processes. For the general case, we obtain an upper-bound on the entropy in terms of the Blumenthal-Getoor index β of the Lévy process. More precisely, we show that
lim sup t→0 + h(L t ) log(1/t) ≤ − 1 β .(3)
A direct consequence of this result is that for Lévy processes with β = 0, the entropy diverges super-logarithmically as t → 0 + , making them more compressible than any locally symmetric and self-similar Lévy process.
These two results together reveal a new entropy-based compressibility hierarchy of Lévy processes determined by their Bluementhal-Getoor index. Among Lévy processes with absolutely continuous marginals, the ones with Blumenthal-Getoor index β = 0 are the most compressible, and more generally the smaller β, the more compressible L. We moreover exemplify our main contributions on several classes of Lévy processes.
Related Work
Entropy & Compressibility. The notion of entropy introduced in 1948 by Claude Shannon in two successive groundbreaking papers [68,69] is a fundamental measure of the quantity of information of discrete random sources and exactly captures their compressibility as formalized by the source coding theorem [16,Theorem 5.4.1]. Initially defined for discrete random variables, it can be generalized to continuous random variables as the differential entropy, whose relation to discrete entropy via quantization was studied by Alfréd Rényi [58]. Note however that contrary to the discrete entropy, there is no universal notion for quantifying the information of continuous random sources and the differential entropy is only one of several generalizations, with its strengths and weaknesses [44].
Beyond discrete random vectors, one can consider the entropy of random sequences, defined as the averaged limit of the vector case [16,Section 4.2]. Going one step further, several generalizations to continuous-time random processes have been proposed, as discussed extensively in the monograph of Shunsuke lhara [42]. In this paper, we follow the recent work of Ghourchian et al. [33], which provides a new definition of the entropy of a random process. Their approach is based on a double discretization in time (sampling) and amplitude (quantization), corresponding to (1) for Lévy processes and relying on discrete entropy. While these authors consider general Lévy processes, whose marginals are not necessarily absolutely continuous, we note that among Lévy processes with absolutely continuous marginals, they only quantify the entropy of stable processes.
Sparsity & Lévy Processes. A signal of interest is sparse if it admits a concise representation that captures most of its information. Many naturally-occurring signals are deeply structured and admit such sparse representations, calling for sparse models and sparsity-promoting methods in the analysis and synthesis of signals. This led for example to the introduction of ℓ 1 methods in statistical learning [75,38] which are ubiquitous in the field of compressed sensing [20,14].
Continuous-domain signals present their own challenges since adequate representations should ideally capture the fine details of possibly fractal-type sample paths [52]. It is wellknown that classic Gaussian models fail at modelling sparsity [74,54,25], in the sense that they generate poorly compressible data. Stable models with infnite variance [62,55] and more generally Lévy processes and Lévy fields [76] have been proposed beyond Gaussian models. Those are particularly interesting since they include a wide range of random processes from non-sparse models, such as the Wiener process, to very sparse models, such as compound Poisson processes, whose rate of innovation is finite [79]. There is strong empirical evidence that Lévy processes are useful in modeling sparsity in signal processing [76]. To the best of our knowledge, [33] complemented by the present paper is the first attempt to provide an information-theoretic justification of this observation.
Approximation-theoretic Compressibility. We also mention a distinct area of research which aims at quantifying the sparsity of random models, with a different perspective based on the theory of approximation. It was initiated with the study of independent and identically distributed random sequences [15] whose compressibility is measured by the asymptotic behavior of truncated subsequences [1] and is strongly linked to the heavytailedness of the distribution of the sequence [37,72]. Extensions to stationary ergodic random sequences and beyond have also been proposed [71,70].
A continuous-domain function is compressible if most of its information is captured by a few coefficients in an adequate dictionary. In this regard, wavelet representations are well-known for their excellent compression rate [51]. A natural line of research, has therefore been to study the wavelet approximability of a random model, understood as the convergence rate of its best approximation in wavelet bases. It was shown in a series of work [28,26,3,4], that the approximation error of the best n-term approximation of a Lévy process in wavelet bases behaves asymptotically like n −1/β , where β is the Blumenthal-Getoor of the process, and decays faster than any polynomial for β = 0. This shows a wavelet-based hierarchy of Lévy processes, from the less sparse (the Wiener process) to the sparsest (compound Poisson or Lévy process β = 0) [29]. Remarkably, this coincides with the information-theoretic compressibility hierarchy provided in this paper.
Outline
Section 2 introduces the family of Lévy processes and their Blumenthal-Getoor index. The entropy of a Lévy process is rigorously defined in Section 3, where we also provide existence results. Section 4 contains the main results of the paper: we first characterize the asymptotic behavior of the entropy of locally symmetric and self-similar Lévy processes in Section 4.2, and then deduce a general upper bound for the entropy of any Lévy process in Section 4.3. We apply our theoretical findings to specific classes of Lévy processes by characterizing their small-time entropic behavior in Section 5. Finally, a summary and discussion of our results is provided in Section 6.
Lévy Processes and their Blumenthal-Getoor Index
We introduce the family of Lévy processes in Section 2.1 and the subfamily of symmetric stable processes in Section 2.2. As we shall see in Sections 4 and 5, the entropic compressibility of a Lévy process is captured by its Blumenthal-Getoor index, whose definition and main properties are recalled in Section 2.3.
Lévy Processes
Lévy processes are named after Paul Lévy, who popularized their study in 1937 [48, Chapter VII] 1 . They generalize the Wiener process by relaxing the requirement that increments be Gaussian. A brief presentation is given thereafter; more details can be found in the classic monographs [2,8,63].
Definition 2.1. A Lévy process is a continuous-time random process L = (L t ) t≥0 satisfying:
1. L 0 = 0 almost surely (a.s.).
2. Stationary increments: for all s, t ≥ 0, L s and L t+s − L t have the same law.
3. Independent increments: for all n ≥ 1 and all 0 ≤ t 0 ≤ t 1 ≤ t 2 ≤ . . . ≤ t n , the random variables L t k − L t k−1 1≤k≤n are mutually independent.
4. Sample paths regularity: t → L t is a.s. right-continuous with left limits over R ≥0 .
Recall that a random variable X is infinitely divisible if it can be decomposed as the sum of n i.i.d. random variables for all n ≥ 1 [63]. Lévy processes are intimately linked to infinitely divisible random variables [63,Section 7]. In particular, the marginals L t are infinitely divisible for all t ≥ 0. Indeed, we can write for each t > 0 and n ≥ 1,
L t = n k=1 L k n t − L k−1 n t ,(4)
where (L k n t − L k−1 n t ) 1≤k≤n is i.i.d. since L has independent and stationary increments. Moreover, the law of a Lévy process is completely characterized by the infinitely divisible law of L 1 , its marginal at time t = 1. The characteristic function Φ L 1 of L 1 can be expressed as Φ L 1 (ξ) = exp(Ψ(ξ)), for ξ ∈ R, where Ψ : R → C, the characteristic exponent of L, is a continuous function admitting a Lévy-Khintchine representation [63,Theorem 8
.1] Ψ(ξ) = iµξ − σ 2 ξ 2 2 + R e iξt − 1 − iξt½ |t|≤1 ν(dt)(5)
for any ξ ∈ R, with µ ∈ R, σ 2 ≥ 0, and ν is a Lévy measure. The latter means that ν is a non-negative measure on R such that ν({0}) = 0 and
|t|≤1 t 2 dν(t) + |t|>1 dν(t) < ∞.
We call (µ, σ 2 , ν) the Lévy triplet of L, which uniquely determines L. Then, for all t ≥ 0, the characteristic function of the infinitely divisible random variable L t is Φ Lt (ξ) = exp (tΨ(ξ)) , ∀ξ ∈ R.
Symmetric-α-Stable Lévy Processes
Among Lévy processes, we now define the subfamily of symmetric-α-stable processes, which will play an important role in our study, as potential limits in law of rescaled Lévy processes (see Section 4.1). More information can be found in the classical monograph by Gennady Samorodnitsky and Murad Taqqu [62]. A random variable X is stable if, for all n ≥ 1, there exists c n > 0 and d n ∈ R such that X 1 + · · · + X n
(L) = c n X + d n ,(6)
where the X k are independent copies of X and = stands for equality in law. From the definition, we readily see that a stable random variable is infinitely divisible. The entire family of stable random variables is described by four parameters [35,Section 34]. If X is moreover symmetric (i.e., X and −X have the same law), then its characteristic function takes the form Φ X (ξ) = exp(−γ|ξ| α ), ∀ξ ∈ R,
where γ > 0 and α ∈ (0, 2]. In this case, Eq. (6) is satisfied with c n = n 1/α and d n = 0 [31, Theorem 1, Section VI.1]. The parameter γ is simply a scale parameter, while the parameter α deeply influences the properties of X. This justifies the terminology symmetric-α-stable laws. For instance, the pth moment E[|X| p ] of X is finite if and only if α = 2 (in which case X is a Gaussian random variable) or 0 < p < α < 2 [62, Proposition 1.2.16]. For α ∈ (0, 2], a symmetric-α-stable Lévy process (or SαS process) is a Lévy process L for which L 1 is a SαS random variable. The characteristic exponent of L is thus given by
Ψ(ξ) = −γ|ξ| α , ∀ξ ∈ R(7)
and, for each t ≥ 0, L t is SαS with characteristic function Φ Lt (ξ) = exp(−γt|ξ| α ).
The Blumenthal-Getoor Index of Lévy Processes
The Blumenthal-Getoor index was introduced by Robert M. Blumenthal and Ronald K. Getoor in 1961 to characterize the small-time behavior of Lévy processes [9]. Since then, it has been recognized as a key quantity to characterize the local Besov regularity [65,64,3] or the variations [60] of the sample paths, the Hausdorff dimension of the image set [43,21], moment estimates [49,19,26,47], the local self-similarity [27], or the local wavelet compressibility [29] of Lévy processes and their generalizations. We demonstrate in this paper that it also quantifies the asymptotic behavior of the entropy of Lévy processes. A Lévy process satisfies the sector condition if there exists a constant C > 0 such that
|ℑΨ(ξ)| ≤ C|ℜΨ(ξ)|, ∀ξ ∈ R,(8)
with Ψ its characteristic exponent, and where ℜ and ℑ respectively stand for the real part and imaginary part of complex numbers. This condition ensures that no drift is dominating the process. It typically excludes the pure drift (deterministic) process L = (µt) t≥0 with µ ∈ R\{0}, for which Ψ(ξ) = iµξ and is automatically satisfied by symmetric Lévy processes, for which L 1 and −L 1 have the same law and whose characteristic exponent Ψ is therefore real. See [12] for additional discussions about the sector condition. We shall always assume that the sector condition is satisfied in this paper without further mention.
Definition 2.2. The Blumenthal-Getoor index β of a Lévy process L with characteristic exponent Ψ is defined by
β = inf p > 0, |Ψ(ξ)| |ξ| p −→ |ξ|→∞ 0 .(9)
The Blumenthal-Getoor index lies in [0,2]. This follows directly from the fact that there exists a constant C > 0 such that |Ψ(ξ)| ≤ C|ξ| 2 for any |ξ| ≥ 1 [23, Proposition 2.4]. The characteristic exponent of a SαS process being given by (7), we deduce that its index is β = α. The characteristic exponent of the Laplace process, for which L 1 is a Laplace random variable, is given by Ψ(ξ) = − log(1 + ξ 2 ) [45]. We therefore have β = 0 in this case. This is also the case for the gamma process (see Section 5.2).
Remark. It comes as no surprise, in light of the seminal work of Blumenthal and Getoor [9], that the local behavior of a Lévy process L is captured (via the index β) by the asymptotic behavior of its characteristic exponent. The latter is indeed the logarithm of the characteristic function of the law of L 1 , whose local properties are known to be linked to asymptotic properties of its Fourier transform.
Entropy of Lévy Processes
Quantization and Entropy of Random Variables
We briefly review the definitions of discrete and differential entropy. The former applies to random variables taking values in a countable space, and the latter applies to absolutely continuous random variables, i.e., variables whose distribution is absolutely continuous with respect to the Lebesgue measure and thus admit a probability density function.
Definition 3.1. Let Y be a discrete random variable taking values into a countable space Y. The discrete entropy of Y is defined by H(Y ) = − y∈Y P(Y = y) log(P(Y = y)) = −E log P(Y ) ,(10)
with the usual convention 0 log(0) = 0. Since the summand in (10) is non-negative, H(Y ) is either +∞ or a non-negative real. In the latter case, we say that Y has finite entropy. Let X be an absolutely continuous real-valued random variable with probability density function p X . The differential entropy of X is
h(X) = − R p X (x) log p X (x) dx = −E log p X (X) ,(11)
which is well-defined in [−∞, ∞] provided that either the positive or negative part of the integrand in (11) is integrable. We say that the real random variable X has finite differential entropy when it is absolutely continuous and the integrand in (11) is absolutely integrable.
The quantization of order m (or m-quantization) of a real x ∈ R is defined by
[x] m = 1 m ⌊mx⌋. That is, [x] m is the unique element of Z/m = {n/m | n ∈ Z} such that [x] m ≤ x < [x] m + 1 m .
For a real random variable X, the study of the entropy of its quantization H([X] m ), and its relation to h(X) (when it exists) was initiated by Alfréd Rényi in [58]. In particular, he established the following result (see also [33,Corollary 1]).
Proposition 3.2 ([58, (11) and Thm. 1]). Let X be a real random variable. If H(⌊X⌋) < ∞, then H([X] m ) < ∞ for all m ≥ 1.
If furthermore X has finite differential entropy, then
H([X] m ) = log m + h(X) + o m (1) .(12)
Remark. Finiteness of the differential entropy h(X) does not necessarily imply finiteness of H(⌊X⌋), as exemplified by Rényi in the remark following [58, Theorem 1].
For an absolutely continuous random variable X, it is known that finiteness of the logmoment E[log(1+|X|)] implies that h(X) < ∞ (see e.g. [59,Proposition 1]). This is proved by a direct application of Gibbs' inequality to the Kullback-Leibler divergence KL(X Y ) for a Cauchy variable Y . A simple adaption of the proof shows that this condition also implies H(⌊X⌋) < ∞. We thus obtain the following proposition, proved in Appendix A.
Proposition 3.3. Let X be a real random variable such that E[log(1 + |X|)] < ∞, then H(⌊X⌋) is finite.
If moreover X is absolutely continuous with bounded probability density function ( e.g., if Φ X is integrable) then X has finite differential entropy and (12) holds.
Definition and Existence of the Entropy of Lévy Processes
The notion of m-quantization is extended to finite-dimensional vectors by writing
[x] m = ([x 1 ] m , . . . , [x n ] m ) ∈ (Z/m) n for x = (x 1 , . . . , x n ) ∈ R n .
Next, we give the definition of the entropy of a Lévy process.
H n,m (L) = H [X n (L)] m = n H L 1/n m ∈ [0, ∞],(13)
where the second equality is because the coordinates of X n (L) are independent and distributed as L 1/n by definition of a Lévy process.
Remark. Although the presentation is slightly different, our definition is essentially equivalent to the one of Ghourchian et al. [33,Eq. (6)]. There, the authors consider the entropy of Lévy white noises, that are weak derivatives of Lévy processes [18]. Thus, the entropy of a Lévy white noise W in the sense of [33] is equal to the entropy of the corresponding Lévy process L such that L ′ = W in Definition 3.4 up to the following minor difference. Ghourchian et al. define the quantization of a random variable X as 1 m ⌊mx + 1/2⌋ instead of 1 m ⌊mx⌋ [33,Definition 6]. This is purely a matter of convention and will not affect the results; we prefer to follow the quantization convention of Rényi [58].
Intuitively, H n,m (L) measures the expected length of the most efficient encoding of L over [0, 1], after approximating it both temporally (via sampling) and in amplitude (via quantization). In [33], the authors argue that the asymptotic behavior of H n,m (L) as n and m → ∞ can be used as a measure of the compressibility of L.
We now provide some useful inequalities for the entropy of Lévy processes, resulting in a characterization of Lévy processes with finite entropy. Some of these inequalities are already known but restated here for the sake of completeness.
Proposition 3.5. Let L be a Lévy process, then for all integers n, m ≥ 1 we have
0 ≤ H n,1 (L) ≤ H n,m (L) ≤ H n,1 (L) + n log m(14)
and
H 1,m (L) − log n ≤ H n,m (L) ≤ nH 1,m (L) + n log n.(15)
In particular, the three following statements are equivalent:
• ∃n, m ≥ 1, H n,m (L) < ∞; • ∀n, m ≥ 1, H n,m (L) < ∞; • H(⌊L 1 ⌋) < ∞.
It follows from Proposition 3.5 that the entropy H n,m (L) of a Lévy process L is always finite or always infinite depending on the finiteness of H(⌊L 1 ⌋). This justifies the following definition.
Definition 3.6. We say that a Lévy process L has finite entropy if H(⌊L 1 ⌋) < ∞ and infinite entropy otherwise.
Proof of Proposition 3.5. Rényi proved in [58,Eq. (11) and (13)] that, for a random variable Y and any integer m ≥ 1,
0 ≤ H([Y ] 1 ) ≤ H([Y ] m ) ≤ H([Y ] 1 ) + log m.
Applying this relation to Y = L 1/n and using (13), we obtain (14).
The infinitely divisible random variable X = L 1 can be decomposed as a sum X = X 1 + · · · + X n of n i.i.d. random variables whose common law is the one of L 1/n . Let y i ∈ R for i = 1 . . . n and set y = n i=1 y i . Then, we have that
n i=1 ⌊y i ⌋ − 1 ≤ y − 1 < ⌊y⌋ ≤ y < n i=1 ⌊y i ⌋ + n ,
which implies that e := ⌊y⌋− n i=1 ⌊y i ⌋ ∈ {0, . . . , n−1}. Applying this relation to y i = mX i and
y = mX = m n i=1 X i , we deduce that E n,m := [X] m − n 1=1 [X i ] m ∈ 0, 1 m , . . . , n − 1 m .
In particular, as a discrete random variable that can take values among a set of cardinal at most n, we have that H(E n,m ) ≤ log n [16, Theorem 2.6.4].
We then remark that
H 1,m (L) = H([X] m ) = H n i=1 [X i ] m + E n,m ≤ H ([X 1 ] m , . . . , [X n ] m , E n,m ) ≤ n i=1 H ([X i ] m ) + H(E n,m ) = n H([L 1/n ] m ) + H(E n,m ) ≤ n H([L 1/n ] m ) + log n = H n,m (L) + log n,
where we used the subadditivity of the discrete entropy [16, 2.6.6]. This shows the leftmost inequality in (15). For the rightmost inequality, we use the argument of [33, Section IV-B, Proof of item 2)] that H([L 1/n ] 1 ) ≤ H([L 1 ] 1 ) + log n and adapt it to quantification m ≥ 1. We have that
H n,m (L) n = H([L 1/n ] m ) = H([X 1 ] m ) = H n i=1 [X i ] m [X 2 ] m , . . . , [X n ] m ≤ H n i=1 [X i ] m = H ([X] m − E n,m ) ≤ H([X] m , E n,m ) ≤ H([X] m ) + H(E n,m ) ≤ H([L 1 ] m ) + log n = H 1,m (L) + log n,
where we used again the subadditivity of the discrete entropy. Then, the rightmost inequality in (15) clearly follows. Finally, (14) and (15) together easily imply that, for all n, m ≥ 1,
H 1,1 (L) − log n ≤ H n,m (L) ≤ nH 1,1 (L) + n log(nm),(16)
from wich the equivalent statements in Proposition 3.5 follow since H(⌊L 1 ⌋) = H 1,1 (L).
We now limit ourselves to Lévy processes whose marginals have finite differential entropy. A direct corollary of Proposition 3.2 is the following result, implicitly stated in [33].
Corollary 3.7 (implicit in [33]). Let L = (L t ) t≥0 be a Lévy process with finite entropy (see Definition 3.6). If L t has finite differential entropy for all t > 0, then
H n,m (L) = n log m + h(L 1/n ) + ε n,m .(17)
where ε n,m vanishes when m → ∞ for all n ≥ 1. In particular, there exists a non-decreasing function m : N → N such that lim n→∞ ε n,m(n) = 0.
Proof of Corollary 3.7. According to Proposition 3.5, H(⌊L 1/n ⌋) < ∞ for all n ≥ 1. Applying Proposition 3.2 for n ≥ 1 fixed, we deduce that
H n,m (L) n = H([L 1/n ] m ) = log m + h(L 1/n ) + ε n,m
where ε n,m → 0 when m → ∞, hence (17) follows. Fix any vanishing sequence (u n ) n≥1 of non-negative real numbers. Then, for any n ≥ 1, there exists m(n) such that |ε n,m(n) | ≤ u n . Moreover, m(n) can be chosen so that m(n) is non-decreasing. Then, |ε n,m(n) | ≤ u n vanishes when n → ∞.
Understanding the set of functions n → m(n) such that lim n→∞ ε n,m(n) = 0 is crucial in understanding the asymptotic behavior of H n,m (L) as n and m grow to infinity. In particular, since h(L 1/n ) → −∞ as n → ∞ (see Proposition 3.10 below) and the entropy of L is nonnegative, it follows from (17) that lim n→∞ ε n,m(n) = +∞ whenever log m(n) ∈ o h(L 1/n ) , in which case the expansion (17) is non-informative.
In contrast, when 2 log m(n) ∈ ω h(L 1/n ) and lim n→∞ ε n,m(n) = 0, then (17) provides the first two terms of the asymptotic expansion of H n,m(n) (L). More generally, the smallest rate of growth of n → m(n) such that ε n,m(n) ∈ o log m(n) + h(L 1/n ) quantifies how the amplitude quantization's order needs to be adjusted to increasing time quantization's orders. This "critical" rate of growth depends on L and needs to be determined on a caseby-case basis. It follows from our discussion that we expect this critical rate to be at least of the order of exp − h(L 1/n ) and in fact, [33,Prop. 1] shows that for SαS Lévy processes for which h(L 1/n ) ∼ − 1 α log n, we have lim n→∞ ε n,m(n) = 0 whenever m(n) ∈ ω(n 1/α ). Remark. In [58], Rényi introduced the dimension and d-dimension associated to a random variable X as follows.
Assume that there exist d ∈ [0, 1] and h ∈ R such that H([X] m ) = d log m + h + o m (1). Then, d is called the dimension and h the d-dimension of X.
Similarly for a Lévy process L, we denote by d n and h n the dimension and d-dimension of L 1/n , assuming they exist, and call (d n ) n≥1 and (h n ) n≥1 the dimension sequence and the d-dimension sequence of L. Corollary 3.7 then shows that the sequences (d n ) n≥1 and (h n ) n≥1 exist as long as L has finite entropy and marginals with finite differential entropy, in which case d n = 1 and h n = h(L 1/n ) for all n ≥ 1.
It is worth noting that it is possible to consider Lévy processes whose marginals are not absolutely continuous and to study their dimension and d-dimension sequences. For instance, Ghourchian et al. considered compound Poisson processes L with rate λ > 0 and jump distribution with finite differential entropy for which the dimension sequence of L is d n = (1 − e −λ/n ) < 1 [33,Proposition 2]. Investigating further the dimension sequences of Lévy processes whose marginals are not absolutely continuous is an interesting question for future work.
Combining Proposition 3.3 with Corollary 3.7 we immediately obtain the following corollary identifying a mild condition under which L has finite entropy and (17) holds. The proof is provided in Appendix A.
Corollary 3.8. Let L be a Lévy process such that E[log(1 + |L 1 |)] < ∞, then L has finite entropy. If moreover the characteristic function Φ Lt is integrable for all t > 0, then L t has finite differential entropy for all t > 0 and (17) holds.
Remark. Finiteness of the log-moment E[log(1 + |L 1 |)] is a condition which also appears when studying Lévy-driven CARMA processes, for which it is equivalent to the existence of a stationary solution [13,7]. Note that this condition is in particular implied by the finiteness of the absolute moment E[|L 1 | p ] < ∞ for some p > 0, which is an equivalent characterization of tempered (i.e., bounded by a polynomial) Lévy processes [24,18].
It is possible to construct Lévy processes whose entropy is infinite, even when the marginals L t are absolutely continuous for t > 0. This is stated in the following proposition, whose proof is provided in Appendix A. Proposition 3.9. There exists a Lévy process L whose marginals L t are absolutely continuous for any t > 0 and such that H(⌊L 1 ⌋) = ∞. In particular, the entropy of L is infinite in the sense of Definition 3.6.
Since all Lévy processes considered in this paper have marginals with finite differential entropy, we will henceforth focus on characterizing the asymptotic behavior of h(L t ) as t → 0 + . By (17), this informs the asymptotic behavior of H n,m (L) (see the discussion after Corollary 3.7) and thus quantifies the compressibility of L. We start with the following observations, valid for all Lévy processes with finite differential entropy, and which will be refined in subsequent sections by considering the Blumenthal-Getoor index.
Proposition 3.10. Let L be a Lévy process such that L t has finite differential entropy for all t > 0, then:
(i) the function t → h(L t ) is non-decreasing, (ii) h(L t ) ≤ h(L 1 ) + 1
2 log t 1−t for t ∈ (0, 1). Proof. (i) Let X and Y be independent random variables with finite differential entropy and h(X|Y ) be the conditional differential entropy (see [16,Section 8.4]). Then, we recall that
h(X) = h(X|Y ) = h(X + Y |Y ) ≤ h(X + Y ),(18)
where the second equality in (18) is the conditional version of [16,Theorem 8.6.3].
Since L has independent increments we obtain, applying (18)
with (X, Y ) = (L t 1 , L t 2 − L t 1 ) for 0 < t 1 ≤ t 2 , that h(L t 1 ) ≤ h(L t 1 + L t 2 − L t 1 ) = h(L t 2 ).
(ii) For mutually independent variables (X 1 , . . . , X n ) with finite differential entropy, we have by the entropy-power inequality 2h( n k=1 X k ) ≥ log n k=1 e 2h(X k ) (see e.g., [78, Eq. (2)]). Applying this inequality to (4) with X k = L k/n − L (k−1)/n , for which h(X k ) = h(L 1/n ), we obtain for all n ≥ 1 that
2h(L 1 ) ≥ log n + 2h(L 1/n ) .(19)
Consider t ∈ (0, 1), and define n = ⌊1/t⌋ so that 1 n+1 < t ≤ 1 n . Then,
h(L t ) ≤ h(L 1/n ) ≤ h(L 1 ) − 1 2 log n ≤ h(L 1 ) + 1 2 log t 1 − t ,
where the first inequality follows from (i), the second from (19), and the last is due to n > (1 − t)/t by definition of n.
Remark. The asymptotic behavior of the upper bound in (ii) is h(
L 1 ) + 1 2 log t + o t (1) as t → 0 + . Since h(W t ) = h(W 1 ) + 1
2 log t for a Wiener process W , the bound in (ii) is asymptotically tight for general Lévy processes. In Section 4, we will obtain the tighter bound h(L t ) ≤ 1 β log t + O t (1), for Lévy processes with Blumenthal-Getoor index β.
Entropy of Lévy Processes at Small Times
As per the discussion below Corollary 3.8, the entropic compressibility of a Lévy process L with finite differential entropy is governed by the small-time asymptotics of h(L t ), that we quantify in this section in terms of the Blumenthal-Getoor index β. We first consider the case of locally symmetric and self-similar Lévy processes, defined in Section 4.1, for which we show in Section 4.2 that h(L t ) ∼ 1 β log t when t → 0 + . We then consider general Lévy processes in Section 4.3, for which we obtain an upper bound on h(L t ). As an important consequence, we deduce that when β = 0, h(L t ) diverges super-logarithmically when t → 0 + .
Locally Self-Similar Lévy Processes
Definition 4.1. We say that a random process X = (X t ) t≥0 is • symmetric if the X and −X = (−X t ) t≥0 have the same probability law;
• self-similar of order H ∈ R if the rescaled random process a H X ·/a = (a H X t/a ) t≥0 and X have the same probability law for all a > 0;
• locally self-similar (LS) of order H ∈ R if a H X ·/a converges in law to a non-trivial random process Y when a → ∞. If moreover the limiting process Y is symmetric, we say that X is locally symmetric and self-similar (LSS).
Self-similar processes have been studied extensively [73,22] and include the fractional Brownian motion [53] and generalizations of SαS processes [62,Chapter 7]. Locally selfsimilar processes were introduced in [27, Definition 4.3]. Higher values of a > 0 in a H X ·/a corresponds to zooming in the process at the origin. The limiting process Y in Definition 4.1 is known to be self-similar of order H [27, Proposition 4.4], which explains the terminology.
The following result, proved in Appendix B, characterizes symmetric self-similar and locally symmetric and self-similar random processes among the family of Lévy processes.
Proposition 4.2. A Lévy process is symmetric and self-similar if and only if it is a SαS process. In this case, the self-similarity order is H = 1/α. A Lévy process with Blumenthal-Getoor index β is LSS if and only if β > 0 and its characteristic exponent Ψ satisfies
Ψ(ξ) ∼ |ξ|→∞ −γ|ξ| β(20)
for some constant γ > 0. The local self-similarity order is then H = 1/β and the limiting process Y is SαS with α = β.
Remark.
A locally symmetric self-similar process is not necessarily symmetric itself, but its potential skewness vanishes at small times. Proposition 4.2 can be extended to the nonsymmetric case, but we will not cover it in this paper (see Section 6 for a short discussion regarding this point).
Remark. According to Proposition 4.2, the Blumenthal-Getoor index β of a LSS Lévy process is necessarily positive. In fact, a Lévy process L with β = 0 is such that a H X ·/a converges in law to 0 for all H ∈ R when a → ∞ [27, Proposition 4.7].
A Small-time Limit Theorem for the Entropy of LSS Lévy Processes
In this section, we consider the differential entropy h(L t ) of an LSS Lévy process L, for which we obtain an exact asymptotic expansion as t → 0 + up to a vanishing term. and such that L 1 has a finite absolute moment E[|L 1 | p ] < ∞ for some p > 0. Then, the differential entropy h(L t ) is finite for all t > 0 and
h(L t ) = 1 β log t + h(Y β ) + o t (1),(21)
where Y β is a SαS random variable with α = β and o t (1) → 0 when t → 0 + .
We shall use the following lemma in the proof of Theorem 4.3.
Lemma 4.4. Let L be a Lévy process with characteristic exponent Ψ, Blumenthal-Getoor index β > 0, and satisfying E[|L 1 | p ] < ∞ for some p > 0. We assume moreover that Ψ is asymptotically dominated by |·| β , i.e., there exists a constant C > 0 such that |Ψ(ξ)| ≤ C|ξ| β for all |ξ| ≥ 1. Then, for all 0 < q < min{1, p, β}, there exists a constant M q > 0 such that
E[|t −1/β L t | q ] ≤ M q , ∀0 < t ≤ 1.(22)
Upper bounds such as (22) are known as moment estimates and play a crucial role when studying the local smoothness of the sample paths of Lévy processes [3,26,12]. Moment estimates and extensions thereof have been studied by several authors [49,19,46]. Lemma 4.4 can be deduced from known estimates. In particular, it is a consequence of [47, Theorem 2.9], which considers the more general class of Lévy-type processes. For completeness, a self-contained proof in the case of Lévy processes is provided in Appendix B. Note that the hypothesis that Ψ is asymptotically dominated by |·| β is automatically satisfied for LSS processes (by Proposition 4.2) but is not true for arbitrary Lévy processes.
Proof of Theorem 4.3. Fix a sequence (t n ) n≥1 of positive reals such that lim n→∞ t n = 0 and define X n = t −1/β n L tn for n ≥ 1. We will prove that lim n→∞ h(X n ) = h(Y β ) which implies that h(t −1/β L t ) → h(Y β ) as t → 0 + since the sequence (t n ) n≥1 is arbitrary. Equation (21) then follows since h(aX) = log a + h(X) for a > 0.
Due to the sector condition (8), we have that
c · |Ψ(ξ)| = c · (ℜΨ(ξ)) 2 + (ℑΨ(ξ)) 2 ≤ |ℜΨ(ξ)| = −ℜΨ(ξ),
where we defined c = (1 + C 2 ) −1/2 . Hence for all n ≥ 1 and ξ ∈ R,
|Φ Xn (ξ)| = |Φ Lt n (t −1/β n ξ)| = | exp t n Ψ(t −1/β n ξ) | = exp t n ℜΨ(t −1/β n ξ) ≤ exp −c · t n |Ψ(t −1/β n ξ)| ,(23)
where the third equality uses that | exp z| = exp(ℜz) for z ∈ C. By Proposition 4.2, the characteristic exponent Ψ of L = (L t ) t≥0 satisfies |Ψ(ξ)| ∼ γ|ξ| β as |ξ| → ∞. In particular, there exists B ≥ 0 such that |Ψ(ξ)| ≥ γ|ξ| β /2 for |ξ| > B. Let us now define the function g : R → R by g(ξ) = 1 for |ξ| ≤ B and g(ξ) = e −cγ|ξ| β /2 for |ξ| > B. For n large enough such that t n ≤ 1 we have, for all ξ ∈ R, exp −c · t n |Ψ(t −1/β n ξ)| ≤ g(ξ).
Indeed, (24) is obvious for |ξ| ≤ B since exp(−x) ≤ 1 for x ≥ 0. For |ξ| > B, we also have |t −1/β n ξ| > B (since 0 < t n ≤ 1), hence c · t n |Ψ(t −1/β n ξ)| ≥ c · t n γ|t −1/β n ξ| β /2 = c · γ|ξ| β /2 and (24) readily follows by definition of g.
By assumption on L, X n converges in distribution to Y β which implies the pointwise convergence Φ Xn (ξ) → Φ Y β (ξ) for all ξ ∈ R as n → ∞. Equation (23) together with (24) implies that Φ Xn is uniformly dominated by the integrable function g for n large enough. Therefore, Φ Xn → Φ Y β in L 1 -norm by Lebesgue's dominated convergence theorem.
Denote by q β the pdf of Y β and by p n the pdf of X n for n ≥ 1 (those are well-defined since Φ Xn and Φ Y β are integrable). Writing p n and q β as the inverse Fourier transforms of Φ Xn and Φ Y β and using the triangle inequality, we obtain for all x ∈ R and n ≥ 1,
2π p n (x) − q β (x) = R Φ Xn (ξ)e −ixξ − Φ Y β (ξ)e −ixξ dξ ≤ R Φ Xn (ξ) − Φ Y β (ξ) dξ .
Since Φ Xn → Φ Y β in L 1 -norm, this implies the pointwise-in fact, uniform-convergence p n (x) → p(x) for all x ∈ R as n → ∞ (this fact is well-known, see for instance [77,Corollary 1.2.4]). By Scheffé's lemma [80,Section 5.10], this in turns implies the convergence p n → p in L 1 -norm.
Writing p n as the inverse Fourier transform of Φ Xn as above, we get 2π p n ∞ ≤ R |Φ Xn | ≤ R |g| for n large enough and similarly for q β . Finally, according to Lemma 4.4, for 0 < q < min{1, β, p}, there exists M q > 0 such that E[|X n | q ] ≤ M q for n large enough such that t n ≤ 1. We then conclude using [34,Theorem 1]
that lim n→∞ h(X n ) = h(Y β ).
Example. If L α is an SαS process, it is obviously LSS and Theorem 4.3 then implies that
h(L α t ) = 1 α log t + h(L α 1 ) + o t (1),
where o t (1) → 0 when t → 0 + . This specific case can be proved directly using the stability property as was done by Ghourchian et al. in [33,Proposition 1]. Theorem 4.3 thus significantly generalizes this result to all LSS processes.
Remark (Link with local limit theorems). The key step in the proof of Theorem 4.3 is to prove the convergence of the sequence (X n ) n≥1 to Y β not only in law but in the (stronger) sense that the sequence of probability density functions (p n ) n≥1 converges uniformly to the density p of Y β . This is closely related to the literature on local limit theorems, whose goal is to characterize the convergence of normalized sums of i.i.d. random variables in terms of the convergence of the pdfs, and includes notably uniform convergence [35], convergence in total variation [57], and convergence in relative entropy, which we further discuss in the next remark. See also [10] for a recent unifying perspective on local limit theorems.
Remark (A local entropic CLT). The study of entropic central limit theorems (CLT) has been initiated by Andrew Barron for the Gaussian case [6] and further generalized to non-Gaussian stable limits by Sergey Bobkov, Gennadiy Chistyakov, and Friedrich Götze [11]. For instance, Barron's contribution has been to show that for a sequence (X n ) n≥1 of finitevariance i.i.d. random variables with finite differential entropy, the normalized sum S n = n k=1 X k / √ n is such that h(S n ) → n→∞ h(G) with G ∼ N (0, σ 2 ) where σ 2 the common variance of the X n . Let L be a Lévy process and let X k = L k − L k−1 be the increments at integral times for k ≥ 1. Then, we have that L n = n k=1 X k is a sum of i.i.d. random variables and entropic CLTs readily apply to the study of the asymptotic behavior of L t for t → ∞. It is worth noting that sequences of i.i.d. random variables studied in the context of entropic CLTs (or "standard" CLTs [31, Chapter VIII]) are usually assumed to be in the basin of attraction of a stable law [6,11]. In the context of Lévy processes, this is equivalent to the fact that the Lévy process L is asymptotically self-similar, as defined in [27,Definition 4.3], or equivalently that the characteristic exponent Ψ of L satisfies Ψ(ξ) ∼ |ξ|→0 −γ|ξ| α for some 0 < α ≤ 2 and γ > 0.
In contrast, Theorem 4.3 characterizes the asymptotic behavior of the differential entropy of L t at small times t → 0 + and our work can thus be seen as the local counterpart of classic entropic central limit theorems. The assumption of local self-similarity, equivalent to Ψ(ξ) ∼ |ξ|→∞ −γ|ξ| α , thus plays the same role for characterizing the asymptotic behavior of L t for small t → 0 + as the self-similarity assumption for t → ∞. Further discussion about the local versus asymptotic properties of Lévy processes and their generalizations can be found in [23,Chapter 7].
An Upper Bound on the Small-Time Entropy of Lévy Processes
The specific case of LSS Lévy processes studied in the previous section excludes both Lévy processes with β > 0 which are not LSS and Lévy processes with β = 0. In this section, we consider a general Lévy process L and upper bound the decay of h(L t ) at small times. Specifically, the following theorem gives an upper bound on the limit superior of h(L t )/ log(1/t) when t → 0 + in terms of the Blumenthal-Getoor index β ∈ [0, 2]. This is achieved by "approximating" L with an LSS process and applying Theorem 4.3.
Theorem 4.5. Let L = (L t ) t≥0 be a Lévy process with Blumenthal-Getoor index β ∈ [0, 2] and such that L 1 has a finite absolute moment E[|L 1 | p ] < ∞ for some p > 0. Then,
lim sup t→0 + h(L t ) log(1/t) ≤ − 1 β ,(25)
with the convention 1/0 = ∞. In particular, for β = 0, we have that
lim t→0 + h(L t ) log(1/t) = −∞.(26)
Proof. The case β = 2 is covered by Proposition 3.10 (ii) so we henceforth assume that β ∈ [0, 2). For α ∈ (β, 2], let L α be a SαS process with parameter α, independent of L, and define L = L + L α . We denote by Ψ, Ψ, and Ψ α the characteristic exponents of L, L, and L α respectively. By definition, there exists γ > 0 such that Ψ α (ξ) = −γ|ξ| α . Moreover, since β < α and by definition of β, we have that |Ψ(ξ)|/|ξ| α → 0 when |ξ| → ∞. This implies that Ψ(ξ) ∼ −γ|ξ| α when |ξ| → ∞, hence L is LSS by Proposition 4.2.
For two independent absolutely continuous random variables X, Y such that (X + Y ) has finite differential entropy, we have that
3 h(X) = h(X + Y |Y ) ≤ h(X + Y ). Hence h(L t ) ≤ h(L t + L α t ) = h( L t ) = − 1 α log(1/t) + O t (1),(27)
where the last equality is by Theorem 4.3 applied to L (the assumptions of Theorem 4.3 are easily satisfied from the assumptions on L). Dividing by log(1/t) for t < 1 and taking the limit superior on both sides of (27) yields
lim sup t→0 + h(L t ) log(1/t) ≤ lim sup t→0 + − 1 α + o t (1) = − 1 α ,
which concludes the proof after letting α → β.
Examples and Applications
We apply our previous findings to specific subfamilies of Lévy processes. We consider nonstable Lévy processes with positive Blumenthal-Getoor indices in Section 5.1 and gamma processes, for which β = 0, in Section 5.2.
Layered and Tempered Stable Processes
Layered stable processes and tempered stable processes were respectively introduced by Christian Houdré and Reiichiro Kawai [39] and by Jan Rosinski [61]. In both cases, we characterize the asymptotic behavior of the entropy, thus demonstrating the applicability of Theorem 4.3 to Lévy processes that were not covered by previous results. For simplicity, we restrict ourselves to symmetric Lévy processes.
Layered Stable Processes
Definition 5.1. A symmetric Lévy process L is said to be a layered stable process with indices (α 0 , α ∞ ) ∈ (0, 2) × (0, ∞) if it has no Gaussian part (σ 2 = 0 in (5)) and if its Lévy measure ν can be written ν(dt) = q(|t|)dt for a continuous function q :
R + \ {0} → R + with q(t) ∼ t→0 a 0 t α 0 +1 and q(t) ∼ t→∞ a ∞ t α∞+1(28)
for come constants a 0 , a ∞ > 0.
Definition 5.1 is adapted from [39, Definition 2.1] to the ambient dimension d = 1. Stable processes correspond to the specific case where q(t) = c |t| α+1 for some constant c > 0 and α ∈ (0, 2). As was studied in details in [39], the index α 0 governs the shortterm behavior of the Lévy process, while α ∞ captures long-terms structures. One should therefore not be surprised to see in the next proposition that the entropy of a layered stable process is asymptotically determined by its index α 0 .
Proposition 5.2. Let L be a layered stable process with parameters (α 0 , α ∞ ) ∈ (0, 2) × (0, ∞). Then, L is LSS with Blumenthal-Getoor index β = α 0 and
h(L t ) = 1 α 0 log t + h + o t (1)(29)
for some constant h ∈ R, where o t (1) vanishes for t → 0 + .
Proof. From (5), we easily deduce that the characteristic exponent of a symmetric Lévy process without Gaussian part is given by
Ψ(ξ) = R (cos(tξ) − 1)ν(dt), ∀ξ ∈ R.(30)
We decompose Ψ = Ψ 0 + Ψ ∞ with Ψ 0 (ξ) = |t|≤1 (cos(tξ) − 1)ν(dt) and Ψ ∞ (ξ) = |t|>1 (cos(tξ) − 1)ν(dt).
Then, after the change of variable u = tξ for ξ = 0, we get
Ψ 0 (ξ) = − R (1 − cos u) q(|u|/ξ) |ξ| α 0 +1 ½ |u|≤|ξ| du |ξ| α 0 := − R g ξ (u)du |ξ| α 0 .
Then, due to (28) for t → 0 + , we have that g ξ (u) → (1 − cos u)/|u| α 0 +1 for all u ∈ R when |ξ| → ∞. Moreover, again according to (28) and using the continuity of q over R \ {0}, there exists M > 0 such that q(t) ≤ M/|t| α 0 +1 for all |t| ≤ 1. Hence, we deduce that
0 ≤ g ξ (u) ≤ ½ |u|≤|ξ| M (1 − cos u) |u| α 0 +1 ≤ M (1 − cos u) |u| α 0 +1 ,
the latter being integrable on R. Lebesgue's dominated convergence theorem then ensures that R g ξ (u)du → γ := R
(1−cos u) |u| α 0 +1 du when |ξ| → ∞, and finally,
Ψ 0 (ξ) ∼ |ξ|→∞ −γ|ξ| α 0 .(31)
Moreover, (28) and the continuity of q implies the existence of M > 0 such that q(|t|) ≤ M/|t| α∞+1 for all |t| > 1. This implies that
|Ψ ∞ (ξ)| = |t|>1 1 − cos(ξt) q(t)dt ≤ 2M |t|>1 dt |t| α∞+1 ,(32)
which is a finite constant independent of ξ ∈ R.
Combining (31) and (32), we deduce that Ψ(ξ) ∼ −γ|ξ| α 0 as |ξ| → ∞ which implies by Proposition 4.2 that L is LSS with Blumenthal-Getoor index β = α 0 .
Finally, E[|L 1 | p ] < ∞ for all p < α ∞ by [39, Proposition 2.1]. Hence L satisfies all the assumptions of Theorem 4.3 and we conclude that (29) holds.
Proper Tempered Stable Processes
Definition 5.3. A symmetric Lévy process is said to be a proper tempered stable process with parameters p > 0 and 0 < α < 2 if it has no Gaussian part and if its Lévy measure ν can be written as ν(dt) = q(|t| p ) dt |t| α+1 (33) for some completely monotone function 4 q : (0, ∞) → (0, ∞) such that
1 0 x 1−α q(x)dx < ∞, ∞ 1 x −1−α q(x)dx < ∞, q(x) −→ x→∞ 0, and q(x) −→ x→0 + c (34)
for some constant 0 < c < ∞.
For this definition, we follow the monograph [36], in particular Definitions 3.1 and 3.2 specialized to the ambient dimension d = 1. Stable processes correspond to the case where q is identically equal to a positive constant, although this is excluded by the condition that q vanishes at infinity. The effect of q is to temper the asymptotic behavior of ν, which impacts the heavy-tailedness of the Lévy process. We refer the interested reader to [36,Chapter 1] for practical motivations and for historical references on tempered stable processes.
Proposition 5.4. Let L be a proper tempered stable process with parameters p > 0 and 0 < α < 2. Then, L is LSS with Blumenthal-Getoor index β = α and
h(L t ) = 1 α log t + h + o t (1) (35) for some constant h ∈ R, where o t (1) vanishes for t → 0 + .
The proof of Proposition 5.4 is very similar to the one of Proposition 5.2 and is given in Appendix C.
Gamma Processes
In the case where β = 0, Theorem 4.5 guarantees that h(L t ) diverges to −∞ superlogarithmically. The decay rate can in fact be significantly faster as illustrated in this section by the gamma process, for which we show that it is asymptotically equivalent to − 1 τ t for some constant τ > 0. The gamma distribution with scale parameter θ > 0 and shape parameter τ > 0 has characteristic function given by ξ → 1/(1−iθξ) τ and is thus infinitely divisible. It defines a Lévy process L = (L t ) t≥0 , called a gamma process, whose characteristic exponent satisfies |Ψ(ξ)| = |τ log(1 − iθξ)| ∼ τ log |ξ| as |ξ| → ∞. Hence its Blumenthal-Getoor index is zero.
Proposition 5.5. Let L = (L t ) t≥0 be a gamma process with scale parameter θ > 0 and shape parameter τ > 0. Then we have as t → 0 + ,
h(L t ) = − 1 τ t − log t + log θ τ − γ + 1 + o t (1).(36)
Proof. For all t > 0, L t is gamma-distributed with scale parameter θ and shape parameter τ t, hence its differential entropy is given by (see e.g., [17,
h(L t ) = τ t + log θ + log Γ(τ t) + (1 − τ t)ψ(τ t),(37)
where Γ and ψ are the gamma and digamma functions respectively. Recall that for all x > 0, xΓ(x) = Γ(x + 1) and ψ(x) + 1/x = ψ(x + 1). Hence as t → 0 + ,
log Γ(τ t) = log Γ(1 + τ t) τ t = − log(τ t) + o t (1) and ψ(τ t) = ψ(τ t + 1) − 1 τ t = − 1 τ t − γ + o t (1) ,
where we used that Γ and ψ are continuous at 1 with Γ(1) = 1 and ψ(1) = −γ, for γ the Euler-Mascheroni constant. We thus obtain from (37) the asymptotic expansion (36).
6 Discussion and Conclusion
The Entropic Compressibility Hierarchy of Lévy processes
In this paper, we studied the entropy H n,m (L) of a Lévy process L, as introduced in [33]. We restricted our study to Lévy processes whose marginals are absolutely continuous with finite differential entropy. In this case, we used the fact that (see Corollary 3.7)
H n,m (L) n − log m = h(L 1/n ) + ǫ n,m with ǫ n,m → 0 when m → ∞ for fixed n ≥ 1 to focus our attention to the small-time behavior of the differential entropy h(L t ), which satisfies h(L t ) → −∞ when t → 0 + . The behavior of h(L t ) has an interpretation in terms of compressibility: the faster it diverges, the more compressible L is. Thus, Theorems 4.3 and 4.5 reveal an entropy-based compressibility hierarchy for Lévy processes with absolutely continuous marginals, determined by the Blumenthal-Getoor index, therefore generalizing [33,Thm. 4] in a substantial manner. This hierarchy is summarized in Figure 1.
Future Directions
We now discuss the assumptions made in the present manuscript and suggest open questions which may be investigated in future work.
Symmetry. We only considered locally symmetric and self-similar Lévy processes in Section 4. This choice was mostly made for convenience and significantly simplifies the exposition. For instance, symmetric stable processes are parameterized by two parameters, their characteristic exponent being of the form ξ → −γ|ξ| α . In contrast, the characterization of Wiener processes and SαS stable processes were already covered in [33].
all stable laws requires two additional parameters, including a skewness parameter, and is thus more involved (see [62,Definition 1.1.6]). It should be possible to generalize Theorem 4.3 to locally self-similar processes with possibly non-symmetric local limit, but this requires extending the currently used results from [27] to non-symmetric limits. Note however that the asymptotic entropy of any (possibly non-symmetric) stable random process was obtained in [33], and that Theorem 4.5 is valid for arbitrary Lévy processes.
Local Self-similarity. For β > 0, we only quantified the small-time entropy of Lévy processes that are locally self-similar. It is however possible to consider Lévy processes that are not, and thus whose characteristic exponent violates (20) We remark however that Theorem 4.5 applies to Lévy processes that are not self-similar, and implies in particular that a Lévy process with absolutely continuous marginals and Blumenthal-Getoor index β ∈ (0, 2) is at least as compressible as SαS processes with α ≥ β.
The case β = 0. We obtained an exact asymptotic expansion for the entropy of gamma processes in Propositions 5.5. Beyond this specific example, our current hierarchy does not distinguish between Lévy processes with β = 0 and simply states that the entropy h(L t ) diverges super-logarithmically when t → 0 + . A possible research direction would be to refine the compressibility hierarchy for Lévy processes with β = 0.
A Deferred Proofs from Section 3
Proof of Proposition 3.3. As already discussed above Proposition 3.3, for an absolutely continuous random variable X, the condition E[log(1 + |X|)] < ∞ implies that h(X) < ∞. This is a direct application of Gibbs' inequality applied to KL(X Y ) where Y has a Cauchy distribution, as detailed in [59,Proposition 1]. Furthermore, if the probability density function p X of X is bounded by some constant a, then h(X) > log 1/a. Indeed, in this
case −h(X) = R p X (x) log p X (x) dx ≤ R p X (x) log a dx = log a .
Finally, by Fourier inversion [31, XV.3 Theorem 3], a random variable X such that Φ X is integrable admits a bounded (and continuous) probability density function. This concludes the proof of the second claim of the proposition. We now adapt [59, Proposition 1] to the discrete case. For k ∈ Z, define p n = P(⌊X⌋ = k) and q k = c 1+k 2 where c = k∈Z (1+k 2 ) −1 is such that k∈Z q k = 1. By Gibbs' inequality,
0 ≤ H(⌊X⌋) = k∈Z p k log(1/p k ) ≤ k∈Z p k log(1/q k ) = k∈Z p k − log c + log(1 + k 2 ) ≤ − log c + k∈Z p k log (1 + |k|) 2 = − log c + 2E log(1 + |⌊X⌋|) .(38)
Finally, using |⌊X⌋| ≤ |X| + 1 ≤ 2|X| + 1, we upper bound the last term in (38)
E[log(1 + |⌊X⌋|)] ≤ E[log(2 + 2|X|)]) = log 2 + E[log(1 + |X|)],
which is finite by assumption. Therefore, H(⌊X⌋) < ∞, which concludes the proof.
Proof of Proposition 3.9. The key argument is to construct a symmetric infinitely divisible and absolutely continuous random variable X such that p X (x) ≥ c |x| log 2 |x| for some constant c > 0 and |x| large enough and to consider the Lévy process L whose law at time t = 1 is the one of X. We reproduce here for the sake of completeness the construction of p X proposed by Iosif Pinelis on Mathoverflow [56].
Let f (x) = α½ x≥e
x log 2 x where 0 < α < ∞ is such that f is a pdf with R f (x)dx = 1. For n ≥ 0, we denote f * n = f * · · · * f the nth fold convolution with the convention that f * 0 = δ. We define the functions, which are easily shown to be pdf,
f t (x) = e −t ∞ n=0 t n f * n (x) n! , g t (x) = e − x 2 2t √ 2πt ,
and p t = f t * g t . Then, we observe that, for s, t > 0, by expanding (s + t) n ,
f s+t = e −(s+t) ∞ n=0 (s + t) n f * n n! = e −s ∞ n=0 s n f * n n! * e −t ∞ n=0 t n f * n n! = f s * f t
We also have that g s+t = g s * g t and therefore p s+t = p s * p t . This shows that there exists an infinitely divisible random variable X whose pdf is p X = p 1 . Moreover, p 1 is absolutely continuous as the convolution of a Gaussian pdf. Then, we have for x ≥ 1 + e that
p X (x) = (f 1 * g 1 )(x) ≥ e −1 (f * g 1 )(x) ≥ e −1 1 0 f (x − y)g 1 (y)dy ≥ e −1 f (x) 1 0 g 1 (y)dy,
where we used that f 1 ≥ e −1 f in the first inequality and that f is decreasing on (e, ∞) for the last one. Finally, the choice of f implies that p X (x) ≥ c x log 2 x for some constant 0 < c < ∞ and every x ≥ e + 1, as desired.
Recall that
p k = P(⌊X⌋ = k) = k+1 k p X (x)dx. Moreover, for k ≥ 1, we have k+1 k dx x log 2 x = 1 log k − 1 log(k + 1) = log(1 + 1/k) log k log(k + 1) ∼ k→∞ 1 k log 2 k
Hence, using that p X (x) ≥ c/(x log 2 x) for x large enough, we deduce that, for k large enough,
p k ≥ c/2 k log 2 k .
Using that x → −x log x is increasing on (0, 1/e], we therefore deduce that, for k large enough (in particular such that p k ≤ 1/e),
−p k log(p k ) ≥ c/2 k log 2 k log k log 2 k c/2 ∼ k→∞ c/2 k log k .
This shows that H(⌊L 1 ⌋) = − k∈Z p k log(p k ) = ∞ since k≥2 1 k log k = ∞.
B Deferred Proofs from Section 4
Proof of Proposition 4.2. The characterization of self-similar Lévy processes is classic and proved for instance in [63,Proposition 13.5].
Assume that the characteristic exponent of L satisfies (20). Then, it is LSS according to [27,Proposition 5.8] (applied to d = 1). The local self-similarity order is then given by 1/β according to [27,Theorem 4.6]. For the converse, assume that L is a LSS Lévy process. First, this implies that β > 0 due to [27,Proposition 4.7]. Let Y be the limit in law of b H L ·/b when b → ∞, which is symmetric by assumption. Denoting by Moreover, Y ·/a is the limit in law of b H L ·/ab when b → ∞ and we therefore have that (ab) H L ·/ab = a H b H L ·/ab
(L) −→ b→∞ a H Y ·/a .
The unicity of the limit implies that, for all a > 0, a H Y ·/a and Y are equal in law, hence Y is self-similar of order H. Moreover, Y is a Lévy process: limiting arguments show that Y 0 = 0 a.s., and that its increments are stationary and independent. The first part of Proposition 4.2 therefore implies that the symmetric and self-similar Lévy process Y is SαS with α = 1/H. Its characteristic exponent is therefore given by Ψ Y (ξ) = −γ|ξ| 1/H for some γ > 0 and all ξ ∈ R. Let Ψ be the characteristic exponent of L. Then, the convergence in law of a H L ·/a to Y implies in particular the pointwise convergence of the characteristic functions of a H L 1/a to the one of Y 1 when a → ∞. This implies that
Setting a H = ξ in (39), we deduce that Ψ(ξ)∼ − γξ 1/H when ξ → ∞. We show similarly that Ψ(ξ)∼ − γ|ξ| 1/H when ξ → −∞. Finally, using again [27,Theorem 4.6], we know that 1/H = β is the Blumenthal-Getoor index of L and Proposition 4.2 is proved.
Proof of Lemma 4.4. Define r = min{1, p, β}. Since E[|L 1 | p ] < ∞ by assumption, we also have E[|L 1 | r ] < ∞. By [63,Thm. 25.3] applied to g(t) = max(|t| r , 1), this is equivalent to |t|≥1 |t| r ν( dt) < ∞. We can therefore apply [23,Prop. 2.4] with r ≤ 1 and deduce that |Ψ(ξ)| ≤ C 1 |ξ| r for some C 1 > 0 and any |ξ| < 1. Moreover, |Ψ(ξ)| ≤ C 2 |ξ| β for some C 2 > 0 and any |ξ| ≥ 1 by assumption. Then, for all ξ ∈ R, we have that |Ψ(ξ)| ≤ C |ξ| β + |ξ| r (40) with C = max(C 1 , C 2 ). For all q ∈ (0, 2) and random variable X such that E[|X| q ] < ∞, we have the classic 5 expression of E[|X| q ] in terms of the characteristic function Φ X (see e.g., [77,Thm 1.5.9]):
E[|X| q ] = c q R 1 − ℜΦ X (ξ) |ξ| q+1 dξ,(41)
where c q > 0 is a constant depending only on q. Moreover, if X is infinitely divisible with characteristic exponent Ψ, there exists a constant M > 0 such that for all ξ ∈ R,
1 − ℜΦ X (ξ) = ℜ(1 − Φ X (ξ)) ≤ |1 − Φ X (ξ)| ≤ M (1 − exp(−|Ψ(ξ)|),(42)
where the last inequality uses that the function f : z → |1 − e z |/(1 − e −|z| ) is bounded by some constant M > 0 for all z ∈ C with ℜz ≤ 0. Indeed, when |z| ≤ 1 we use the inequality |1 − e z | ≤ e |z| − 1 (easily proved using the power series expansion of e z and the triangle inequality) which implies f (z) ≤ e |z| −1 1−e −|z| = e |z| ≤ e. When |z| ≥ 1, we use |1 − e z | ≤ 1 + |e z | = 1 + e ℜz ≤ 2 for z with ℜz ≤ 0, which implies f (z) ≤ 2 1−e −|z| ≤ 2e e−1 . The bound (42) together with (41) implies
E[|X| q ] ≤ c q M R 1 − exp(−|Ψ(ξ)|) |ξ| q+1 dξ.(43)
For X = t −1/β L t whose characteristic exponent is ξ → tΨ(t −1/β ξ) we have by (40) tΨ(t −1/β ξ) ≤ C |ξ| β + t 1−r/β |ξ| r ≤ C |ξ| β + |ξ| r for all 0 < t ≤ 1, where the second inequality uses that t 1−r/β ≤ 1 since r ≤ β. Using that y → 1 − exp(−y) is increasing, we get from (43) that for all 0 < t ≤ 1,
E[|X| q ] ≤ c q M R 1 − e −C(|ξ| β +|ξ| r ) |ξ| q+1 dξ.(44)
Finally, for 0 < q < r, the integral in (44) is finite since the integrand is equivalent to 1/|ξ| q+1 when |ξ| → ∞, and to C|ξ| r−q−1 with r > q, when ξ → 0. Thus, E[|X| q ] is upperbounded by a constant independent of t that we denote by M q . This implies (22).
C Deferred Proof from Section 5
Proof of Proposition 5.4. The Blumenthal-Getoor index β of L is characterized in [36, Lemmas 3.22 and 3.23] and β = α. We study the asymptotic behavior of the characteristic exponent Ψ. Tempered stable processes being symmetric, Ψ is given by (30). As we did for layered stable processes, we use the decomposition Ψ = Ψ 0 + Ψ ∞ with Ψ 0 (ξ) = |t|≤1 (cos(tξ) − 1)ν(dt) and Ψ ∞ (ξ) = |t|>1 (cos(tξ) − 1)ν(dt).
We then observe that, by a simple change of variable and using (33),
Ψ 0 (ξ) = − R (1 − cos u) q(|u/ξ| p ) |u| α+1 ½ |u|≤|ξ| du |ξ| α := − R h ξ (u)du |ξ| α .(45)
Then, we have that h ξ (u) → c(1−cos u) |u| α+1
when |ξ| → ∞ for any u ∈ R where c > 0 is given by (34). Moreover, the function q being continuous, it is in particular bounded over [0, 1], let say by M > 0. Then, we have that
h ξ (u) ≤ q u ξ p ½ |u|≤|ξ| 1 − cos u |u| α+1 ≤ M 1 − cos u |u| α+1 ,
which is integrable over R. The Lebesgue dominated convergence theorem therefore applies and R h ξ (u)du converges to γ = R c(1−cos u) |u| α+1 du > 0. Hence, (45) implies that Ψ 0 (ξ) ∼ |ξ|→∞ −γ|ξ| α .
Moreover, the function q is continuous and vanishes at infinity. It is therefore bounded and we deduce that
|Ψ ∞ (ξ)| = |t|>1 1 − cos(ξt) q(|t| p ) |t| α+1 dt ≤ 2 q ∞ |t|>1 dt |t| α+1 ,
which is a constant independent from ξ.
We have therefore shown that Ψ(ξ) ∼ |ξ|→∞ −γ|ξ| α , hence L is LSS with of order H = 1/α by Proposition 4.2. Moreover, for r < α, we have that |t|>1 |t| r ν(dt) = |t|>1 q(|t| p ) |t| 1+(α−r) dt ≤ q ∞ |t|>1 dt |t| 1+(α−r) < ∞.
This implies that E[|L 1 | r ] < ∞ for some r > 0 and the conditions of Theorem 4.3 apply from which we obtain (35).
H
n,m (L) = H([∆ 1 ] m , . . . , [∆ n ] m )
Definition 3 . 4 .
34Let L be a Lévy process and n, m ≥ 1 be integers. First, let X n (L) = (L 1/n , L 2/n − L 1/n , . . . , L 1 − L (n−1)/n ) be the random vector of sampled values of L. Then, the entropy of L with time-quantization n and amplitude-quantization m is defined as
Theorem 4. 3 .
3Let (L t ) t≥0 be a LSS Lévy process with Blumenthal-Getoor index β ∈ (0, 2]
Figure 1 :
1Entropic compressibility of Lévy processes with absolutely continuous marginals. The expressions shown are upper-bounds for general Lévy processes and equal to the dominant term of the asymptotics of h(L t ) for locally symmetric and self-similar processes.
a H ) = log E[e ia H L 1/a ] −→ a→∞ log E[e iY 1 ] = Ψ Y (1) = −γ.
Table 7 .
72]):
by Proposition 4.2. Examples of such processes have been constructed by Walter Farkas, Niels Jacob, and René Schilling in [30, Examples 1.1.14 and 1.1.15]. It would be interesting to quantify the small-time evolution of h(L t ) for such Lévy processes.
Lévy processes had been previously introduced by Bruno de Finetti[32]. Paul Lévy generalized existing results to the class of additive processes, that is, processes satisfying Definition 2.1 with the exception of stationarity. We refer the interested reader to[50] for a detailed historical exposition of this matter.
Recall that the notation f (n) ∈ ω g(n) expresses that f (n) = g(n) · h(n) for some function h satisfying h(n) → ∞ as n → ∞.
The finiteness of h(X + Y ) implies that h(X) is well-defined in [−∞, ∞). This is therefore the case for X = Lt in the proof. Note that the theorem is trivially true when h(Lt) = −∞.
See[66] for a general reference on completely monotone functions.
This condition appears apparently for the first time in the study of fractional absolute moments of random variables by Pao-Lu Hsu in[40].
AcknowledgmentsThe authors are grateful to Shayan Aziznejad and Arash Amini for fruitful discussions in the early days of this project and to Iosif Pinelis for the construction of an infinitely divisible random variable with infinite differential entropy (cf. Proposition 3.9). Julien Fageot was supported by the Swiss National Science Foundation (SNSF) under Grant P2ELP2 181759. Alireza was partially supported by the MathWorks Engineering Fellowship.
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Probability with martingales. D Williams, Cambridge university press17D. Williams. Probability with martingales. Cambridge university press, 1991. 17
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arxiv |
TRANSITIVE CHARACTERISTICALLY SIMPLE SUBGROUPS OF FINITE QUASIPRIMITIVE PERMUTATION GROUPS
22 Jan 2019
Pedro H P Daldegan
Csaba Schneider
TRANSITIVE CHARACTERISTICALLY SIMPLE SUBGROUPS OF FINITE QUASIPRIMITIVE PERMUTATION GROUPS
22 Jan 2019If the socle of G, denoted by soc(G), is nonabelian, then H lies in soc(G). An explicit description is given for the possibilities of H under the condition that H does not contain a nontrivial normal subgroup of soc(G).
The first main result of this paper is that a finite nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian transitive characteristically simple subgroups H of finite quasiprimitive permutation groups G.
Introduction
The characterization of transitive subgroups of finite permutation groups is an important problem with several applications in algebraic combinatorics and algebraic graph theory. Without the ambition of being exhaustive, we recall some results in this research area. Finite primitive permutation groups with a transitive abelian subgroup were classified by Li [Li03], while finite permutation groups with a transitive cyclic subgroup were described by Li and Praeger [LP12]. Regular subgroups of finite primitive permutation groups were classified by Liebeck, Praeger and Saxl [LPS10] and by Baumeister [Bau07]. A general description of transitive subgroups of finite primitive permutation groups was given by Liebeck, Praeger and Saxl [LPS00].
In this paper we focus on transitive nonabelian characteristically simple subgroups of finite quasiprimitive permutation groups. Transitive simple subgroups of wreath products in product action were classified by the second author in collaboration with Baddeley and Praeger in [BPS04]. The examples of [BPS04] consisted of two isolated groups and of two infinite families and the scarcity of such examples gave the hope that the results of [BPS04] could be extended to finite characteristically simple groups. This goal is achieved in this paper. It is an easy consequence of the divisibility relations of Lemma 5.1 that if T is a transitive nonabelian simple subgroup of a wreath product in product action, then T is always contained in the base group of the wreath product. Our first theorem generalizes this result to transitive characteristically simple groups.
Theorem 1.1. Let Γ be a finite set such that |Γ| 2, let r 2, and let W = Sym(Γ) wr S r be considered as a permutation group on Ω = Γ r in product action. If H is a transitive nonabelian characteristically simple subgroup of W , then H is a subgroup of the base group; that is, H Sym(Γ) r .
Since finite quasiprimitive permutation groups are often contained in wreath products in product action, Theorem 1.1 leads to the following more general result.
Theorem 1.2. Let G be a finite quasiprimitive permutation group acting on Ω with nonabelian socle S and let H be a transitive nonabelian characteristically simple subgroup of G. Then H soc(G).
Once we know by Theorem 1.2 that H lies in the socle of G, a more detailed description of the possibilities of H can be given considering the possible O'Nan-Scott classes of G. For the description of the O'Nan-Scott classes of quasiprimitive permutation groups see Section 3, while for the terminology related to strips see Section 2. Given a natural number r, the symbol r denotes {1, . . . , r}. Theorem 1.3. Let G be a finite quasiprimitive permutation group acting on Ω and let S = Q 1 × · · · × Q r be the socle of G, where each Q i is isomorphic to a simple group Q. Assume that T is a nonabelian finite simple group, and let H ∼ = T 1 × · · · × T k ∼ = T k be a transitive subgroup of G with k 2. If H does not contain a minimal normal subgroup of S, then, after possibly relabeling the T i , one of the following holds.
(0) G has type HA, T ∼ = SL(3, 2), |Ω| = 8 k .
(1) G has type As, S = A n and G = A n or G = S n acting naturally on n points with n 10.
(2) G has type Sd or HS, k = r = 2 and T i < Q i , for i = 1, 2. Moreover, the groups Q and T are described in one of the rows of Table 2. Furthermore, if S α = {(q, qα) | q ∈ Q 1 } for some isomorphism α : Q 1 → Q 2 , then Q 1 = T 1 (T 2 α −1 ).
T Q 1 A 6 A 6 2 M 12 M 12 3
M 12 A 12 4 PΩ + 8 (q) PΩ + 8 (q) 5 PΩ + 8 (q) A n where n = | PΩ + 8 (q) : Ω 7 (q)| 6 PΩ + 8 (q) Sp 8 (2) 7 Sp 4 (2 a ), a 2 Sp 4b (2 a/b ) where b | a 8 Sp 4 (2 a ), a 2 A n where n = |Sp 4 (2 a ) : Sp 2 (2 2a ) · 2| Table 1. (3) G has type Pa and one of the following is valid.
(a) k = r and T i < Q i for all i ∈ r. (b) The T i are pairwise disjoint strips in S with | supp(T i )| ∈ {1, 2} for all i, with | supp(T i )| = 2 for at least one i, and the groups T and Q are as in one of the rows of Table 1. (c) S ∼ = (A n ) r , where |Ω| = n r and n 10.
(4) G has type Cd or HC, k = r and T i < Q i for all i; the groups Q and T are described in one of the rows of Table 2. Further, a point stabilizer S α is the direct product of pairwise disjoint strips D with | supp(D)| = 2. If
D = {(q, qα) | q ∈ Q i } is a strip in S α with some isomorphism α : Q i → Q j , then Q i = T i (T j α −1 ).
Note that we stated Theorem 1.2 under the condition that H contains no minimal normal subgroup of soc(G), which is the same condition as the one applied in [LPS00].
If we allow that H can contain a minimal normal subgroup of soc(G), then the simple components of H and soc(G) must be isomorphic and H is the direct product of full strips of soc(G). Such examples do arise and their classification can be achieved by considering the factorization soc(G) = H(soc(G) α ) where soc(G) α is a point stabilizer in soc(G). We also note that the group G in Theorem 1.3 is actually primitive, except perhaps in part (3). This shows that a finite imprimitive quasiprimitive group G very rarely contains a transitive characteristically simple subgroup that does not contain a minimal normal subgroup of soc(G).
In the proofs of Theorems 1.2 and 1.3 we could have relied more on the descriptions of transitive subgrops of primitive permutations groups presented in [LPS00]. However, since we are interested only in transitive characteristically simple subgroups, our arguments are simpler that those in [LPS00], and so we preferred a more direct approach.
Using Theorem 1.3, we obtain a characterization of regular characteristically simple subgroups of quasiprimitive permutation groups.
Corollary 1.4. Suppose that G and H are as in Theorem 1.3. If H is regular then the O'Nan-Scott type of G is either As or Pa.
If we allow that H may contain a minimal normal subgroup of soc(G), then we obtain several examples of regular characteristically simple subgroups of G with G having other O'Nan-Scott types. For example, if G has type Sd and H is a maximal normal subgroup of soc(G), then H is regular and characteristically simple.
Regular subgroups occur in the automorphisms groups of Cayley graphs. Quasiprimitive subgroups of automorphisms of Cayley graphs of finite simple groups were studied in [FPW02]. Our results allow us to state the following corollary concerning quasiprimitive subgroups of the automorphism groups of Cayley graphs of characteristically simple groups.
Corollary 1.5. Let H be a finite nonabelian characteristically simple group and let Γ be a noncomplete Cayley graph of H. Suppose that G is a quasiprimitive subgroup of Aut(Γ) with nonabelian socle such that H G. Then either H contains a minimal normal subgroup of soc(G) or the O'Nan-Scott type of H is Pa.
Theorems 1.1 and 1.2 are proved in Sections 5 and 6, respectively. The proofs of Theorem 1.3, and Corollaries 1.4 and 1.5 can be found in Section 7. Examples are described in Section 4 to show that all possibilities described in Theorem 1.3 arise.
Subgroups of characteristically simple groups and factorizations
If A and B are proper subgroups of a group G such that G = AB, then we call this expression a factorization of G. In a transitive permutation group G, a subgroup H G is transitive if and only if G = HG α where G α is the stabilizer of a point. Hence factorizations play a natural role in the study of transitive subgroups of permutation groups and in this section we collect some auxiliary results concerning subgroups and factorizations of finite characteristically simple groups.
Lemma 2.1. Let Q be a finite simple group, and suppose that Q = AB is a factorization of Q where A ∼ = T s 1 and B ∼ = T s 2 for some finite nonabelian simple group T and positive integers s 1 and s 2 . Then s 1 = s 2 = 1 and Q and T are as in Table 2.
Proof. For a group G, let p(G) denote the set of primes that divide |G|. We claim that p(Q) = p(A) = p(B). Clearly, p(A) ⊆ p(Q) and p(B) ⊆ p(Q). On the other hand, given a prime number p ∈ p(Q), since |Q| = |A||B|/|A ∩ B| and |A| and |B| are powers of |T |, the prime p divides |T |, and so p divides |A| and |B|. Thus p(Q) = p(A) = p(B), as claimed.
In the terminology of [BP98], the expression Q = AB is a full factorization of the simple group Q and such factorizations are described in [BP98, Theorem 1.1]. Keeping in mind that Sp 6 (2) ∼ = Ω 7 (2), inspection of [BP98, Table I] shows that the only options where |A| and |B| are powers of the same finite simple group occur when A ∼ = B ∼ = T and Q and T are as in Table 2.
Next we introduce some terminology to describe diagonal subgroups in wreath products.
Definition 2.2. Let Q 1 , . . . , Q r be groups and set S = Q 1 × · · · × Q r . Consider, for i ∈ r, the projections π i : S → Q i (q 1 , . . . , q r ) → q i , and assume that P S.
(i) P is a strip of S if P = 1 and, for each i ∈ r, either the restriction of π i to P is injective or P π i = 1. We define the support of P as
supp(P ) = {Q i | P π i = 1}.
(ii) A strip P of S is said to be nontrivial if | supp(P )| > 1.
(iii) A strip P of S is said to be a full strip if P π i = Q i for all Q i ∈ supp(P ).
(iv) Two strips P and Q are said to be disjoint if supp(P ) ∩ supp(Q) = ∅.
(v) P is a subdirect subgroup of S if P π i = Q i for each i ∈ r.
(vi) P is a diagonal subgroup of S if the restriction of π i to P is injective for each i ∈ r. (vii) P is a full diagonal subgroup of S if P is both a subdirect and a diagonal subgroup of S.
Note that the definitions above depend on the given direct decomposition of the group S. However, we will usually assume that S is the direct product of pairwise isomorphic nonabelian simple groups and, unless explicitly stated otherwise, the conditions in Definition 2.2 will be interpreted with respect to the unique finest direct decomposition of S. The concepts introduced in Definition 2.2 are studied in more depth in [PS18b, Section 4.4]. If P is a strip of S, as above, and, for some i ∈ r, the restriction π i | P is injective, then P π i ∼ = P .
The first part of the following lemma appeared in Scott's paper [Sco80, Lemma p. 328], and is known as Scott's Lemma (see also [PS18b,Section 4.6] for several generalizations). It describes the structure of the subdirect subgroups of a direct product of nonabelian simple groups. The second part can be found, for example, in [KL90, Proposition 5.2.5(i)]. Note that Lemma 2.3 does not assume that the groups should be finite.
Lemma 2.3. Consider S = Q 1 × · · · × Q r , where each Q i is a nonabelian simple group, and let P be a nontrivial subgroup of S.
(1) If P is a subdirect subgroup of S, then P is the direct product of full strips whose supports form a partition of {Q 1 , . . . , Q r }.
(2) If P is a normal subgroup of S, then P = j∈J Q j , where J ⊆ r.
The next result characterizes the factorizations S = HD of finite nonabelian characteristically simple groups S, in which H is a nonabelian characteristically simple subgroup and D is a full diagonal subgroup of S.
Lemma 2.4. Let Q and T be nonabelian finite simple groups, let S = Q 1 × · · · × Q r where r 2 and Q i ∼ = Q for all i, and let H ∼ = T k be a nonabelian characteristically simple subgroup of S. Consider, for i ∈ r, the projections π i : S → Q i and suppose that 1 < Hπ i < Q i for all i ∈ r. Assume, for i = 2, . . . , r, that α i : Q 1 → Q i is an isomorphism and define D = {(q, qα 2 , . . . , qα r ) | q ∈ Q 1 }.
If DH = S, then r = k = 2 and H = T 1 × T 2 where T 1 = H ∩ Q 1 , T 2 = H ∩ Q 2 and T 1 ∼ = T 2 ∼ = T . Further, in this case, Q 1 = T 1 (T 2 α −1 2 ) and the groups Q and T are as in one of the rows of Table 2.
Proof. By [PS18b,Lemma 8.16], r 3. Further, if r = 3, then the same lemma implies that Q 1 = Hπ 1 (Hπ 2 α −1 2 ∩ Hπ 3 α −1 3 ) = Hπ 2 α −1 2 (Hπ 1 ∩ Hπ 3 α −1 3 ) = Hπ 3 α −1 3 (Hπ 1 ∩ Hπ 2 α −1 2 ). Hence, in the terminology of [BP98], the set {Hπ 1 , Hπ 2 α −1 2 , Hπ 3 α −1 3 } is a strong multiple factorization of Q 1 . Noting that Hπ i ∼ = T s i with s i 1, and considering the classification of strong multiple factorizations of finite simple groups in [BP98, Table V], we obtain that r = 2. Furthermore, Q 1 = (Hπ 1 )(Hπ 2 α −1 2 ) where Hπ 1 ∼ = T s 1 and Hπ 2 α −1 2 ∼ = Hπ 2 ∼ = T s 2 with some s 1 , s 2 1. Now Lemma 2.1 implies that Hπ 1 ∼ = Hπ 2 ∼ = T and that the groups Q and T are as in Table 2. Finally, note that H ∩ Q i Hπ i holds for i = 1, 2.
Hence either H ∩ Q i = Hπ i or H ∩ Q i = 1. Thus, if H = (H ∩ Q 1 ) × (H ∩ Q 2 )
, then H ∼ = Hπ 1 ∼ = Hπ 2 ∼ = T , and hence the factorization S = DH is impossible, since Lemma 2.5. Let T be a nonabelian finite simple group and let X and Y be direct products of pairwise disjoint nontrivial full strips in T r with r 2. Then XY = T r .
|D||H| = |Q||T | < |Q| 2 = |S|. Thus H = (H ∩ Q 1 ) × (H ∩ Q 2 ) must hold. Setting T 1 = H ∩ Q 1 and T 2 = H ∩ Q 2 ,
The following technical lemma, which characterizes certain factorizations related to the Mathieu group M 12 , will be used in the last section.
Lemma 2.6. Let S = Q 4 and A 1 , A 2 , A 3 , A 4 Q such that Q = M 12 and each A i ∼ = M 11 . Let X = {(p, p, q, q) | p, q ∈ Q} and Y = {(a 1 , a 2 , a 2 ψ, a 4 ) | a i ∈ A i } be subgroups of S, where ψ : A 2 → A 3 is an isomorphism. Then S = XY .
Proof. We have that X = D 1 × D 2 ∼ = Q 2 is the direct product of two full strips of S and Y ∼ = (M 11 ) 3 is the direct product of three strips of S, where the second strip is a diagonal subgroup of Q 2 .
Assume that Q 4 = XY and consider the projections π 12 : S → Q 2 and π 34 : S → Q 2 , where π 12 projects onto the first two coordinates and π 34 projects onto the last two coordinates. Applying these projections to Q 4 = XY , we obtain that
Q 2 =D 1 (A 1 × A 2 ), (1) Q 2 =D 2 (A 3 × A 4 ). (2) It follows from Lemma 2.4 that Q = A 1 A 2 = A 3 A 4 . Thus, |A 1 ∩ A 2 | = |A 3 ∩ A 4 | = |M 11 | 2 /|M 12 | = 660. Set C 1 = A 1 ∩ A 2 and C 2 = (A 3 ∩ A 4 )ψ −1 . Note that X ∩ Y = {(c, c, cψ, cψ) | c ∈ C 1 ∩ C 2 } ∼ = C 1 ∩ C 2 . Since Q 4 = XY , |X ∩ Y | = |X||Y | |Q| 4 = |M 11 | 3 |M 12 | 2 = 55. As X ∩ Y ∼ = C 1 ∩ C 2 A 2 and |C 1 C 2 | = |C 1 ||C 2 | |X ∩ Y | = 660 2 55 = 7920 = |A 2 |, we find that A 2 = C 1 C 2 .
Since M 11 has a unique conjugacy class of subgroups of order 660 [CCN + 85, p.18], C 1 and C 2 are conjugate in A 2 , which contradicts to A 2 = C 1 C 2 . Then Q 4 = XY and the result is proved.
Quasiprimitive permutation groups
A permutation group is quasiprimitive if all its nontrivial normal subgroups are transitive. For example, primitive permutation groups are quasiprimitive. Since a transitive simple group is always quasiprimitive, but not necessarily primitive, the class of quasiprimitive permutation groups is strictly larger than the class of primitive permutation groups.
Finite primitive and quasiprimitive groups were classified by the respective versions of the O'Nan-Scott Theorem; see [PS18b,Chapter 7]. In this classification, we distinguish between 8 classes of finite primitive groups, namely HA, HS, HC, SD, CD, PA, AS, TW, and 8 classes of finite quasiprimitive groups, namely HA, HS, HC, Sd, Cd, Pa, As, Tw.
Here we only give a brief summary of these classes; the interested reader can find a more detailed treatment in [PS18b,Chapter 7]. The type of a primitive or quasiprimitive group G can be recognized from the structure and the permutation action of its socle, denoted soc(G). Let G Sym(Ω) be a quasiprimitive permutation group, let M be a minimal normal subgroup of G, and let ω ∈ Ω. Note that M is a characteristically simple group. If M is nonabelian, then by a subdirect subgroup of M we mean one that is subdirect with respect to the unique finest direct decomposition of M (see Definition 2.2). For a group G, the holomorph Hol G is the semidirect product G ⋊ Aut G viewed as a permutation group acting on the set G; see [PS18b, Section 3.3] for more details. The main characteristics of G and M in each primitive and quasiprimitive type are as follows. Sd: M is nonabelian and nonsimple; M ω is a simple subdirect subgroup of M and C G (M) = 1. If, in addition, G is primitive, then the type of G is SD.
Cd: M is nonabelian and nonsimple; M ω is a nonsimple subdirect subgroup of M and C G (M) = 1. If, in addition, G is primitive, then the type of G is CD.
Pa: M is nonabelian and nonsimple; M ω is not a subdirect subgroup of M and M ω = 1; C G (M) = 1. If, in addition, G is primitive, then the type of G is PA.
As: M is nonabelian and simple; C G (M) = 1. If, in addition, G is primitive, then the type of G is AS.
Tw: M is nonabelian and nonsimple; M ω = 1; C G (M) = 1. If, in addition, G is primitive, then the type of G is TW.
Note that if G is a primitive permutation group of type HS or HC, then there exists a primitive group G of type SD or CD, respectively, that contains G as a subgroup of index 2 (see [PS18b,Corollary 3.11]). Hence in the proof of the inclusion H G in Theorem 1.2, the cases when G has type HS or HC can be reduced to the cases when G has type SD or CD, respectively.
Examples of transitive characteristically simple subgroups
In this section we describe some examples to show that all the possibilities described in Theorem 1.3 arise. In Examples 4.1-4.5, T is a nonabelian finite simple group.
Example 4.1 (G has type HA). Suppose that R = SL(3, 2). Then R acts irreducibly on V = F 3 2 and set G = V ⋊ R. It is well-known (see for example [Pra90, Proposition 5.2]) that G has a transitive simple subgroup T isomorphic to R. Letting W = G wr S r with r 2, the direct product T r is a transitive characteristically simple subgroup in the primitive permutation group W of type HA acting on Ω = F 3r 2 . Hence T r is contained in any primitive subgroup of F 3r 2 ⋊ GL(2, 3r) that contains W . These inclusions are as in Theorem 1.3(0). Example 4.3 (G has type Sd or HS). Suppose that T and Q are as in one of the rows of Table 2. Set D to be the diagonal subgroup
D = {(q, q) | q ∈ Q}
of Q 2 and let Ω denote the right coset space [Q 2 : D]. The normalizer N Sym(Ω) (Q 2 ) is a primitive group of type SD whose abstract group structure is (Q 2 · Out(Q)) ⋊ C 2 . Suppose that G N Sym(Ω) (Q 2 ) such that G contains Q 2 . Then G is a primitive permutation group of type SD or HS depending on the projection of G on C 2 . Assume that T 1 and T 2 are subgroups of Q such that T 1 ∼ = T 2 ∼ = T and T 1 T 2 = Q and H = T 1 × T 2 . Viewing H as a subgroup of Q 2 , easy calculation shows that DH = Q 2 and so H is a transitive characteristically simple subgroup of G. These examples are as in Theorem 1.3(2).
Example 4.4 (G has type Pa). Examples for Theorem 1.3(3) can be constructed by wreathing smaller examples. Suppose that T is a simple transitive subgroup of a nonabelian simple group Q acting on a set Γ. Then, for r 2, the wreath product W = Q wr S r contains the transitive characteristically simple subgroup T r . Furthermore, the inclusion T r W is as in Theorem 1.3(3/a).
Suppose that H and Q are as in Example 4.2. Then, for r 2, H r is contained in Q wr S r and this inclusion is as in Theorem 1.3(3/c).
Finally, let Q and T be as in one of the rows of Table 1. Then T can be embedded as a transitive subgroup of G = Q wr S 2 , and so, for r 2, T r is a transitive characteristically simple subgroup of W = G wr S k . The inclusion T r W is as in Theorem 1.3(3/b).
Example 4.5 (G has type Cd or HC). Suppose that G and H are as in Example 4.3. If r 2, then the wreath product W = G wr S r is a primitive group of type CD or HC (depending on the type of G) that contains the transitive characteristically simple subgroup H r = T 2r . The inclusion H W is as in Theorem 1.3(4).
The proof of Theorem 1.1
This section contains the proof of Theorem 1.1. We start by stating a number theoretic result which is a corollary of Legendre's Formula for the largest prime-power that divides n!.
Lemma 5.1. Given natural numbers p, n 2, the following are valid.
(1) p n ∤ n!.
(2) If p n−1 | n!, then p = 2 and n is a power of 2.
(3) 4 n−1 ∤ n!.
Next we review some concepts related to subgroups of wreath products in product action. Let Γ be a finite set such that |Γ| 2, let r 2, and let W = Sym(Γ) wr S r be considered as a permutation group on Ω = Γ r in product action. An element of W is written as (a 1 , . . . , a r )b where a i ∈ Sym(Γ) for all i ∈ r, and b ∈ S r . Suppose that X is a subgroup of W . For j ∈ r, we define the j-th component X (j) of X as follows. Suppose that W j is the stabilizer in W of j under the permutation representation π : W → S r . Then
(3) W j = Sym(Γ) × (Sym(Γ) wr S r−1 ),
where the first factor of the direct product acts on the j-th coordinate, while the second factor acts on the other coordinates. In particular, 'S r−1 ' is taken to be the stabilizer of j in S r . We define X (j) as the projection of X j = X ∩ W j onto the first factor of W j . We view X (j) as a subgroup of Sym(Γ We turn to the proof of Theorem 1.1.
Proof of Theorem 1.1. Suppose that H = T 1 × · · · × T k = T k for some nonabelian finite simple group T . Suppose, as above, that π : W → S r is the natural projection. Let B be the base group Sym(Γ) r of W . Then B = ker π. Assume for contradiction that H B; that is Hπ = 1. First we assume that Hπ is transitive on r. The case when Hπ is intransitive will be treated afterwards. Set Proof of Claim. Suppose, for j ∈ r, that σ j : Sym(Γ) r → Sym(Γ) denotes the j-th coordinate projection. Then H B . Let j ∈ r. Since H B is transitive on r, there is some element g = (g 1 , . . . , g r )h of H B such that 1(gπ) = 1h = j. Then m g = (1, m 2 , . . . , m r ) g = (1, m g 2 2 , . . . , m gr r ) h , and so the j-th coordinate of m g is 1. Hence m g ∈ ker σ j ∩ H B , and then (ker σ 1 ∩ H B ) g ker σ j ∩ H B . Analogously, the same argument above shows that ker σ j ∩ H B (ker σ 1 ∩ H B ) g , and so ker σ j ∩ H B = (ker σ 1 ∩ H B ) g . On the other hand, ker σ 1 ∩ H B is a subgroup of H B and H B centralizes H B , and so ker σ 1 ∩ H B = ker σ j ∩ H B for all j. Thus ker σ 1 ∩ H B acts trivially on Ω, and so ker σ 1 ∩ H B = 1, which gives ker σ j ∩ H B = 1 for all j. Therefore, H Thus the restrictions to H B of the projection maps σ j are monomorphisms. Then
β j = σ −1 1 σ j : H (1) B → H (j)
B is an isomorphism for all j. As a consequence, every element m ∈ H B can be expressed uniquely as m = (y, yβ 2 , . . . , yβ r ), for some y ∈ H Claim. For all j ∈ r, there is some element x j ∈ Sym(Γ) such that yβ j = y x j for all y ∈ H
(1) B .
Proof of Claim. Suppose that y ∈ H
(1) B . Then m = (y, yβ 2 , . . . , yβ r ) ∈ H B . Let j ∈ r and, using the transitivity of H B on r, suppose that g = (g 1 , . . . , g r )h ∈ H B is such that 1(gπ) = 1h = j. Then g centralizes m and hence (y, yβ 2 , . . . , yβ r ) = m g = (y g 1 , (yβ 2 ) g 2 , . . . , (yβ r ) gr ) h .
Comparing the j-th coordinates in the two sides of the last equation, we find that yβ j = y g 1 . Taking x j = g 1 , thus we have yβ j = y x j .
Claim. If Σ is a H B -orbit in Ω, then |Σ| = |Γ|.
Proof of Claim. Since H is transitive on Ω and H B
H, all the H B -orbits have the same size. Hence it suffices to show the claim for just one H B -orbit. Choose the elements 1, x 2 , . . . , x r as in the previous claim, let γ ∈ Γ and consider the element ω = (γ, γx 2 , . . . , γx r ). Suppose that m ∈ H B . By the previous claim, m has the form m = (y, y x 2 , . . . , y xr ) for some y ∈ H
|ω H B | = |H B | |(H B ) ω | = |H (1) B | |(H (1) B ) γ | = |Γ|,
as desired.
Claim. The case when Hπ is transitive is impossible.
Proof of Claim. H B is a normal subgroup of H and every H B -orbit has size |Γ|. Hence the number of H B -orbits is |Γ| r−1 . Since H is transitive on Ω, H B is transitive on the set of H B -orbits and hence |Γ| r−1 | |H B |. Since H B has a faithful action on r, this leads to |Γ| r−1 | r!. Now, since Γ is an orbit for the characteristically simple group H
(1) B , we find that |Γ| 5. Hence |Γ| is divisible by p, where p is either an odd prime or p = 4, which is a contradiction by Lemma 5.1.
This completes the proof for the case when Hπ is a transitive subgroup of S r . Let us now turn to the case when Hπ is intransitive. Recall that B is the base group of W . Assuming that H B implies that there exists an Hπ-orbit ∆ in r with size at least 2. Set ∆ = r \ ∆ and r 1 = |∆|. Then H can be embedded into the direct product W 1 × W 2 = (Sym(Γ) wr S r 1 ) × (Sym(Γ) wr S r−r 1 ) such that the projection H 1 of H into W 1 acts transitively on r 1 . Now, since H is transitive on Γ r , H 1 is also transitive on Γ r 1 . Further, as H is characteristically simple, so is H 1 . Hence using the theorem in the case when Hπ is transitive gives a contradiction. Therefore, H B.
Since in several classes of quasiprimitive permutation groups, the individual groups are subgroups in wreath products in product action, Theorem 1.1 leads to the following corollary.
Corollary 5.3. Let G Sym(Ω) be a finite quasiprimitive permutation group with nonabelian socle S and let H be a transitive nonabelian characteristically simple subgroup of G. Let α ∈ Ω and assume that S = Q 1 × · · · × Q r , where r 2 and the Q i are pairwise G-conjugate normal subgroups of S such that (4) S α = (Q 1 ∩ S α ) × · · · × (Q r ∩ S α ).
Then H N G (Q i ) for all i ∈ r. Further, if the Q i are simple groups, then H S.
Proof. Set Γ to be the right coset space [Q 1 : Q 1 ∩ S α ]. By [PS18b,Theorem 4.24], we may assume without loss of generality that G is a subgroup of W = Sym(Γ) wr S r acting in product action on Γ r , and so H can also be viewed as a transitive subgroup of W . By Theorem 1.1, H is a subgroup of the base group Sym(Γ) r of W . Now [PS18b, Theorem 4.24] also implies that the conjugation action of G on the set {Q 1 , . . . , Q r } is equivalent to its action on r induced by the natural projection π : W → S r . Since ker π = Sym(Γ) r , we have that H acts trivially on {Q 1 , . . . , Q r } by conjugation, and so H N G (Q i ) holds for all i.
Suppose now that the Q i are simple. Since C G (soc(G)) = 1 and since H N G (Q i ) for all i, the group H can be viewed as a subgroup of
Aut S ∩ r i=1 N Aut S (Q i ) = i (Aut Q i ). Therefore H/(H ∩ S) ∼ = (HS)/S i (Aut Q i )/Q i .
Since Schreier's Conjecture holds, the group on the right-hand side of the last display is soluble. On the other hand, if H ∩S = H, then H/(H ∩S) is nonabelian characteristically simple, which is impossible. Hence H ∩ S = H, which is equivalent to H S.
The proof of Theorem 1.2
In this section we prove Theorem 1.2. Suppose that G Sym(Ω) is a finite quasiprimitive permutation group with nonabelian socle S and let H be a transitive nonabelian characteristically simple subgroup of G. We are required to show that G S. Our strategy is to analyze each of the possible O'Nan-Scott classes of G.
G has type AS: S is a simple group and S G Aut(S). Then
H/(H ∩ S) ∼ = (HS)/S Aut(S)/S = Out(S).
As S is simple, it follows from Schreier's Conjecture that Out(S) is soluble. Since H is nonabelian and characteristically simple, H ∩ S = H. Therefore H S = soc(G), as desired.
G has type HS: S = Q 1 × Q 2 where Q 1 and Q 2 are simple minimal normal subgroups of G. Furthermore, S G Hol(Q 1 ) and H/(H ∩ S) ∼ = (HS)/S G/S Hol(Q 1 )/S ∼ = Out(Q 1 ).
Therefore, arguing as we did for the type AS, it follows from Schreier's Conjecture that H ∩ S = H, and so H S.
G has type TW: S is regular, and so S clearly satisfies equation (4) with respect to its finest direct decomposition into the direct product of simple groups. Thus, by Corollary 5.3, H soc(G).
For the next four O'Nan-Scott types, namely for HC, PA, SD, and Cd, we will assume that S = Q 1 × · · · × Q r where r 2 and the Q i are pairwise isomorphic nonabelian simple groups. We let Q denote the common isomorphism type of the Q i . We define π i as the coordinate projection π i : S → Q i .
G has type HC: S = S 1 × S 2 where S 1 and S 2 are nonsimple, regular minimal normal subgroups of G. Then r is even and, after possibly reordering the Q i , a point stabilizer in S is a direct product D 1 × · · · × D r/2 where each D i is a full diagonal subgroup in Q i × Q r/2+i . Since r 4, Corollary 5.3 applies to the direct decomposition S = (Q 1 × Q r/2+1 ) × · · · × (Q r/2 × Q r ) and this gives that H normalizes Q i × Q r/2+i for all i ∈ r/2. Since C G (S) = 1, the conjugation action of H on S embeds H into X = Aut(Q 1 × Q r/2+1 ) × · · · × Aut(Q r/2 × Q r ). Now X/S is a subgroup of ((Out Q) wr C 2 ) r , which is a soluble group by Schreier's Conjecture. Now the usual argument implies that H soc(G).
G has type PA: for all α ∈ Ω, S α is not a subdirect subgroup of S and S is not regular. In general, the groups of this class do not satisfy equation (4) in Corollary 5.3, but it is well-known, for α ∈ Ω, that G has a faithful quotient action on the right coset space Ω = [S : P ] where P = (S α π 1 ) × · · · × (S α π r ). Since H also acts transitively on Ω and P is a point stabilizer in S under this action, equation (4) holds for P with the finest direct product decomposition of S, and so H S must hold.
G has type SD: this is the most difficult O'Nan-Scott type to deal with. In this case, for α ∈ Ω, S α is a simple subdirect subgroup of S. Since S is transitive on Ω and S α ∼ = Q, |Ω| = |Q| r−1 . In this case, G can be considered as a subgroup of G = (S · Out(Q)) ⋊ S r , where S r permutes by conjugation the factors of S and Out(Q) acts on S ∼ = Q r diagonally; see [PS18b,Section 7.4].
Consider the extension S = S · Out(Q). We have that G permutes the elements in Σ = {Q 1 , . . . , Q r } and the kernel of this action is precisely S. If we denote by H 0 the kernel of H acting on Σ, we obtain that H 0 = H ∩ S. Since H is characteristically simple, we have by Lemma 2.3(2) that H 0 ∼ = T k 0 for some integer k 0 , and there exists a normal subgroup H 1 of H such that H = H 0 × H 1 . It follows from the Isomorphism Theorem and from the definition of S that
H 0 /(H 0 ∩ S) ∼ = (H 0 S)/S S/S ∼ = Out(Q).
Since Out(Q) is soluble by Schreier's Conjecture, and H 0 is nonabelian and characteristically simple, we conclude that H 0 = H 0 ∩ S, which means that H 0 S.
If H 1 = 1, then, since H 0 S, we obtain at once that H S. Hence it suffices to prove that H 1 = 1. Suppose that H 1 = 1. Since H 1 ∩ S = 1, H 1 permutes the elements in Σ faithfully, and so |H 1 | | r!. In particular, since the size of the smallest nonabelian simple group is 60, we have r 5.
We claim that H 0 = 1. In fact, if that is not the case, then H = H 1 , so H 1 is transitive on Ω. Then applying the Orbit-Stabilizer Theorem and the transitivity of S, we obtain
|H 1 | |(H 1 ) α | = |Ω| = |Q| r−1 , so |Q| r−1 | |H 1 |. Since |H 1 | | r!, we obtain that |Q| r−1 | r!.
Since the order of every finite nonabelian simple group is divisible by four, this implies that 4 r−1 | r!, which is not possible by Lemma 5.1. Therefore, H 0 = 1. Let us analyze the action of H 0 on Ω. Since H 0 H and H is transitive, the orbits of H 0 form a block system for H. In particular, the H 0 -orbits have the same size. By the Orbit-Stabilizer Theorem, it follows that |(H 0 ) α | is independent of α. Therefore, the number of H 0 -orbits on Ω is equal to
(5) |Q| r−1 |α H 0 | = |Q| r−1 |T | k 0 |(H 0 ) α |.
Since H = H 0 × H 1 and H is transitive on Ω, we have that H 1 is transitive on the set of H 0 -orbits. Then the Orbit-Stabilizer Theorem gives that the number of H 0 -orbits divides |H 1 |. Therefore, the number of H 0 -orbits divides r!. From the Isomorphism Theorem we have that
(6) Q i H 0 π i ∼ = H 0 /(ker π i ∩ H 0 ) ∼ = T s i , where s i 0 for all i ∈ r.
We claim that s i 1 for all i. Suppose, on the contrary, that there exists i ∈ r such that s i 2. In particular, Q contains a subgroup isomorphic to T 2 . As every finite simple group has a cyclic Sylow subgroup (see [KLST90,Theorem 4.9]), we can choose a prime p such that the Sylow p-subgroups of Q are cyclic. Since a Sylow p-subgroup of T 2 is contained in a Sylow p-subgroup of Q, we have that p ∤ |T |. In particular, p = 2. Considering the number of H 0 -orbits in (5), we obtain that p r−1 divides the number of H 0 -orbits, and so p r−1 divides r!. However, this contradicts Lemma 5.1. Therefore, s i 1 for all i, as claimed.
Since H 0 = 1, s i = 1 for some i ∈ r. Thus (6) gives that Q has a subgroup isomorphic to T . Since each T i H 0 is simple, each T i H 0 is a strip of S. We assert that if i = j, then supp(T i ) ∩ supp(T j ) = ∅. In fact, if there is m ∈ r such that T i π m ∼ = T and T j π m ∼ = T , then (T i × T j )π m ∼ = T 2 . However, this is impossible, since s m 1. Therefore
(7) H 0 = T 1 × · · · × T k 0 , where each T i ∼ = T is a diagonal subgroup of Q j ∈ supp(T i ) Q j ,
in such a way that supp(T i ) ∩ supp(T j ) = ∅ for all i = j. As a consequence, we obtain that k 0 r. Assume first that k 0 < r; we will treat the case k 0 = r separately. Since |T | | |Q|, we have by (5) that |Q| r−k 0 −1 divides the number of H 0 -orbits in Ω. As H 1 is transitive on the set of H 0 -orbits, this implies that |Q| r−k 0 −1 divides |H 1 |.
Let d i = | supp(T i )|. Moreover, let m 1 be the number of factors T i for which d i 5, and let m 2 be the number of factors T i such that d i < 5. So m 1 + m 2 = k 0 . Relabeling if necessary, we can write H 0 = T 1 × · · · × T m 1 × T m 1 +1 × · · · × T m 1 +m 2 , such that d i 5 if and only if i m 1 . Set m 3 = r − k 0 i=1 d i ; that is, m 3 is the number of factors Q i such that H 0 π i = 1.
Since H 1 centralizes H 0 , we have that H 1 centralizes each T i . for each i k 0 . As, for h 1 ∈ H 1 and i ∈ k 0 , we have
(supp(T i )) h 1 = supp(T i h 1 ) = supp(T i ),
we conclude that each supp(T i ) is H 1 -invariant. In particular, H 1 acts by conjugation on supp(T i ) for all i ∈ k 0 , and, since H 1 is a nonabelian characteristically simple group, this action is trivial whenever | supp(T i )| < 5. Hence H 1 acts trivially on supp(T m 1 +1 )∪ · · ·∪ supp(T k 0 ).
Since the action of H 1 on Σ is faithful, we obtain that
(8) |H 1 | | (d 1 !) · · · (d m 1 !)(m 3 !).
As we assumed H 1 = 1, we have from (8) that either m 1 = 0 and m 3 5, or m 1 = 0. In both cases we conclude that
(9) d 1 + · · · + d m 1 + m 3 − m 1 − 1 > 0.
Recall that |Q| r−k 0 −1 divides |H 1 |. So (8) implies that
(10) |Q| r−k 0 −1 | (d 1 !) · · · (d m 1 !)(m 3 !).
On the other hand, we have that r d 1 + d 2 + · · · + d m 1 + m 2 + m 3 . So |Q| d 1 +d 2 +···+dm 1 +m 2 +m 3 | |Q| r .
Since (9) is valid and k 0 = m 1 + m 2 , we obtain
|Q| (d 1 −1)+···+(dm 1 −1)+m 3 −1 = |Q| d 1 +d 2 +···+dm 1 +m 2 +m 3 −m 1 −m 2 −1 | |Q| r−k 0 −1 .
Therefore, using equation (10), we obtain that
|Q| (d 1 −1)+···+(dm 1 −1)+m 3 −1 | (d 1 !) · · · (d m 1 !)(m 3 !).
Since 4 | |Q|, the previous line gives that
2 d 1 · · · 2 dm 1 2 m 3 | (d 1 !) · · · (d m 1 !)(m 3 !),
which is a contradiction by Lemma 5.1. This implies that H 1 = 1, which means that if k 0 < r, then H S. Now consider the case where k 0 = r. Then (7) implies that d i = 1 for all i ∈ r. Since each supp(T i ) is H 1 -invariant, we have that H 1 acts faithfully and trivially on Σ, thus H 1 = 1. Therefore if k 0 = r, we also obtain H = H 0 S. Therefore, if G has type Sd, then H S.
G has type CD: for α ∈ Ω, S α is a subdirect subgroup of S, but it is not simple. So Lemma 2.3 implies that there exist sets Σ = {S 1 , . . . , S q } and {D 1 , . . . , D q }, where q 2, each D i is a full diagonal subgroup of S i and S i = Q j ∈supp(D i ) Q j , such that S = S 1 ×· · ·×S q , G α acts transitively by conjugation on Σ and, considering the projections π i : S → S i , we have that
(11) S α = S α π 1 × · · · × S α π q ,
where each S α π i = D i . By [PS18b,Theorem 11.13], G is permutationally isomorphic to a subgroup of a wreath product of the form W = G 0 wr S q acting on Γ q in product action where G 0 is a quasiprimitive permutation group on Γ of type Sd. Further, setting S = S 1 , soc(G 0 ) = S and soc(G) = soc(W ) = S q = S 1 × · · · × S q . By Corollary 5.3, H N G (S i ) for all i, and hence H lies in the base group (G 0 ) q of W . For i ∈ q, let σ i : (G 0 ) q → G 0 denote the i-th projection map. Then Hσ i is a transitive characteristically simple subgroup of G 0 acting on Γ. Since G 0 is quasiprimitive of Sd type, we find that Hσ i soc(G 0 ) = S. Since this is true for all i, we obtain that H S q = soc(G).
This concludes the proof of Theorem 1.2. We note that the cases when G has type HS or HC could have been reduced to the types Sd and Cd, respectively, as explained at the end of Section 3. However, we chose not to do this, since these arguments are significantly easier than the one given for the type Sd. Though the group X is transitive on Ω, it may be imprimitive. Suppose that ∆ is a maximal proper X-block in Ω (if X is primitive, then ∆ is a singleton) and let B denote the corresponding X-invariant block system. For a subgroup Y of Sym(Ω) preserving B, let Y B denote the permutation group of Sym(B) induced by Y . By the maximality of ∆, X B is a primitive group in which V B is a transitive abelian normal subgroup and H B is a transitive nonabelian characteristically simple subgroup. In particular, V B and H B are nontrivial and V B is regular. Since X = V H, we obtain that X B = V B H B . Furthermore, V B ∩ H B is an elementary abelian normal subgroup in the nonabelian characteristically simple group H B , which gives that X B = V B ⋊ H B . As X B is primitive of HA type, the conjugation action of H B induces an irreducible linear group on V B . If |V B | = p d 0 , then the group H B has a transitive permutation representation of degree p d 0 and also a faithful irreducible representation of degree d 0 over the field of p elements. Now [Bau00, Theorem 1.1] implies that H B ∼ = T ∼ = SL(3, 2), p = 2, d 0 = 3 and the stabilizer K in H B = T is a subgroup of index eight 1 . Since, up to conjugacy, K is the unique subgroup of T = SL(3, 2) with 2-power index, we find that the stabilizer of a point α ∈ Ω in H must be of the form K k , and so |Ω| = 8 k .
In the rest of this proof, let G be a finite quasiprimitive permutation group on Ω of type HS, HC, AS, TW, PA, SD or CD, and let H ∼ = T k be a transitive nonabelian characteristically simple subgroup of G where k 2 and T is a nonabelian finite simple group. Assume that S = Q 1 × · · · × Q r is the socle of G, where each Q i ∼ = Q for a nonabelian simple group Q. Consider the projections π i : soc(G) → Q i of soc(G) onto its direct factors. According to Theorem 1.2, H S and we assume that H does not contain a nontrivial normal subgroup of soc(G). If G had type Tw, then S would be regular, and no proper subgroup of S would be transitive. Therefore, under these conditions, the type of G cannot be Tw. The rest of the proof of Theorem 1.3 is by considering each possible O'Nan-Scott type for G.
(1) In this case, S ∼ = Q is a nonabelian simple group. For α ∈ Ω, the factorization S = HS α holds. Now [BP03, Theorem 1.4] implies that S = A n with n 10 acting naturally on n points and hence G = A n or G = S n must follow.
(2) G has type Sd or HS. Since S is transitive on Ω and S α ∼ = Q, |Ω| = |Q| r−1 . For a fixed j ∈ r, denote Q j = Q 1 × · · · × Q j−1 × Q j+1 × · · · × Q r .
Given i 0 ∈ r, a priory, we have three options: Hπ i 0 = 1, Hπ i 0 = Q i 0 or 1 < Hπ i 0 < Q i 0 .
We claim, for all i 0 ∈ r, that 1 < Hπ i 0 < Q i 0 . First we assume that Hπ i 0 = 1 for some i 0 ∈ r. Without loss of generality, assume that i 0 = 1. Then H is a transitive subgroup of Q 1 , which is a regular subgroup. Since no proper subgroup of Q 1 is transitive, H would have to be equal to Q 1 , which is impossible, as we assume that H contains no nontrivial normal subgroup of S. Hence Hπ i 0 = 1 holds for all i 0 ∈ r.
Consider now the case in which Hπ i 0 = Q i 0 for some i 0 ∈ r. In this case Q is a composition factor of H, which means by the Jordan-Hölder Theorem that Q ∼ = T , and so k r. Further, the transitivity of H implies that k = r − 1 or k = r. If k = r, then H = S, which is impossible in our conditions. Assume now that k = r − 1. By the previous paragraph, Hπ i 0 = 1 for all i 0 , and so Hπ i 0 = Q i 0 must hold for all i 0 . Therefore H is a subdirect subgroup of S. By Lemma 2.3, H must be the direct product of full strips with pairwise disjoint supports. Since k = r − 1, either r = 2 and k = 1 or one of these strips must be equal to a direct factor of S. The latter possibility cannot hold by our conditions. In the former case, Q 1 × Q 2 = S α H is a factorization with two full diagonal subgroups, which is not possible, by Lemma 2.5.
Therefore 1 < Hπ j < Q j must hold for all j ∈ r as claimed. We have that S α is a full diagonal subgroup of S. Since H is transitive, S = HS α . Now Lemma 2.4 implies that k = r = 2, H = T 2 = T 1 × T 2 where T 1 < Q 1 and T 2 < Q 2 and Q and T are described in one of the rows of Table 2. Assuming, as we may, that
S α = {(q, qα) | q ∈ Q 1 }
for some isomorphism α : Q 1 → Q 2 , the same lemma implies that Q 1 = T 1 (T 2 α −1 ).
(3) G has type Pa. As S α is nontrivial and is not a subdirect subgroup of S, 1 < S α π i < Q i holds for all i and the projections S α π i are permuted by G α transitively. Furthermore, since Hπ i is a homomorphic image of H, we have that Hπ i = T s i with some s i 0. Since H is transitive, we have that S α H = S, which implies that s i 1 for all i ∈ r. Further, each minimal normal subgroup T i of H is a strip in S. Thus we may consider the support supp(T i ). The transitivity of H implies that
(12) Q i = (HS α )π i = (Hπ i )(S α π i ) = T s i (S α π i ).
Case 1: Suppose that s i 0 2 for some i 0 ∈ r. In this case, the factorization in (12) and [BP03, Theorem 1.4] implies Q ∼ = A n and S α π i 0 ∼ = A n−1 where n 10. Since the S α π i are pairwise isomorphic, S α π i ∼ = A n−1 holds for all i ∈ r. Set P = S α π 1 × · · · × S α π r . In particular, P ∼ = (A n−1 ) r .
We claim that S α = P . Assume that S α = P . Since P is a nonabelian characteristically simple group and S α is a subdirect subgroup of P , by Scott's Lemma (Lemma 2.3), S α is the direct product of diagonal subgroups S α = D 1 × · · · × D l for some l r. Renumbering if necessary and using that S α = P , we can assume that supp(D 1 ) = {Q 1 , . . . , Q m } with 2 m r. Consider the projection π : S → Q 1 × Q 2 . Since S = S α H, we have Q 1 × Q 2 = Sπ = (S α π)(Hπ), where S α π = {(x, xα) | x ∈ S α π 1 } for an isomorphism α : S α π 1 → S α π 2 . Since n 10, the automorphisms of A n are induced by conjugations by elements of S n [Wil09, 2.4.1]. Thus we can extend the isomorphism α to an isomorphism α between Q 1 and Q 2 . Now
Q 1 × Q 2 = D(Hπ) = D(Hπ 1 × Hπ 2 ),
where D = {(x, xα) | x ∈ Q 1 } is a full diagonal subgroup in Q 1 × Q 2 . By Lemma 2.4, the possibilities for Q and T are in Table 2. Since Q ∼ = A n with n 10, we obtain a contradiction. Therefore S α = P , as desired.
Applying the Orbit-Stabilizer Theorem, we see that
|Ω| = |S| |S α | = n r .
So if s i 2 for some i ∈ r, then S ∼ = (A n ) r , S α ∼ = (A n−1 ) r and |Ω| = n r , where n 10. Hence in this case we obtain Theorem 1.3(3/c).
Case 2: Suppose that s i = 1 for all i ∈ r. Since T i is simple, we have that T i is a strip of S for all i ∈ r. Moreover, the supports of the T i are pairwise disjoint. In fact, if for some l we have T i π l ∼ = T ∼ = T j π l for distinct i, j ∈ r, then T 2 ∼ = (T i × T j )π l Hπ l ∼ = T , which is absurd. Then the supports supp(T i ) are pairwise disjoint and we can write
T 1 Q 1 × · · · × Q l 1 , T 2 Q l 1 +1 × · · · × Q l 1 +l 2 , . . . . . . . . . T k Q l 1 +l 2 +···+l k−1 +1 × · · · × Q l 1 +l 2 +···+l k .
First suppose that l i 2 for some i ∈ r. Renumbering, if necessary, assume that l 1 2. Write l = l 1 and consider the projection map π : S → Q 1 × · · · × Q l . As S = HS α , it follows that (Hπ)(S α π) = Q 1 × · · · × Q l . Write L = S α π 1 × · · · × S α π l . Since Hπ = T 1 and S α π L, T 1 L = Q 1 × · · · × Q l . Therefore, T 1 is a transitive subgroup of Q 1 × · · · × Q l under its faithful action by right multiplication on the right coset space [Q 1 × · · · × Q l : L]. On the other, hand Q 1 × · · · × Q l can be embedded into the quasiprimitive permutation group W = Q 1 wr S l acting in product action on [Q 1 : S α π 1 ] l . Hence T 1 is a transitive simple subgroup of a wreath product in product action. According to [BPS04, Theorem 1.1] and [BPS06, Theorem 1.1(b)], l = 2 and T and Q are as in one of the rows of Table 1. Thus, in this case, Theorem 1.3(3/b) is valid. Now suppose that l i = 1 for all i ∈ r. Then k = r and T i < Q i for all i ∈ r. In this case, Theorem 1.3(3/a) holds.
(4) G has type Cd or HC. Lemma 2.3 gives two sets Σ = {S 1 , . . . , S l } and D = {D 1 , . . . , D l } where l 2, each D i is a full diagonal subgroup of S i and S i = Q j ∈supp(D i ) Q j , such that S = S 1 × · · · × S l and S α = D 1 × · · · × D l ∼ = Q l .
We claim that 1 < Hπ j < Q j for all j ∈ r. Suppose that this is not the case and assume first that Hπ i 0 = Q i 0 for some i 0 ∈ r. In this case Q is a composition factor of H, which means by the Jordan-Hölder Theorem that Q ∼ = T , and so k r. Therefore H is the direct product of disjoint full strips in S. Since H is transitive, S = S α H. Now Lemma 2.5 implies that H must contain a direct factor of S. In our case, this is impossible.
Suppose now that Hπ i 0 = 1 for some i 0 ∈ r. Let, for i ∈ l,π i denote the coordinate projectionπ i : S → S i . We may assume that Q i 0 ∈ supp(D 1 ). Applyingπ 1 to the factorization S = S α H, we obtain that S 1 = D 1 (Hπ 1 ) and Hπ 1 is contained in X = Q i ∈supp(D 1 )\{Q i 0 } Q i . Now S 1 = D 1 X is a factorization with D 1 ∩X = 1, which gives that Hπ 1 = X. Therefore Hπ j = Q j must hold for some j = i 0 . Hence by the analysis in the previous paragraph, this is also impossible. Therefore 1 < Hπ j < Q j holds for all j ∈ r, as claimed.
Considering the factorization
S i = D i (Hπ i )
we obtain, as in part (2), that | supp(D i )| = 2 for all i ∈ l. That is, r is even and S i ∼ = Q 2 , Hπ i ∼ = T where and Q and T are as in one of the rows of Table 2. Renumbering, if necessary, assume that
S = Q 1 × Q 2 S 1 × Q 3 × Q 4 S 2 × · · · × Q r−1 × Q r S r/2 .
We claim that H = Hπ 1 × · · · × Hπ r . It suffices to prove that Hπ i H for all i ∈ r. So assume the opposite, that is, Hπ i H for some i ∈ r. Then H has a nontrivial strip X ∼ = T such that | supp(X)| 2.
We claim that supp(D i ) supp(X) for each i ∈ l. To prove the claim, assume that Q 1 , Q 2 ∈ supp(X) and consider the projection π 1 : S → S 1 . As S = S α H, S 1 = Sπ 1 = (S α π 1 )(Hπ 1 ) = D 1 (Xπ 1 ).
However, we obtain, analyzing the orders, that
|S 1 | = |D 1 (Xπ 1 )| = |D 1 ||Xπ 1 | |D 1 ∩ Xπ 1 | = |Q||T | |D 1 ∩ Xπ 1 | < |Q| 2 ,
which is a contradiction. Thus, supp(D i ) supp(X) for all i ∈ l. Again, renumbering if necessary, we may assume that Q 2 , Q 3 ∈ supp(X). Considering the projection π : S → S 1 × S 2 , we have (13) S 1 × S 2 = Sπ = (S α π)(Hπ) = (D 1 × D 2 )(Hπ),
where Hπ is contained in a subgroup H of S 1 × S 2 that is isomorphic to T 3 . Set P = (D 1 × D 2 ) ∩ H. Then |Q| 4 = |Q| 2 |T | 3 /|P |, thus (14) |P | = |T | 3 |Q| 2 .
Suppose that Q ∼ = A 6 and T ∼ = A 5 . Then, by (14), |P | = 5/3, which is impossible. If Q ∼ = M 12 and T ∼ = M 11 , then the claim follows from Lemma 2.6. Assume that Q ∼ = PΩ + 8 (q) and T ∼ = Ω 7 (q). Then by (14) |P | = q 3 · (q 6 − 1) gcd(4, q 4 − 1) 2 (q 2 + 1) gcd(2, q − 1) 3 .
We prove that there exists an odd prime that divides q 2 +1 but does not divide q 3 ·(q 6 −1). If q is even, then q 2 + 1 is odd, and so there exists an odd prime p that divides q 2 + 1. On the other hand, if q is odd, then q 2 + 1 ≡ 2 (mod 4). Thus q 2 + 1 is even but it is not a 2-power, so there exists an odd prime p that divides q 2 + 1. Therefore, in both cases, there exists an odd prime p that divides q 2 + 1. We claim that p does not divide q 3 · (q 6 − 1). Since p is odd, p does not divide q 2 − 1. As q 6 − 1 = (q 2 − 1)(q 4 + q 2 + 1), and p does not divide q 2 − 1 but divides q 2 + 1, we find that p does not divide q 6 − 1. Hence p is a prime we are looking for. This means that also |P | is not an integer, which is impossible. Therefore, H = Hπ 1 × · · · × Hπ r as asserted. Thus 1 < Hπ i < Q i for all i ∈ r, k = r and we may assume after possibly reordering the T i that T i < Q i for all i. The proof of Theorem 1.3 is now complete. Now it only remains to prove Corollaries 1.4 and 1.5.
The proof of Corollary 1.4. In part (0) of Theorem 1.3, H is not regular. In part (2), H = T 1 × T 2 Q 2 such that T 1 T 2 = Q. The stabilizer of H is isomorphic to T 1 ∩ T 2 which, as can be verified in each of the lines of Table 2, is nontrivial. Hence H is nonregular. Since, in case (4), the group G is a blow-up of a group that appears in case (2), H is nonregular in case (4) also. Hence G and H must be as either in Theorem 1.3(1) or in Theorem 1.3(3) and the O'Nan-Scott type of G is either As or Pa.
The proof of Corollary 1.5. Let H, G and Γ be as in Corollary 1.5. Then H is a regular nonabelian and characteristically simple subgroup of the quasiprimitive group G acting on the vertex set of Γ. Noticing that if G is as in Theorem 1.3(1), then G is either an alternating or a symmetric group, and so Γ, in this case, would be a complete graph, the result follows from Corollary 1.4.
HA: M is abelian, C G (M) = M and G Hol M. The group G is always primitive. HS: M is nonabelian, simple, and regular; soc(G) = M × C G (M) ∼ = M × M and G Hol M. The group G is always primitive. HC: M is nonabelian, nonsimple, and regular; soc(G) = M × C G (M) ∼ = M × M and G Hol M. The group G is always primitive.
Example 4.2 (G has type As). Set H = T k with k 2. Suppose that X is a corefree subgroup of H and consider the right coset space Ω = [H : X]. Then H can be viewed as a transitive subgroup of Q = Alt(Ω) which is a primitive permutation group of type AS. These examples satisfy the conditions of Theorem 1.3(1).
∼
H B = H ∩ B. Then H B is a normal subgroup of H and, by Lemma 2.3(2), it is of the form T s , with some s. Further, H = H B × H B where similarly H B = T k−s , and H B acts transitively and faithfully by π on r. For j ∈ r, consider the component H = H B for all j.
∼
= H B /(ker σ j ∩ H B ). Let m be an element of ker σ 1 ∩ H B . Thus m = (1, m 2 , . . . , m r ) with m j ∈ H (j)
B
. Hence ω m = (γy, γyx 2 , . . . , γyx r ). Thus m stabilizes ω if, and only if, y ∈ Sym(Γ) stabilizes γ. Thus (H B ) ω = (H (1) B ) γ . So by applying the Orbit-Stabilizer Theorem twice, and using that |H B | = |H
The proof of Theorem 1.3. (0) Suppose that G Sym(Ω) has type HA. Then G has a unique minimal normal subgroup V which is elementary abelian of order p d . Suppose that H = T k is a nonabelian transitive characteristically simple subgroup of G. Then H ∩ V is a normal subgroup of H which is elementary abelian, and so, by Lemma 2.3(2), H ∩ V = 1. Hence one may consider the group X = V ⋊ H. Since both V and H are transitive on Ω, so is X.
Table for Theorem
for1.3(3/b)
T
Q
1 A 5
A 6
2 M 11
M 12
3 Ω 7 (q) PΩ +
8 (q) with q 2
Table 2. Table for Theorem 1.3(2)-(4)
the rest of the lemma follows. The following result was originally proved in [BP03, Lemma 2.2]; see also [PS18a, Theorem 1.2] and [PS18b, Section 4.8] for generalizations.
Theorem 5.2. If X is a transitive subgroup of W , then each component of X is transitive on Γ. Moreover, if X acts transitively on r, then each component of the intersection X ∩ (Sym(Γ)) r is transitive on Γ.). The following theorem was stated in [PS12,
Theorem 1.2]; see also [PS18b, Corollary 5.17].
The cited theorem of Baumeister contains a misprint and SL(3, 3) is written instead of SL(3, 2).
Pedro H. P. Daldegan and Csaba Schneider
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Composition factors from the group ring and Artin's theorem on orders of simple groups. Wolfgang Kimmerle, Richard Lyons, Robert Sandling, David N Teague, Proc. London Math. Soc. 603Wolfgang Kimmerle, Richard Lyons, Robert Sandling, and David N. Teague. Composition factors from the group ring and Artin's theorem on orders of simple groups. Proc. London Math. Soc. (3), 60(1):89-122, 1990.
The finite primitive permutation groups containing an abelian regular subgroup. Li Cai Heng, Proc. London Math. Soc. 873Cai Heng Li. The finite primitive permutation groups containing an abelian regular subgroup. Proc. London Math. Soc. (3), 87(3):725-747, 2003.
On finite permutation groups with a transitive cyclic subgroup. Heng Cai, Cheryl E Li, Praeger, J. Algebra. 349Cai Heng Li and Cheryl E. Praeger. On finite permutation groups with a transitive cyclic subgroup. J. Algebra, 349:117-127, 2012.
Transitive subgroups of primitive permutation groups. Martin W Liebeck, Cheryl E Praeger, Jan Saxl, J. Algebra. 2342Special issue in honor of Helmut WielandtMartin W. Liebeck, Cheryl E. Praeger, and Jan Saxl. Transitive subgroups of primitive per- mutation groups. J. Algebra, 234(2):291-361, 2000. Special issue in honor of Helmut Wielandt.
Regular subgroups of primitive permutation groups. Martin W Liebeck, Cheryl E Praeger, Jan Saxl, Mem. Amer. Math. Soc. 20395274Martin W. Liebeck, Cheryl E. Praeger, and Jan Saxl. Regular subgroups of primitive permu- tation groups. Mem. Amer. Math. Soc., 203(952):vi+74, 2010.
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Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type. Cheryl E Praeger, Csaba Schneider, Israel J. Math. 2282Cheryl E. Praeger and Csaba Schneider. Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type. Israel J. Math., 228(2):1001-1023, 2018.
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MG, Brazil, [email protected], www.mat.ufmg.br/∼csaba (P. Daldegan) Departamento de Matemática. Schneider) Departamento De Matemática, Av. Antônio Carlos. 6627Instituto de Ciências Exatas, Universidade Federal de Minas GeraisAv. AmazonasSchneider) Departamento de Matemática, Instituto de Ciências Exatas, Universi- dade Federal de Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG, Brazil, [email protected], www.mat.ufmg.br/∼csaba (P. Daldegan) Departamento de Matemática, Centro Federal de Educação Tecnológ- ica de Minas Gerais, Av. Amazonas 7675, Belo Horizonte, MG, Brazil, phpdalde- [email protected], sig.cefetmg.br/sigaa/public/docente/portal.jsf?siape=1395846
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arxiv |
Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory
Loc V Tran
Department of Mechanical Construction and Production
Faculty of Engineering and Architecture
Ghent University
9000GhentBelgium
Department of Architectural Engineering
Sejong Unviersity
98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea
Jaehong Lee
Department of Architectural Engineering
Sejong Unviersity
98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea
H Nguyen-Van
Faculty of Civil Engineering
Ho Chi Minh City University of Architecture
196 Pasteur Street, District 3, Viet NamHo Chi Minh City
H Nguyen-Xuan
Department of Architectural Engineering
Sejong Unviersity
98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea
Department of Computational Engineering
Vietnamese-German University
Binh Duong New CityVietnam
M Abdel Wahab
Department of Mechanical Construction and Production
Faculty of Engineering and Architecture
Ghent University
9000GhentBelgium
Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory
1Laminated composite plateIsogeometric analysisHigher-order Shear Deformation Theorynonlinear analysis * Corresponding author Email address: jhlee@sejongackr (Jaehong Lee) 2
In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors. IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the von-Karman strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.
Introduction
Various plate theories have been addressed since long time. Pagano [1] initially investigated the analytical three-dimensional (3D) elasticity method to predict the exact solution of simple static problems. Noor [2] had further developed the 3D elasticity solution formulas for stress analysis of composite structures. It is well known that the exact 3D approach is a potential tool to obtain the true solutions of plates. Nevertheless, it is not often easy to solve the practical problems that consider the complex (or even slightly complicated) geometries, arbitrary boundary conditions, and lamination schemes or nonlinearities. In addition, each layer in the 3D elasticity theory is modeled as a 3D solid so that the computational cost of laminated composite plate analyses is increased significantly. Hence, many equivalent single layer (ESL) plate theories with suitable assumptions [3] have been then proposed to transform the 3D problems to 2D problems.
Among the ESL plate theories, the classical laminate plate theory (CLPT) based on the Love -Kirchhoff assumption was proposed. However, the laminated plates stacking from many laminae are very susceptible to the transverse shear deformation effect due to the significantly smaller effective transverse shear modulus as compared to the effective elastic modulus along the fiber direction [4]. Thus, first order shear deformation theory (FSDT) based on Reissner-Mindlin theory [5,6], which takes into account the shear effect, was therefore developed. In the FSDT model, with the linear in-plane displacement through plate thickness assumption, the obtained shear strains/stresses distribute inaccurately and do not satisfy the traction free boundary conditions at the plate surfaces. It is hence required to amend the unrealistic shear strain energy part by the shear correction factors (SCF). To overcome the limitation of the FSDT, various kinds of higher order shear deformable theory (HSDT) have then been devised (see Ref. [7] for a brief review). The HSDT models, which include higher-order terms in the approximation of the displacement field, ensure non-linear distributions of the shear strains/stresses with traction-free boundary condition at the plate surfaces. As a result, the HSDT models provide better results and yield more accurate and stable solutions (e.g. inter-laminar stresses and displacements) [8,9] than the FSDT ones without the SCF. The HSDT requires the C 1continuity of generalized displacement field leading to the second-order derivative of the stiffness formulation. However, the enforcement of even C 1 continuity across the inter-3 element boundaries in standard finite element method is not a trivial task. In attempts to overcome this difficulty, some kinds of methods have been developed such as C 0 continuous elements [10][11][12], nonconforming plate bending elements based on Hermite polynomial functions [3] or mixed finite elements [4,13]. It is known that there exists an algebraically complicated requirement in the construction of these elements. Furthermore, some extra unknown variables are needed to form their formulations which require much storage and computational cost. This shortcoming motivates us to develop in this paper a novel computational approach based on isogeometric analysis (IGA).
Isogeometric approach [14] firstly proposed by Hughes fulfills a seamless bridge link between computer aided design (CAD) and finite element analysis (FEA). IGA uses same B-Spline or non-uniform rational B-Spline (NURBS) functions in describing the exact geometry of problem and constructing finite approximation for analysis. Being thankful to higher order continuity of NURBS, IGA naturally verifies the C 1 -continuity of plates based on the HSDT assumptions. IGA has been widely applied to the plate structures with various plate models such as CLPT [15], FSDT [16,17], HSDT [7,[18][19][20], four unknown variables refined plate theory (RPT) [21,22], layerwise [23,24], etc. The literatures mentioned above, however, did not take into account geometric nonlinearity. So far, there are very few published materials related to geometrically nonlinear plate models using IGA, except two recent papers [25,26] based on the FSDT. Apparently, there are no researches on geometrically nonlinear isogeometric analysis for the plates based on the HSDT model. Therefore, our goal in this paper is for the first time using HSDT model in study both geometrically nonlinear bending and transient analysis of the laminated composite plates. Based on the von-Karman strain which considers small strain and moderate rotation assumptions, the nonlinearity of the plates is formulated using total The paper is outlined as follows. The next section introduces the generalized higherorder shear deformation theory for laminated composite plate. In section 3, the geometrically nonlinear formulations of plate based on IGA are described. Section 4 presents the solution scheme to solve the nonlinear problems. The numerical results and 4 discussions are provided in section 5. Finally, this article is closed with some concluding remarks.
Generalized higher-order shear deformation theory for laminated composite plate
According to a generalized higher-order shear deformation theory [8], the displacements of an arbitrary point in the plate can be expressed in the general form [27].
1 2 3 () z f z u u u u (1) where 1 0 0 T u v w u is the axial displacement , 2 , ,0 T
For a bending plate, the Green strain vector is expressed by 11 22
j i k k ij j i i j u u u u x x x x (2)
Using the von-Karman assumptions, the nonlinear straindisplacement relation adopts here by neglecting second-order terms of u 0 and v (4) and the nonlinear component of in-plane strain can be rewritten as
0 displacements 12 () 0 ( ) 0 m z f z fz κ γ β (3) where 2 0,, 2 0, , 0, 0, , 1 2 25 1 2 NL A θ where , ,, , ,, 0 0Q Q Q Q Q Q Q Q Q QQ QQ (6)
where material constants are given by 1 ,,
,,
E E E Q Q Q Q G Q G Q G (7)
in which E 1 , E 2 are the Young modulus in the 1 and 2 directions, respectively, and G 12 , G 23 , G 13 are the shear modulus in the 1-2, 2-3, 3-1 planes, respectively, and ij are Poisson's ratios.
The laminate is usually made of several orthotropic layers in which the stress-strain relation for the k th orthotropic lamina with the arbitrary fiber orientation compared to the reference axes is given by [3]
Q Q Q Q Q Q Q Q Q QQ QQ (8)
The inplane force, moments and shear force are defined as
Q f z z Q (9)
It is noted that the function
N A B E 0 M B D F 0 σ Dε E F H 0 P 0 0 0 D Q β (10) where /2 22 /2 , , , , , 1, , , ( ), ( ), ( ) d , , 1,2,6 h ij ij ij ij ij ij ij h A B D E F H z z f z zf z f z Q z i j /2 2 /2 ( ) d , , 4,5 h S ij ij h D f z Q z i j (11)
and the generalized strain is divided into the linear and nonlinear strain components
12 T LL β and T NL NL , respectively ˆˆL NL (12)
Neglecting damping effect, the equation of motion obtained from Lagrange's equation using Hamilton's variation principle can be briefly expressed as [28] ˆd d d
T T T s σ u mu f u (13)
where m and f s are the consistent mass matrix (detailed in [8]) and the mechanical surface loads, respectively and
1 2 3 T u u u u(14)
7
Isogeometric formulation for nonlinear analysis of plate
A brief of NURBS basic functions
A knot vector 1 2 1 , ,..., np Ξ is a non-decreasing sequence of parameter values i , 1,... i n p , where i R called i 1 ,0 1 if 0 otherwise ii i N (15) as 1 p the basis functions are obtained from 1 , , 1 1, 1 11 ip i i p i p i p i p i i p i N N N (16)
An example of B-spline basis is illustrated in Figure 1. Using two open knot vectors Bivariate.
To present exactly some conic sections, e.g., circles, cylinders, spheres, etc., nonuniform rational B-splines (NURBS) need to be used. Being different from B-spline, each control point of NURBS has an additional value called an individual weight 0
A . , 1 d A i p NN ξξ (17) 9 , , AA A mn AA A N R N (18)
NURBS basic functions also inherit all of features of B-spline basic functions and become B-spline basic functions when the individual weight of control points is constant.
NURBS formulation for nonlinear bending of composite plates
Using NURBS basis functions, the displacement field of the plate is approximated as
,, mn h AA A R uq (19) where 00 T A A A xA yA A u v w q
is the vector of nodal degrees of freedom associated with the control point A.
Substituting Eq. (19) into Eq. (12), the generalized strains can be rewritten as: (22) Substituting Eq. (20) into Eq. (13), and after eliminating the virtual displacement, the equation of motion is written in the following matrix form ext Kq Mq F (23) where K and M are the global stiffness and mass matrices, respectively (25) in which (26) and the external force vector under the transverse load f 0 is computed by
1 1 2 mn L NL A A A A B B q (20) where 12 T T T T T L m b b s A A A A A B B B B B in which ,, 1 ,, , , , 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 A x A xx mb A A y A A yy A y A x A xy RR RR R R R B , B ,, 2 , ,, 0 0 00 0 0 0 0 0 0 0 0 , 0 0 0 0 000 Ax A bs A A y A A A y A x R R R R RR BB (21) while strain matrix NL A B is still dependent upon displacement gradient 10 () NL g AA A B q B 0 where , , 0 0 0 0 0 0 0 0 Ax g A Ay R R B ( ) 0.5 d T L NL L NL K q B B D B B (24) d T M N mN1 2 3 , A AA A N NN N 1 0 0 0 0 0 0 0 0 ; 0 0 0 0 A AA A R R R N , 23 , 0 0 0 0 0 0 0 0 0 0 0 0 ; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A x A A A y A A RR RR NN 0 ( ) 0 0 0 0 d T ext AA f t R F(27)
Solution scheme
From Eq. (23), it is observed that the equation of dynamic system is dependent upon both time domain and unknown displacement vector. To discretize this problem, the Newmark's integration scheme association with the iteration methods is employed.
Time integration
The dynamic problem is solved in step-by-step of a number of equal time intervals, t with zero displacement, velocity and acceleration at initial time, t = 0. And the first and second derivative of displacement are sought implicitly at time ( 1) mt as below 11 11 (29) where and are constant and equal to 0.25 and 0.5, respectively [30].
Substituting Eq. (28) into Eq. (23), we obtain q , this equation must be computed as an iterative processing such as the Picard algorithm or the Newton-Raphson method which can be presented in the next section.
1 1 1m m m K q F(30)
Iteration methods
At time step ( 1) mt , Eq.(30) can be rewritten in term of the residual force as below
1 1 1 1m m m m φ K q F(32)
The residual force presents the error in this approximation and tends to zero during iteration. If 1 i m q , an approximate trial solution at the i th iteration, makes unbalance residual force, an improved solution is then proposed 1 11 ii mm
q q q(33)
where the incremental displacement is calculated by equaling to zero curtailed Taylor's series expansion of 1 (34) in which T K is called tangent stiffness matrix. If it takes the same form of the effective stiffness matrix given in Eq. (31), this iterative method is so-called the direct iteration method or Picard method. This method is simple in concept and implementation. But sometimes, it does not work because it is too hard to get invertible form of the unsymmetric stiffness matrix. Another way, T K could be computed following to the Newton-Raphson method
1 i m φ [31] 1 / i mT q φK( ) / i T K φ q q(35)
It is known that this matrix T K is always symmetric for all structure problems and it helps this method converges faster for most applications than the Picard one [32].
At each time step, the process in Eq.(33) is repeated until the displacement error between two consecutive iterations reduces to the desired error tolerance.
Numerical results
In this section, we show performance of IGA for several geometrically nonlinear plate problems using HSDT. In previous works [8,9,21], it is found that just with 11 11 cubic NURBS elements, IGA using HSDT produces an ultra-accurate solutions for plate analysis. In this study, this mesh therefore is employed. Some sets of the material properties are used for numerical investigations:
Material I:
qa P Eh , w w h , 2 2 L Eh
Geometrically nonlinear bending analysis
In this section, the geometrically nonlinear bending analysis of plate is studied by computing Eq. (23) without the mass matrix effect ext Kq F (37) Herein, the global stiffness matrix K is still nonlinear relation with unknown variable q because of dependence on nonlinear strain matrix NL B . To solve this equation, Newton-Raphson methods is employed to calculate the tangent stiffness as follow: (38) where ,, L NL g K K K are the linear, nonlinear and geometric stiffness matrix, respectively
T L NL g K K K K 0 dˆ+ d d T LL L T T T L NL NL N NL NL NL T gg g K B DB K B DB B DB B DB K B N B(39)
in which 00 0 00
x xy xy y NN NN
N is a matrix related to the in-plane forces.
Isotropic plates
Let us consider a clamped square thin plate under a uniformly distributed load. This is a benchmark problem which is often tested by many researchers [31,35,36] for the geometrically nonlinear validation of thin plate formulation. Table 1 provides the results for central deflection w and axial stress (0,0, / 2)
x h .
The obtained solutions are compared with the analytical ones using a double Fourier series by Levy [35] and finite element ones using nine node element [31], and mixed finite element solution [36]. Their relations with load parameter are also depicted in Figure 2. As seen, the present results are in excellent agreement with those from the literature and gain the best axial stress as compared to analytical Kirchhoff solution. Table 2 are compared with the analytical Kirchhoff solution [37], that of Kirchhoff-based elements such as DKT [38], RNEM [39] and that of Mindlin-based elements: nine-node Lagrangian quadrilateral element (QL) [31], mixed interpolation smoothing quadrilateral element with 20DOF (MISQ20) [40]. As seen, the present method produces the most accurate solution. [37] are shown in parenthesis.
Symmetric laminated plates
Firstly, the benchmark problems with the experimental results of Zaghloul and Kenedy are studied for validating the present method. Figure 3 reveals comparison between the present solutions based on the HSDT model with others according to CPT [41], FSDT [36] and the experimental results [41]. Figure 3a Next, the effect of span to thickness ratio on the central deflection of the symmetric laminated composite plates is revealed in Table 3 and Table 4. Herein, material III is set. In Table 3
The obtained results are compared with those of nine-node Lagrangian quadrilateral element with 9DOF based on C 0 HSDT [42], mesh free method based on SSDT [43]; and that of the FSDT model using MISQ20 [40], four-node quadrilateral isoparametric plane element with 20 and 24 DOFs [44]. It is interesting to note that just using 5 DOFs, present method gains very good agreement with other published solutions. Figure 4 shows that the present results match very well with the solutions by Kant [42] for various span to thickness ratios L/h = 10, 20, 40. In Table 4, the linear and geometrically nonlinear solutions for simply supported (SSSS) of the [0/90/90/0] and [0/90/0] laminated plates are presented. In case of linear problem, Pagano has given the exact solutions by using 3D elasticity model [1]. However, the nonlinear one is not available. For a comparison purpose, the MITC element [45] is, thus, also used to compute the displacement for the plates based on the FSDT with SCF equals to 5/6. It can be seen that the HSDT model gets higher results than the FSDT with better accuracy as compare with Pagano's results. And the discrepancy between them increases according to increase in the thickness to length ratio or the magnitude of applied load. The clearer observation is found in Figure 5.
Furthermore, in thick plate the geometrical nonlinearity is pronouncedly observed with more highly curved load-displacement line than that of the thin plate. Figure 6
Antiymmetric laminated plate
This subsection deals with the analysis of the simply supported square laminated plate with L/h=10 and material IV. considering the bending-stretching coupling, the antisymmetric laminated plate obtains the lower deflection than the symmetric one and their discrepancy increases according to increase in angle from 0 to 45; (4) nonlinear deflection parameter reduces following to increase in applied load and always lower than that of linear results.
Geometrically nonlinear transient analysis
In the geometrically nonlinear transient analysis, fully Lagrange equation motion in Eq. Figure 11. It is observed that present method predicts the very close deflection response as compared with finite strip method (FSM) by J. Chen [33]. It also clearly exhibits that the magnitude and wavelength of the non-linear response are lower than that of linear behavior with the same loading intensity.
Conclusions
An effectively numerical procedure based on IGA and HSDT has been presented for geometrically nonlinear analysis of the laminated composite plates. Herein, using cubic approximation functions, the present method naturally satisfies the C 1 continuity across inter-element boundaries without any additional variables. It is believed that utilizing NURBS basic functions helps present method to eliminate the error of geometric approximation.
Lagrange approach and solved by the Newmark time integration association with the iteration methods. Several numerical examples are given to show the effectiveness of the present formulation in comparison with other available procedures in the literature.
traction-free boundary condition is automatically satisfied at the top and bottom plate surfaces. Furthermore, the transverse shear forces are described parabolically through the plate thickness. Hence, the shear correction factors are not required in this model.
th knot lies in the parametric space, p is the order of the B-spline and n is number of basis functions. In the so-called open knot vector, the first and the last knots are repeated p+1 times and have the value of 0 and 1, respectively. Using Cox-de Boor algorithm, the univariate B-spline basis functions , ip N are defined recursively [29] on the corresponding knot vector start with order p = 0
CFigure 1 .
1two sets of univariate quadratic and cubic B-splines are plotted in Figure 1a and b, respectively. The important properties of B-Spline functions are summarized as below: Being piecewise polynomial functions. continuity at the interior knot with k is repeated time of this knot. By a simple wayso-called tensor product of univariate B-splines, the multivariate B-B-splines basic functions: a) Univariate quadratic; b) Univariate cubic and c)
Figure 3 .
3shows the relation between the central deflection and the uniform load intensity for an orthotropic plate (material set I) clearly seen that considering the shear deformation effect, the IGA results with 5 DOFs/control points match well with those of FSDT [36] using MIXFEM with 8 DOFs/control points and get good agreement with experimental ones. The load-deflection curves of: (a) the simply supported plate ( 12in, 0.138in) Lh and (b) the clamped plate ( 12in, 0.096in) Lh † .
the central deflection versus the load parameter is depicted for the plate subjected to uniform load and simply supported constraint (
plots the stress distributions through the plate thickness of the four cross-ply plate (L/h=10) via the change of load intensity. Regarding the nonlinear part in in-plane strain, through the mid-plane. And its magnitude at the bottom reduces faster than that at the top surface according to increase in the load intensity. Being different from the axial stress, reduces according to increase in load parameter. Furthermore, the HSDT model enables to obtain the parabolic shape of the shear stresses, which naturally satisfy the traction-free boundary condition at the plate surfaces. To close this subsection, effect of the boundary conditions on the nonlinear behavior of [to change of boundary condition from SSSS to SSSS2 and CCCC because of increase in constraint at the boundary, which leads to stiffen the plate structure.
Figure 5 .Figure 6 .Figure 7 .
567Load-displacement curve of symmetric laminated [0/90/90/0] plate: HSDT and FSDT results according to solid and dash lines, respectively. Their discrepancy increases by increasing in applied load or the length to thickness ratios. Effect of the load parameter P on the distributed stresses: the magnitude of load is 50, 100, 200 and 300 for the red, purple, green and blue curves, respectively. Effect of the boundary conditions on the nonlinear behavior of [0/90/90/0] laminated plate under uniform pressure: (a) Central deflection and (b) Axial stress.
Figure 8
8reveals the effect of number of layers on the nonlinear behavior of the [0/90] N cross-ply plates. It is observed that with the same total thickness, increase in number of layers N helps the plate stiffer with deflection reduction. Moreover, it also reduces the nonlinear effect on the laminated plate. It means that the load-deflection curve becomes closer to a straight line as the linear solution.
Figure 8 . 21 Figure 9
8219Effect of number of layers on deflection of the [0/90] N laminated composite plate. shows the loaddeflection curves of angle-ply [-//-/] plate via the fiber orientation angle which changes from 0 to 15, 30, 45. As seen, the plate has the most stiffness at = 45. Figure 10 reveals the effect of fiber angle on the deflection behavior of both symmetric and anti-symmetric plates under various loading levels. The general observations are: (1) deflections are symmetric about angle = 45; (2) the plate behavior is the weakest in case of one orthotropic layer ( = 0 or 90) and the stiffest at = 45; (3)
Figure 9 .Figure 10 .
910Effect of fiber orientation angle on loaddeflection curves of the angle-ply [-//-/] plate. 22 +: [-///-] symmetric laminated plate, o: [-//-/] antisymmetric laminated plate Behavior of angle-ply plates under various loading levels.
( 23 )
23is computed by a combined technique between Newmark's integration and Picard method. An orthotropic plate with set of material V and dimensions as length L=250 mm, thickness h=5 mm is firstly studied for validation. For this problem, the fully simply supported plate is subjected to a uniform step loading of 1 MPa. Its transient response according to the normalized central deflection w under both linear and nonlinear analysis is shown in
Figure 11 . 24 Figure 12 .Figure 13 .
11241213Time history of the transverse displacement of an orthotropic plate under step uniform load with intensity 1MPa. ‡ Next, the dynamic response of three layer [0/90/0] thick plate is investigated. The material set VI is used for this plate (L/h = 5, h = 0.1526m). The transverse load is sinusoidally distributed in spatial domain and is assumed to vary with time as q and value of force 0 () Ft depicted in Figure 12 depends on loading types: step, triangular, sinusoidal and explosive blast, respectively. Once again the observation is that nonlinear analysis takes the lower central deflection and higher frequency than that of the linear one. ‡ The results by FSM is cited from [33]. Time history of applied load F Effect of different loadings on the deflection respond of the cross-ply [0/90/0] square laminated plate: (a) step; (b) triangular; (c) sine and (d) explosive blast loading.
are the rotations in the x, y and z axes, respectively. The function ()xy
ww
u
and
3
0
T
xy
u
fz is the so-called
distributed function which is set
2
4
3
3
()
h
f z z
z
follow to the famous Reddy's plate
theory
Table 1 :
1Central deflection and axial stress of a clamped square plate under uniform loadLoad
P
Anal. Sol. [35]
MXFEM[36]
FEM Q9 [31]
IGA
w
x
w
x
w
x
w
x
17.8
0.237
2.6
0.2392
2.414 0.2361
2.614
0.2367
2.5626
38.3
0.471
5.2
0.4738
5.022 0.4687
5.452
0.4693
5.3273
63.4
0.695
8.0
0.6965
7.649 0.6902
8.291
0.6910
8.0998
95.0
0.912
11.1
0.9087
10.254 0.9015
11.066
0.9025 10.8273
134.9
1.121
13.3
1.1130
12.850 1.1050
13.789
1.1061 13.5223
184.0
1.323
15.9
1.3080
15.420 1.2997
16.456
1.3009 16.1806
245.0
1.521
19.2
1.5010
18.060 1.4916
19.178
1.4928 18.9069
318.0
1.714
21.9
1.6880
20.741 1.6775
21.938
1.6786 21.6797
402.0
1.902
25.1
1.8660
23.423 1.8545
24.713
1.8555 24.4700
a)
b)
Figure 2. Comparison of the non-dimensional central deflection (a) and normal stress (b) of
isotropic plate.
Table 2 :
2Normalized central deflection of a clamped circular plate under uniform load LoadP
Normalized central deflection
w
MISQ20 [40] QL [31]
DKT[38] RNEM [39] IGA
Anal. Sol. [37]
1
0.170
0.1682
0.172
0.1664
0.1669
0.169
(0.59)
(0.47)
(1.78)
(1.54)
(1.24)
2
0.327
0.3231
0.330
0.3179
0.3208
0.323
(1.24)
(0.03)
(2.17)
(1.58)
(0.68)
3
0.465
0.4591
0.470
0.4514
0.4562
0.457
(1.75)
(0.46)
(2.84)
(1.23)
(0.18)
6
0.780
0.7702
0.791
0.7637
0.7671
0.761
(2.50)
(1.21)
(3.94)
(0.35)
(0.80)
10
1.067
1.0514
1.082
1.0544
1.0487
1.035
(3.09)
(1.58)
(4.54)
(1.87)
(1.32)
15
1.320
1.3007
1.342
1.3164
1.2989
1.279
(3.21)
(1.70)
(4.93)
(2.92)
(1.56)
The errors compared to analytical Kirchhoff solution
Table 3 :
3Central deflection w of a simply supported (SSSS2) [0/90/90/0] square plate.L/h
P
HOST
[42]
MISQ20
[40]
RDKQ-
NL20[44]
RDKQ-
NL24 [44]
MQRBF
[43]
Present
40
50
0.293
0.296
0.291
0.294
0.2654
0.2936
100
0.464
0.473
0.461
0.467
0.446
0.4643
150
0.582
0.592
0.577
0.587
0.616
0.5798
200
0.664
0.683
0.667
0.679
0.7355
0.6683
250
0.738
0.759
0.74
0.754
0.8355
0.7407
20
50
0.320
0.312
0.323
0.327
0.3004
0.3126
100
0.493
0.487
0.487
0.494
0.5085
0.4807
150
0.592
0.603
0.597
0.608
0.6591
0.5928
200
0.680
0.691
0.682
0.695
0.778
0.6784
250
0.752
0.765
0.751
0.766
0.8771
0.7486
† The data is redrawn according to Urthaler's work [36].
Table 4 :
4Central deflection w of simply supported symmetric laminated composite plates
[0/90/90/0]
[0/90/0]
L/h
P
Linear
Nonlinear
Linear
Nonlinear
HSDT FSDT 3D [1] HSDT FSDT HSDT FSDT 3D [1] HSDT FSDT
4
50 0.947 0.856
0.977 0.7198 0.6791 0.961 0.889
1.003 0.7262 0.6948
100 1.894 1.712
1.1214 1.0788 1.922 1.778
1.1284 1.0974
200 3.787 3.423
1.6555 1.6111 3.844 3.556
1.6606 1.6316
300 5.681 5.135
2.0447 1.9877 5.765 5.335
2.0475 2.0078
10
50 0.357 0.331
0.372 0.3474 0.3236 0.356 0.334
0.370 0.3462 0.3264
100 0.715 0.662
0.6501 0.6121 0.712 0.669
0.6478 0.6162
200 1.430 1.324
1.1148 1.0667 1.425 1.338
1.1116 1.0713
300 2.144 1.986
1.4612 1.4100 2.137 2.006
1.4586 1.4154
20
50 0.253 0.245
0.259 0.2504 0.2428 0.252 0.246
-
0.2494 0.2432
100 0.506 0.490
0.4872 0.4734 0.504 0.491
0.4849 0.4737
200 1.012 0.980
0.8960 0.8763 1.008 0.982
0.8921 0.8752
300 1.518 1.470
1.2255 1.2024 1.513 1.473
1.2190 1.1999
100
50 0.217 0.216
0.217 0.2159 0.2150 0.217 0.216
-
0.2158 0.2149
100 0.434 0.432
0.4243 0.4226 0.434 0.432
0.4238 0.4222
200 0.868 0.865
0.7993 0.7967 0.868 0.865
0.7969 0.7945
300 1.303 1.297
1.1146 1.1117 1.303 1.297
1.1101 1.1074
FSDT results using MITC element [45]
The nonlinearity of the plates based on the von-Karman strain assumptions including nonlinear bending and transient problems which are solved by the Newmark time integration associated with the iteration methods. Numerous numerical examples have been carried out for isotropic, symmetric and unsymetric laminated plates subjected to constant or dynamic loads. In all cases, the present results are in good agreement with available solutions in the literature. Furthermore, just utilizing five degrees of freedom, the present plate model remedies the shear locking phenomenon and shows very good performances for both thick and thin plates even though it is based on the thick plate theory. Especially, for clamped circular plate, present method gets the best results as compared with analytical Kirchhoff solution.
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| {'fraction_non_alphanumeric': 0.06624217340549719, 'fraction_numerical': 0.06252785737026424, 'mean_word_length': 3.942929080990348, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 61, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors. IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the von-Karman strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.', 'arxivid': '1411.3508', 'author': ['Loc V Tran \nDepartment of Mechanical Construction and Production\nFaculty of Engineering and Architecture\nGhent University\n9000GhentBelgium\n\nDepartment of Architectural Engineering\nSejong Unviersity\n98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea\n', 'Jaehong Lee \nDepartment of Architectural Engineering\nSejong Unviersity\n98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea\n', 'H Nguyen-Van \nFaculty of Civil Engineering\nHo Chi Minh City University of Architecture\n196 Pasteur Street, District 3, Viet NamHo Chi Minh City\n', 'H Nguyen-Xuan \nDepartment of Architectural Engineering\nSejong Unviersity\n98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea\n\nDepartment of Computational Engineering\nVietnamese-German University\nBinh Duong New CityVietnam\n', 'M Abdel Wahab \nDepartment of Mechanical Construction and Production\nFaculty of Engineering and Architecture\nGhent University\n9000GhentBelgium\n'], 'authoraffiliation': ['Department of Mechanical Construction and Production\nFaculty of Engineering and Architecture\nGhent University\n9000GhentBelgium', 'Department of Architectural Engineering\nSejong Unviersity\n98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea', 'Department of Architectural Engineering\nSejong Unviersity\n98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea', 'Faculty of Civil Engineering\nHo Chi Minh City University of Architecture\n196 Pasteur Street, District 3, Viet NamHo Chi Minh City', 'Department of Architectural Engineering\nSejong Unviersity\n98 Kunja Dong, Kwangjin Ku143-747SeoulSouth Korea', 'Department of Computational Engineering\nVietnamese-German University\nBinh Duong New CityVietnam', 'Department of Mechanical Construction and Production\nFaculty of Engineering and Architecture\nGhent University\n9000GhentBelgium'], 'corpusid': 18901745, 'doi': '10.1016/j.ijnonlinmec.2015.02.007', 'github_urls': [], 'n_tokens_mistral': 18341, 'n_tokens_neox': 14878, 'n_words': 7686, 'pdfsha': 'e4dfaebaa6293031b176d55b204401b8ef54cd48', 'pdfurls': ['https://arxiv.org/pdf/1411.3508v1.pdf'], 'title': ['Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory', 'Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory'], 'venue': []} |
arxiv |
Critique of Fermionic RνMDM and its Scalar Variants
25 Jul 2012 May 22, 2014
Krešimir Kumerički
Department of Physics
Faculty of Science
University of Zagreb
P.O.B. 331HR-10002ZagrebCroatia
Ivica Picek
Department of Physics
Faculty of Science
University of Zagreb
P.O.B. 331HR-10002ZagrebCroatia
Branimir Radovčić
Department of Physics
Faculty of Science
University of Zagreb
P.O.B. 331HR-10002ZagrebCroatia
Critique of Fermionic RνMDM and its Scalar Variants
25 Jul 2012 May 22, 20141460Pq1460St9535+d1480Cp Keywords: Neutrino massExotic leptonsDark matterNon-standard scalar fields
We examine the stability of minimal dark matter (MDM) particlecandidates in the setup in which they participate in radiative neutrino (Rν) masses. We first point out the existence of an additional renormalizable term in recently proposed RνMDM Lagrangian, which violates the claimed accidental Z 2 symmetry and spoils the stability of the fermionic MDM quintuplet component. We then explore the viability of RνMDM variants based on scalar MDM multiplets. There are ubiquitous super-renormalizable terms in the scalar potential which make these scalar multiplets unstable.
Introduction
There are two experimental pieces of evidence [1] pressing us to extend the standard model (SM): the evidence for neutrino masses and for the dark matter (DM) content of the Universe. The ongoing experiments at the Large Hadron Collider (LHC) may provide us with new insights on new heavy particles enabling Weinberg's dimension-five operator LLHH [2] as an explanation of the smallness of masses of observed neutrinos.
The known tree level realizations of Weinberg's operator proceed by adding to the particle content of the SM a single, GUT-scale multiplet: a heavy fermion singlet for the type-I [3], a scalar triplet for the type-II [4] and a fermion triplet for the type-III [5] seesaw mechanism. On the other hand additional discrete symmetries are instrumental [6,7,8] in bringing the seesaw mechanism to the TeV scale accessible at the LHC.
Alternative lowering of the seesaw scale, without discrete symmetries, can be achieved either by going beyond dimension-five operator or by generating the neutrino masses at the loop level. The approach used by two recent models is to employ non-zero hypercharge seesaw mediators: the dimension-seven mechanism with fermion triplets in [9] and the dimension-nine mechanism with fermion quintuplets in [10,11]. Both of these mechanisms offer the spectacular triply-charged states verifiable at the LHC.
The approach in [12], which we follow here, is to link a DM candidate to the scale of radiatively generated neutrino (Rν) masses. The authors of [13] exemplify it by employing a 10 TeV scale fermionic quintuplet of zero hypercharge. Let us stress that an isolated Σ ∼ (5, 0) fermion field or an isolated septuplet Φ ∼ (7, 0) scalar field have been singeld out as viable DM particle candidates within the so-called minimal dark matter (MDM) model [14]. The dubbed RνMDM model [13] claims to achieve the stability of the neutral MDM quintuplet component without introducing a symmetry beyond the SM gauge symmetries, on account of an accidental Z 2 symmetry arising from the choice of the field content. It has been adopted in more recent paper [15] aimed at successful leptogenesis.
From the point of generating the neutrino masses, the fermion quintuplet has to be accompanied by an appropriate scalar multiplet. High enough scalar field multiplet [13] avoids the terms in the scalar potential which may lead to and induced vacuum expectation value (vev) of the new scalar field, and there are only radiative neutrino masses. This RνMDM is realized by accompanying the zero hypercharge quintuplets Σ ∼ (5, 0) by a hypercharge one sextuplets, Φ ∼ (6, 1), proposing the neutral component Σ 0 as fermionic DM candidate.
In Section 2 we give a brief description of this attempt and explicate a previously omitted term in the scalar potential, which spoils the stability of Σ 0 particle. In Section 3 we analyze two new seesaw models, the first in which a tentative DM particle appears as a neutral component of the scalar quintuplet Φ ∼ (5, 0), accompanied by a quadruplet seesaw mediator Σ ∼ (4, 1). In the second model we show that originally viable MDM scalar septuplet Φ ∼ (7, 0) candidate also becomes unstable in the context of generating radiative neutrino masses. We summarize our results in the concluding Section.
Instability of fermionic quintuplet RνMDM
To allow a Yukawa coupling of the isospin two fermion Σ ∼ (5, 0) to the lepton doublet L L ∼ (2, −1), necessary for a mass generation mechanism,
L Y = L L Y ΦΣ + H.c. ,(1)
one has to introduce a scalar field Φ of a half-integer (3/2 or 5/2) isospin in addition to the SM higgs H ∼ (2, 1). In order to avoid a vev of Φ which might come from quartic H 3 Φ term, the fermionic quintuplet Σ ∼ (5, 0) has to be accompanied by a sextuplet scalar Φ ∼ (6, −1) as in [13]. These authors have argued that the neutral component of the fermionic quintuplet, Σ 0 , can be a DM candidate on account of an accidental Z 2 symmetry of the Lagrangian. We show that it is a viable DM candidate only if one forbids or finetunes a renormalizable HΦ 3 term with the quartic coupling λ, omitted in [13]. In their tensor notation, the omitted term reads
λΦ * Φ * ΦH * + H.c. , Φ * Φ * ΦH * = Φ * iabcd Φ * pqrst Φ abpqr H * n ǫ in ǫ cs ǫ dt ,(2)
and it can induce a rapid decay of a DM particle at the loop level. For example, the loop diagram on Fig. 1 generates an effective dimension-six operator
O 6 = L L i Σ jklm W l a W m b H * n ǫ ja ǫ kb ǫ in ,(3)
which leads to decays of the DM candidate Σ 0 . In particular, we estimate the decay amplitude for Σ 0 → νW + W − H arising from it,
A ∼ g 2 16π 2 Y λ m Σ m 2 Φ m 2 Σ m 2 W .(4)
It produces the decay width into four final-state particles, which is given by
Γ ∼ 1 2m Σ |A| 2 dLIPS 4 ,(5)Γ ∼ g 4 192π 1 16π 2 4 Y 2 λ 2 m 9 Σ m 4 Φ m 4 W .(6)
The inverse of it, a lifetime of a DM candidate, should be longer then the age of the Universe given by the Hubble time H −1 ≈ 10 17 s. For Σ 0 to have the lifetime τ DM > 10 17 s, the quartic coupling λ has to be tiny. By taking g = 0.65, m Σ = m Φ = 10 TeV and Y = 10 −1 , the values adopted from [13], we obtain λ < 10 −20
as an upper limit. In the context of the decaying DM [16] the bound on the τ DM can be nine orders of magnitude larger (τ DM > 10 26 s), implaying even stronger limit, λ < 10 −24 . This means that Σ 0 is a viable DM candidate only if the coupling in Eq. (2) has an extremely small value. This is in contradiction with the claim in [13], and in order to avoid the fine-tuning one has to impose the Z 2 symmetry forbidding the renormalizable term in Eq. (2), by hand. Without such discrete symmetry a model more minimal than in [13] is possible with scalar quadruplet Φ ∼ (4, −1) replacing previous sextuplet field. Like in case of previously studied non-zero hypercharge fermion [10,11], this field Φ develops an induced vev from the quartic ΦH 3 term. The Yukawa terms result in a tree-level dimension-nine seesaw operator. These and additional contributions to the neutrino masses from radiative loopsuppressed diagrams are presented in a separate paper [17].
RνMDM Variants with Scalar MDM
In the light of the described blow to fermionic MDM candidate in the radiative neutrino masses setup, let us now reconsider the scalar MDM multiplets [14,18,19]. The models involving hypercharge-zero scalar field in radiatively generating neutrino masses through diagrams displayed on Fig. 2 have two essential features: (i) the scalar field is accompanied with appropriate fermion multiplet; (ii) there are two scalar fields of different hypercharge in play, the one with hypercharge zero being the DM candidate.
ν Σ Σ ν Φ Φ
Model with scalar quintuplet DM
A minimal possibility to realize the diagram on Fig. 2 through the scalar quintuplet Φ 1 ∼ (5, 0) as MDM candidate is to accompany it with another scalar quintuplet Φ 2 ∼ (5, 2) in conjunction with non-zero hypercharge fermionic quadruplet Σ ∼ (4, 1) as the seesaw mediator. With two scalars Φ 1 ∼ (5, 0) and Φ 2 ∼ (5, 2), there are, up to our knowledge previously unconsidered, dimension-three Z 2 noninvariant operators in the Lagrangian
L S = µ 1 Φ 1 Φ 1 Φ 1 + µ 2 Φ 1 Φ 2 Φ * 2 ,(8)
which in tensor notation read
Φ 1 Φ 1 Φ 1 = Φ 1ijkl Φ 1mnpq Φ 1rstu ǫ im ǫ jn ǫ kr ǫ ls ǫ pt ǫ qu , Φ 1 Φ 2 Φ * 2 = Φ 1ijkl Φ 2mnpq Φ * klpq 2 ǫ im ǫ jn .(9)
These terms make the DM candidate Φ 0 1 unstable through loop diagrams like the one on Fig. 3. We estimate the amplitude for the decay Φ 0
1 → W + W − A ∼ g 2 16π 2 (µ 1 + µ 2 ) m 2 Φ 1 m 2 W(10)
and the decay width
Γ ∼ g 4 16π 1 16π 2 2 (µ 1 + µ 2 ) 2 m 3 Φ 1 m 4 W .(11)
To have the lifetime of a DM candidate τ DM > 10 17 s, the parameters µ 1,2 in Eq. (8) have to have very small values, µ 1,2 < 10 −10 eV.
W + W − Φ 1 Φ 1,2 Φ 1,2
Model with scalar septuplet DM
The second possibility to realize the diagram on Fig. 2 is through the scalar septuplet Φ 1 ∼ (7, 0) as MDM, together with another scalar septuplet Φ 2 ∼ (7, 2), now in conjunction with fermionic sextuplet Σ ∼ (6, 1). For a single scalar septuplet Φ 1 , the term Φ 3 1 is forbidden by Bose statistics [18], but in presence of another septuplet Φ 2 there is the dimension-three Z 2 noninvariant operator
L S = µΦ 1 Φ 2 Φ * 2 , Φ 1 Φ 2 Φ * 2 = Φ 1ijklmn Φ 2pqrstu Φ * lmnstu 2 ǫ ip ǫ jq ǫ kr ,(12)
making the DM candidate Φ 0 1 unstable. These terms make the DM candidate Φ 0 1 unstable through loop diagrams, like the one on Fig. 4. We estimate the amplitude for the decay Φ 0
1 → W + W − Z 0 A ∼ g 3 16π 2 µ m Φ 1 m 3 Φ 1 m 3 W ,(13)
and the decay width To have the lifetime of a DM candidate τ DM > 10 17 s, the parameter µ in Eq. (12) is restricted to very small value, µ < 10 −11 eV. To conclude, a major blow to scalar RνMDM variants comes from superrenormalizable dimension-three Φ 3 operators. Besides these renormalizable Z 2 violating operators, there are possible effects from the Planck scale considered in [14,16] and references therein.
Γ ∼ g 6 32π 1 16π 2 3 µ 2 m 5 Φ 1 m 6 W .(14)W + W − Φ 1 Φ 2 Φ 2 Z 0 Φ 2
Conclusions
We address in a common framework the open questions of the neutrino masses and the dark matter content of the Universe. In view of the immense DM possibilities, a minimality of the DM model appears as a criterion essential for its predictivity and testability. The minimality is satisfied by adding only one extra SU(2) L multiplet in the MDM setup. However, by imposing additional seesaw mission to a given MDM multiplet, it has to be accompanied by additional scalar field introducing new obstacles to the DM stability. It would be appealing if the DM candidate, a neutral component of a certain higher weak-isospin multiplet, could be stabilized without introducing a symmetry beyond the SM gauge symmetries, as claimed in [13]. We have shown that this is not possible for already selected MDM particles, if the neutrino masses are generated through simple one-loop diagrams on Fig. 2. The obstacle for the fermionic quintuplet MDM candidate in [13] comes from renormalizable quartic term omitted there. Concerning the scalar RνMDM variants, we have shown that the stability of the scalar quintuplet and septuplet MDM candidate is violated by ubiquitous super-renormalizable terms that can be tamed only by imposing extra discrete symmetry.
Figure 1 :
1An example of decays of the heavy lepton Σ at the loop level.
Figure 2 :
2Diagram for radiatively generated neutrino masses where one of the scalar fields contains the MDM component.
Figure 3 :
3An example of the loop diagram for quintuplet Φ 1 decays.
Figure 4 :
4An example of the loop diagram for septuplet Φ 1 decays.
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| {'fraction_non_alphanumeric': 0.06362621591452719, 'fraction_numerical': 0.05862967097220007, 'mean_word_length': 3.8943808532778355, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 19, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We examine the stability of minimal dark matter (MDM) particlecandidates in the setup in which they participate in radiative neutrino (Rν) masses. We first point out the existence of an additional renormalizable term in recently proposed RνMDM Lagrangian, which violates the claimed accidental Z 2 symmetry and spoils the stability of the fermionic MDM quintuplet component. We then explore the viability of RνMDM variants based on scalar MDM multiplets. There are ubiquitous super-renormalizable terms in the scalar potential which make these scalar multiplets unstable.', 'arxivid': '1204.6597', 'author': ['Krešimir Kumerički \nDepartment of Physics\nFaculty of Science\nUniversity of Zagreb\nP.O.B. 331HR-10002ZagrebCroatia\n', 'Ivica Picek \nDepartment of Physics\nFaculty of Science\nUniversity of Zagreb\nP.O.B. 331HR-10002ZagrebCroatia\n', 'Branimir Radovčić \nDepartment of Physics\nFaculty of Science\nUniversity of Zagreb\nP.O.B. 331HR-10002ZagrebCroatia\n'], 'authoraffiliation': ['Department of Physics\nFaculty of Science\nUniversity of Zagreb\nP.O.B. 331HR-10002ZagrebCroatia', 'Department of Physics\nFaculty of Science\nUniversity of Zagreb\nP.O.B. 331HR-10002ZagrebCroatia', 'Department of Physics\nFaculty of Science\nUniversity of Zagreb\nP.O.B. 331HR-10002ZagrebCroatia'], 'corpusid': 118732499, 'doi': '10.1007/jhep07(2012)039', 'github_urls': [], 'n_tokens_mistral': 6967, 'n_tokens_neox': 5740, 'n_words': 3237, 'pdfsha': '9ab9a849c0fd0a900cc2e77592877695f48f7153', 'pdfurls': ['https://arxiv.org/pdf/1204.6597v2.pdf'], 'title': ['Critique of Fermionic RνMDM and its Scalar Variants', 'Critique of Fermionic RνMDM and its Scalar Variants'], 'venue': []} |
arxiv |
GENERALIZED CALIBRATIONS
Sep 1999
Jan Gutowski [email protected]
DAMTP
University of Cambridge
Silver StreetCB3 9EWCambridge
GENERALIZED CALIBRATIONS
Sep 1999arXiv:hep-th/9909096v1 14
We present a generalization of calibrations in which the calibration form is not closed. We use this to examine a class of supersymmetric p-brane worldvolume solitons.As an example we consider M5-brane worldvolume solitons in an AdS background.
Introduction. There has been considerable progress recently in the classification of p-brane worldvolume solitons. In a flat background with no Born-Infeld type fields the worldvolume dynamics are governed by the Nambu-Goto action
S N G = d p+1 x − det g (1.1)
where g denotes the pull-back of the background metric to the worldvolume. For such configurations this Lagrangian is equal to the energy density of the p-brane and solutions to the equations of motion minimize both the volume and the energy. A large number of solutions have been classified in terms of calibrated geometries [1], [2], [3], [4]. The solutions are described by calibration forms defined on the target space, and the brane worldvolumes correspond to calibrated sub-manifolds of the target space. Many examples have been constructed, and the types of geometry encountered are typically Kähler, special Lagrangian and exceptional. The resulting solitons may be seen to arise from multiple intersections of p-branes which preserve some proportion of the supersymmetry depending on the geometry of the contact set. However there are limitations to the methods used here, they do not allow treatment of configurations with non-vanishing Born-Infeld fields and they do not describe solitons in curved backgrounds 1 whose worldvolume actions are modified by the presence of Wess-Zumino terms.
In this article we present a generalization of the concept of a calibrated geometry which enables us to describe a class of solitons in a curved background but with vanishing Born-Infeld fields. It is necessary to modify the definition of the calibration form as in a curved background the energy is not equal to the p-brane volume.
Generalized Calibrations. Let (M, g) be an n-dimensional Riemannian manifold with metric g, and x ∈ M . Suppose that G(p, T x (M )) is the Grassmannian of (oriented) p-planes in T x (M ). Then for χ ∈ G(p, T x (M )), there exists an orthonormal basis with respect to g,
{e 1 ...e n } of T x (M ) such that {e 1 ...e p } is a basis of χ. The co-volume of χ is then defined as → χ = e 1 ∧ ... ∧ e p . (1.2)
For our definition of a generalized calibration (referred to from now on as simply a calibration), we drop the standard requirement of closure for the calibration form.
A calibration of degree p on an open subset U ⊂ T x (M ) is a p-form φ such that, at each x ∈ U , φ x ( → χ) ≤ 1 for all χ ∈ G(p, T x (M )
). It is also required that the contact set G(φ) should be non-empty, where
G(φ) = {χ ∈ G(p, T x (M )) : φ( → χ) ≤ 1}. (1.3)
Suppose now that N is a p-dimensional submanifold of M . Then N is a calibrated submanifold (or calibration for short) of degree p if
φ x ( → N x ) = 1 (1.4)
for all x ∈ N ,where φ is a calibration of degree p in T (M ).
→ N x is the co-volume of the tangent space T x N .
As we have removed the requirement that dφ = 0 should hold, it is not the case generally that N is volume minimizing. However another quantity is minimized. Suppose N is calibrated and U is an open submanifold of N and V is an open sub-manifold of M such that ∂U = ∂V . Then let L be a manifold with oriented boundary ∂L = U − V . We may then write
µ U = φ( → U )µ U = U φ = V φ + L dφ ≤ µ V + L dφ (1.5)
where we have used Stokes' theorem and µ denotes the volume form. So the new minimized quantity is
d p x det(g) − B dφ (1.6)
where B is a manifold whose boundary is N and g is the pull-back of the metric to N. This quantity is of considerable interest in theories for which the Born-Infeld fields vanish, and there is a non-trivial Wess-Zumino term; such as p-branes in a curved supergravity background, which we consider in the next section. For these theories, N is identified with some spatial submanifold of the worldvolume and ∂B = N .
Examples. We consider worldvolume solitons on a M5-brane in the background of a stack of parallel M5-branes in the near horizon limit, so the background metric and 4-form are given by
ds 2 = r R ds 2 (E 5,1 ) + R 2 r 2 (dr 2 + r 2 ds 2 (S 4 )) (1.7) G 4 = µ S 4 . (1.8)
R is a positive constant and the above geometry is AdS 7 × S 4 . We work in the static gauge and consider solutions which depend only on the 5 − q worldvolume co-ordinates {x i : i = 1, . . . , 5 − q}. The worldvolume action may then be written as
S = λ d 5−q x det(g ij ) − F (1.9)
for constant λ and
g ij = r R q+1 5−q r R δ ij + R 2 r 2 δ ab ∂ i y a ∂ j y b . (1.10)
The y a are the transverse scalars. We remark that by adopting an anzatz in which we set y 5 = 0 we may set the pull-back of the background 3form to the M5-brane worldvolume to zero so the effective worldvolume action is indeed (1.9). We therefore consider solitons with 2,3 or 4 active transverse scalars. The definition of the calibration forms proceeds in exactly the same manner as for the flat computations with the co-ordinate basis replaced by an orthonormal basis defined with respect tog. Thus for example the SU (4) Kahler calibration generalizes to a SU (4) Hermitian calibration again preserving 1 16 of the supersymmetry. The calibration form is φ = r 3 R 3 dx 1 ∧dx 2 ∧dx 3 ∧dx 4 +dx 1 ∧dx 2 ∧dy 1 ∧dy 2 +dx 3 ∧dx 4 ∧dy 1 ∧dy 2 .
(1.11) It is required that X 1 + iX 2 and X 3 + iX 4 should be holomorphic functions of x 1 +ix 2 and x 3 +ix 4 . More interesting examples may be obtained by considering generalizations of special Lagrangian and exceptional geometries [5]. In all cases it is straightforward to verify that dφ =F so that the equations of motion are satisfied. All of the examples in this background are supersymmetric, i.e.
Γǫ = ǫ (1.12)
where for a p-brane with vanishing Born-Infeld fields
Γ = 1 (p + 1)! ǫ µ 1 ...µ p+1 γ µ 1 . . . γ µ p+1 (1.13)
and ǫ is a Killing spinor [6]. It has been shown that just as for the flat background the calibration form may be constructed from these Killing spinors satisfying appropriate constraints [7]. Moreover the relation dφ =F may be seen to arise as a consequence of the supersymmetry algebra.
The methods outlined here may be applied to p-brane configurations in a large number of backgrounds for which the Born-Infeld type fields vanish. In addition, an extension of this treatment has been presented in [8] which includes these fields.
AcknowledgmentsI thank EPSRC for a studentship and the organizers for an excellent school.
Calibrated Geometries. R Harvey, H B Lawson, Acta. Math. 14847R.Harvey and H.B. Lawson, Calibrated Geometries, Acta. Math. 148 (1982) 47.
Calibrations and Intersecting Branes. G W Gibbons, G Papadopoulos, Commun. Math. Phys. 202593G.W. Gibbons and G. Papadopoulos, Calibrations and Intersecting Branes, Commun. Math. Phys. 202 (1999) 593.
Branes and Calibrated Geometries. J P Gauntlett, N D Lambert, P C West, Commun. Math. Phys. 202571J.P.Gauntlett, N.D.Lambert and P.C. West, Branes and Calibrated Geometries, Commun. Math. Phys. 202 (1999) 571.
Branes at Angles and Calibrations. B S Acharya, J M Figueroa-O'farrill, B Spence, JHEP. 0412B.S. Acharya, J.M. Figueroa-O'Farrill and B.Spence, Branes at Angles and Cal- ibrations, JHEP 04:012 (1998).
AdS Calibrations. J B Gutowski, G Papadopoulos, hep-th/9902034J.B. Gutowski and G.Papadopoulos, AdS Calibrations, hep-th/9902034.
κ-symmetry, Supersymmetry and Intersecting Branes. E Bergshoeff, R Kallosh, T Ortín, G Papadopoulos, Nucl. Phys. 502149E. Bergshoeff, R. Kallosh, T. Ortín and G. Papadopoulos, κ-symmetry, Super- symmetry and Intersecting Branes, Nucl. Phys. B502 (1997) 149.
Supersymmetry and Generalized Calibrations. J B Gutowski, G Papadopoulos, P K Townsend, hep-th/9905156J.B. Gutowski, G.Papadopoulos and P.K. Townsend, Supersymmetry and Gen- eralized Calibrations, hep-th/9905156.
O Baerwald, N D Lambert, P C West, hep-th/ 9907170A Calibration Bound for the M-Theory Fivebrane. O. Baerwald, N.D. Lambert and P.C. West, A Calibration Bound for the M- Theory Fivebrane, hep-th/ 9907170.
| {'fraction_non_alphanumeric': 0.061627347135291284, 'fraction_numerical': 0.030572941742898412, 'mean_word_length': 3.7317767653758542, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We present a generalization of calibrations in which the calibration form is not closed. We use this to examine a class of supersymmetric p-brane worldvolume solitons.As an example we consider M5-brane worldvolume solitons in an AdS background.', 'arxivid': 'hep-th/9909096', 'author': ['Jan Gutowski [email protected] \nDAMTP\nUniversity of Cambridge\nSilver StreetCB3 9EWCambridge\n'], 'authoraffiliation': ['DAMTP\nUniversity of Cambridge\nSilver StreetCB3 9EWCambridge'], 'corpusid': 15353543, 'doi': '10.1007/978-94-010-0852-5_19', 'github_urls': [], 'n_tokens_mistral': 2807, 'n_tokens_neox': 2402, 'n_words': 1445, 'pdfsha': '1b6ea381343f8a9510513b4a9556f8038f71ea3c', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/9909096v1.pdf'], 'title': ['GENERALIZED CALIBRATIONS', 'GENERALIZED CALIBRATIONS'], 'venue': []} |
arxiv |
New Families of Large Band Gap 2D Topological Insulators in Ethynyl-Derivative Functionalized Compounds
Lauryn Wu
Thomas Jefferson High School for Science and Technology
22312AlexandriaVAUnited States of America
Department of Electrical and Computer Engineering
George Mason University
22030FairfaxVAUnited States of America
Kunming Gu
Department of Electrical and Computer Engineering
George Mason University
22030FairfaxVAUnited States of America
Key Laboratory of Advanced Functional Material
Material School of Shenzhen University
518060Shenzhen, ShenzhenGuangdongPeople's Republic of China
Qiliang Li
Department of Electrical and Computer Engineering
George Mason University
22030FairfaxVAUnited States of America
New Families of Large Band Gap 2D Topological Insulators in Ethynyl-Derivative Functionalized Compounds
The search for large band gap systems with dissipationless edge states is essential to developing materials that function under a wide range of temperatures. Two-dimensional (2D) topological insulators (TIs) have recently attracted significant attention due to their dissipationless transport, robust properties and excellent compatibility with device integration. However, a major barrier of 2D TIs is their small bulk band gap, which allows for applications only in extremely low temperatures. In this work, first principle calculations were used to analyze the geometric, electronic, and topological properties of PbC2X and BiC2X (X = H, Cl, F, Br, I) compounds. The band gap values are remarkably large, ranging from 0.79eV to 0.99eV. The nanoribbons of these compounds exhibited nontrivial topological order in the simulation, thus proving ethynylderivative functionalized Pb and Bi films to be new classes of giant band gap 2D TIs. In addition, these findings indicate that chemical functionalization with ethynyl-derivatives is an effective method to tune the band gap and preserve the nontrivial topological order. These novel materials that are applicable at both room temperature and high temperatures open the door to a new generation of electronics.
I. Introduction
Topological insulators (TIs), with the remarkable quantum spin Hall (QSH) effect, have recently shown great potential in revolutionizing the field of materials science. [1,2,3] While behaving like ordinary insulators in the bulk, QSH insulators have metallic states on the edge and are protected by time-reversal symmetry. One of their remarkable characteristics is the absence of backscattering, so electronic currents can flow without dissipation. Furthermore, TIs retain their distinctive properties even when diluted with impurities. Due to their versatility, the QSH insulators are promising in technological applications, such as quantum computing and spintronics. [4] Despite the fascinating applications of TIs, the lack of large band gap in the bulk is hindering the advancement. Thus far, many three dimensional (3D) compounds, such as Bi2Se3, Bi2Te3, and Sb2Te3, have already been experimentally confirmed to be QSH insulators. [1,5,6,7] On the other hand, the small bulk band gap of TI materials impedes their realistic application.
The first material anticipated to be Two dimensional (2D) TI was graphene [8], but weak spinorbit coupling (SOC) and a small band gap caused TI properties to only be observable in temperatures below 0.01 K. [9,10] The first experimentally observed and tested 2D TI materials, the quantum wells HgTe/CdTe and InAs/GaSb were similarly observable only at extremely low temperatures. [11,12,13] However, compared to 3D materials, 2D TIs have more advantageous features, including better flexibility and being easier to integrate into current electronics. [14] 2D materials can be readily integrated by the well-developed microfabrication technologies for high performance and high-density logic and memory devices. Furthermore, the surface of 3D TIs is not protected against backscattering in any direction other than 180°, whereas 2D TIs have robust edge states that prevent backscattering. [2] In addition, the chemical bonding in 2D structures can be easily modified in post synthesis processes to tune the band gap or achieve certain properties. However, the tiny band gaps in 2D TIs are impeding their progress. Recently, many studies have attempted to investigate 2D TIs with large band gaps for room-temperature applications. [15,16,17,18,19,20,21,22,23,24] Some graphene-like 2D honeycomb structures have been proposed to be QSH insulators, including silicene [25], germanene [26], and stanene. [2] However, the quest for 2D TIs that possess a large band gap is a difficult challenge in order to realize the QSH effect at room temperature.
Several methods can increase the band gap in 2D TI systems, such as placing the materials on a substrate and chemical functionalization. By placing the structure on a substrate, the electronic structure is modified by the interaction between the material and substrate, possibly destroying the topological order. Chemical functionalization of TI monolayers, on the other hand, is a very effective method to widen the band gap and improve structural stability, while preserving the nontrivial topological order. Pristine stanene, for example, has a calculated band gap of 0.1 eV, but with the addition of functional groups, the band gap reached 0.3 eV. [2] Furthermore, the bonds can be easily modified, and therefore the materials can be experimentally realized. [19,2,27,28] Materials that use heavy atoms, such as Pb and Bi, are favorable in TIs because heavier atoms generally have stronger SOC, leading to a larger band gap. Pb is the heaviest element in group-IV, so it may drive a nontrivial topological order and a large band gap. Despite its huge capability, lead films have not been extensively studied, especially compared to other 2D group-IV honeycomb structures such as graphene and stanene. [29,2,30,31,8,32,33] On the other hand, Bi is the largest element in group-V and has an unusually low toxicity for a heavy metal. It is also known for its strong SOC and its ability to drive a material to a nontrivial topological state. [20] For these reasons, Pb and Bi were chosen to be the base atoms in the study. Moreover, recent studies found that hydrogenation and fluorination of materials have led to quickly increasing lattice disarrangement and defects. [34] Thus, focus has shifted to decorating 2D films with small molecules. [35,36,37] Ethynyl-derivatives (C2H, C2F, C2Cl, C2Br, C2I) are excellent options for decorating the surface of 2D structures to enhance the geometric stability, increase the band gap, and preserve a nontrivial topology.
In this work, the structural, electronic, and topological properties of PbC2X and BiC2X (X = H, F, Cl, Br, I) monolayers are investigated based on first principle calculations. New classes of QSH insulators in ethynyl-derivative functionalized Pb and Bi monolayers were identified.
The analyzed 2D materials achieve substantial band gaps that are large enough to operate in room temperature and high temperature applications. Furthermore, nanoribbon calculations confirm the nontrivial topological order and conducting edge states of the materials. These newly discovered robust TIs are capable for practical use in quantum and electronic devices.
II. Methods
To investigate the structural and electronic properties, first principle calculations were performed based on the Density Functional Theory (DFT), as implemented in the Virtual Nanolab Atomistix ToolKit (ATK) package. The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) pseudopotential method with the SG15 Optimized Norm-Conserving Vanderbilt (ONCV) basis was used. [38,39,40,41] The mesh energy cutoff was 75
Ha and the total energy convergence criteria was 10 -6 Ha. For unit cell calculations, the integration over the Brillouin zone was sampled with a 11×11×1 -centered Monkhorst-Pack grid. The nanoribbon calculations were completed with a 1×11×1 k-mesh. The vacuum region was set to 20 Å to minimize artificial interactions between periodic layers. SOC was considered in the self-consistent calculations. Structural relaxation was performed until the forces on each atom were less than 0.01 eV/Å. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method was employed for geometry optimization and all structures were fully relaxed. The two center atoms are bonded with ethynyl-derivatives alternating on each side. Like graphene, the hexagonal lattice structure has a threefold rotational symmetry, and therefore has inversion symmetry. However, unlike graphene, the chemically functionalized systems prefer a low buckled configuration, with the two center (Pb or Bi) atoms in the unit cell on different planes.
III. Results and Discussion
For each material, a fully relaxed structure was achieved in the calculation, ensuring structural stability for all systems. Table 1 The band structures of all Bi based monolayers, as shown in Figure 3, were also calculated to examine the electronic and topological properties. Like the Pb film, the Bi monolayer has no band gap, but unlike the Pb monolayer, there is a large gap at the point in the Bi band structure. In addition, at the K point, the bands cross twice linearly, once above and once below the Fermi level, in the pure Bi film. With the inclusion of SOC, the energy-degeneracy of the bands around the Fermi level is lifted. Due to the strong SOC of Bi atoms, the band gap of the pure Bi system increases to 0.53eV. The band gap becomes indirect, with the valence band maximum and conduction band minimum near the K point.
Like the functionalized Pb monolayer films, the decorated Bi monolayers have no band gap without SOC. In the absence of SOC, the bands cross linearly once at the K point, forming a Dirac cone with the Fermi level crossing the Dirac point. When SOC is induced, the band gaps at the K point increases considerably. The huge band gaps of the functionalized Pb systems range from 0.93eV to 0.99eV, which exceeds the thermal energy at room temperature. Thus, these materials are predicted to not only be sustainable in room temperature devices, but also viable in high temperature electronics. The specific band gap values can be found in Table 1.
Figure 1
1displays the geometric structure of functionalized Pb and Bi monolayer films.
lists the structural parameters, including the optimized lattice constants, buckling heights, bond lengths, and band gaps. The lattice parameter of all the ethynyl-derivative functionalized materials is larger than the pure monolayer. The buckling height for the functionalized Pb films is decreased from the pristine Pb monolayer, but the buckling height increases in the functionalized Bi films, compared to the free-standing Bi monolayer. The bond length between the central atom and C2X does not have a strong correlation to the X atom, and the bond length in all the Pb systems was 2.22 Å.To analyze the electronic properties of the systems, band structures with and without SOC were calculated. All Pb based materials in this study have direct band gaps of 0.0 eV without SOC, located at the point. Thus, they all display a gapless semiconductor-like nature in the absence of SOC. This is also referred to as a semimetal, with a degenerate conduction band minimum and valence band maximum at the Fermi level. As confirmed byFigure2, with SOC, the band gaps open significantly for all the materials in the family, and this can be attributed to the strong SOC of Pb. The SOC effect lifts the energy-degeneracy at the point, with the conduction band minimum upshifted and the valence band maximum downshifted. The specific values of the band gap for each system can be found in Table 1. The SOC induced band gap opening at the Fermi level is an indicator for a topologically nontrivial material. Furthermore, in the PbC2X materials, band inversions are present. Band inversions appear with the inclusion of SOC, which is another strong evidence of nontrivial topology. Without SOC, the band gaps of all the materials are direct, but the band gaps become indirect in the presence of SOC due to the band inversions. Compared to the free-standing Pb monolayer, the chemically functionalized Pb systems do not have energy-band cross at the Fermi level at the K point, in cases with and without SOC. The functionalized structures also have significantly larger band gaps when SOC is turned on, varying from 0.79eV to 0.87eV, compared to the band gap of 0.37 eV in pure Pb. This shows that chemical functionalization of 2D structures using ethynyl-derivatives is an effective method to tune the band gap. Because these band gap values are massive in comparison to the thermal energy at room temperature (≈ 0.026 eV), ethynyl-derivative functionalized Pb systems are excellent TI material candidates for practical applications.
Interestingly, with SOC, the CBM is downshifted at the point and the VBM remains at the K point, causing an indirect band gap in all BiC2X materials. Although there is no band inversion in the functionalized Bi systems, the band gap opening in the presence of SOC indicates a TI nature. The hallmark of 2D QSH insulators is their conducting edge states. The nontrivial topological order of these materials can be proved by the existence of protected gapless edge states. To confirm the nontrivial topology of the systems in the study, the electronic structures of zigzag nanoribbons were calculated. The unit cell of the ribbons was created by transforming the previously calculated slab structures, as seen in Figure 4. The same computational methods used for the bulk structure were employed, with the exception of the k mesh. The vacuum region of 20 Å in the Z direction was maintained, and additional vacuum regions of about 10 Å beside each edge of the nanoribbon were considered to avoid interactions between periodic images. The unit cell of the ribbon had a width of approximately 8nm to prevent interactions between the two edges and all dangling bonds were passivated by hydrogen atoms. The zigzag nanoribbon band structures were calculated with the inclusion of SOC. The BiC2I nanoribbon band structure is shown in Figure 4. The the valence and conduction bands are connected, crossing linearly at the point. The nanoribbon exhibits a Dirac cone, which verifies the topologically protected conducting edge states. One pair of edge states traverses the bulk band gap, so the valence and conduction bands are connected. Therefore, the Dirac point at the point in the BiC2I nanoribbon signifies that it is a QSH insulator with the key feature of conducting edge states. The presence of the gapless edge states in the nanoribbon agrees with the SOC induced band gap and band inversion in the bulk band structures, proving that these materials are QSH insulators. As shown in Figure 5, the calculated current vs. bias (I-V) curve of the nanoribbon is quite linear with excellent conductivity, further demonstrating the metallic conduction. Such edge states allow for dissipationless electron transport, without backscattering, along the boundary of a material.IV. ConclusionsFirst principles calculations were performed on PbC2X and BiC2X (X = H, Cl, F, Br, I) compounds using DFT. Based on the calculations, we predict new series of 2D TIs and find that chemical functionalization using ethynyl-derivatives is an effective mechanism for tuning the band gap of 2D TIs. The band gap of all functionalized materials, ranging from 0.79eV to 0.99eV, are substantially larger than the band gap of the corresponding pristine monolayer. The largest band gap obtained in this study, 0.99eV, was found in BiC2Br. All studied systems have giant band gap values that are large enough to be sustainable both in room temperature and high temperatures. The SOC induced band gap and band inversion observed in the bulk band structures indicate a nontrivial topological order, and the nanoribbon band structure calculations further confirm the TI properties. The presence of a Dirac point in the bulk gap of the nanoribbon implies conductive, topologically protected edge states and proves a nontrivial phase. As an actively researched topic in the recent materials science community, TIs have a huge potential in future technology. Their applications include quantum computers, spintronic devices, thermoelectric devices, infrared detectors, exotic particles, image monopoles, and the realization of Majorana Fermions. The 2D materials in this research have several advantageous characteristics, including easy integration into electronic devices, robust edge states without backscattering, easily modifiable bonds, strong SOC, and huge band gaps. This study on 2D TIs offers new and promising options for dissipationless electronic applications at room temperature. Captions: Table 1. The lattice constants a (Å), buckling heights h (Å), central atom to C2X bond lengths d (Å), band gaps without SOC Eg (eV), and SOC induced band gaps Eg-SOC (eV) of all 2D structures in this study.
Figure 1 .
1The hexagonal lattice structure of ethynyl-derivative functionalized 2D systems from the top and side views. The gray atoms represent the center atoms (Pb or Bi); the green atoms represent C, and the red atoms represent H, F, Cl, Br, or I.
Figure 2 .
2Calculated band structures of functionalized Pb films are shown. The first, third, and fifth columns show band structures without SOC, while the second, fourth, and sixth columns show band structures with SOC. The Fermi level is indicated by the dotted green line.
Figure 3 .
3Calculated band structures of functionalized Bi films are shown. The first, third, and fifth columns show band structures without SOC, while the second, fourth, and sixth columns show band structures with SOC. The Fermi level is indicated by the dotted green line.
Figure 4 .
4(a) Calculated zigzag nanoribbon band structure of a 2D BiC2I monolayer. (b) The zigzag nanoribbon geometric structure that was used to calculate the edge states and current vs. bias characteristics.
Figure 5 .
5The calculated current vs. bias (I-V) curve of the BiC2I zigzag nanoribbon.
Figure 1 .
1The hexagonal lattice structure of ethynyl-derivative functionalized 2D systems from the top and side views. The gray atoms represent the center atoms (Pb or Bi); the green atoms represent C, and the red atoms represent H, F, Cl, Br, or I.
Figure 2
2Figure 2
Figure 2 .
2Calculated band structures of functionalized Pb films are shown. The first, third, and fifth columns show band structures without SOC, while the second, fourth, and sixth columns show band structures with SOC. The Fermi level is indicated by the dotted green line.
Figure 3
3Figure 3
Figure 3 .
3Calculated band structures of functionalized Bi films are shown. The first, third, and fifth columns show band structures without SOC, while the second, fourth, and sixth columns show band structures with SOC. The Fermi level is indicated by the dotted green line.
Figure 4
4Figure 4
Figure 4 .
4(a) Calculated zigzag nanoribbon band structure of a 2D BiC2I monolayer. (b) The zigzag nanoribbon geometric structure that was used to calculate the edge states and current vs. bias characteristics.
Figure 5
5Figure 5
Figure 5 .
5The calculated current vs. bias (I-V) curve of the BiC2I zigzag nanoribbon.
Table Table 1
Table. The lattice constants a (Å), buckling heights h (Å), central atom to C2X bond lengths d (Å), band gaps without SOC Eg (eV), and SOC induced band gaps Eg-SOC (eV) of all 2D structures in this study.System
a(Å)
h(Å)
d(Å)
E g (eV)
E g-SOC (eV)
Pb
5.00
1.01
-
0.00
0.37
PbC 2 H
5.25
0.60
2.22
0.00
0.88
PbC 2 F
5.26
0.69
2.22
0.00
0.87
PbC 2 Cl
5.27
0.58
2.22
0.00
0.79
PbC 2 Br
5.23
0.59
2.22
0.00
0.84
PbC 2 I
5.23
0.60
2.22
0.00
0.84
Bi
5.30
0.00
-
0.00
0.53
BiC 2 H
5.54
0.01
2.22
0.00
0.96
BiC 2 F
5.54
0.06
2.20
0.00
0.93
BiC 2 Cl
5.54
0.04
2.20
0.00
0.93
BiC 2 Br
5.52
0.10
2.20
0.00
0.99
BiC 2 I
5.52
0.08
2.12
0.00
0.97
Figures
Figure 1
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| {'fraction_non_alphanumeric': 0.06350346809627767, 'fraction_numerical': 0.03921297703595214, 'mean_word_length': 4.119977362761743, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The search for large band gap systems with dissipationless edge states is essential to developing materials that function under a wide range of temperatures. Two-dimensional (2D) topological insulators (TIs) have recently attracted significant attention due to their dissipationless transport, robust properties and excellent compatibility with device integration. However, a major barrier of 2D TIs is their small bulk band gap, which allows for applications only in extremely low temperatures. In this work, first principle calculations were used to analyze the geometric, electronic, and topological properties of PbC2X and BiC2X (X = H, Cl, F, Br, I) compounds. The band gap values are remarkably large, ranging from 0.79eV to 0.99eV. The nanoribbons of these compounds exhibited nontrivial topological order in the simulation, thus proving ethynylderivative functionalized Pb and Bi films to be new classes of giant band gap 2D TIs. In addition, these findings indicate that chemical functionalization with ethynyl-derivatives is an effective method to tune the band gap and preserve the nontrivial topological order. These novel materials that are applicable at both room temperature and high temperatures open the door to a new generation of electronics.', 'arxivid': '1810.01060', 'author': ['Lauryn Wu \nThomas Jefferson High School for Science and Technology\n22312AlexandriaVAUnited States of America\n\nDepartment of Electrical and Computer Engineering\nGeorge Mason University\n22030FairfaxVAUnited States of America\n', "Kunming Gu \nDepartment of Electrical and Computer Engineering\nGeorge Mason University\n22030FairfaxVAUnited States of America\n\nKey Laboratory of Advanced Functional Material\nMaterial School of Shenzhen University\n518060Shenzhen, ShenzhenGuangdongPeople's Republic of China\n", 'Qiliang Li \nDepartment of Electrical and Computer Engineering\nGeorge Mason University\n22030FairfaxVAUnited States of America\n'], 'authoraffiliation': ['Thomas Jefferson High School for Science and Technology\n22312AlexandriaVAUnited States of America', 'Department of Electrical and Computer Engineering\nGeorge Mason University\n22030FairfaxVAUnited States of America', 'Department of Electrical and Computer Engineering\nGeorge Mason University\n22030FairfaxVAUnited States of America', "Key Laboratory of Advanced Functional Material\nMaterial School of Shenzhen University\n518060Shenzhen, ShenzhenGuangdongPeople's Republic of China", 'Department of Electrical and Computer Engineering\nGeorge Mason University\n22030FairfaxVAUnited States of America'], 'corpusid': 119206601, 'doi': '10.1016/j.apsusc.2017.08.131', 'github_urls': [], 'n_tokens_mistral': 11873, 'n_tokens_neox': 9932, 'n_words': 5749, 'pdfsha': 'd045a5ade6ecd10ab5a1f276efc4a4c2984a15a9', 'pdfurls': ['https://arxiv.org/pdf/1810.01060v1.pdf'], 'title': ['New Families of Large Band Gap 2D Topological Insulators in Ethynyl-Derivative Functionalized Compounds', 'New Families of Large Band Gap 2D Topological Insulators in Ethynyl-Derivative Functionalized Compounds'], 'venue': []} |
arxiv |
MOMENT BOUNDS FOR NON-LINEAR FUNCTIONALS OF THE PERIODOGRAM
31 Jul 2008
Gilles Faÿ
MOMENT BOUNDS FOR NON-LINEAR FUNCTIONALS OF THE PERIODOGRAM
31 Jul 2008arXiv:0807.5096v1 [math.ST]linear processesdiscrete Fourier transformperiodogramlong range dependenceGeweke and Porter-Hudak (GPH) estimator
In this paper, we prove the validity of the Edgeworth expansion of the Discrete Fourier transforms of some linear time series. This result is applied to approach moments of non linear functionals of the periodogram. As an illustration, we give an expression of the mean square error of the Geweke and Porter-Hudak estimator of the long memory parameter. We prove that this estimator is rate optimal, extending the result ofGiraitis, Robinson, and Samarov [1997]from Gaussian to linear processes.
Introduction
Many estimators in time series analysis involve non-linear functionals of the periodogram. Examples include the estimation of the innovation variance [Chen and Hannan, 1980, Lee, Cho, Kim, and Park, 1995, Deo and Chen, 2000, Ginovian, 2003, log-periodogram regression [Taniguchi, 1979, 1991, Shimotsu and Phillips, 2002, robust non-parametric estimation of the spectral density [von Sachs, 1994, Janas andvon Sachs, 1995]. Non-linear functionals of the periodogram also play a predominant role in the analysis of long-memory time-series: one of the much widely used estimator of the memory parameter is based on the regression of the log-periodogram ordinates on the log-frequency [Geweke and Porter-Hudak, 1983, see also Robinson 1995b, Moulines andSoulier 1999].
The statistical analysis of such functionals has proved to be a very challenging problem, due to the intricate dependence structure of periodogram ordinates. The first attempts to study these statistics were made under the additional assumption that the underlying process is Gaussian. Because the Fourier transform coefficients are in this case also Gaussian, one may then apply results on non-linear transforms of Gaussian random variables; see for example Taqqu [1977], Taniguchi [1980] and Arcones [1994].
These techniques do not extent to non-Gaussian processes. A first step to weaken this assumption was taken by Chen and Hannan [1980] who proved the consistency of an additive functional of the log-periodogram of a linear stationary process, with an application to the estimation of the innovation variance. These techniques were based on the so-called Bartlett [1955] expansion; this technique was later improved by Faÿ, Moulines, and Soulier [2002] who proved a central limit theorems for these functionals. It used byVelasco [2000] to establish the weak consistency of the log-periodogram regression estimate of the long memory parameter for long range dependent linear time series. Edgeworth expansions are used to estimates moments of the functional of the unobservable periodogram of the innovation sequence. Remainder Date: July 31, 2008. I am very grateful and indebted to Eric Moulines and Philippe Soulier for their help and the many fruitful discussions we had on this subject some years ago.
terms can be bounded in probability. The Bartlett expansion is indeed useful to establish limit theorems but does not in general allow to determine the moments of these functionals.
An alternative approach has been considered by von Sachs [1994], Janas and von Sachs [1995]. These authors prove the mean-square consistency of general additive functional of nonlinear transforms of the (tapered) periodogram, using Edgeworth expansions of the discrete Fourier transform of the observed time series itself. Janas and von Sachs [1995] apply these results to prove the mean-square consistency of an Huberized (peak insensitive) non-parametric spectral estimator. These results rely on the Edgeworth expansion of a triangular array of strongly mixing process with geometrically mixing coefficient established by Götze and Hipp [1983]. The mixing conditions herein are rather stringent, and thus the conclusions reached by Janas and von Sachs [1995] are proved under a set of restrictive assumptions, precluding for instance their use in a long-memory context.
The main objective of this paper is to develop a method allowing to compute the moments of functionals of non-linear transforms of the (possibly tapered) periodogram of a linear process. These results are based on Edgeworth expansion of a (possibly infinite) triangular array of i.i.d. random variables obtained earlier in Faÿ, Moulines, and Soulier [2004] and recalled in Appendix A. The linearity of the process is then crucial. Our results cover both short-memory and long-memory processes.
The remaining of the paper is organized as follows. In Section 2 we give the assumptions on the linear structure of the time series and define the cumbersome notations related to Edgeworth expansions. In Section 3, we formulate the validity of Edgeworth expansions and moment bounds under short memory set of hypotheses. As an application, we derive the mean-square consistency of additive functionals of non-linear transform of the periodogram for a short-memory linear time-series. In Section 4, we follow the same lines but in a long-range dependence framework, and apply the moment bounds we obtained to control the meansquare error of the Geweke and Porter-Hudak [1983] estimator of the fractional difference parameter for a non-Gaussian linear long-memory process. This extends the rate optimality property of the Geweke and Porter-Hudak (hereafter, GPH) estimator obtained earlier by Giraitis, Robinson, and Samarov [1997] for Gaussian processes. A small Monte-Carlo experiment is run to confirm our results for finite-sample observations. Proofs are postponed to the appendices.
Notations and assumptions
Assume that X = (X t ) t∈Z is a covariance stationary process that have a spectral density f . For any integer r ≥ 0, we define the tapered discrete Fourier transform (DFT) and periodogram of order r as
d r,n (λ) def = (2πna r ) −1/2 n t=1 h r t,n X t e itλ , I r,n (λ) def = |d r,n (λ)| 2 (2.1)
where h t,n def = 1 − e 2iπt/n is the data taper introduced in Hurvich and and a r def = n −1 n t=1 |h t,n | 2r = 2r r is a normalization factor. Denote d r,n,k = d r,n (λ k ) and I r,n,k = I r,n (λ k ) the tapered DFT and tapered periodogram evaluated at the Fourier frequencies λ k def = 2πk n , k = 1, . . . , [(n − 1)/2]. Define for r ∈ N, D r,n (λ) the normalized kernel function
D r,n (λ) def = (na r ) −1/2 n t=1 h r t,n exp(itλ) = (na r ) −1/2 r k=0 r k (−1) k D n (λ + λ k ) (2.2)
where D n (λ) def = n t=1 e −iλt denotes the non-symmetric Dirichlet kernel. The latter relation implies that D r,n (λ k ) = 0 for k ∈ {1, · · · ,ñ}, withñ def = ⌊(n − 2r − 1)/2⌋, so that the tapered Fourier transform is invariant to shift in the mean. As shown in Hurvich and Chen [2000], the decay rate of the kernel in the frequency domain increases with the kernel order, namely
∀λ ∈ [−3π/2, 3π/2] , |D r,n (λ)| ≤ Cn 1/2 (1 + n|λ|) r+1 (2.3)
This property means that higher order kernels are more effective to control frequency leakage. If X is a white noise and r = 0, the DFT ordinates at different Fourier frequencies are uncorrelated. This property is lost by tapering. More precisely, for 1 ≤ k = j ≤ñ, E[d r,n,k d r,n,j ] = 0,
and E[d r,n,k d r,n,j ] def = (2π) −1 ς r (k − j)
wherez denotes the complex conjugate of z and ς r defined in (3.6).
Many statistical applications (see the references given in the Introduction) require to study weighted sums of non-linear functionals of the periodogram ordinates
T n (X, φ) = K k=1 β n,k φ I r,n,k f (λ k ) , (2.4)
where (β n,k ) k∈{1,...,K} is a triangular array of real numbers. If X is a Gaussian white noise, then (I r,n,k ) are i.i.d and the moments of the sum T n (X, φ) can be calculated explicitly. In any other case, the random variables (I r,n,k ) k∈{1,...,K} are not independent, and the calculation of the moments of T n (X, φ) is a difficult problem. The only attempt to solve it has been made by Janas and von Sachs [1995], who proposed a technique to compute moment of order 1 and 2. As already outlined, their results are based on mixing conditions, precluding their use for long-memory processes.
Remark. Sometimes the periodogram ordinates are averaged along blocks of adjacent frequencies. This technique is known as pooling and is appropriate to reduce asymptotic variance of the estimators of non linear functionals of the periodogram (see Robinson, 1995b,a). For simplicity, we will not present any explicit result or application with the pooled periodogram, but the Edgeworth expansion results that follow allow to derive moment bounds on functionals of tapered and pooled periodogram as well.
In this contribution, we focus on non-Gaussian strict sense linear processes, i.e. it is assumed that
X t = j∈Z ψ j Z t−j , j∈Z ψ 2 j < ∞ , (2.5)
where (Z j ) j∈Z is a sequence of i.i.d random variables such that E[Z 1 ] = 0, E[Z 2 1 ] = 1. In addition, for some s ≥ 3, p ≥ 1 and p ′ ≥ 0,
(A1) E[|Z 1 | s ] < ∞ and R |t| p ′ |E[e itZ 1 ]| p dt < ∞ .
Remark. Apart from a classical moment condition, (A1) suppose that the distribution of the i.i.d. noise is smooth; for example, lattice distributions are forbidden. This condition is stronger than the usual Cramér condition. It ensures that the distributions of the Fourier coefficients of Z are eventually continuous. We need this continuity to bound moments of singular functionals of the periodogram. Note that this condition could be dispensed with, were we concerned with smooth functionals.
Define ψ(λ) = j∈Z ψ j e ijλ (the convergence holds in L 2 ([−π, π], dx)) the transfer function of the linear filter (ψ j ) j∈Z and f (λ) = (2π) −1 |ψ(λ)| 2 the spectral density of the process X.
For an integer k ∈ {1, · · · ,ñ} such that f (λ k ) = 0, define the normalized DFT ω r,n,k def = √ 2π d r,n,k /|ψ(λ k )|. Let k 1 < k 2 < . . . < k u be an ordered u-tuple of such integers in the range 1, . . . ,ñ and write k = (k 1 , . . . , k u ). Define (the reference to r is suppressed in the notation) S n (k) def = [Re(ω r,n,k 1 ), Im(ω r,n,k 1 ), · · · , Re(ω r,n,ku ), Im(ω r,n,ku )] .
(2.6)
With those definitions,
I n,k,r = f (λ k )|ω r,n,k | 2 = f (λ k ) S n (k) 2 . (2.7)
Since X admits the linear representation (2.5), S n (k) can be further expressed as a 2udimensional infinite triangular array in the variables (Z t ) t∈Z . Precisely
S n (k) = j∈Z U n,j (k)Z j , (2.8) with U n,j (k) def = (na r ) −1/2 F −1 n (k) n t=1 ψ t−j C n,t (k), (2.9) C n,t (k) def = r p=0
(−1) p r p cos(tλ k 1 +p ), sin(tλ k 1 +p ), . . . , cos(tλ ku+p ), sin(tλ ku+p ) ′ and F n (k) def = diag |ψ(λ k 1 )|, |ψ(λ k 1 )|, . . . , |ψ(λ ku )|, |ψ(λ ku )| .
To formulate our results, some notations related to Edgeworth expansions are required, which we take from the monograph of Bhattacharya and Rao [1976]. For u a positive integer, ν = (ν 1 , . . . , ν u ) ∈ N u and z = (z 1 , . . . , z u ) ∈ C u , denote |ν| = u i=1 ν i , ν! = ν 1 !ν 2 ! · · · ν u ! and z ν = z ν 1 1 z ν 2 2 · · · z νu u . If 1 ≤ |ν| ≤ s, denote χ n,ν (k) the cumulants of S n (k). Then
χ n,ν (k) = κ |ν| j∈Z U ν n,j (k) where κ r denotes the r-th cumulant of Z 1 , r ≤ s. Let V n (k) def = cov[S n (k)] = j∈Z U n,j (k)U ′ n,j (k). Let χ = {χ ν ; ν ∈ N u }
be a set of real numbers. For any integer r ≥ 2 and z ∈ C u , define χ r (z) def = r! |ν|=r χν z ν ν! . The polynomialsP r (z, χ) are formally defined for r ≥ 1 by the identities
1 + ∞ r=1P r (z, χ)t r = exp ∞ r=3 χ r (z) r! t r−2 = 1 + ∞ m=1 1 m! ∞ r=3 χ r (z) r! t r−2 m ,
and we setP 0 ≡ 0. Denote ϕ V the density of a Gaussian r.v in R u with zero mean and non-singular covariance matrix V. Define P r :
R u → R by P r (x, V, χ) = P r (−D, χ) ϕ V (x)
where, for any polynomial P (z) = ν a ν z ν , P (−D) is interpreted as a polynomial in the differentiation operator D, P (−D) = ν a ν (−1) |ν| D ν , with D ν = ∂ |ν | ∂x ν 1 1 ...∂x νu u , ν = (ν 1 , . . . , ν u ) ∈ N u . By construction P r andP r do not depend on the coefficient χ ν if |ν| > r + 2, and P r (it, χ)e −t ′ Vt/2 is the Fourier transform of P r (x, V, χ). Let ξ Γ be a centered a-dimensional Gaussian vector with covariance matrix Γ and g : R a → R a measurable mapping. Define N s (g) = R a (1 + x s ) −1 |g(x)| dx and g 2 Γ = E[g 2 (ξ Γ )]. The Hermite rank of g, g 2 Γ < ∞, with respect to Γ is defined as the smallest integer τ such that there exists a polynomial P of degree τ with E[g(ξ Γ )P (ξ Γ )] = 0. We denote τ (g, Γ) the (positive) Hermite rank of g − E[g(ξ Γ )] with respect to Γ.
Moment bounds: short memory case
In this section we consider short-range dependent processes. For any reals α, δ > 0 and β < ∞, denote by G(α, β, δ) the set of real sequences (ψ j ) j∈Z such that
|ψ 0 | + j∈Z |j| 1/2+δ |ψ j | ≤ β , (3.1) α ≤ inf λ∈[−π,π] ψ(λ) . (3.2)
Theorem 1. Assume (A1) with some integer s ≥ 3, p ≥ 1 and p ′ = 0 and assume that (ψ j ) j∈Z ∈ G(α, β, δ) for some α, δ > 0 and β < ∞. Then, there exists constants C and N (depending only on s, p, α, β, δ, u and the distribution of Z 0 ) such that, for all n ≥ N , and all u-tuple k of distinct integers, the distribution of S n (k) has a density q n,k with respect to Lebesgue's measure on R 2u and sup x∈R 2u
(1 + x s ) q n,k (x) − s−3 r=0 P r (x, V n (k), {χ n,ν (k)}) ≤ Cn −(s−2)/2 . (3.3)
Several interesting consequences can be derived from this result. A straightforward integration of the expansion (3.3) yields the following corollary which gives an Edgeworth expansion of some moment E[g(S n (k))] around the centered Gaussian distribution with covariance matrix V n (k).
Corollary 2. There exists a constant C and an integer N (depending only on s, p, α, β, δ, u and the distribution of Z 0 ) such that, for any u-tuple of distinct integers k, n ≥ N and measurable function g satisfying N s (g) < ∞,
E[g(S n (k))] − s−3 r=0 R 2u g(x)P r (x, V n (k), {χ n,ν (k)}) dx ≤ C N s (g) n −(s−2)/2 . (3.4)
One can also use Theorem 1 to develop the same moment around the limiting Gaussian distribution of S n . Recalling that ω r,n,k = a −1/2 r r s=0 r s (−1) s ω 0,n,k+s , we have lim
n→∞ V n (k) = V(k) under short memory conditions, where V(k) is the 2u × 2u matrix defined component-wise by [V(k)] 2i−1,2j−1 = [V(k)] 2i,2j = 1 2 ς r (k i − k j ) , [V(k)] 2i−1,2j = [V(k)] 2i,2j−1 = 0 , (3.5) for i, j = 1, · · · , u, with ς r (l) def = 0 if |l| > r , a −1 r (−1) l 2r r+l if |l| ≤ r . (3.6) Note that V(k) = 1 2 I 2u if r = 0. Corollary 3.
There exists a constant C and N (depending only on s, p, α,β,δ,u and the distribution of Z 0 ), such that for all measurable function g on R 2u such that N 3 (g) < ∞, all u-tuple of distinct integers k, and any n ≥ N ,
E [g(S n (k))] − R 2u g(x)ϕ V(k) (x) dx ≤ C n −1/2 N 3 (g) + n −τ (g,V(k))/2 g V(k) . (3.7)
For some functions g, it is possible to sharpen this result by considering higher-order (s > 3) expansions and approximating the terms appearing in these expansions. We shall consider mappings g : R 2u → R such that
g(x 1 , . . . , x 2u ) = u j=1 g j (x 2j−1 , x 2j ) with g j (x, y) = g j (y, x) = g j (−x, y), j = 1, . . . , u.
(3.8)
Recalling (2.7), products of functionals of the periodogram are included in this particular case. Better bounds are obtained by considering frequencies k 1 , . . . , k u separated by r, so that the asymptotic decorrelation is achieved, V(k) = 1 2 I 2u as in the r = 0 case. Under those conditions, the O(n −1/2 ) of Corollary 3 can ben improved to O(n −1 ).
Corollary 4.
Under the hypothesis that s ≥ 4, there exists a constant C and N (depending only on s, p, α, β, δ, u and the distribution of Z 0 ), such that for all measurable function g satisfying (3.8) and such that N q (g) < ∞, all u-tuple of ordered integers k such that k i < k i+1 − r, and any n ≥ N ,
E [g(S n (k)] − R 2u g(x)ϕ I 2u /2 (x) dx ≤ C n −(s−2)/2 N s (g) + n −1 (1 + x s )g(x) I 2u .
(3.9)
The proofs of Corollaries 3 and 4 are postponed to the Appendix E.
Remark. Pushing to higher orders s ≥ 4 in Corollary 4 is sometimes necessary to have N s (g) < ∞ (see the applications below). But it does not improve the O(n −1 ) bound.
To illustrate the results above, we compute bounds for the mean-square error of plug-in estimators of non-linear functionals of the spectral density Λ(f ) = π 0 w(λ)G(f (λ)) dλ where w is a function of bounded variation and G is a function such that there exists a function H satisfying, for any x > 0,
∞ 0 |H(xv)|e −v dv < ∞ and v>0 H(xv)e −v dv = G(x), i.e.
H is the inverse Laplace transform of the function t → G(1/t)/t. We consider the following estimator
Λ n = (π/ñ)ñ k=1 w(λ k )H(I n,k )
and put Λ n = (π/ñ) ñ k=1 w(λ k )G(f (λ k )). Here, r = 0 and I n,k def = I 0,n,k is the ordinary periodogram. We assume that the approximation error Λ n − Λ is neglectable in comparison with the mean-square error E(Λ n − Λ n ) 2 . These functionals have been studied in Taniguchi [1980] in the Gaussian case and Janas and von Sachs [1995] for non-Gaussian linear process, under rather stringent assumptions [see also Deo and Chen, 2000, and the references therein] . The moment bounds we have established allow to extend Janas and von Sachs [1995]'s result, by relaxing the conditions on the dependence (from |ψ j | < Cρ |j| for some ρ ∈ (0, 1) to j∈Z |j| 1/2 |ψ j | < ∞). Proposition 5. Let (X t ) t∈Z be sequence satisfying the assumptions of Theorem 1 with some
s ≥ 4. Put H 1 (x 1 , x 2 ) = H(x 2 1 + x 2 2 ), H 2 (x 1 , x 2 , x 3 , x 4 ) = H 1 (x 1 , x 2 )H 1 (x 3 , x 4 ) and assume that N 3 (H 2 1 ) < ∞ and N 5 (H 2 ) < ∞. Then, uniformly in f ∈ G(α, β, δ) E[(Λ n − Λ n ) 2 ] ≤ Cn −1 .
Sketch of the proof. Applying Corollary 3 to the function
g k,f (x 1 , x 2 ) = H[f (λ k )(x 2 1 + x 2 2 )]) and Corollary 4 to g k,j,f (x 1 , x 2 , x 3 , x 4 ) = H[f (λ k )(x 2 1 +x 2 2 )]H[f (λ j )(x 2 3 +x 2 4 )
] yield asymptotic expansions for the moments E[H 2 (I n,k )] and E[H(I n,k )H(I n,j )], which are sufficient to derive the result. The uniformity of the constant C follows from the existence of bounds on N 3 (g k,f ) and N 4 (g k,j,f ) which are uniform in ψ ∈ G(α, β, δ).
Moment bounds : Long memory case
4.1. Assumptions and main results. We consider two sets of assumptions, depending on available information on the behavior of the spectral density outside a neighborhood of the zero frequency. Recall that a real valued function φ defined in a neighborhood of zero is regularly varying at zero with index ρ ∈ R if, for all x and all t > 0, lim
x→0 φ(tx)/φ(x) = t ρ . If ρ = 0, the function φ is said slowly varying at zero. Let ϑ ∈ (0, π), 0 < δ < 1/2, ∆ < δ. We say that the linear filter (ψ j ) j∈Z belongs to the set F(ϑ, δ, ∆, µ) if ∞ j=−∞ ψ 2 j < ∞ and if there exists d ∈ [∆, δ] such that ψ(λ) is regularly varying at zero with index −d and that π 0 λ 2d |ψ(λ)| 2 dλ min 0≤|λ|≤ϑ λ 2d |ψ(λ)| 2 ≤ µ , (4.1) ∀j ≥ 0, |ψ j | + ∞ |t|≥j |ψ t+1 − ψ t | min 0≤λ≤ϑ λ d |ψ(λ)| ≤ µ(1 + j) d−1 , (4.2) An example is provided by ψ(λ) def = 1 − e iλ −d the transfer function of the causal fractional integration filter, ψ t = Γ(t + d)/(Γ(d)Γ(t + 1)), t ≥ 0.
Local-to-zero assumptions. We first consider local-to-zero assumptions for which nothing is required outside a neighborhood of the zero frequency, apart from integrability of the spectral density (see Robinson [1995b]). For β > 0, we say that the sequence (ψ j ) j∈Z belongs to the
set F local (ϑ, β, δ, ∆, µ) if (ψ j ) j∈Z ∈ F(ϑ, δ, ∆, µ) and ∀λ ∈ (0, ϑ] , |ψ * (λ) − ψ * (0)| min λ∈(0,ϑ] |ψ * (λ)| ≤ µλ β (4.3) with ψ * (λ) = (1 − e iλ ) d ψ(λ)
where d is the index of regular variation of ψ. This class is quite general and includes the impulse response of FARIMA filters [see Doukhan, Oppenheim, and Taqqu, 2002, and the references therein] but also processes whose spectral density may exhibit singularity outside the zero frequency, such as the Gegenbauer's processes. As seen below, under local-to-zero assumptions, the validity of the Edgeworth expansion can only be established for the DFT coefficients in a degenerating neighborhood of zero frequency. This is enough for, say, semi-parametric estimation of the long-memory index by the GPH method.
Global assumptions. In some situations, it is possible to formulate regularity assumptions over full the frequency range [−π, π] or a subset of it. These assumptions allow to prove the validity of the Edgeworth expansion for all the frequency ordinates. We say that the sequence
(ψ j ) belongs to the set F global (ϑ, β, δ, ∆, µ) if (ψ j ) ∈ F local (ϑ, β, δ, ∆, µ) and if in addition, for all (λ, λ ′ ) ∈ (0, ϑ] × (0, ϑ], ψ * (λ) − ψ * (λ ′ ) ≤ µ |ψ * (λ)| ∨ |ψ * (λ ′ )| |λ| ∧ |λ ′ | |λ − λ ′ | (4.4)
Under those assumptions and as in the short-memory case, we are able to prove the validity of the Edgeworth expansion for the DFT's (Theorem 6) and deduce some moment bounds (Corollaries 7, 9 and 10). In comparison with short memory results, note that tapering (r > 0) and (A1) with s ≥ p ′ are required.
Theorem 6. Assume (A1) with some integer s ≥ 3, p ≥ 1 and p ′ ≥ s. Let r be a positive integer and β, δ, ∆, µ, ϑ be constants such that 0 < δ < 1/2, −r + 1/2 < ∆ ≤ 0, µ > 0 and ϑ ∈ (0, π]. Let (m n ) n≥0 be a non-decreasing sequence. Assume either
(ψ j ) j∈Z ∈ F local (ϑ, β, δ, ∆, µ) and lim n→∞ 1 m n + m n n = 0 (4.5) or (ψ j ) j∈Z ∈ F global (ϑ, β, δ, ∆, µ) and m n ≤ ϑñ. (4.6)
Then there exist a constant C and positive integers K 0 , N 0 which depends only on ϑ, β, δ, ∆, µ, the distribution of Z 1 and the sequence (m n ), such that for any n ≥ N 0 and k = (k 1 , . . . , k u ) of integers in the range {K 0 , . . . , m n }, the distribution of S n (k) has a density q n,k with respect to Lebesgue measure on R 2u which satisfies
sup x∈R 2u (1 + x s ) q n,k (x) − s−3 r=0 P r (x, V n (k), {χ n,ν (k)}) ≤ Cn −(s−2)/2 . (4.7)
If u = 1, one can take K 0 = 1.
Integrating some function g against the density q n,k and using (4.7) yields the following corollary.
Corollary 7. Under the assumptions of Theorem 6, there exists a constant C and an integer N depending only on ϑ, β,δ, ∆, µ, u, r and such that, for all u-tuple of distinct integers k satisfying K 0 ≤ min(k), max(k) ≤ m n and any n ≥ N , and all measurable function g such that N s (g) < ∞,
E[g(S n (k))] − s−3 r=0 R 2u g(x)P r (x, V n (k), {χ n,ν (k)}) dx ≤ C N s (g) n −(s−2)/2 . (4.8)
Similarly to the short-memory case, one could approximate E[g(S n (k))] using the limiting distribution of S n (k) in place of the Gaussian approximation as Corollary 7. Under long-range dependence and for fixed k, the limiting covariance matrix of S n (k) fully depends on k and not only on (k 2 − k 1 , . . . , k u − k u−1 ). This behavior at "very-low frequencies" as been studied for instance by Hurvich and Beltrao [1993]. However, one can control the covariance of the standardized DFT coefficients and then the difference V n (k) − V(k) thanks to the following lemma.
Lemma 8. For 1 ≤ k ≤ j ≤ ϑn/π − r and r ≥ 1, there exists a constant C depending only on ϑ, β, δ, ∆, µ such that |E(ω r,n,k ω r,n,j )| + |E(ω r,n,kωr,n,j ) − ς r (k − j)| ≤ Cp(k, j, n, β) (4.9)
with p(k, j, n, β) = (jk) −1/2 + j∨k n β under (4.5) (jk) −1/2 under (4.6).
(4.10)
Thus, we can develop the moments around the Gaussian distribution with covariance matrix V(k) as in the short-memory context. The two following corollaries prove sufficient for our applications. The next corollary is useful for moment bounds on one frequency k.
Corollary 9. Under the assumptions of Theorem 6, there exist a constant C and a positive integer N 0 which depend only on ϑ, β, δ, ∆, µ, the distribution of Z 1 and the sequence (m n ), such that for any n ≥ N 0 , for any integer k in the range {1, . . . , m n } and any measurable function g on R 4 such that
N 3 (g) < ∞ E [g(S n (k))] − R 2 g(x)ϕ I 2 /2 (x)dx ≤ C n −1/2 N 3 (g) + p(k, k, n, β) −τ (g,V(k))/2 g(x) I 2u .
The next corollary is useful for moment bounds on two frequencies k < j − r.
Corollary 10. Under the assumptions of Theorem 6, there exist a constant C and positive integers K 1 ≥ K 0 , N 0 which depends only on ϑ, β, δ, ∆, µ, the distribution of Z 1 and the sequence (m n ), such that for any n ≥ N 0 and for any couple k = (k, j) of integers in the range {K 0 , . . . , m n } such that k < j − r and any measurable function g on R 4 verifying (3.8) and such that N 4 (g) < ∞ E [g(S n (k))] − R 4 g(x)ϕ I 4 /2 (x)dx ≤ C n −(s−2)/2 N s (g) + n −1/2 p 2 (k, j, n, β) (1 + x s )g(x) I 4 /2 . 4.2. GPH estimation of the long memory parameter.
Theoretical results.
A very widely used estimator of the memory parameter d was introduced by Geweke and Porter-Hudak [1983]. It is obtained from the linear regression of the log-periodogram of the observations using the logarithm of the frequencies as explanatory variable. In contrast with the Whittle estimator, the GPH is defined explicitly in terms of the logperiodogram ordinates, see Eq. (4.11) below. Many theoretical work has been achieved on this estimator, in stationary or non-stationary contexts (see Faÿ, Moulines, Roueff, and Taqqu, 2008 for a survey of the main results). For instance, Giraitis et al. [1997] proved that the GPH of Gaussian X is rate optimal for the quadratic risk and over some classes of spectral densities that is included in our F local . To compute the risk of the GPH estimator, one need to compute or approximate moments of the log-periodogram. The log-periodogram is a non-smooth function of the Fourier transform of the observation, which are Gaussian if X is Gaussian. The proof by Giraitis et al. [1997] relies on moment bounds of non-linear function of Gaussian variables [see Arcones, 1994, Soulier, 2001; this technique does not extend naturally to non-Gaussian time series. Here, we shall apply the Edgeworth approximations obtained in preceding section to extend this result to the case of strong sense linear process.
For the sake of simplicity of exposition, we only consider a taper of order r = 1 and write I k = I 1,n,k . The GPH estimator is obtained by an ordinary least square regression of log(I k ) on log |2 sin(λ k /2)| [see Geweke andPorter-Hudak, 1983, Robinson, 1995b]. With the frequency spacing and taper order r, one regresses on every r + 1 frequency. For r = 1 it writes
(d m ,Ĉ) = arg min d ′ ,C ′ m k=1 log(I 2k+1 ) + 2d ′ log |2 sin(λ 2k+1 /2)| − C ′ 2 . where m = m(n) is a bandwidth parameter. Explicitlŷ d m = s −2 m m k=1 ν k log(I 2k+1 ), (4.11) with ν k = −2 log |2 sin(λ 2k+1 /2)|− 1 m m j=1 log |2 sin(λ j /2)| and s 2 m = m k=1 ν 2 k . We consider E[(d m − d) 2 ]
the mean square error (MSE) of the GPH estimator. Theorem 11 gives a bound on the MSE which is uniform over a class of long-range dependent linear processes, from which rate optimality can be deduced.
Theorem 11. Under the assumptions of Theorem 6 with s ≥ 5 and conditions (4.5), there exists a constant C which depends only on β, δ, ∆, ϑ, µ and the distribution of Z 1 such that
E[(d m − d) 2 ] ≤ C m n 2β + 1 m . With m proportional to n 2β/(2β+1) , E[(d m − d) 2 ] ≤ Cn −2β/(2β+1) .
Remark. The condition s ≥ 5 seems slightly stronger than necessary for bounding the MSE ofd. But it is allows the function h(x 1 , . . . , x 4 ) = g(x 1 , x 2 )g(x 3 , x 4 ) with g(x) = log( x 2 ) −η to have finite N s (h) norm (see Corollary 4 and the remark that follows.
Monte Carlo results.
Assuming more stringent global condition on the regularity of the spectral density allows one to evaluate the bias term in the decomposition of the mean squared error. For comparison, using the specific set of assumptions Hurvich, Deo, and Brodsky [1998], we can prove that the leading terms in the MSE are of the form am 4 /n 4 + b/m for bandwidth such that lim n→∞ 1/m + m log(m)/n = 0. The constant a and b can be made uniform in the class of spectral densities under consideration. It shows that the MSE of the GPH estimator is asymptotically insensitive to the distribution of the innovation as soon as this distribution satisfies some moment and regularity conditions. Finite sample implications of this statement is illustrated here by the results of a Monte Carlo study. For sample sizes n = 250, 500, 1000, 2500, 5000, we have simulated 1000 realizations of a FARIMA(1, d, 0) processes defined by
(1 − B) 0.3 (1 − 0.3B)X t = Z t
where B is the back-shift operator and (Z t ) t∈Z is a zero mean unit variance i.i.d sequence with the following marginal distributions (a) Gaussian (b) Laplacian (c) zero-mean (shifted) Pareto, with P(Z 0 ≤ u) = (1 − (u + 7/6) −7 )1 u≥−7/6 .
Whereas it is possible to simulate exactly a Gaussian FARIMA(p, d, q) process (e.g. computing the covariance structure and using Levinson-Durbin algorithm), there is no general way to do it for non Gaussian processes. In the Monte-Carlo experiment, the process (X t ) is obtained using a truncated MA(∞) representation. For each realization of each process, we evaluate the squared error (d m − d) and define the Monte Carlo MSE as the average of those errors. We have focused on the sensitivity with respect to the distribution of Z of the bandwidth m which is optimal in the MSE sense. Figure 1 and Table 1 show that for sample size n = 250 the MSE is minimized at m = 37 or 38 which means that the optimal bandwidth is about the same for those three linear processes. Figure 2 represents the box-and-whiskers plot of the GPH estimator for two different sample sizes and the three models we are concerned with. Here again, the sensitivity with respect to the distribution of the driving noise is hardly discernible. In Table 1 we displayed the value of the bias and of the mean square error of the GPH at this estimated optimal bandwidth. In this section we recall the theorem established in Faÿ, Moulines, and Soulier [2004]. Let (Z t ) t∈Z be an i.i.d sequence and (U n,j ) j∈Z,n∈N an array of vectors in R u , where u is an integer. Define S n = j∈Z U n,j Z j and let V n = j∈Z U n,j U ′ n,j . For ν ∈ N u , 2 ≤ |ν| ≤ s, denote χ n,ν the cumulants of S n . Then χ n,ν = κ |ν| j∈Z U ν n,j . where κ r denotes the r-th cumulant of Z 1 , r ≤ s. Consider the following assumptions.
(B1) There exist positive constants v * and v * such that
v * ≤ lim inf n v min [V n ] ≤ lim sup n v max [V n ] ≤ v * where v min [V n ] (resp. v max [V n ])
is the smallest (resp. the largest) eigenvalue of V n . Theorem 12 (Faÿ, Moulines, and Soulier, 2004). Let s ≥ 3, and p ′ ≥ 0 be integers and p ≥ 1 be a real number. Assume (A1)(s, p, p ′ ), (B1) and (B2). Assume in addition either (B3) or p ′ ≥ s in (A1)(s, p, p ′ ). Then, there exist a constant C and an integer N (depending only on the distribution of Z 1 , and the constants appearing in the assumptions) such that, for all n ≥ N , the distribution of S n has a density q n with respect to Lebesgue measure on R u which satisfies
sup x∈R u (1 + x s ) q n (x) − s−3 r=0 P r (x, V n , {χ n,ν }) ≤ C j∈Z U n,j s (A.4) Appendix B. Proof of Theorem 1
The proof consists in checking that assumptions (B1), (B2) and (B3) hold uniformly with respect to ψ ∈ G(α, β, δ) and k for U n,j 's of the form (2.9). To prove (B1), write Hannan [1960, p. 54], we have under (3.1)
V n (k) = V(k) + W n (k), with V(k) defined in (3.5). Define W 1 = max 1≤i≤v v j=1 |w i,j | for any matrix W = (w i,j ) 1≤i,j≤v . Similarly toW n (k) 1 ≤ C(α, β)n −1 . (B.1)
The matrices V(k) have the following algebraic property.
Lemma 13. There exist two positive constants v * and v * such that
2v * ≤ inf v min [V(k)] ≤ sup v max [V(k)] ≤ 2v * (B.2)
where the infimum and supremum are taken over all the u-tuples of distinct integers in N u .
Proof. Noting that trace[V(k)] = u, v max [V(k)] ≤ trace[V(k)]/2u = 1/2. (B.3)
Take v * = 1/4. Recall that k 1 < · · · < k u . Note that, for any n ≥ 2k u + 2r + 1, V(k) is the covariance matrix of √ 2π(c Y r,n,k 1 , s Y r,n,k 1 , . . . , c Y r,n,ku , s Y r,n,ku ) with c Y r,n,k = (2πa r n) −1/2 n t=1 h r t,n Y t cos(tλ k ) and s Y r,n,k = (2πa r n) −1/2 n t=1 h r t,n Y t sin(tλ k ) the sine and cosine transform of a unit-variance zero-mean Gaussian white noise (Y n ) n∈Z . Recall that c Y r,n,k = a −1/2 r r l=0 (−1) l ( r l ) c Y 0,n,k+l and s Y r,n,k = a −1/2
r r l=0 (−1) l ( r l ) s Y 0,n,k+l . (B.4)
The random variables c 0,n,k and s 0,n,k , k = 1, . . . , [(n − 1)/2] are centered i.i.d Gaussian with variance 1/4π. Assume that V(k) is not invertible. It yields that for some 2u-tuple of reals (α 1 , β 1 , · · · , α u , β u ) = (0, 0, · · · , 0, 0), u j=1 (α j c r,n,k j + β j s r,n,k j )
L 2 = 0.
Then by (B.4), there exists a linear combination of c 0,n,k 's and s 0,n,k 's that is equal to zero. c 0,n,ku+r and s 0,n,ku+r appear in this combination with coefficients a −1/2 r (−1) r α u and a −1/2 r (−1) r β u , respectively. It follows from the independence and non-degeneracy of the c 0,n,k 's and s 0,n,k 's that α u = β u = 0. Iterating the argument yields the contradiction α u = β u = α u−1 = β u−1 = · · · = α 1 = β 1 = 0. Thus for any u-tuple k of distinct integers v min [V(k)] > 0.
(B.5)
It remains to prove that v min [V(k)] is bounded away from zero uniformly in k. Define
K u = {k = (k 1 , · · · , k u ′ ) ∈ N u ′ , 1 ≤ u ′ ≤ u, 0 < k i+1 − k i ≤ r}.
Note now that by (3.5), v min [V(k)] is a function of the vector (k 2 − k 1 , k 3 − k 2 , · · · , k u − k u−1 ) thus taking finitely many different values on K u . From the this remark and (B.5),
v 1 def = inf k∈Ku v min [V(k)] > 0 (B.6)
since the infimum is taken on a finite set of positive values. Consider now a u-tuple k that does not belong to K u ; In this case, for some i ∈ {1, · · · , u − 1}, k i+1 − k i > r, and then k may be partitioned as L ≥ 2 blocks of indexes (k 1 , . . . , k L ) such that all the k i 's belong to K u and, for all i ∈ {1, · · · , L − 1}, min k i+1 − max k i > r. Let l i denotes the length of the block k i , i = 1, · · · , L. By this construction and (3.5), the matrix V(k) has a block-diagonal structure
V(k) = V(k 1 ) 0 . . . 0 V(k L ) . Using (B.6), v min [V(k)](v max [V(k)]) 2u−1 ≥ det[V(k)] = L i=1 det[V(k i )] ≥ L i=1 v 2l i 1 = v 2u 1 . (B.7)
We conclude from (B.7) and (B.3) that For any ǫ > 0 and large enough n, |j|≥n U n,j (k) 2 ≤ ǫ uniformly in k and ψ ∈ G(α, β, δ). (A.3) follows from (B.9), (B.10) and (B.11). Finally, j∈Z U n,j (k) ≤ C(r)α −1/2 n −1/2 n t=1 j∈Z |ψ t+j | = C(r)α −1/2 n 1/2 j∈Z |ψ j | ≤ C(r)α −1/2 βn 1/2 = C(r) 2 α −1 β 2 M −1 n so that (B3) holds with ζ = 1.
Appendix C. Proof of Lemma 8
The proof is an adaptation of Lang and Soulier [2002] to fit our need of uniformity of the bounds with respect to the function ψ whether it belongs to F global or F local only. For sake of brevity, the proof is omitted and we refer the interested reader to their paper. It derives from their more general analytical lemma that we recall here.
Lemma 14 (Lang and Soulier [2002]). Let q ∈ N, K ≥ 1, ϑ ∈ (0, π]. Let ψ be an integrable function on [−π, π], such that for all x ∈ (0, ϑ] \ {0}, ψ(−x) =ψ(x) and
|ψ(x) − ψ(y)| ≤ K |ψ(x)| + |ψ(y)| x ∧ y |x − y|, for all(x, y) ∈ (0, ϑ] × (0, ϑ] (C.1)
Assume that |ψ| is regularly varying at zero with index ρ and that
ψ(x) = x ρ c(x) exp ϑ x g(s) s ds with (i) lim x→0 g(x) = 0; (ii) lim x→0 c(x) exists in (0, ∞). Let D n be such that for x ∈ [−π, π],
|D n (x)| ≤ C n 1/2 (1 + n|x|) q+1 .
(C.2)
• If ρ ∈ (−1, 2q + 1), there exists a constant C such that, for all n ≥ 1 and all k such
that 0 < x k ≤ ϑ/2, π −π ψ(x) ψ(x k ) − 1 |D n (x k − x)| 2 dx ≤ C log ν(q) (k)k −1 . (C.3)
with ν(0) = 1 and ν(q) = 0 if q ≥ 1. • If ρ ∈ (−1/2, q + 1/2), there exists a constant C such that, for all n ≥ 1 and all integers k, j such that
0 < x k = x j ≤ ϑ/2, π −π ψ(x) ψ(x k ) − 1 D n (x k − x)D n (x j − x) dx + π −π ψ(x) ψ(x k ) − 1 D n (x k − x)D n (x + x j ) dx ≤ C(1 + |ψ(x j )/ψ( k )|)|k − j| −q (j ∨ k) −1 ≤ C(jk) −1/2 (q > 0); C(jk) −1/2 log(j ∨ k) (q = 0). (C.4)
• For any β > 0, if ρ ∈ (−1, 2q + 1), for any integer k such that 0 < x k ≤ ϑ/2,
π −π ψ(x)|x| β ψ(x k ) |D n (x k − x)| 2 dx ≤ C k −2q−1 + (k/n) β . (C.5)
• If ρ ∈ (−1/2, q + 1/2), for any integers j, k such that
0 < x k = x j ≤ ϑ/2, π −π ψ(x)|x| β ψ(x k ) |D n (x k − x)D n (x j − x)| dx + π −π ψ(x)|x| β ψ(x k ) |D n (x k − x)D n (x j + x)| dx ≤ C(1 + |ψ(x j )/ψ(x k )|) (jk) −q (j ∨ k) −1 + |j − k| −q−1 ((j ∨ k)/n) β log ν(q) (j ∨ k) ≤ C (jk) −1/2 + ((j ∨ k)/n) β log ν(q) (j ∨ k) . (C.6)
Appendix D. Proof of Theorem 6
The proof of Theorem 6 consists in checking that assumptions (B1), (B2) hold uniformly.
Lemma 15. There exist integers N 0 , K 0 , and v * > 0, v * > 0 (depending only on ϑ, β, δ, ∆, µ) such that, for all n ≥ N 0 , we have,
(1) for all u-tuple k of distinct integers, 1 ≤ min k ≤ max k ≤ m n , v max [V n (k)] ≤ v * ; (D.1) (2) for all integer k, 1 ≤ k ≤ m n v * ≤ v min [V n (k)] ; (D.2) (3) for all u-tuple k of distinct integers, K 0 ≤ min k ≤ max k ≤ m n , v * ≤ v min [V n (k)] . (D.3)
Proof. As in Appendix B, we put V n (k) = V(k) + W n (k) where V(k) is defined in (3.5).
Applying Lemma 8, we obtain W n (k) 1 ≤ C(ϑ, β, δ, ∆, µ) 2), it remains to prove that for any integer k, 1 ≤ k ≤ K 0 , V n (k) converges to a positive definite matrix V(k) and that this convergence is uniform w.r.t to ψ, for ψ ∈ F local (ϑ, β, δ, ∆, µ) or ψ ∈ F global (ϑ, β, δ, ∆, µ). What follows is an adaptation of [Iouditsky, Moulines, and Soulier 2001, Lemma 7.3]. Write
E[|ω r,n,k | 2 ] = 1 f (λ k ) |λ|≤ϑπ + |λ|>ϑπ |D r,n (λ − λ k )| 2 f (λ)dλ =: A 1 + A 2 (D.5)
where D r,n is defined in (2.2). For n ≥ 4πK 0 /ϑ, 1 ≤ k ≤ K 0 and |λ| ≥ ϑπ, |n(λ−λ k )| ≥ nϑ/2. Using (2.3) and (4.1), we get
A 2 ≤ Cλ 2d k λ 2d k f (λ k ) n −2r−1 |λ|>ϑπ λ 2d f (λ)dλ ≤ Cn −2r (D.6)
By change of variable,
A 1 = n 2d |1 − e −iλ k | 2d f * (λ k ) |λ|≤nϑ n −1/2 D r,n (λ/n − λ k ) 2 n −2d |1 − e −iλ/n | −2d f * ( λ n )dλ.
Write lim n→∞ n −1/2 D r,n (λ/n) = 1 √ 2πar 1 0 (1 − e 2iπs ) r e −isλ ds =:ĥ r (λ). By Riemann approximation, it can be seen that |n −1/2 D r,n (λ/n) −ĥ r (λ)| ≤ C(1 + |λ|)/n. Note also that |ĥ r (λ)| ≤ C|λ| −r−1 . Then
A 1 − n 2d |1 − e iλ k | 2d f * (λ k ) × nϑ −nϑ ĥ r (λ − 2πk) 2 n −2d |1 − e iλ/n | −2d f * ( λ n )dλ ≤ Ck 2d n −r ≤ Cn −r . (D.7)
Here and in the following, C is a generic constant which depends only on ϑ, β, δ, ∆, µ, r and K 0 . For |λ| ≤ nϑ, using (4.5),
f * (0) f * (λ k ) n −2d |1 − e iλ/n | −2d − |λ| −2d + |f * ( λ n ) − f * (0)| f * (λ k ) n −2d |1 − e iλ/n | −2d ≤ C n −2d |1 − e iλ/n | −2d − |λ| −2d + n −2d |1 − e iλ/n | −2d | λ n | β ′ (D.8) with β ′ = β ∧ 1. For x ∈ [−π, π], 2 π |x| ≤ |e ix − 1| = |2 sin x 2 | ≤ |x| and ||e ix − 1| − |x|| ≤ x 2 /2. Also, for any υ ∈ R, x > 0, y > 0, |x υ − y υ | ≤ |υ|(x υ−1 ∨ y υ−1 )|x − y|. Using those relations, write, for λ ∈ [−nπ, nπ], n −2d |1 − e iλ/n | −2d − |λ| −2d ≤ Cn −2d | λ n | −2d−1 |1 − e iλ/n | − | λ n | ≤ Cn −1 |λ| −2d+1 Then nϑ −nϑ |ĥ r (2πk − λ)| 2 n −2d |1 − e iλ/n | −2d − |λ| −2d dλ ≤ Cn −1 nϑ −nϑ |ĥ r (2πk − λ)| 2 |λ| −2d+1 dλ ≤ Cn −1 nϑ −nϑ |ĥ r (2πk − λ)| 2 |λ| 2r dλ ≤ Cn −1 (D.9) and nϑ −nϑ |ĥ r (2πk − λ)| 2 n −2d |1 − e iλ/n | −2d | λ n | β ′ dλ ≤ n −β ′ nϑ −nϑ |ĥ r (2πk − λ)| 2 |λ| −2d+β ′ dλ ≤ Cn −β ′ .(E[|ω r,n,k | 2 ] − (2π) 2d k 2d f * (0) f * (λ k ) +∞ ∞ ĥ r (λ − 2πk) 2 |λ| −2d dλ ≤ Cn −β ′
Similar arguments leads to
E[ω 2 r,n,k ] − (2π) 2d k 2d f * (0) f * (λ k ) +∞ ∞ĥ r (λ − 2πk)ĥ r (λ + 2πk)|λ| −2d dλ ≤ Cn −β ′
Defining the scalar product (u, v) d = R u(λ)v(λ)|λ| −2d dλ, Then det V n (k) is uniformly approximated by the Gram determinant of the functionsĥ r (λ− 2kπ) andĥ r (λ+ 2kπ) associated with the product (·, ·) d and then is a continuous function of η k (d) := lim n→∞ E[|ω r,n,k | 2 ] and η ′ k (d) := lim n→∞ E[ω 2 r,n,k ].. The whole set of functionsĥ r (λ + 2jπ), j ∈ Z is linearly independent, so that those determinant are positive. Using continuity of η k and η ′ k w.r.t. d, the infimum on the compact set [−∆, δ] and the minimum over k = 1, . . . , K 0 is positive too, which concludes the proof.
Lemma 16. There exists a constant C (depending only on ϑ, β, δ, ∆, µ,r) such that for all k ∈ {1, . . . ,ñ},
1 nf (λ k ) n t=1 h r t,n ψ t+j e itλ k ≤ Cn −1/2 . (D.11)
Proof. The main tool of the proof is the bound (2.3) and the technique are the same as the one used in the proof of Lemma 8. Decompose
|ψ(λ k )| −1 1 √ 2πa r n n t=1 h r t,n ψ t−j e itλ k = |ψ(λ k )| −1 π −π ψ(λ)e ijλ D n,r (λ k − λ)dλ into A 1 = |ψ(λ k )| −1 −ϑ −π + π ϑ ψ(λ)e ijλ D n,r (λ k − λ)dλ, A 2 = |ψ(λ k )| −1 ψ * (0) ϑ −ϑ (1 − e iλ ) −d e ijλ D n,r (λ k − λ)dλ, A 3 = |ψ(λ k )| −1 ϑ −ϑ (1 − e iλ ) −d (ψ * (λ) − ψ * (0))e ijλ D n,r (λ k − λ)dλ.
By Eq. (2.3), if |λ| ∈ [ϑ, π], |D n,r (λ k − λ)| ≤ Cn −1/2−r . Note that n −1 λ d k = n −1 λ −1 k λ d+1 k ≤ 1/(2πk). (4.1) implies that |A 1 | ≤ Cn 1/2−r k −1 ≤ Cn −1/2 . Consider A 2 . Since π −π D n,r (λ)dλ = 0,
A 2 = ϑ −ϑ ∆(λ, λ k )D n,r (λ k − λ)dλ, ∆(λ, λ k ) = (1 − e iλ ) −d − (1 − e iλ k ) −d e ijλ |ψ(λ k )| −1 . Decompose this integral on the intervals [−ϑ, −λ k /2], [−λ k /2, λ k /2], [λ k /2, 2λ k ] and [2λ k , ϑ]. If λ ∈ [−λ k /2, λ k /2], then |D n,r (λ k − λ)| ≤ C √ nk −r−1 and |∆(λ, λ k )| ≤ C |λ| −d λ d k + 1 . Hence : λ k /2 −λ k /2 ∆(λ, λ k )D n,r (λ k − λ)dλ ≤ Ck −r n −1/2 . If λ ∈ [λ k /2, 2λ k ], then |∆(λ, λ k )| ≤ C λ −1 k |λ − λ k | + 1 . Since λ k −λ k /2 (1 + n|λ|) −r−1 dλ ≤ Cn −1 , we have 2λ k λ k /2 ∆(λ, λ k ) D n,r (λ k − λ)dλ ≤ Cn −1/2 .
If λ ∈ [2λ k , ϑ] (and similarly on [−ϑ, −λ k /2]), we use that |∆(λ, λ k )| ≤ C(λ −d λ d k + 1) and |D n,r (λ − λ k )| ≤ n −1/2−r |λ − λ k | −1−r . Hence,
ϑ 2λ k ∆(λ, λ k )D n,r (λ k − λ)dλ ≤ Cn −1/2−r ∞ λ k λ d λ −d k + 1 λ −1−r dλ ≤ Cn −1/2 k −r .
Consider A 3 . Under (4.3), we have
|A 3 | ≤ Cλ d k ϑ −ϑ |λ| −d+β |D n,r (λ − λ k )|dλ.
Decompose this integral as above. If λ ∈ [−λ k /2, λ k /2], proceeding as above:
λ d k λ k /2 −λ k /2 |λ| −d+β |D n,r (λ − λ k )|dλ ≤ Cn −1/2 k −r λ β k . If λ k ∈ [λ k /2, 2λ k ], λ d k |λ| −d+β ≤ Cλ β k , and 2λ k λ k /2 |D n,r (λ − λ k )|dλ ≤ Cn −1/2 . Hence: λ d k 2λ k λ k /2 |λ| −d+β |D n,r (λ − λ k )|dλ ≤ Cn −1/2 λ β k .
Finally, if λ ∈ [2λ k , ϑ] (and similarly, if λ ∈ [−ϑ, −λ k /2]), we have as above:
λ d k ϑ 2λ k λ −d+β λ −1−r dλ ≤ Cλ d k n −1/2−r ϑ λ k λ −d−1−r dλ = Cn −1/2 k −r .
Lemma 17. There exists a constant C (depending only on ϑ, β, δ, ∆, µ,r) such that for all k ∈ {1, . . . ,ñ},
1 nf (λ k ) n t=1 h r t,n ψ t+j e itλ k ≤ Cn −1/2 λ d−1 k (1 + |j|) d−1 ≤ Cn −1/2 ((1 + |j|)/n) d−1 . (D.12)
Proof. By applying the definition of the weights h r t,n and summation by parts, we have: For all y ∈ (0, π) and all ℓ ∈ N * , ℓ u=1 e iuy ≤ 2/y. The proof follows from condition (4.2).
Proceed now with the proof of Theorem 6. If |j| ≥ n, then ((1 + |j|)/n) d−1 ≤ 1. Hence by Lemma 16, for some constant C which depends only on β, δ, ∆, ϑ, µ, r and the distribution of Z 1 , ∀ j, n, k, M n,j def = Cn −1/2 1 ∧ ((1 + |j|)/n) δ−1 ≥ U n,j (k) . Finally, define for any γ ≥ 1 the set J n = {j ∈ Z, |j| ≤ γn}. Then card(J n ) ≤ c 0 M −2 n and j∈Z\Jn U n,j (k) 2
Note that
j∈Z U n,j (k) 2 ≤ j∈Z\Jn M 2 n,j j∈Z U n,j (k) 2 ≤ C(v * ) −1 n 1−2δ |j|≥γn j 2δ−2 ≤ C(v * ) −1 γ 2δ−1 .
Choosing γ large enough yields (A.3) uniformly.
Appendix E. Proofs of Corollaries 3, 4, 9 and 10
Proof of Corollary 3. By the triangle inequality, the LHS of inequality (3.7) is bounded by
E [g(S n (k))] − R 2u g(x)ϕ Vn (k) (x) dx + R 2u g(x) ϕ Vn(k) (x) − ϕ V(k) (x) dx .
By Corollary 2 with s = 3, the first term of the previous display is bounded by Cn −1/2 N 3 (g). For A a matrix, denote ρ(A) its spectral radius. Denote I a the a-dimensional identity matrix.
To bound the second term, note that ρ(V n (k) − V(k)) ≤ C(α, β)n −1 by (B.1) and that τ (g, V(k)) ≥ 1 by definition, then apply the following lemma which is an easy adaptation of Soulier [2001, Theorem 2.1].
Lemma 18. Let Γ be a u-dimensional positive matrix. There exists ǫ > 0 and a constant C such that, for all symmetric positive matrix Γ ′ verifying ρ(Γ ′−1 − Γ −1 ) < ǫ, and for all measurable functions g on R u satisfying g 2 Γ < ∞, we have
R u g(x) {ϕ Γ ′ (x) − ϕ Γ (x)} dx ≤ Cρ τ (g,Γ)/2 (Γ ′ − Γ) g Γ .
Proof of Corollary 4. The LHS of (3.9) is bounded by
A 1 + A 2 + A 3 + A 4 with A 1 = E [g(S n (k))] − R 2u g(x) s−3 r=0 P r (x, V n (k), {χ n,ν (k)})} dx , A 2 = R 2u g(x) ϕ Vn(k) (x) − ϕ I 2u /2 (x) dx , A 3 = R 2u g(x)P 1 (x, V n (k), {χ n,ν (k)}) dx , A 4 = R 2u g(x) s−3 r=2 P r (x, V n (k), {χ n,ν (k)})} dx ,
A 4 = 0 if s = 4. Using (3.8), we get τ (g, I 2u /2) = 2. It follows, as in the proof of Corollary 3 that A 2 is bounded by Cn −1 g V(k) , whereas A 1 is bounded by Cn −(s−2)/2 N s (g). Write shortly P r (x, V n (k), {χ n,ν (k)}) = R r (x)ϕ Vn(k) , where R r is a polynomial of order r + 2 (the dependence w.r.t V n (k) and {χ n,ν (k)} is ommited in this notation). Note also that
|χ n,ν (k)| ≤ |κ |ν| | j∈Z U n,j (k) |ν| ≤ |κ |ν| |M |ν|−2 n ( j∈Z U n,j (k) 2 ) ≤ |κ |ν| |M |ν|−2 n trace[V n (k)] ≤ C|κ |ν| |M |ν|−2 n (E.1)
where M n ≤ C(α, β)n −1/2 . Then, the coefficients of R r are O(n −(r/2) ) uniformly in k and ψ since they involve χ n,ν (k)'s with |ν| = r and elements of V −1 n (k) [for details, see Bhattacharya and Rao, 1976]. Let F n (k) def = (V −1 n (k) − V −1 (k))/2 and write (E.2) By (B.1), F n (k) 1 ≤ Cn −1 and | det(V n (k)) −1/2 − det(I 2u /2) −1/2 | ≤ Cn −1 uniformly so that A 4 ≤ Cn −1 (1 + x s )g(x) 2I 2u /3 . We can derive this way that A 3 ≤ Cn −1/2 which is not enough. Improving this bound requires some care and uses the symmetries of g. Actually, R 1 is a sum of polynomials which are odd with respect to one or three components. Write | exp{−x ′ F n (k)x} − 1 + x ′ F n (k)x| ≤ Cn −2 x 4 exp Cn −1 x 2 (E.3) and notice that {1 − x ′ F n (k)x}R 1 (x) is a sum of polynomials of the form i r i (x 2i−1 , x 2i ), each of them being odd with respect to at least one variable. Consider a typical term odd with respect to x 1 , say. Using (3.8)
g(x)ϕ Vn(k) (x)R r (x) dx = det V(k) det V n (k) 1/2 g(x)R r (x) exp{−x ′ F n (k)x}ϕ V(k) (x) dxR 2u g(x) i r i (x 2i−1 , x 2i )ϕ I 2u /2 (x) dx = R 2 g 1 (x 1 , x 2 )r 1 (x 1 , x 2 )ϕ I 2 /2 (x 1 , x 2 ) dx 1 dx 2 × R 2u−2 i>1 g i (x 2i−1 , x 2i )r i (x 2i−1 , x 2i )ϕ I 2u−2 /2 (x) dx 3 · · · dx 2u = 0,
since the first integral vanishes. Hence, R 4 h(x)R 1 (x)(x ′ F n (k)x)ϕ I 4 (x) dx = 0. Gathering (E.1), (E.2) and (E.3), A 3 ≤ Cn −2 .
Proofs of Corollaries 9 and 10. As those corollaries are the counterparts of Corollaries 3 and 4 in a long memory context, we only gives the necessary adaptations from the preceding proofs. From Lemma 8, ρ(V n (k) − V(k)) ≤ Cp(k, k, n, β), F n (k) 1 ≤ Cp(k, j, n, β) and
| det(V n (k)) −1/2 − det(V(k)) −1/2 | ≤ Cp(k, j, n, β).
The LHS of (E.3) is now bounded by p 2 (k, j, n, β) x 4 exp Cp(k, j, n, β) x 2 . The term A 3 is then bounded by Cn −1/2 p 2 (k, j, n, β)
R 4
x 5 h(x) exp{− x 2 (1 + Cp(k, j, n, β))} dx.
If m = o(n) and K 1 > 2C, then for large enough n and K 1 ≤ k < j − r ≤ m − r, the integral is uniformly bounded. Thus A 3 ≤ Cn −1/2 p 2 (k, j, n, β) whereas A 1 ≤ Cp 2 (k, j, n, β).
Appendix F. Proof of Theorem 11
In the sequel, C denotes a constant which depends only on β, δ, ϑ, µ and the distribution of Z 1 and whose value may change upon each appearance. Note first that |ν k | = O(log(k)), s 2 m /m → C > 0 (see for instance Robinson [1995b], or Hurvich, Deo, andBrodsky [1998]). Define f * (λ) = |1−e −iλ | −2d f (λ) and L(λ) = log(f * (λ)/f * (0)). Since ψ ∈ F(ϑ, β, δ, ∆, µ), there exists a constant C such that ∀k ∈ {1, · · · , m}, |L(λ k )| ≤ C|λ k | β .
(F.1)
Letη denote E(log Y 2 ) where Y is a centered Gaussian random vector with covariance matrix I 2 /2. Define η k = log(I k /f (λ k )) −η, 1 ≤ k ≤ m. With these notations and since For x ∈ R 2 , define g(x) = log( x 2 ) −η. Then η k = g(S n,k ) and N 3 (g 2 ) < ∞. For (x 1 , . . . , x 4 ) ∈ R 2 , define h(x 1 , . . . , x 4 ) = g(x 1 , x 2 )g(x 3 , x 4 ). Then η k η j = h(S n,(k,j) ), h has property (3.8) and
N 5 (h) = R 4 g((x 1 , x 2 ))g((x 3 , x 4 )) 1 + x 5 ≤ 4(N 5/2 (g)) 2 dx < ∞ where we have used 4(1+(a 2 +b 2 ) s/2 ) ≥ (1+|a| s/2 )(1+|b| s/2 ). Note that N 4 (h) = N 2 (g) = +∞, which motivates the expansion up to order s = 5. Let σ 2 def = var(log Y 2 ) = π 2 /6. Applying Corollaries 9 and 10 respectively to the functions g, h, we get for some integer l 0 and any k, j such that l 0 ≤ k < j ≤ m,
|E[η 2
k ] − σ 2 | ≤ C(β, δ, ϑ, µ) k −1 + (k/n) β + n −1/2 (F.5) |E[η k η j ]| ≤ C(β, δ, ϑ, µ) k −2 + (j/n) 2β + n −1 . ν 2 k k −1 + (k/n) β + n −1/2 + C(β, δ, ϑ, µ)s −4 m ℓ<k<j≤m ν k ν j k −2 + (j/n) 2β + n −1 =C(β, δ, ϑ, µ)s −2 m 1 + O ℓ −1/2 + m 1/2 l −3/2 + m 2β+1 n −2β + m/n . (F.7)
Figure 1 .Figure 2 .
12Comparisons of the MSE versus the bandwidth for the FARIMA processes (a),(b) and (c). Sample size n Box-plot of the GPH estimator for processes (a),(b) and (c), sample size n = 250, 2500 Appendix A. Edgeworth expansion for triangular arrays
Table 1 .
1MC (dm opt − d) -0.02813 -0.01871 -0.02778 E MC ((dm opt − d) MC (dm opt − d) -0.01984 -0.02377 -0.01737 E MC ((dm opt − d) MC (dm opt − d) -0.01492 -0.00967 -0.02037 E MC ((dm opt − d) MC (dm opt − d)-0.01097 -0.00937 -0.01269 E MC ((dm opt − d) Optimal bandwidth, bias and MSE for processes (a), (b) and (c) and different sample sizes n. All those values are estimated by Monte Carlo
( B2 )( B3 )
B2B3There exist positive constants η, c 0 , a sequence (M n ) n∈N of positive numbers, and a sequence (J n ) n∈N of subsets of Z, such that, for all n There exist ζ ≥ 1 and a sequence (M n ) n∈N satisfying (A.
from (B.6) and (B.8) with v * = 1 2 min(v 1 , v 2 ). Proof of Theorem 1. By (B.1) and Lemma 13, (B1) holds with v * and v * of Lemma 13, for some N 0 , n ≥ N 0 and uniformly in k, α and β.With (B.3), j∈Z U n,j (k) 2 − u = |trace[V n (k)] − u| ≤ C(α, β)n −1/2 . (B.9)Prove now that (B2) is verified. Since f is bounded away from zero and j∈Z |ψ j | ≤ β < ∞, (A.1) and (A.2) are verified with M n def = C(r)βα −1/2 n −1/2 . Put J n = {j, |j| < 2n}. Then card(J n ) ≤ c 0 M −2 n for some c 0 depending only on r, α, β and |j|∈Z\Jn U n,j (k) 2 ≤ C(r)α −2 , |ψ j | ≤ β|j| −1/2−δ so that |j|≥n |j|ψ 2 j ≤ βn −2δ |j|≥n |j| 1/2+δ |ψ j | ≤ β 2 n −2δ (B.11)
follows immediately. The proof of (D.3) follows by picking N 0 , K 0 large enough. For (D.
e
iuλ k (ψ t+j − ψ t+j+1 ) + n u=1 e iuλ k ψ n+j .
M n,j = Cn −1/2 .Then (A.1) and (A.2) hold uniformly in k. By Lemma 15, Eq. (D.3) or (D.2), we have j U n,j (k) 2 = trace[V n (k)] ≥ v * > 0.
L(λ k ) =: d + W m + b m . (F.2)The mean-square error of the GPH writes E((d m − d) 2 ) = EW 2 m + 2b m EW m + b 2 m . Applying (F.1) and the Cauchy-Schwartz inequality,
Thus, to prove Theorem 11, we only need to show that E[W 2 m ] ≤ Cm −1 . We now compute E[W 2 m ]. Let ℓ = ℓ(m) be a non decreasing sequence of integers such that 1 ≤ ℓ ≤ m and define W 1,m = s −2 m ℓ k=1 ν k η k and W 2,m = W m − W 1,m . We first give a bound for E[W 2 1,m ]. Note thatE[W 2
1,m ] ≤ ℓs −4
m
ℓ
k=1
ν 2
k E[η 2
k ].
(F.4)
F.6) (F.4) and (F.5) yield E[W 2 1,m ] ≤ Cℓ 2 m −2 . We now bound E[W 2 2,m ]:ν k ν j E[η k η j ].E[W 2
2,m ] = s −4
m
m
k=ℓ+1
ν 2
k E[η 2
k ] + 2s −4
m
ℓ<k<j≤m
Using (F.5) and (F.6), we obtain
E[W 2
2,m ] − s −2
m σ 2 ≤C(β, δ, ϑ, µ)s −4
m
m
k=ℓ+1
Choosing ℓ ≤ m such that ℓ 2 = o(m) and m = o(ℓ 3 ) (for instance ℓ =. m η ] with 1/3 < η <Choosing ℓ ≤ m such that ℓ 2 = o(m) and m = o(ℓ 3 ) (for instance ℓ = [m η ] with 1/3 < η <
This bound and (F.3) conclude the proof of Theorem 11. = O(m −1 ). This bound and (F.3) conclude the proof of Theorem 11.
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Laboratoire Paul-Painlevé, Current address: APC, Université. Université Lille-1, 59655 Villeneuve-d'Ascq Cedex, France; Paris-7, 10, rue Alice Domonet Léonie Duquet, 75205 Paris Cedex 13, France E-mail address: [email protected] Paul-Painlevé, Université Lille-1, 59655 Villeneuve-d'Ascq Cedex, France. Current address: APC, Université Paris-7, 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France E-mail address: [email protected]
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arxiv |
Published as a Tiny Paper at ICLR 2023 TOPEX: TOPIC-BASED EXPLANATIONS FOR MODEL COMPARISON
2 Jun 2023
Shreya Havaldar [email protected]
Department of Computer Science
University of Pennsylvania
Adam Stein [email protected]
Department of Computer Science
University of Pennsylvania
Eric Wong [email protected]
Department of Computer Science
University of Pennsylvania
Lyle Ungar [email protected]
Department of Computer Science
University of Pennsylvania
Published as a Tiny Paper at ICLR 2023 TOPEX: TOPIC-BASED EXPLANATIONS FOR MODEL COMPARISON
2 Jun 202310.1145/35465771
Meaningfully comparing language models is challenging with current explanation methods. Current explanations are overwhelming for humans due to large vocabularies or incomparable across models. We present TopEx, an explanation method that enables a level playing field for comparing language models via model-agnostic topics. We demonstrate how TopEx can identify similarities and differences between DistilRoBERTa and GPT-2 on a variety of NLP tasks.1 Note that our approach works with any feature-based explanation. 2 When a word in our vocabulary is not in any topics, (e.g. punctuation, LDA stopwords or words not in LIWC) we naively treat it as a different topic. We leave other approaches, such as clustering, for future work.
HOW DO WE COMPARE LANGUAGE MODELS?
Language models (LMs) often exhibit differences in behavior even when trained on the same dataset. Architecture, pre-training, and hyperparameter choices can all lead to varying behaviors in the LM.
However, understanding these differences beyond comparing performance metrics is challenging. Existing post-hoc interpretability approaches primarily focus on explaining individual models as opposed to comparing models. These explanations can be generally categorized by how behavior is explained: (a) feature-based, using feature attributions (Shapley et al., 1953;Sundararajan et al., 2017); (b) example-based, using previously observed samples or generated counterfactuals; or (c) concept-based, using concepts extracted from a model's latent space (Madsen et al., 2022). Methods that fall under (b) and (c) cannot be used for comparison, as the examples and concepts are model-specific and not easily comparable across models. Methods under (a) can be used to compare models, but the tens of thousands of unique tokens renders such comparisons uninterpretable.
In order to meaningfully explain and compare LMs, we propose TopEx -a topic-based explanation method. TopEx condenses feature attributions into a model-independent explanation using topic modeling, a popular statistical method that assigns words to meaningful categories.
TOPIC-BASED EXPLANATIONS (TOPEX)
In this section, we outline our approach for generating topic-based explanations, which consists of two main steps: (1) calculation of feature-based scores followed by (2) aggregation into topics. Given two LMs trained on the same dataset, we first generate word-level importance scores for each model. We extract Shapley values 1 (Lundberg & Lee, 2017) for all instances and aggregate these scores into global importance scores g w for each word w. We then map these word-level importance scores g w to topic-level importance scores G t as follows:
Gt = w∈topic t P (topic t |w)gw(1)
Specifically, for all words in a given topic t, we sum over word importance scores, weighted by word membership in each topic P (topic t |w). 2 Here, the weight comes from the topic model, which could come from an existing topic lexicon such as LIWC Pennebaker et al. (2001) or be automatically learned with e.g. Latent Dirochlet Allocation (LDA) (Blei et al., 2003). Details on token-to-word aggregation and topic weighting schemes are in Appendix B and C respectively. Figure 1: Generating a global explanation via TopEx. We extract an importance score for each token using Shapley values, aggregate to average global word importances, and map these importance scores to the corresponding topics for each word.
Figure 1 demonstrates our method on an example sentence. The first step of TopEx computes wordlevel importance scores. For example, the word "tasty" gets a an aggregate score of 1.01 as an average of its feature attribution scores 0.73 and 1.29. The second step of TopEx computes topiclevel importance scores. For example, scores for food-related words such as "burgers" and "fries" are aggregated with "tasty" to get the final "food" topic score. The resulting topic-based explanation is a concise summary of the model that can be used to directly compare with other models.
TOPEX EXPLAINS DIFFERENCES BETWEEN MODELS
We compare fine-tuned DistilRoBERTa (Sanh et al., 2019) and GPT-2(Radford et al., 2019) on the Yelp Reviews dataset (Zhang et al., 2015) and the GoEmotions dataset (Demszky et al., 2020). From our topic-based explanations of these models, G BERT and G GPT , we calculate the distance between explanations as G ∆ = ∥G BERT − G GPT ∥ 1 . The two topics with the most different and most similar importance scores are highlighted in Figure 2. We can see that DistilRoBERTa focuses more than GPT-2 on descriptions of dining when classifying a 5-star rating, while GPT-2 looks more at negativity than DistilRoBERTa. In this case, GPT-2 may be determining bad reviews through negative words, while DistilRoBERTa has learned to better recognize descriptions of dining experiences characteristic of 5-star reviews. Experiment details and additional results are given in Appendix D and E.
Conclusion. The vast array of possible LM architectures and training schemes motivates the need to deeper understand differences in model behavior beyond performance metrics. In this work, we present TopEx, a method that enables direct model comparisons via model-agnostic topics that can reveal why and how models behave differently.
Yelp (5-star)
GoEmo (fear) AFFECT WE PRONOUNS NEG. EMOTION LIWC TOPIC Figure 2: We explain differences in behavior of DistilRoBERTa and GPT-2 via G ∆ . We show the two topics with most different (max(|G ∆ |)) and most similar (min(|G ∆ |)) importances between models, using LDA topics for Yelp and LIWC topics for GoEmotions. Topic visualizations in blue indicate G ∆ t > 0 (i.e. the topic is more important for DistilRoBERTa), while red indicates G ∆ t < 0 (i.e. the topic is more important to GPT-2).
A APPENDIX
We include additional information detailing TopEx. Appendix B explains token-level to word-level importance score aggregation, Appendix C details topic membership weighting, Appendix D provides experiment details, and Appendix E shows additional results and topic visualizations.
B TOKEN-TO-WORD AGGREGATION
This section details the aggregation from token-level Shapley values,
[v i 1 , v i 2 , . . . , v i Ki ] to global word-level importance scores, [g 1 , g 2 , . . . , g V ],
where K i is the total number of tokens in the ith input and V is the size of our vocabulary.
For each of our models, we extract Shapley values
, v i = [v i 1 , v i 2 , . . . , v i Ki ], computed with Partition SHAP (Lundberg & Lee, 2017) for each token x i k in the ith input s i = [x i 1 , x i 2 , . . . , x i Ki ].
We then calculate the Shapley values for each wordx i w by summing over its constituent tokens. We write the ith word-level input asŝ
i = [x i 1 , . . . ,x i Wi ] with corresponding word-level Shapley valueŝ v i = [v i 1 , . . . ,v iv i j = x k ∈xj v i k (2) g w = C(w) N i=1 |ŝ i | j=1 1 [xj =w] |v i j |(3)
The weighting C(w) in Equation 3 can be set to balance between the impact of word frequency and word importance when aggregating local to global explanations. One choice for C(w) is simply C(w) = 1, which is the traditional way of aggregating local explanations through summation.
An alternative is the inverse of the number of times a word appears in the dataset,
C(w) = 1 N i=1 |ŝ i | j=1 1 [xj =w] ,(4)
which removes the effect of word frequency from the global word importance. Results in Appendix E use the above weighting and thus do not take into account word frequency when mapping to word-level importance scores.
B.1 PRESERVING ADDITIVITY
Note that we can slightly modify the above equations to first compute local topic-level importance scores that preserve Shapley additivity, and then aggregate from local to global topic-based explanations, generating a faithful explanation.
Equation 5 describes aggregation from local word-level Shapley valuesv
i = [v i 1 , . . . ,v i
Wi ] to a local word-level importance score, l i w for word w in the ith input. We then aggregate from the local word-level importance score to a local topic-level importance score, l i t for the tth topic, shown in Equation 6. Lastly, Equation 7 details how to aggregate local topic-level importance scores, L i = [L i 1 , . . . , L i T ] to compute a global topic-level explanation.
l i w = |ŝ i | j=1 1 [xj =w]v i j (5) L i t = w∈topic t P (topic t |w)l i w (6) G t = N i=1 |L i t | (7)
This modified aggregation provides a way to compute a topic-based explanation for a single instance that preserves the additive property of Shapley values, as local topic-based explanations for an instance will sum to the model's output for that instance. Additionally, the sum of the topic importances equals the total effect.
C TOPIC MEMBERSHIP WEIGHTING
We describe the value of P (topic t |w) in Equation 1 and 6 for various topic modeling methods. When using LDA topics, this weight is learned by the topic model, and comes from P (w|topic t ).
To derive a topic distribution, P (topic t |w), from the topic model's word distibution, P (w|topic t ), for each topic, we simply renormalize the word distributions:
P (topic t |w) = P (w|topic t )P (topic t ) P (w) ∝ P (w|topic t ) w P (w|topic t ) .(8)
This works under the assumption that P (topic t ) is equal for all topics, which holds for LDA due to the use of the symmetric Dirichlet distribution.
For LIWC and other unweighted topic lexicons, this membership weight is 1/T w , where T w is the number of topics a word appears in.
D EXPERIMENT DETAILS
Datasets We fine-tune DistilRoBERTa and GPT-2 on three classification tasks: the Yelp Reviews dataset (Zhang et al., 2015), a polarity detection task based on the number of stars associated with a text review; the GoEmotions dataset (Demszky et al., 2020), an emotion classification task where we scope to only the six Ekman emotions; and the Blog Authorship Corpus (Schler et al., 2006), an authorship attribution task where we scope to age and gender 3 .
Models Both models were trained with Adam using 3 epochs, a learning rate of 5.00e−5, and half precision. DistilRoBERTa was trained with a batch size of 64 and GPT-2 was trained with a batch size of 32. The maximum token length was always set to 512 so that both DistilRoBERTa and GPT-2 were trained on the same data. For GoEmotions, we add an output layer of size 6 use binary cross entropy loss for multilabel classification. Similarly, for Blog Authorship we use binary cross entropy loss and an output layer of size 5 (for two genders and 3 age groups). For Yelp we use an output layer of size 5 and train using cross entropy loss for single label classification into 1-5 stars. Table 1 shows the resulting accuracies and F1 scores for these tasks. For multilabel classification datasets (Blog and GoEmotions), the table contains the average accuracy and F1 score across all classes. For Yelp Reviews, we combine 1-2 star classifications (negative reviews) and 3-4 star classifications (positive reviews) and report performance metrics on polarity classification.
Topic Modeling To get our LDA topics, we run MALLET (Graham et al., 2012) with 30 topics, α = 5.0, and β = 0.01. We use the 100 most frequent words in each dataset as stopwords. We use the standard LIWC lexicon (Pennebaker et al., 2001) and treat each category as a unique topic with identical weighting for words within each category.
E FULL RESULTS
For models trained on Yelp Reviews and the Blog Authorship Corpus, we use TopEx with Partition SHAP Lundberg & Lee (2017) for feature attribution and LDA for topic modeling. For models trained on GoEmotions, we use TopEx with Partition SHAP for feature attribution and LIWC topics. Table 2 shows the three most important topics and least important topics for each model, along with the corresponding topic importance score. These scores are L1 normalized for direct numerical comparison between models. We also calculate ||G BERT || 1 − ||G GPT || 1 = G ∆ to measure topicwise differences in importance. The topics with the greatest magnitude in this residual explanation are shown in the rightmost column of Table 2. Specifically, we show the three topics with the most different importance scores (max(|G ∆ |)) and the most similar importance scores (min(|G ∆ |)) between models. Table 2. Topics are named based on manual evaluation of the top 15-20 words within each topic. We find all topics had some unifying theme and were easy to name. Figure 3 contains the Yelp Reviews topics and Figure 4 contains the Blog Authorship Corpus topics.
Figure 3 :Figure 4 :
34Visualizations Visualizations for LDA topics on the Blog Authorship Corpus E.1 TOPIC VISUALIZATIONS We show word clouds to visualize all LDA topics shown in
Wi ] for the W i words in the ith input. This calculation of local word-level Shapley values is shown in Equation 2. From local word-level Shapley values, we derive global word-level importance scores by aggregating the absolute value of local word-level Shapley values over each word as shown in Equation 3. Note that we choose to take the absolute value to aggregate over magnitude of importance.
Table 1 :
1Accuracy and F1 score for trained models on the three benchmark datasets. For multilabel classification datasets (Blog and GoEmotions), we report the average accuracy and F1 score across all classes as well as per class metrics. For the Yelp dataset, we report the accuracy on the standard polarity task as well as accuracy for predicting 5-star reviews.DistilRoBERTa
GPT-2
F1
Accuracy
F1
Accuracy
Blog (Avg.)
69.2
46.9
69.0
45.9
Blog (Female)
69.2
69.4
70.3
69.8
Blog (23-33)
72.6
72.7
71.4
72.1
Yelp (Polarity)
93.3
91.9
92.5
90.9
Yelp (5-star)
76.2
78.1
74.1
76.2
GoEmotions (Avg.) 53.5
87.2
58.7
88.0
GoEmotions (Joy)
60.4
97.7
62.3
97.9
GoEmotions (Fear) 69.8
99.1
68.4
99.1
Table 2 :
2TopEx on three benchmark datasets. We do a manual evaluation of LDA results to name topics, and show further topic visualizations Appendix E.1G BERT G GPT G BERT − G GPTTopic
Importance Topic
Importance Topic
Importance
Blog
(Female)
technology
4.82e−2 technology
4.46e−2 animals
−4.91e−3
texting
4.20e−2 animals
4.35e−2 technology
3.57e−3
travel
3.90e−2 texting
4.29e−2 "xbubzx"
−3.31e−3
descriptors
1.02e−2 descriptors
9.33e−3 games
−7.79e−6
longing
1.07e−2 longing
1.20e−2 school/work
1.28e−4
temporal
1.48e−2 temporal
1.46e−2 temporal
2.04e−4
Blog
(23-33)
technology
5.59e−2 technology
5.01e−2 texting
−6.54e−3
animals
4.06e−2 animals
4.54e−2 technology
5.87e−3
books/movies
4.01e−2 texting
4.26e−2 animals
−4.87e−3
descriptors
1.01e−2 descriptors
8.95e−3 politics
−4.17e−5
longing
1.08e−2 longing
1.20e−2 music
−2.85e−4
temporal
1.41e−2 temporal
1.30e−2 economy
3.20e−4
Yelp
(5-star)
atmosphere
6.51e−2 review
6.18e−2 dining
1.21e−2
review
5.60e−2 atmosphere
6.10e−2 negativity
−1.19e−2
entertainment
5.55e−2 entertainment
4.74e−2 conversation
−1.17e−2
time
1.31e−2 visiting
1.75e−2 american food −5.14e−5
visiting
1.50e−2 time
1.81e−2 labor
−4.84e−4
location
2.03e−2 breakfast
2.20e−2 french
−5.82e−4
GoEmo
(Fear)
AFFECT
0.172 AFFECT
8.92e−2 NEGEMO
9.01e−2
NEGEMO
0.161 NEGEMO
7.11e−2 AFFECT
8.27e−2
ANX
0.102 ADJ
4.84e−2 ANX
6.32e−2
FILLER
1.57e−5 WE
1.11e−4 PPRON
−1.10e−5
WE
3.93e−5 FILLER
1.43e−4 WE
−7.21e−5
YOU
5.31e−5 YOU
2.56e−4 SHEHE
−1.10e−4
GoEmo
(Joy)
AFFECT
0.174 AFFECT
0.122 POSEMO
5.38e−2
POSEMO
0.161 POSEMO
0.107 AFFECT
5.16e−2
ADJ
6.38e−2 ADJ
4.59e−2 ADJ
1.80e−2
WE
8.89e−5 WE
6.97e−5 WE
1.92e−5
SHEHE
1.68e−4 THEY
2.04e−4 ARTICLE
1.96e−5
YOU
1.99e−4 FILLER
2.33e−4 INGEST
−3.17e−5
Gender was measured by a binary label, and we note this does not cover the entire population of possible bloggers.
The authors acknowledge that the first and second authors of this work meet the URM criteria of ICLR 2023 Tiny Papers Track.
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Character-level convolutional networks for text classification. Xiang Zhang, Junbo Zhao, Yann Lecun, Xiang Zhang, Junbo Zhao, and Yann LeCun. Character-level convolutional networks for text classification.
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| {'fraction_non_alphanumeric': 0.0570141065830721, 'fraction_numerical': 0.044572884012539185, 'mean_word_length': 4.1493064312736445, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 3, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Meaningfully comparing language models is challenging with current explanation methods. Current explanations are overwhelming for humans due to large vocabularies or incomparable across models. We present TopEx, an explanation method that enables a level playing field for comparing language models via model-agnostic topics. We demonstrate how TopEx can identify similarities and differences between DistilRoBERTa and GPT-2 on a variety of NLP tasks.1 Note that our approach works with any feature-based explanation. 2 When a word in our vocabulary is not in any topics, (e.g. punctuation, LDA stopwords or words not in LIWC) we naively treat it as a different topic. We leave other approaches, such as clustering, for future work.', 'arxivid': '2306.00976', 'author': ['Shreya Havaldar [email protected] \nDepartment of Computer Science\nUniversity of Pennsylvania\n\n', 'Adam Stein [email protected] \nDepartment of Computer Science\nUniversity of Pennsylvania\n\n', 'Eric Wong [email protected] \nDepartment of Computer Science\nUniversity of Pennsylvania\n\n', 'Lyle Ungar [email protected] \nDepartment of Computer Science\nUniversity of Pennsylvania\n\n'], 'authoraffiliation': ['Department of Computer Science\nUniversity of Pennsylvania\n', 'Department of Computer Science\nUniversity of Pennsylvania\n', 'Department of Computer Science\nUniversity of Pennsylvania\n', 'Department of Computer Science\nUniversity of Pennsylvania\n'], 'corpusid': 258999858, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 6785, 'n_tokens_neox': 6003, 'n_words': 3102, 'pdfsha': 'ec5660aeca844379dff83d1e90b93e2dace7bbf7', 'pdfurls': ['https://export.arxiv.org/pdf/2306.00976v2.pdf'], 'title': ['Published as a Tiny Paper at ICLR 2023 TOPEX: TOPIC-BASED EXPLANATIONS FOR MODEL COMPARISON', 'Published as a Tiny Paper at ICLR 2023 TOPEX: TOPIC-BASED EXPLANATIONS FOR MODEL COMPARISON'], 'venue': []} |
arxiv |
Kinematic Jacobi Identity is a Residue Theorem
Sebastian Mizera
Institute for Advanced Study
Einstein Drive08540PrincetonNJUSA
Kinematic Jacobi Identity is a Residue Theorem
We give a geometric interpretation of color-kinematics duality between tree-level scattering amplitudes of gauge and gravity theories. Using their representation as intersection numbers we show how to obtain Bern-Carrasco-Johansson numerators in a constructive way as residues around boundaries of the moduli space. In this language the kinematic Jacobi identity between each triple of numerators is a residue theorem in disguise.
INTRODUCTION
Computation of scattering amplitudes in gravitational theories has traditionally posed a formidable task-even for tree-level processes-due to a proliferation of Feynman diagrams involved. This fact has changed with the introduction of color-kinematics duality [1,2] by Bern, Carrasco, and Johansson (BCJ), which provides a shortcut in computing gravitational observables by extracting the relevant information from gauge theory. It has since found applications in a spectrum of topics ranging from the study of ultraviolet properties of gravity [3][4][5][6][7][8], through the construction of classical solutions [9][10][11][12][13][14][15][16], to gravitational-wave physics [17][18][19][20][21][22].
Working at tree level, let us make the statement of colorkinematics duality more precise. Scattering amplitudes of n gauge bosons can be expressed as
A gauge n = Γ n Γ c Γ e∈Γ p 2 e ,(1)
where the sum goes over all (2n−5)!! trivalent trees Γ with propagators p 2 e associated to each internal edge e of Γ. Here c Γ denotes the color structure attached to each diagram, while n Γ is the remaining part of the numerator involving kinematic information such as contractions of momenta and polarization vectors.
Let us isolate triples of terms in (1) with graphs denoted by (Γ s , Γ t , Γ u ) differing only by a single subdiagram as follows:
Γ s Γ t Γ u(2)
Color structures associated to such triples satisfy the Lie algebra Jacobi identity, c Γs + c Γt + c Γu = 0. Suppose that for every (Γ s , Γ t , Γ u ) we enforce a similar condition on the kinematic numerators, n Γs + n Γt + n Γu = 0,
known as the kinematic Jacobi identity. Since the numerators coming from Feynman diagram expansion do not naturally satisfy (3), it is typically a difficult task to bring them into such a form by reshuffling terms in (1). Assuming this can be done BCJ proposed [1] that scattering amplitudes in gravity theory can be written, up to normalization, as
A gravity n = Γ n ΓñΓ e∈Γ p 2 e ,(4)
where n Γ 's andñ Γ 's are two (possibly distinct) sets of Jacobi-satisfying numerators. This statement is now proven [23] and can be extended to loop level [1,2,[24][25][26][27][28][29][30][31][32][33][34][35], gauge and gravity theories with different supersymmetry and matter content [36][37][38][39][40][41][42][43][44], as well as various other theories [45][46][47][48][49][50][51][52]. Kinematic algebras leading to (3) have been investigated in [53][54][55]. For a comprehensive review of color-kinematics duality see [56]. At this stage one can ask if the kinematic Jacobi identity (3) has a geometric interpretation, and whether there exists a representation of scattering amplitudes that manifests this fact. These questions turn out to have a common answer, whose elucidation is the goal of this letter.
It has recently emerged that a natural framework for addressing such problems is that of intersection theory [57]. It was previously used to provide a geometric interpretation of Kawai-Lewellen-Tye (KLT) [58] relations between string-and field-theory amplitudes in terms of intersections of associahedra [59][60][61]; write down higherloop monodromy and BCJ [1,62] relations for loop integrands [63]; understand precise conditions under which the low-energy limit of string-theory amplitudes localizes on scattering equations [57,61]; as well as give a new perspective on differential equations, dimensional recurrence relations, and integration-by-parts identities for multi-loop Feynman integrals [64][65][66][67], among other applications [68][69][70][71][72][73][74]. At the same time this line of research unraveled connections between scattering amplitudes and more formal topics including Morse theory [57], Euler characteristics [61,64], Landau-Ginzburg models [67], and Yang-Baxter relations [61].
The central role in this theory is played by the socalled intersection numbers, which provide a geometric representation of tree-level amplitudes in various quantum field theories [57,61]. Selecting a theory amounts to specifying two differential forms, ϕ − and ϕ + , on the moduli space of Riemann spheres with n punctures, M 0,n . In the low-energy limit intersection numbers are computed by [61] Γ Res vΓ (ϕ − ) Res vΓ (ϕ + )
e∈Γ p 2 e .(5)
Here the sum is of exactly the same form as in (1) and (4), and the role of numerators-both color and kinematic ones-is played by the residues around maximalcodimension boundaries of the moduli space, v Γ , which are in one-to-one map with trivalent diagrams Γ. We will prove that the numerators in (5) always satisfy the kinematic Jacobi identity (3) as a consequence of a residue theorem, thus providing a manifestly colorkinematics dual representation of amplitudes.
In this language the problem of finding numerators for various theories translates to different choices of ϕ ± . After reviewing a known catalog of such forms for gauge and gravity theories we give explicit examples of computing Jacobi-satisfying numerators.
BOUNDARIES AND RESIDUES
Let us briefly review the factorization structure of the moduli space M 0,n provided by its compactification [75]. When a subset R of punctures collides on the Riemann sphere, the surface should be thought of as "bubbling" into two new spheres, where an emergent puncture I separates the set R from the complementary set L (with sizes 2 ≤ |L|, |R| ≤ n−2):
I ∼ = L R (6)
It is a codimension-one component of the boundary divisor ∂M 0,n . We can make this procedure concrete on the level of differential forms. Take ϕ to be a top (degree n−3) holomorphic form on M 0,n , i.e., proportional to the SL(2, C)-covariant measure
dµ n = (z p −z q )(z q −z r )(z p −z r ) n i=1 i =p,q,r dz i ,(7)
where (z p , z q , z r ) denote the positions of three arbitrary punctures fixed by the action of SL(2, C). For massless scattering we must require that ϕ is invariant under
SL(2, C) transformations z i → (Az i +B)/(Cz i +D) with AD−BC = 1 for all z i 's.
A standard way of modeling the above factorization is to embed the original sphere CP 1 as a conic in CP 2 with a new parameter , such that it factors into CP 1 ×CP 1 as → 0, see, e.g., [76]. In coordinates, we perform the change of variables
z i = /x i for i ∈ L, y i / for i ∈ R,(8)
where x i 's and y i 's are positions of punctures on the new spheres with exactly two x i 's and two y i 's fixed. Since the boundary lies along { 2 =0} we can simply take
Res 2 =0 (ϕ) = ϕ L ∧ ϕ R ,(9)
where ϕ L (x i ) and ϕ R (y i ) are now top (degree |L|−2 and |R|−2) holomorphic forms on the moduli spaces M 0,|L|+1 and M 0,|R|+1 of the left and right sphere respectively. From the perspective of the particles on the left sphere the emergent puncture is at x I = 0, while from the right sphere it is at y I = 0. Repeating this procedure exactly n−3 times one obtains maximal-codimension components (vertices) v Γ of ∂M 0,n , which are in one-to-one map with trivalent graphs Γ, as all punctures are fixed by the action of SL(2, C), e.g.,
↔ 2 I 1 I 2 5 1 2 3 4 1 5 4 3 I 1 I 2(10)
The corresponding numerator n Γ = Res vΓ (ϕ) is a function computed by applying (9) consecutively n−3 times. There exists an alternative way of computing Res vΓ (ϕ), based on the dihedral extension of M 0,n employing crossratio coordinates suited for each v Γ [77], which is particularly useful for planar amplitudes, see, e.g., [61,68,78].
KINEMATIC JACOBI IDENTITY
Let us consider the stage at which bubbling already happened n−4 times, i.e., when we are only one residue away from a trivalent factorization. It means there is exactly one sphere with four punctures:
R z t z s z u z(11)
This leaves us with a one-form ϕ M on the moduli space of the "middle" sphere, which was computed as an (n−4)fold residue of the original form ϕ. Let us call the unfixed puncture z and the fixed ones (z s , z t , z u ), such that z colliding with z i leads to a trivalent graph Γ i , as in (2). By definition of the numerators entering (5) we have:
n Γs = Res z=zs (ϕ M ), n Γt = Res z=zt (ϕ M ),
n Γu = Res z=zu (ϕ M ),(12)
which are residues around the boundaries of the remaining moduli space:
z zs zt zu Γ s Γ t Γ u(13)
Since there are no other poles the residue theorem reads
n Γs + n Γt + n Γu = 0,(14)
which is precisely the kinematic Jacobi identity (3). Given that we could have started with any configuration (11), this identity is satisfied for all possible triples (Γ s , Γ t , Γ u ). 1
BUILDING BLOCKS
At this stage we have demonstrated that any rational form ϕ on M 0,n leads to Jacobi-satisfying numerators, however it does not yet mean that the resulting (5) is a scattering amplitude. We need to learn how to pick differential forms of physical relevance, which is a domain of intersection theory.
The first step is to realize that such forms should be really treated as elements of cohomology (equivalence) classes labeled by a ± sign,
ϕ ± ∼ ϕ ± + (d ± dW ∧)ξ(15)
for any rational (n−4)-form ξ. Here W is a potential given by
W = 1 Λ 2 i<j 2p i ·p j log(z i −z j ).(16)
with a mass scale Λ. This is precisely how ϕ ± "know" about physics through the kinematic invariants p i ·p j . To distinguish them from ordinary differential forms we call ϕ ± twisted forms. Their space is (n−3)!-dimensional [80], in contrast with the space of ordinary forms, which is (n−2)!-dimensional [79]. In order to make the statements below non-trivial we typically impose that twisted forms 1 The identity (14) means that for each triple only two out of three numerators are Z-independent. One can ask how these relations combine for subdiagrams with m≥4 external legs by considering the "middle" sphere (11) to have m points. The results of [79] show that all residue theorems must reduce the number of Zindependent numerators down to dim H m−3 (M 0,m , Z) = (m−2)! from the total of (2m−5)!!.
have no kinematic poles, which in turn implies that the numerators n Γ are local. One can construct a bilinear of ϕ − and ϕ + called their intersection number, ϕ − |ϕ + dW , given by integrating the two forms over the moduli space. While such invariants have been known in mathematics for decades [81][82][83], only recently they were identified as representations of tree-level scattering amplitudes in various massive and massless quantum field theories in arbitrary space-time dimension [57], see [61] for a comprehensive introduction. We focus on massless external states, p 2 i =0, from now on. There exists a catalog of twisted forms, which can be mixed and matched to compute different amplitudes [61]. For theories with color degrees of freedom T ci we have
ϕ color ± = dµ n Tr(T c1 T c2 · · · T cn ) (z 1 −z 2 )(z 2 −z 3 ) · · · (z n −z 1 ) + perm. ,(17)
where the symmetrization involves (n−1)! cyclic permutations (the definition is the same for both ±). By construction the associated numerator is precisely the color structure of a given diagram, i.e., Res vΓ (ϕ color ± ) = c Γ , as in (1). For theories with polarization vectors ε µ i we can use
ϕ gauge ± = dµ n n i=1 dθ i dθ i θ k θ z k −z exp i =j Φ ij ,(18)
(the choice of k and is arbitrary) with
Φ ij = − θ i θ j p i ·p j +θ iθj ε i ·ε j + 2(θ i −θ j )θ i ε i ·p j z i −z j ∓ Λ 2 θ i θ j .(19)
For conciseness we wrote it in terms of Grassmann integrals over θ i andθ i , which can be expanded as a degree-
ϕ bosonic ± = dµ n (±Λ) n−2 n i=1 dθ i dθ i exp i =j Ξ ij , (20) where Ξ ij = ± 1 Λ 2θ jθj p i ·ε j z i −z j + θ iθi θ jθj ε i ·ε j (z i −z j ) 2 .(21)
Upon the identification Λ 2 =1/α , (18) and (20) are in fact the same objects as those in super-and bosonic string perturbation theory respectively [84], but-surprisinglynow appear in a purely field-theoretic context. A partial list of theories whose amplitudes are known to have an interpretation as intersection numbers is given below [ [42,86] Even though (18) depends on Λ, this dependence drops out from the resulting amplitudes in the first three cases (it is not true for the last two) [61,87]. Since amplitudes are written as bilinears in this representation, KLT relations between the above theories become simply a consequence of linear algebra. The total differential 0 ∼ (d±dW ∧)ϕ color ±,n−1 implies the fundamental BCJ relation [1], as an extension of the arguments in [88]. Twisted forms for states lying in the low-energy spectrum of string theory, such as those involving fermions or mixed Einstein-Yang-Mills interactions, can be readily written down using the techniques discussed in [61], but we will not pursue it here. Below we will extend the table with a few additional entries.
Scattering amplitudes in such a representation can be computed exactly using recursion relations [61], however the resulting numerators do not come in a Jacobisatisfying way. Instead, the localization formula (5) is known to arise as the Λ 0 order in the low-energy (Λ → ∞) expansion of intersection numbers [61], (22) when ϕ ± are independent of Λ. 2 However, with the exception of (17), twisted forms given above are polynomials in Λ 2 , which leads to mixing of different orders in (22). To consistently extract the leading order Λ 0 with (22) one needs to first remove the Λ-dependence from twisted forms by a repeated use of (15). Given that Yang-Mills and Einstein gravity amplitudes are independent of Λ, once this is done the terms of order O(Λ −2 ) are not present and the numerators are exact.
ϕ − |ϕ + dW = Γ Res vΓ (ϕ − ) Res vΓ (ϕ + ) e∈Γ p 2 e + O(Λ −2 ),
EXAMPLES
We proceed with two illustrative examples. In order to contain expressions within the margins of this letter we focus on the case n = 4, where amplitudes with color degrees of freedom take the form
A 4 = n s c s s + n t c t t + n u c u u ,(23)
with s = (p 1 +p 2 ) 2 , t = (p 2 +p 3 ) 2 , u = (p 1 +p 3 ) 2 and a single triple. Fixing the punctures (z 1 , z 2 , z 3 ) leaves us with a single coordinate z 4 on M 0,4 . Evaluating color 2 In the massless limit (Λ → 0) intersection numbers have another localization formula on the so-called scattering equations, dW =0, which at the leading order Λ 0 gives the Cachazo-He-Yuan (CHY) [85,89] formulation of massless amplitudes, see [57,61] for details. Since Yang-Mills and Einstein gravity amplitudes are independent of Λ to begin with, this limit is exact. Subleading corrections O(Λ 2p≥2 ) are given by higher residue pairings [67,90]. numerators using (17) for n = 4 amounts to computing the residues:
c s = Res z4=z3 (ϕ color −,4 ) = f c1c2b f bc3c4 , c t = Res z4=z1 (ϕ color −,4 ) = f c2c3b f bc1c4 ,(24)c u = Res z4=z2 (ϕ color −,4 ) = f c3c1b f bc2c4 ,
with the convention f abc = Tr(T a [T b , T c ]). In this case the residue theorem implies the usual Jacobi identity c s +c t +c u = 0. We consider kinematic numerators next.
Non-Linear Sigma Model
Before obtaining numerators in Yang-Mills theory, let us consider a toy model of the color-kinematics duality between massless non-linear sigma model (NLSM) and special Galileon amplitudes [45]. To this end we use replacement rules of [45,91], after which the twisted form ϕ gauge ± undergoes a vast simplification and becomes:
ϕ scalar ± = − dµ n (z k −z ) 2 det P [k ] ,(25)
where the subscript [k ] instructs one to remove columns and rows labeled by k and prior to taking the determinant. Entries of the matrix P are given by
P ij = 2p i ·p j z i −z j for i = j, − l =i 2p i ·p l z i −z l for i = j.(26)
Note that ϕ scalar ± is independent of Λ and (22)
dz 4 ,(27)
where z ij := z i −z j . Since (22) truncates at the leading order, we have the set of kinematic numerators for NLSM: n s = Res z4=z3 (ϕ scalar +,4 ) = s 2 + 2st, n t = Res z4=z1 (ϕ scalar +,4 ) = t 2 ,
n u = Res z4=z2 (ϕ scalar +,4 ) = −u 2 , which satisfy n s +n t +n u = 0. Note that numerators are not unique. For example, different initial choices of (k, ) in (25) lead to distinct sets of numerators. Amplitudes of the special Galileon theory are obtained by replacing c Γ → n Γ in (23).
Gauge Theory
Let us consider numerators in Yang-Mills theory. Choosing (k, ) = (1, 2) for n = 4 the twisted form (18) becomes:
ϕ gauge + = z 13 z 23 Pf Ψ [12] − 4Λ 2 ε 1 ·ε 2 ε 3 ·ε 4 z 12 z 2 34 dz 4 . (29)
Here Ψ is the matrix known from the CHY formalism [89] in the conventions of [61]. In order to fix the issue with Λ-non-homogeneity we use (15) with
ξ = 4Λ 2 ε 1 ·ε 2 ε 3 ·ε 4 z 13 z 24 z 12 z 34 ,(30)
obtained by integrating minus the final term in (29). Adding (d+dW ∧)ξ to (29) gives us a form cohomologous to (29), but independent of Λ: Therefore the leading order in (22) computes the full Yang-Mills amplitude. Using this representation we find: n s = Res z4=z3 (ϕ gauge +,4 ) = 8ε 1,µ ε 2,ν ε 3,ρ ε 4,τ [ p 1 ·p 2 (η µρ η ντ −η µτ η νρ ) (32)
ϕ gauge +,4 = z 13 z 23 Pf Ψ [12] dz 4(31)−p 2 ·p 3 η µν η ρτ + (p ρ 1 p τ 2 −p ρ 2 p τ 1 )η µν + p ν 1 p τ 3 η µρ −p ν 1 p ρ 4 η µτ −p µ 2 p τ 3 η νρ +p µ 2 p ρ 4 η ντ +(p µ 3 p ν 4 −p µ 4 p ν 3 )η ρτ ], n t = Res z4=z1 (ϕ gauge +,4 ) = 8ε 1,µ ε 2,ν ε 3,ρ ε 4,τ [ p 1 ·p 2 η µτ η νρ + p 2 ·p 3 η µν η ρτ (33) + p ρ 2 p τ 1 η µν − p ν 3 p τ 1 η µρ + (p ν 1 p ρ 4 −p ν 4 p ρ 1 )η µτ + (p µ 2 p τ 3 −p µ 3 p τ 2 )η νρ − p µ 4 p ρ 2 η ντ + p µ 4 p ν 3 η ρτ ], n u = Res z4=z2 (ϕ gauge +,4 ) = 8ε 1,µ ε 2,ν ε 3,ρ ε 4,τ [ −p 1 ·p 2 η µρ η ντ − p ρ 1 p τ 2 η µν(34)+ (p ν 3 p τ 1 −p ν 1 p τ 3 )η µρ + p ν 4 p ρ 1 η µτ + p µ 3 p τ 2 η νρ + (p µ 4 p ρ 2 −p µ 2 p ρ 4 )η ντ − p µ 3 p ν 4 η ρτ ].
One can check that n s +n t +n u = 0 and the resulting amplitude (23) is gauge invariant. Scattering amplitude of four gravitons is obtained by replacing c Γ →ñ Γ (withε i instead of ε i ) followed by a symmetrization of polarization tensors, ε µν i = ε (µ iε ν)
i .
CONCLUSION
In this letter we introduced a representation of tree-level scattering amplitudes that manifests color-kinematics duality. The problem of finding theories with Jacobisatisfying numerators translates to a classification of twisted forms, which motivates further extension of their available catalog.
The amplitudes computed with (18) have a remarkable property of being Λ-independent, as expected for massless theories, despite the fact ϕ gauge ± is not. On the other hand, it was previously shown that intersection numbers of logarithmic forms are independent of Λ [57,83]. Thus, one might suspect that once ϕ gauge ± is brought into a Λ-independent form (perhaps using the algorithms of [29,87,[92][93][94][95][96][97]) it would become logarithmic, as is the case for the examples (31) and (27). 3 The answer has to be proportional to Pf Ψ [k ] plus corrections polynomial in ∂W/∂z i since the latter ought to vanish after taking the Λ → 0 limit which, by (15), imposes scattering equations dW =0, cf. [94,95]. Finding a closed-form expression for all n remains an open problem, which is of both theoretical and practical importance.
Generalization to higher-loop order consists of two separate steps. The first is writing down the analogue of (5) in terms of (3g+n−3)-fold residues on genus-g moduli spaces, which necessarily satisfies the kinematic Jacobi identity by the same arguments as for g=0. The second is finding appropriate twisted forms generalizing (18) that give rise to loop integrands of gauge and gravity theories. The latter problem needs to be considered in the light of the fact that projectedness of supermoduli spaces (which was implicitly assumed in deriving (18)) breaks down at genus five [99].
ACKNOWLEDGMENTS
The author thanks Ricardo Monteiro, Radu Roiban, and Edward Witten for many useful comments. He gratefully acknowledges the funding provided by Carl P. Feinberg.
The author would like to thank Hadleigh Frost and Lionel Mason for sharing their parallel work [100] containing certain overlap with this letter. * [email protected] 3 Although any twisted form can be written as a logarithmic form [98], it is a non-trivial question whether such a form is independent of Λ and has no kinematic poles. This is true in pure spinor superspace [87,92].
Λ 2 of Pfaffians, see, e.g., [61, eq. (4.8)]. Similarly, we have the forms:
has no O(Λ −2 ) corrections. We can add the following entries to the previous table:ϕ −
ϕ +
theory
ϕ color
−
ϕ scalar
+
NLSM
ϕ scalar
−
ϕ scalar
+
special Galileon [45]
ϕ scalar
−
ϕ gauge
+
Born-Infeld
For instance, choosing (k, ) = (1, 2) for n = 4 we have
ϕ scalar
+,4
=
z 13 z 32
z 12
u
z 31
+
t
z 32
+
s
z 34
t
z 41
+
u
z 42
+
s
z 43
dz 4
− s 2 z 13 z 23
z 12 z 2
34
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arxiv |
Quantum Walk on Orbit Spaces
9 Jan 2023
Satoshi Ohya [email protected]
Institute of Quantum Science
Nihon University
Kanda-Surugadai 1-8-14101-8308ChiyodaTokyoJapan
Quantum Walk on Orbit Spaces
9 Jan 2023(Dated: January 10, 2023)
Inspired by the covering-space method in path integral on multiply-connected spaces, we here present a universal formula of time-evolution kernels for continuous-and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. We show that the time-evolution kernel on Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulae of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on Λ/Γ, all of which are constructed in exactly the same way as for the timeevolution kernel.
Introduction
Quantum walk-a quantum-mechanical analog of classical random walk on lattices or graphs-has been the subject of intense study over the last two decades. Just as in classical random walk, there exist two distinct formulations in quantum walk: continuous-time quantum walk and discrete-time quantum walk, the former is equivalent to tight-binding models in condensed matter physics, while the latter is a natural generalization of classical random walk and formulated without recourse to Hamiltonian operators. These two formulations have their own merits and their applications now appear in many disciplines, including quantum search algorithm [1,2], universal quantum computation [3][4][5][6], and topological phases of matter [7]; see refs. [8][9][10] for reviews. In both formulations, the central object is the probability amplitude for finding particles (walkers), which is given by a matrix element of time-evolution operator in position space-the time-evolution kernel. 1 This time-evolution kernel is normally calculated through spectral decomposition or numerical calculation, which becomes harder as the matrix size becomes larger. It would therefore be desirable if there exists a simpler method.
The purpose of this note is to present such a method by generalizing the Dowker's covering-space method [11] in path integral (see also refs. [12][13][14][15][16]). As is well known, in quantum mechanics on continuous spaces, the time-evolution kernel can be represented by the Feynman path integral, which provides a number of powerful methods to analyze quantum systems nonperturbatively. Among them is the covering-space method: it provides a universal method to construct the time-evolution kernel on multiply-connected spaces of the form =/ 1 (), where is the universal covering space of and 1 () is the fundamental group of . In this method, the path integral on is given by a linear combination of partial amplitudes, where each partial amplitude is given by the path integral on the universal covering space and linear-combination coefficients are given by a one-dimensional unitary representations of the fundamental group 1 (). Inspired by this method, we here present a universal formula for the time-evolution kernel in both continuous-and discrete-time quantum walks where walkers' configuration space can be regarded as the orbit space Λ/Γ. Here Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. A typical example for such configuration spaces is that for a single walker on a periodic lattice. Another typical example is the configuration space for identical walkers on an arbitrary lattice, where the indistinguishability of identical particles makes their configuration space an orbit space [13,[17][18][19][20]. We show that the time-evolution kernel on the orbit space Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ. This universal formula offers a simpler method to construct the time-evolution kernel on Λ/Γ because computation becomes generally much easier on Λ.
In what follows, we first set up the problem and then present our main formula and its proof. We then present a number of examples of the formula in section 3. In section 4, we present several other quantities that can be constructed in exactly the same way as for the time-evolution kernel. Examples include the resolvent kernel, the canonical density matrix, and a unitary representation of arbitrary groups. Section 5 is devoted to conclusion. Appendix A presents some sample computations in continuous-time quantum walk.
Throughout this note we will use the units in which ℏ = = 1, where is a lattice spacing.
Time-evolution kernel
To begin with, let us fix some notation. Let Λ be an arbitrary lattice (i.e., a discrete space spanned by a set of linearly independent vectors in a Euclidean space) and let Γ be a discrete group whose action on Λ has no fixed points. We note that Γ must be a discrete subgroup of the isometry of the Euclidean space, which consists of reflections, translations, and rotations. Let Λ/Γ be the orbit space (quotient space) given by the identification ∼ in Λ, where stands for the action of ∈ Γ on ∈ Λ that satisfies the compatibility condition 1 ( 2 ) = ( 1 2 ) for any 1 , 2 ∈ Γ and ∈ Λ. For the moment, we shall consider continuous-time quantum walk on the lattice Λ/Γ, where the Hilbert space H is the set of square-summable sequences on Λ/Γ, H = 2 (Λ/Γ). (Note, however, that the formula presented below is turned out to be applicable to discrete-time quantum walk as well; see section 4.3.) The action of the time-evolution operator on a state 0 ∈ H is defined by
( 0 )( ) ≔ ∈Λ/Γ ( , ) 0 ( ), ∀ ∈ Λ/Γ,(1)
where ( , ) is the time-evolution kernel and the subscript ∈ R represents the time. The probability for finding a particle at the time and at the position is then given by
( ) = |( 0 )( )| 2 .(2)
In particular, if the particle is initially localized at = 0 (i.e., 0 ( ) = , 0 ), the probability is simply given by ( ) = | ( , 0 )| 2 .
In the following, we shall construct ( , ) in terms of the time-evolution kernel on Λ. The key is the group property of the time-evolution operator.
The formula
The time-evolution operator is a one-parameter family of unitary operators. It satisfies the composition law 1 2 = 1 + 2 , the unitarity † (= −1 ) = − , and the initial condition 0 = , where stands for the identity operator. Correspondingly, the time-evolution kernel ( ⋅ , ⋅ ) must satisfy the following properties:
Here the overline ( ) stands for the complex conjugate. As we shall prove shortly, such a kernel can be constructed as follows:
( , ) = ∈Γ ( )̃ ( , ),(4)
where ∶ Γ → (1) ( ↦ ( )) is a one-dimensional unitary representation of Γ that satisfies the group composition law ( ) ( ′ ) = ( ′ ) and the unitarity ( ) = ( ) −1 = ( −1 ) for any , ′ ∈ Γ.
Herẽ ( ⋅ , ⋅ ) is a time-evolution kernel on Λ that satisfies the following assumptions:
• Assumption 1. (Composition law)
∈Λ̃ 1 ( , )̃ 2 ( , ) =̃ 1 + 2 ( , ), ∀ , ∈ Λ. (5a) • Assumption 2. (Unitarity)̃ ( , ) =̃ − ( , ), ∀ , ∈ Λ.(5b)
• Assumption 3. (Initial condition)̃
0 ( , ) = , , ∀ , ∈ Λ. (5c) • Assumption 4. (Γ-invariance) ( , ) =̃ ( , ), ∀ , ∈ Λ, ∀ ∈ Γ.(5d)
We note that the Γ-invariance (5d) is guaranteed if the Hamiltonian operator on Λ is invariant under the action of Γ. Before giving the proof, let us first present a quick derivation of the formula (4) by following the Dowker's method [11]. To this end, let̃ ( ) be an equivariant function on Λ that satisfies̃ ( ) = ( )̃ ( ) for any ∈ Λ and ∈ Γ. (The reason for using this will be apparent shortly.) Then we havẽ
( ) = ∈Λ̃ ( , )̃ 0 ( ) = ∈Λ/Γ ∈Γ̃ ( , )̃ 0 ( ) = ∈Λ/Γ ∈Γ̃ ( , ) ( )̃ 0 ( ) = ∈Λ/Γ ∈Γ ( )̃ ( , ) ̃ 0 ( ),(6)
where in the second equality we have used the following identity:
∈Λ ( ) = ∈Λ/Γ ∈Γ ( ).(7)
Here ( ) is an arbitrary test function on Λ. This identity just says that first summing over the orbit Γ ⋅ ≔ { ∶ ∈ Γ} of ∈ Λ/Γ and then summing over all ∈ Λ/Γ yields the summation over the whole space Λ. By comparing eq. (6) with the definition (1), we arrive at the formula (4). Now, sincẽ ( ⋅ , ⋅ ) is defined on the lattice Λ, the domain of ( ⋅ , ⋅ ) defined by eq. (4) can be naturally extended from Λ/Γ to Λ. In particular, it satisfies the following equation:
( , ) = ( ) ( , ), ∀ , ∈ Λ, ∀ ∈ Γ.(8)
In fact, a straightforward calculation gives
( , ) = ′ ∈Γ ( ′ )̃ ( , ′ ) = ′ ∈Γ ( −1 ′ )̃ ( −1 , −1 ′ ) = ( ) ′ ∈Γ ( −1 ′ )̃ ( , −1 ′ ) = ( ) ′′ ∈Γ ( ′′ )̃ ( , ′′ ) = ( ) ( , ),(9)
where the second equality follows from the Γ-invariance (5d) and the third equality follows from the group composition law ( −1 ′ ) = ( ) ( −1 ′ ). In the fourth equality, we have changed the summation variable from ′ to ′′ ≔ −1 ′ . It is now obvious from eq. (8) that ( 0 )( ) defined by eq. (1) also satisfies ( 0 )( ) = ( )( 0 )( ) for any ∈ Λ and ∈ Γ; that is, ( 0 )( ) becomes an equivariant function on Λ. This is the reason why we used the equivariant function in the above derivation. As we shall see in section 3, eq. (8) provides boundary conditions on Λ/Γ.
Finally, let us comment on the case where the action of Γ has fixed points. First, the identity (7) does not hold in general if there is a fixed point: if there is a point ∈ Λ that satisfies = for some (≠ ) ∈ Γ, where stands for the identity element of Γ, the right-hand side of eq. (7) leads to an overcounting of the fixed point . 2 Note, however, that if ( ) is subject to the Dirichlet boundary condition at the fixed point, such an overcounting does not occur so that eq. (7) holds true even in the presence of fixed points. 3 Note that the Dirichlet boundary condition ( ) = 0 at = can be deduced from the equivariant property ( ) ( ) = ( ) = ( ) if ( ) ≠ 1. Hence, if ∶ Γ → (1) is not the trivial representation, our formula (4) can be applied equally well to the case in which the action of Γ has fixed points. For the case of the trivial representation, however, the equivariant property does not lead to any definite boundary conditions. For simplicity, in this note we will mainly focus on the case where Γ has no fixed points.
Proof
Now we show that ( ⋅ , ⋅ ) given by the formula (4) satisfies the required properties (3a)-(3c) if is a one-dimensional unitary representation of Γ and if̃ ( ⋅ , ⋅ ) satisfies the assumptions (5a)-(5d). The proof is by direct computation. Each property is proved as follows. (See also refs. [21][22][23] for similar proofs in path integral.) Property 1. (Composition law) Let us first prove the composition law (3a). By substituting eq. (4) into the left-hand side of eq. (3a), we get
∈Λ/Γ 1 ( , ) 2 ( , ) = ∈Λ/Γ 1 ∈Γ 2 ∈Γ ( 1 ) ( 2 )̃ 1 ( , 1 )̃ 2 ( , 2 ) = ∈Λ/Γ 1 ∈Γ 2 ∈Γ ( 1 2 )̃ 1 ( , 1 )̃ 2 ( 1 , 1 2 ) = ∈Γ ( ) ∈Λ/Γ 1 ∈Γ̃ 1 ( , 1 )̃ 2 ( 1 , ) = ∈Γ ( ) ∈Λ̃ 1 ( , )̃ 2 ( , ) = ∈Γ ( )̃ 1 + 2 ( , ) = 1 + 2 ( , ),(10)
where the second equality follows from the group composition law ( 1 ) ( 2 ) = ( 1 2 ) and the Γinvariance (5d). The third equality follows from the change of the summation variable from 2 to ≔ 1 2 , and the fourth equality follows from the formula (7). Finally, the fifth equality follows from the assumption (5a).
Property 2. (Unitarity)
Let us next prove the unitarity (3b). By substituting eq. (4) into the left-hand side of eq. (3b), we get where the second equality follows from the unitarity properties ( ) = ( −1 ) and (5b). The third equality follows from the Γ-invariance (5d), and the last equality follows from the definition (4) (where the summation is over −1 instead of ).
( , ) = ∈Γ ( )̃ ( , ) = ∈Γ ( −1 )̃ − ( , ) = ∈Γ ( −1 )̃ − ( , −1 ) = − ( , ),(11)
Property 3. (Initial condition) Let us finally prove the initial condition (3c). By substituting eq. (4) into the left-hand side of eq. (3c), we get
0 ( , ) = ∈Γ ( )̃ 0 ( , ) = ∈Γ ( ) , = ( ) , = , ,(12)
where the second equality follows from the assumption (5c). The third equality follows from the fact that and cannot be equal for any , ∈ Λ/Γ except for the case = . Finally, the last equality follows from ( ) = 1 for any one-dimensional unitary representations of Γ.
Putting all the above things together, we see that eq. (4) is the sufficient condition to be the timeevolution kernel on the orbit space Λ/Γ. This completes the proof.
Examples
There exist a number of examples in which walkers' configuration space can be regarded as an orbit space. Typical examples are a single walker on a torus, the half space, and a cubic. Another typical example is identical walkers on an arbitrary lattice, where their configuration space always becomes an orbit space. In this section, we shall focus on one spatial dimension for simplicity and present several examples that fit into the formula (4). Let us start with single-walker examples.
A single walker in one dimension
Let̃ ( , ) be a time-evolution kernel on the integer lattice Λ = Z that satisfies the composition law (5a), the unitarity (5b), and the initial condition (5c) as well as the translation invariancẽ ( + , + ) = ( , ) and the reflection invariancẽ ( − , − ) =̃ ( , ) for any , , ∈ Z. A typical example of such a kernel is that of a free particle given bỹ ( ,
) = e 2 | − | | − | ( ),
where is the Bessel function of the first kind and (> 0) is a hopping parameter; see eq. (A.8) in appendix A. (Note, however, that the formulae presented below are not limited to free-particle theories. They are robust against any perturbations unless boundary conditions (8) are changed.) Below we shall construct time-evolution kernels for a single walker on a circle, the half line, and a finite interval by gauging these discrete symmetries.
Example 1. (A single walker on a circle) Let us first consider a single walker on a periodic lattice of sites, {1, 2, ⋯ , (mod )}. This lattice can be constructed from Z by making the identification ∼ + , where is an arbitrary integer. Hence the configuration space is the orbit space Z/ Z, where Z = ⟨ | ∅⟩ is the free group generated by a translation . Its action on Z is defined by
≔ + .(13)
Note that any element of Z can be written as the product , whose action on Z is given by = + . Now we need to find out one-dimensional unitary representations of Z. Since Z is the free group generated by a single generator , we have a one-parameter family of maps [ ] ∶ Z → (1) labeled by an angle parameter :
[ ] ( ) = e ,(14)
where ∈ R/2 R. It then follows from the formula (4) that the time-evolution kernel for a single walker on Z/ Z takes the following form:
[ ] ( , ) = ∞ =−∞ [ ] ( )̃ ( , ) = ∞ =−∞ e ̃ ( , + ).(15)
Just as in the path integral on a circle (see, e.g., section 2.4 of ref. [24]), eq. (15) represents the summation over winding numbers. Physically, eq. (15) describes the situation in which the walker acquires the Aharonov-Bohm phase e every time it winds around the circle, where plays the role of a magnetic flux penetrating through the circle. This is the physical meaning of the weight factor (14) and the summation over the orbit of initial point. Now two remarks are in order. First, it follows from eq. (8)
Z/Z 2 , where Z 2 = ⟨ | 2 = ⟩ is the cyclic group of order 2.
Here is the reflection whose action on Z is defined by
≔ 1 − .(16)
Note that 2 = . Note also that the reflection (16) does not have a fixed point in the integer lattice.
(Its fixed point is = 1/2.) Now, since 2 = , any one-dimensional unitary representation ∶ Z 2 → (1) must satisfy the condition ( ) 2 = 1, whose solution is ( ) = ±1. Hence there exist two distinct maps [ ] given by
[ ] ( ) = e ,(17)
where ∈ {0, (mod 2 )}. Correspondingly, there exist the following two distinct time-evolution kernels for a single walker on Z/Z 2 :
[ ] ( , ) = 1 =0 [ ] ( )̃ ( , ) =̃ ( , ) + e ̃ ( , 1 − ).(18)
Again, just as in the path integral on the half line [21,25,26], eq. (18) represents the summation over bouncing numbers off the boundary: the = 0 term is the contribution from the direct path, while the = 1 term is the contribution from the reflected path off the boundary. The physical meaning of the weight factor (17) is now clear: it plays the role of the reflection amplitude off the boundary. In other words, the walker acquires the phase shift when reflected from the boundary.
Notice that eq.
Z/ ∞ , where ∞ = Z ⋊ Z 2 = ⟨ , | 2 = ,
= −1 ⟩ is the infinite dihedral group generated by a translation and a reflection . 4 The actions of these operators on Z are defined as follows:
≔ + 2 and ≔ 1 − .(19)
Note that any element of ∞ can be written as , where = 0, ±1, ±2, ⋯ and = 0, 1. The action of this operator on Z is given by = + 2 for = 0 and = 2 + 1 − for = 1, respectively. Note also that, in contrast to the previous examples, ∞ is a non-Abelian discrete group. Now, since 2 = and = −1 , any one-dimensional unitary representation ∶ ∞ → (1) must satisfy the conditions ( ) 2 = 1 and ( ) ( ) ( ) = ( ) −1 , which leads to ( ) 2 = 1. Thus we have ( ) = ±1 and ( ) = ±1; that is, there exist 2 2 = 4 distinct maps [ , ] given by
[ , ] ( ) = e and [ , ] ( ) = e ,(20)
where , ∈ {0, (mod 2 )}. Correspondingly, there exist the following four distinct time-evolution kernels for a single walker on Z/ ∞ :
[ , ] ( , ) = ∞ =−∞ 1 =0 [ , ] ( )̃ ( , ) = ∞ =−∞ e ̃ ( , + 2 ) + e e ̃ ( , 2 + 1 − ) .(21)
Once again, just as in the path integral on a finite interval [22,[27][28][29], eq. We note in closing that eq. (21) can also be obtained from the time-evolution kernel on a circle (15) by gauging the reflection invariance at = 0, (mod 2 ). In fact, eq. (21) can be written as
[ , ] ( , ) = ∑ 1 =0 [ ] ( ) [ ] ( , ) = ∑ 1 =0 ∑ ∞ =−∞ [ ] ( ) [ ] ( )̃ ( , ),
where [ ] is the one-dimensional unitary representation of Z 2 given by eq. (17). An important lesson from this example is that there could exist several ways to construct time-evolution kernels on orbit spaces.
Identical walkers in one dimension
Now let us turn to the problem of multiple identical walkers on a lattice. The key to this problem is the indistinguishability of identical particles, where physical observables must be invariant under permutations of multiparticle coordinates. As is well known, this indistinguishability always makes the multiparticle configuration space an orbit space [13,[17][18][19][20]. The basic idea behind this is to regard the permutation invariance as a gauge symmetry (i.e., redundancy in description). From this perspective, the configuration space must be a collection of inequivalent gauge orbits because gauge-equivalent configurations are physically equivalent.
To date, there exist two distinct formulations of this idea in identical-particle problems. The first regards the configuration space of identical particles as the orbit space ( − Δ )/ , where is the -fold Cartesian product of a single-particle configuration space and Δ ⊂ is the set of fixed points under the action of the symmetric group [13,[17][18][19]. On the other hand, the second includes the fixed points and regards the configuration space as the orbit space / [20]. The difference between these two formulations is very subtle (especially in lattices) and beyond the scope of this note. Fortunately, however, we can circumvent this issue and solve the -identical-walker problems as follows.
Suppose that itself is a nontrivial orbit space and takes the form =̃ / , where is a discrete group whose action oñ has no fixed points. In this case, the configuration space can also be written as (̃ −Δ )/( ≀ ) or̃ /( ≀ ). 5 Here ≀ stands for the wreath product defined by the semidirect product ≀ ≔ ⋊ andΔ ⊂̃ is the set of fixed points of . Hence, irrespective of the formulations, once given a time-evolution kernel on Λ =̃ −Δ or̃ , the problem just reduces to the classification of one-dimensional unitary representations of the discrete group Γ = ≀ .
In this section, we shall focus on the cases = Z, Z/ Z, Z/Z 2 , and Z/ ∞ and construct timeevolution kernels for identical walkers on the infinite line, a circle, the half line, and a finite interval. In the following,̃ ( , ) represents a time-evolution kernel on Z −Δ or Z that satisfies the translation invariance, reflection invariance, and permutation invariance.
Example 4. ( identical walkers on the infinite line) Let us first consider identical walkers on the integer lattice Z. In this case, the discrete group Γ = is just the symmetric group of order !, whose presentation is
= ⟨ 1 , ⋯ , −1 | | 2 = , +1 = +1 +1 , = (| − | ≥ 2) ⟩ .(22)
Here = ( , +1) is the adjacent transposition that interchanges and +1. Its action on = ( 1 , ⋯ , ) ∈ Z is defined as follows:
≔ ( 1 , ⋯ , −1 , +1 , , +2 , ⋯ , ).(23)
An arbitrary element ∈ can be written as a product of the generators 1 , ⋯ , −1 . Its action on = ( 1 , ⋯ , ) can be written as = ( (1) , ⋯ , ( ) ), where ( ) stands for the permutation of under . Now, there exist two distinct one-dimensional unitary representations of : the trivial representation and the sign representation. Though this result is well known, let us reproduce it here just for later convenience. Since 2 = and +1 = +1 +1 , any one-dimensional unitary representation ∶ → (1) must satisfy the conditions ( ) 2 = 1 and ( ) ( +1 ) ( ) = ( +1 ) ( ) ( +1 ), whose solutions are ( ) = ±1 and ( ) = ( +1 ). Hence we have ( 1 ) = ⋯ = ( −1 ) = ±1; that is, there exist two distinct maps [±] given by
[±] ( ) = (±1) # ,(24)
where # stands for the number of adjacent transpositions in the permutation . In the standard terminology, [+] is the trivial representation and [−] is the sign representation. 6 Correspondingly, 5 Here is the proof. First, the wreath product ≀ = ⋊ can be written as the set { ∶ ∈ , ∈ } equipped with the group composition law ( )( ′ ′ ) = ( ′ −1 )( ′ ) for any , ′ ∈ and , ′ ∈ . Here ↦ −1 is the automorphism of the -fold direct-product group = × ⋯ × defined by −1 ≔ (1) ⋯ ( ) for any = 1 ⋯ ∈ × ⋯ × . It is now obvious that first making the identification ∼ by ∈ iñ and then making the identification ∼ by ∈ iñ / is equivalent to making the identification ∼ by = ( −1 ) ∈ ≀ iñ . Hence (̃ / )/ is equivalent tõ /( ≀ ). By subtracting the set of fixed points of , we also see that (̃ / − Δ )/ is equivalent to (̃ −Δ )/( ≀ ). See also refs. [20,30] for similar results in continuous spaces. 6 The sign representation can also be written as [−] ( ) = sgn( ), where sgn( ) stands for the signature of . It is defined by sgn( ) = ±1 for even (odd) permutations. there exist the following two distinct time-evolution kernels for identical walkers on Z:
[±] ( , ) = ∈ [±] ( )̃ ( , ) = ∈ (±1) # ̃ ( , ).(25)
Notice that eq. (25) satisfies the identity [±] ( , ) = (±1) # [±] ( , ). The weight factors (24) thus describe particle-exchange phases under the permutation of identical particles. It is now obvious that the two distinct representations [±] correspond to two distinct particle statistics: [+] describes the time-evolution kernel for identical bosons, while [−] describes that for identical fermions.
Example 5. ( identical walkers on a circle) Let us next consider identical particles on the periodic lattice of sites. In this case, the discrete group is the wreath product Γ = Z ≀ , whose presentation is given by
Z ≀ = ⟨ 1 , ⋯ , , 1 , ⋯ , −1 | | | | | | | = , 2 = , +1 = +1 +1 , = (| − | ≥ 2), = +1 , = ( ≠ , + 1) ⟩ .(26)
Here the actions of the generators and are defined by eq. (23) and
≔ ( 1 , ⋯ , −1 , + , +1 , ⋯ , ).(27)
Note that any element of Z≀ can be written as 1 1 ⋯ , where is a permutation and 1 , ⋯ , = 0, ±1, ⋯. Its action on = ( 1 , ⋯ , ) is given by 1
1 ⋯ = ( (1) + 1 , ⋯ , ( ) + ).
The time-evolution kernel for identical walkers on Z/ Z is therefore
[ ,±] ( , ) = ∞ 1 =−∞ ⋯ ∞ =−∞ ∈ [ ,±] ( 1 1 ⋯ )̃ ( , 1 1 ⋯ ) = ∞ 1 =−∞ ⋯ ∞ =−∞ e ( 1 +⋯+ ) [±] ( , 1 1 ⋯ ),(29)
where [±] is given by eq. (25). Notice that the kernel (29) Example 6. ( identical walkers on the half line) Let us next consider identical particles on the semi-infinite lattice. In this case, the discrete group is Γ = Z 2 ≀ , where
Z 2 ≀ = ⟨ 1 , ⋯ , , 1 , ⋯ , −1 | | | | | | | = , 2 = 2 = , +1 = +1 +1 , = (| − | ≥ 2), = +1 , = ( ≠ , + 1) ⟩ .(30)
The actions of the generators are defined by eq. (23) and
≔ ( 1 , ⋯ , −1 , 1 − , +1 , ⋯ , ).(31)
Note that any element of Z 2 ≀ can be written as the product 1 1 ⋯ , where ∈ and 1 , ⋯ , = 0, 1. Its action on = ( 1 , ⋯ , ) is given by 1 1 ⋯ = (⋯ , ( ) , ⋯) for = 0 and 1 1 ⋯ = (⋯ , 1 − ( ) , ⋯) for = 1.
By repeating the same procedure as above, one can show that one-dimensional unitary representation ∶ Z 2 ≀ → (1) must satisfy ( 1 ) = ⋯ = ( ) = ±1 and ( 1 ) = ⋯ = ( −1 ) = ±1. Hence there exist 2 2 = 4 distinct maps [ ,±] given by
[ ,±] ( 1 1 ⋯ ) = e ( 1 +⋯+ ) (±1) # ,(32)
where ∈ {0, (mod 2 )}. The time-evolution kernel for identical walkers on Z/Z 2 is therefore
[ ,±] ( , ) = 1 1 =0 ⋯ 1 =0 ∈ [ ,±] ( 1 1 ⋯ )̃ ( , 1 1 ⋯ ) = 1 1 =0 ⋯ 1 =0 e ( 1 +⋯+ ) [±] ( , 1 1 ⋯ ).(33)
Notice that eq. (33) satisfies
Γ = ∞ ≀ , where ∞ ≀ = ⟨ | | | | | | | | | = , = , 2 = 2 = , = −1 , = ( ≠ ), +1 = +1 +1 , = (| − | ≥ 2), = +1 , = ( ≠ , + 1) ⟩ .(34)
The actions of the generators are given by eqs. (23), (31), and
≔ ( 1 , ⋯ , −1 , + 2 , +1 , ⋯ , ).(35)
We note that any element of ∞ ≀ can be written as the product 1
⋯ ) = e ( 1 +⋯+ ) e ( 1 +⋯+ ) (±1) # ,(36)
where , ∈ {0, (mod 2 )}. Correspondingly, we have the following eight distinct time-evolution kernels for identical walkers on Z/ ∞ :
Physically, [ , ,±] describes the system of identical bosons (fermions) that acquire the phase shifts and + when reflected off the boundaries = 1 and = , respectively.
Asides
Now, there exist several other quantities that can be constructed in exactly the same way as for the time-evolution kernel (4). Examples include the resolvent kernel (Green's function) and the canonical density matrix (density matrix in the canonical ensemble). Another example is a unitary representation of an arbitrary group on a (tensor-product) Hilbert space, which includes the time-evolution kernel in discrete-time quantum walk. In this section, we shall briefly discuss the construction of these quantities on the orbit space Λ/Γ.
Resolvent kernel
Let
( , ) = ∞ 0 ( , ) e for Im > 0, (38a) ( , ) = ∞+ −∞+ 2 ( , ) e − for > 0. (38b)
Hence, by applying the Laplace transform to the formula (4), we find that the resolvent kernel on Λ/Γ takes the following form:
( , ) = ∈Γ ( )̃ ( , ),(39)
where ̃ ( , ) = ∞ 0 ̃ ( , ) e (Im > 0) is the resolvent kernel on Λ.
An immediate application of the above formula is the local density of states given by ( ) = ⟨ | ( − )| ⟩. In fact, by using the identity
lim Im →0 + ( − ) −1 = P( − ) −1 − ( − ),(40)
where P stands for the Cauchy principal value, we find Im ( , ) = Im⟨ |( − ) −1 | ⟩ = − ⟨ | ( − )| ⟩ = − ( ) in the limit Im → 0 + . Thus,
( ) = − 1 Im ∈Γ ( )̃ ( , ) as Im → 0 + .(41)
The density of states = tr ( − ) then takes the form = −(1/ ) Im ∑ ∈Λ/Γ ∑ ∈Γ ( )̃ ( , ).
Canonical density matrix
Let us next consider the canonical density matrix on Λ/Γ. In thermal equilibrium at temperature −1 , the canonical density matrix is given by = − / ( ), where − = e − is the Gibbs operator and ( ) = tr − is the canonical partition function. Note that the Gibbs operator satisfies the composition law
− 1 − 2 = − ( 1 + 2 )
, the hermiticity † − = − , and the initial condition 0 = . Its matrix elements (heat kernel) − ( , ) = ⟨ | e − | ⟩ must then satisfy these conditions as well. Namely, we must have ∑ ∈Λ/Γ − 1 ( , ) − 2 ( , ) = − ( 1 + 2 ) , − ( , ) = − ( , ), and 0 ( , ) = , . Under these conditions, one can again show that − ( , ) can be written as − ( , ) = ∑ ∈Γ ( )̃ − ( , ). Hence the matrix elements of the canonical density matrix is
( , ) = 1 ( ) ∈Γ ( )̃ − ( , ),(42)
where ( , ) = ⟨ | | ⟩. Here ( ) is the canonical partition function given by
( ) = ∈Λ/Γ ∈Γ ( )̃ − ( , ).(43)
We note that the partition function (43) can also be written as ( ) = ∑ ∈Λ/Γ ∑ ∈Γ ( )⟨ | e − ̃ | ⟩ = ∑ ∈Γ ( ) tr(e − ̃ ), wherẽ is the Hamiltonian operator on Λ and is a unitary operator defined by | ⟩ = | ⟩.
Unitary representations of arbitrary groups on a tensor-product Hilbert space
As mentioned in the beginning of section 2, our main formula (4) is also applicable to discrete-time quantum walk, where the time takes discrete values and the one-particle Hilbert space is the tensorproduct of the position and coin Hilbert spaces. In this section, we shall see this from a more general perspective: the construction of matrix elements of a unitary representation of an arbitrary group on a tensor-product Hilbert space. The time-evolution kernel in discrete-time quantum walk just corresponds to the special case = Z (the additive group of integers).
To
Conclusion
Inspired by the covering-space method in path integral on multiply-connected spaces, we have developed a general theory of quantum walk on orbit spaces. In this note, we have proved the universal formulae for time-evolution kernels, resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in continuous-and discrete-time quantum walks on the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. All of these quantities are given by summations over the orbit of initial point on Λ, where each orbit is weighted by a phase factor given by a one-dimensional unitary representations of Λ.
There are several advantages of this orbit-space method. A main advantage is its universality: our formulae are just based on geometric and group-theoretic structures of configuration spaces so that they are robust against any perturbations or interparticle interactions as far as boundary conditions (8) are remained unchanged. Another advantage is its computational simplicity: in our formalism, one just needs to compute matrix elements on Λ, which is generally much easier than computations on Λ/Γ.
Finally, let us comment on one possible future direction of this work. A promising direction would be a generalization of our formulae to the problem of identical walkers on graphs. Recent studies have shown that exotic statistics may show up in many-body problems of identical particles on graphs [31][32][33][34][35]. Such exotic statistics are generalizations of braid-group statistics in two dimensions. Hence, just as in topological quantum computation using anyons [36], they would have potential applications in quantum computer science. Our formalism and its generalization may well serve a basic tool for studying dynamics as well as thermodynamics of such systems.
Acknowledgments
The author would like to thank Naoto Namekata for discussion.
A Sample computations
Continuous-time quantum walk is just equivalent to tight-binding models in condensed matter physics. The advantage of this perspective is that it is straightforward to study many-particle problems by using the second-quantization formalism. In this section, we study tight-binding models for free spinless particles in one dimension and present sample computations that justify the formulae in section 3.
A.1 Tight-binding model on the infinite line
Let us first consider spinless particles on the integer lattice Z only with a nearest-neighbor coupling. In the second-quantization formalism, the Hamiltonian operator is given bỹ
= − 2 ∈Z † +1 + † +1 , (A.1)
where (> 0) is a hopping parameter. and † are annihilation and creation operators for spinless bosons (fermions) and subject to the following (anti-)commutation relations: where | ⟩ = † |0⟩ is the position-space basis in the one-particle sector. It satisfies the orthonormality ⟨ | ⟩ = ⟨0| † |0⟩ = , for both bosons and fermions. In order to calculate the matrix element (A.3), we first diagonalize the Hamiltonian operator, which can be achieved by the following Fourier integral: where = − cos( ) is the single-particle energy eigenvalue. It is now easy to see that the timeevolution kernel (A.3) takes the following form: where the second equality follows from − ( ) = e ( ). By substituting eq. (A.7) into eq. (A.6) and then using the orthogonal relation − 2 e ( − − ) = , − , we obtaiñ
= − 2 ̃ e ,( , ) = ⟨0| e − ̃ † |0⟩ = − 2 − 2 ⟨0|̃ e − ̃ ̃ † |0⟩ e − = − 2 − 2 e − ⟨0|̃ ̃ † |0⟩ e − =( , ) = e 2 | − | | − | ( ), ∀ , ∈ Z. (A.8)
This is the well-known transition amplitude for a single walker on the lattice Z (see, e.g., ref. [37]). Note that eq. (A.8) satisfies the composition law (3a), the unitarity (3b), and the initial condition (3c), which follow from the addition theorem
1 − 2 ( 1 + 2 ) e 2 ( 1 − 2 ) = ∑ ∈Z 1 − ( 1 ) 2 − ( 2 ) e 2 ( 1 − ) e 2 ( 2 − ) ( 1 , 2 ∈ Z)
, the analytic continuation (e ) = e ( ), and (0) = ,0 , respectively. Note also that eq. (A.8) enjoys the translation invariancẽ ( + , + ) =̃ ( , ) and the reflection invariancẽ ( − , − ) =̃ ( , ) for any , , ∈ Z. As we shall see shortly, eq. (A.8) provides the building block for the construction of time-evolution kernels for a free particle on a circle, the half line, and a finite interval.
Several comments are in order.
• Resolvent kernel for a single walker. As discussed in section 4.1, the resolvent kernel (Green's function) is given by the Laplace transform of̃ ( , ). Let be a complex number with Im > 0.
= 2 ∮ | |=1 2 | − | 2 + 2 + 1 , (A.9)
where in the second equality we have substituted the last line of eq. (A.6) and performed the integration with respect to . In the last equality we have changed the integration variable from to = e , where the integration is over the closed loop | | = 1 in the counter-clockwise direction. By using the residue theorem we find
̃ ( , ) = e | − | sin( ) , (A.10)
where we have parameterized the energy as = − cos( ) with Re ∈ (0, ) and Im ∈ (0, ∞).
Eq. (A.10) provides the building block for the construction of single-particle resolvent kernels on a circle, the half line, and a finite interval.
• By substituting this into the first line and using the orthogonal relation − 2 e ( − − ) = , − , we arrive at eq. (A.11). As discussed in section 4.2, eq. (A.11) provides the building block for the construction of canonical density matrices for free particles on a circle and a finite interval.
• Time-evolution kernel for identical walkers. In the second-quantization formalism, it is easy to generalize the above results to many-particle problems. First, the position-space basis in the -particle sector is given by
| 1 , ⋯ , ⟩ ≔ † 1 ⋯ † |0⟩. (A.13)
Notice that eq. (A.13) satisfies the orthonormality condition on the orbit space
(Z − Δ )/ ≅ {( 1 , ⋯ , ) ∈ Z ∶ 1 > ⋯ > }.
In fact, for 1 > ⋯ > and 1 > ⋯ > , we have
⟨ 1 , ⋯ , | 1 , ⋯ , ⟩ = ⟨0| ⋯ 1 † 1 ⋯ † |0⟩ = ∈ (±1) # (1) , 1 ⋯ ( ) , = 1 , 1 ⋯ , ,(A.14)
where the last line follows from the fact that ( (1) , ⋯ , ( ) ) and ( 1 , ⋯ , ) cannot be equal except for = . It is now easy to show that the time-evolution kernels for identical bosons and fermions take the following forms: 7
⟨ 1 , ⋯ , | e − ̃ | 1 , ⋯ , ⟩ = ⟨0| ⋯ 1 e − ̃ † 1 ⋯ † |0⟩ = =1 − 2 − 2 e (cos( 1 )+⋯+cos( )) × ⟨0|̃ ⋯̃ 1̃ † 1 ⋯̃ † |0⟩ e 1 1 +⋯+ − 1 1 −⋯− = ∈ (±1) # =1 e 2 | − ( ) | | − ( ) | ( ). (A.15)
As shown in eqs. (29), (33), and (37), eq. (A.15) can be used to construct the time-evolution kernels for free identical walkers on a circle, the half line, and a finite interval.
A.2 Tight-binding model on a circle
Let us next consider the tight-binding model for free spinless particles on the periodic lattice {1, 2, ⋯ , (mod )} subject to the twisted boundary condition + = e . As we shall see shortly, the following Hamiltonian operator yields the desired results: 16) In the following, we assume that ranges from 0 to 2 .
= − 2 =1 † +1 + † +1 , where +1 ≡ e 1 . (A.
In order to compute the time-evolution kernel, we first have to diagonalize the Hamiltonian operator (A.16), which can be done by using the mode expansion. Under the twisted boundary condition, the annihilation operator can be expanded into the following: where = − cos( 2 + ) is the single-particle energy eigenvalue on the periodic lattice.
Now it is easy to compute the time-evolution kernel in the one-particle sector. A straightforward 7 It should be noted that̃ ( 1 , ⋯ , , 1 , ⋯ , ) = ∏ =1 e 2 | − | | − | ( ) is equivalent to a single-particle time-evolution kernel on Z rather than Z − Δ . As noted in the beginning of section 3.2, in this note we will not touch upon this type of issues related to the fixed points of . which exactly coincides with eq. (15) with̃ ( ⋅ , ⋅ ) given by eq. (A.8). This sample computation implies that there is an equivalence (or duality) between the summation over energy spectrum and the summation over particle's trajectories, which is the heart of the trace formula in harmonic analysis and representation theory (see, e.g., ref. [39]). In this respect, one could say that our formula is a version of the trace formula in lattice geometry. Although we omit the details, it is not difficult to show that the resolvent kernel, the canonical density matrix, and the time-evolution kernel for identical particles all coincide with the universal formulae.
We note in closing that the parameter can be removed from the twisted boundary condition under the gauge transformation ↦ −1 = e , where is a unitary operator given by = exp(− ∑ =1 † ) (see, e.g., ref. [40]). In fact, a straightforward calculation gives The time-evolution kernel in the one-particle sector for this Hamiltonian coincides with eq. (A.20) up to a phase factor e ( − ) and hence is physically equivalent. which exactly coincides with eq. (18). Other quantities can be calculated in a similar way and coincide with the universal formulae.
A.3 Tight-binding model on the half line
We note that the model that satisfies the Dirichlet boundary condition = 0 at = 0 is described by the Hamiltonian operator = −( /2) ∑ ∞ =1 ( † +1 + † +1 ). In this case, the time-evolution kernel coincides with another formula discussed in example 2 in section 3.1. Note that this is the summation over the energy spectrum. However, as was done in appendix A.2, this summation can be rewritten into the following summation over the bouncing numbers off the boundaries: which exactly coincides with the universal formula (21). If one wants to study the model that satisfies the Dirichlet boundary conditions = 0 at = 0 and = + 1, one should use = −( /2) ∑ −1 =1 ( † +1 + † +1 ). In this case, the time-evolution kernel coincides with another formula discussed in example 3 in section 3.1.
A.4 Tight-binding model on a finite interval
•
Property
2
In general, eq. (7) becomes ∑ ∈Λ ( ) = ∑ ∈(Λ−Δ)/Γ ∑ ∈Γ ( ) + ∑ ∈Δ ( ), where Δ stands for the set of fixed points of Γ.3 More generally, such an overcounting does not occur if ∑ ∈Δ ( ) = 0.
that [ ] ( ⋅ , ⋅ ) satisfies the identity [ ] ( + , ) = e [ ] ( , ); that is, it satisfies the twisted boundary conditions [ ] ( + 1, ) = e [ ] (1, ) and [ ] (0, ) = e − [ ] ( , ). Namely, eq. (15) gives the universal formula of the timeevolution kernel for a single walker on a circle subject to these twisted boundary conditions. The second remark is that, under the reflection, eq. (15) satisfies [ ] ( − , − ) = [− ] ( , ). Hence, at = 0 or (mod 2 ), eq. (15) becomes reflection invariant. We can use this invariance for the construction of time-evolution kernels on a finite interval; see example 3. Example 2. (A single walker on the half line) Let us next consider a single walker on a semi-infinite lattice {1, 2, ⋯}. This lattice can be constructed from the integer lattice Z by making the identification ∼ 1 − . Hence the configuration space is the orbit space
(18) satisfies the identity [ ] (1 − , ) = e [ ] ( , ); that is, it satisfies the boundary condition [ ] (0, ) = e [ ] (1, ). Hence, eq. (18) gives the universal form of the time-evolution kernel for a single walker on the half line subject to this boundary condition. We emphasize that, as noted at the end of section 2.1, if one wants a theory subject to the Dirichlet boundary condition at = 0, one should consider the reflection defined by ≔ − and choose the representation = . In this case, one arrives at the formula [ = ] ( , ) =̃ ( , ) −̃ ( , − ) which satisfies [ = ] (0, ) = 0.
Example 3 .
3(A single walker on a finite interval) Let us next consider a single walker on a finite interval of sites, {1, 2, ⋯ , }. This lattice can be constructed from Z by making the identifications ∼ + 2 and ∼ 2 + 1 − , where is an arbitrary integer. Hence, the configuration space is the orbit space
(21) represents the summation over bouncing numbers off the two boundaries. Physically, e and e ( + ) play the roles of the reflection amplitudes off the boundaries = 1 and = , respectively. Now, it follows from eq. (8) that eq.(21) satisfies the identities [ , ] ( + 2 , ) = e [ , ] ( , ) and [ , ] (1 − , ) = e [ , ] ( , ), which implies the boundary conditions [ , ] (0, ) = e [ , ] (1, ) and [ , ] ( + 1, ) = e ( + ) [ , ] ( , ). This means that eq. (21) gives the universal form of the timeevolution kernel for a single walker on the finite interval subject to these boundary conditions. If one wants a theory that satisfies the Dirichlet boundary conditions at = 0 and = + 1, one should redefine the translation and reflection as ≔ + 2( + 1) and ≔ − , respectively, and choose the representation = . In this case, one obtains [ , = ] ( , ) = ∑ ∞ −∞ e [̃ ( , + 2 ( + 1)) − ( , 2 ( + 1) − )] which satisfies [ , = ] (0, ) = 0 and [ , = ] ( + 1, ) = 0.
Now we have to classify one-dimensional unitary representation ∶ Z ≀ → (1). First, the relations 2 = and +1 = +1 +1 imply ( 1 ) = ⋯ = ( ) = ±1. Second, the relation = +1 implies ( ) ( ) ( ) = ( +1 ), which, together with ( ) 2 = 1, leads to ( 1 ) = ⋯ = ( ) = e , where ∈ R/2 R. Thus we have two distinct one-parameter families of the maps [ ,±] given by [ ,±] ( 1 1 ⋯ ) = e ( 1 +⋯+ ) (±1) # .
satisfies the identities [ ,±] ( , ) = (±1) # [ ,±] ( , ) and [ ,±] ( , ) = e [ ,±] ( , ) for any = 1, ⋯ , . Physically, [ ,+] ( [ ,−] ) describes the system of identical bosons (fermions) on a circle with a nonzero magnetic flux.
[ ,±] ( , ) = (±1) # [ ,±] ( , ) and [ ,±] ( , ) = e [ ,±] ( , ) for any = 1, ⋯ , . Hence, [ ,±] describes the system of identical bosons (fermions) that acquire the phase shift when reflected off the boundary.Example 7. ( identical walkers on a finite interval) Let us finally consider identical particles on a finite interval. In this case, the discrete group is
1
, ⋯ , = 0, ±1, ±2, ⋯, and 1 , ⋯ , = 0, 1. Its action is given by ( ) , ⋯) for = 1. Now it is a straightforward exercise to show that there exist 2 3 = 8 distinct one-dimensional unitary representations of the wreath product ∞ ≀ . The result is the following maps:
e
( 1 +⋯+ ) e ( 1 +⋯+ ) [±] ( ,
e
us first start with the resolvent kernel-a matrix element of the resolvent operator in position space. Let be the Hamiltonian operator of the system. Then, the resolvent operator = ( − ) −1 for Im > 0 and the time-evolution operator = e − for > 0 are transformed into one another through the Laplace transform ( − ) − e and the inverse Laplace transform e − = ∞+ −∞+ 2 ( − ) −1 e − , respectively, where is an arbitrary positive real. Consequently, the matrix elements ( , ) = ⟨ | | ⟩ and ( , ) = ⟨ | | ⟩ are mutually related through the following:
begin with, let { ∈ (H) ∶ ∈ } be a unitary representation of on the tensor-product Hilbert space H = H position ⊗ H coin , where (H) stands for the set of unitary operators on H, H position =
[
, † ] ∓ = , and [ , ] ∓ = 0, (A.2) where [ , ] ∓ = ∓ . Let |0⟩ be the Fock vacuum that satisfies |0⟩ = 0 for all . The time-evolution kernel in the one-particle sector of the model is then given bỹ ( , ) = ⟨ | e − ̃ | ⟩, (A.3)
† satisfy [̃ ,̃ † ] ∓ = 2 ( − ) and [̃ ,̃ ] ∓ = 0 for any , ∈ (− , ). By substituting eq. (A.4) into eq. (A.
the third equality we have used e − ̃ † |0⟩ = e − ̃ † |0⟩, which follows from e − ̃ † e = e − ̃ † and e − |0⟩ = |0⟩ (or, equivalently, [ ,̃ † ] = ̃ † and |0⟩ = 0). The fourth equality follows from ⟨0|̃ ̃ † |0⟩ = 2 ( − ) for both bosons and fermions. To evaluate the last integral in eq. (A.6), we note that e cos( ) is a generating function of the Bessel function of the first kind . In fact,
Heat kernel for a single walker. The matrix element of the Gibbs operator e − ̃ can be calculated in exactly the same way as for̃ ( , ). Under the substitution → − in eq. (A.6) we find − ( , ) = − ( ) stands for the modified Bessel function of the first kind. Here in the last line we have used the fact that e cos( ) is the generating function of ( ). In fact,
† satisfy [̃ ,̃ † ] ∓ = , and [̃ ,̃ ] ∓ = 0 for any , ∈ {0, 1, ⋯ , − 1}. By substituting eq. (A.17) into eq. (A.16), we find that the Hamiltonian operator is diagonalized as follows:
have used e − ̃ † |0⟩ = e − ̃ † |0⟩ in the fourth line and ⟨0|̃ ̃ † |0⟩ = , in the last line.Notice that eq. (A.19) is the summation over the energy spectrum. In order to obtain the summation over winding numbers, we therefore have to perform a resummation, which can be done by using eq. (A.7). By substituting e cos( 2 + ) = ∑ ∈Z e 2 | | | | ( ) e − 2 + into eq. (A.19) and using the orthogonal relation 1 ∑ −1 =0 e 2 + ( − − ) = e , − − ( ∈ Z), we find that the time-evolution kernel (A.19) can be put into the following alternative equivalent form
us next consider the tight-binding model on the semi-infinite lattice {1, 2, ⋯} with the boundary condition 0 = 1 , where ∈ {0, }. The Hamiltonian operator that ensures this boundary condition is given by − cos( ) is the single-particle energy eigenvalue. The time-evolution kernel for a single walker is given by [ ] ( , ) = ⟨0| e 2 | − | | − | ( ) + e e 2 | −1+ | | −1+ | ( ), (A.25)
Let us finally quickly study the tight-binding model on the finite lattice {1, 2, ⋯ , } with the boundary conditions 0 = e 1 and +1 = e ( + ) , where , ∈ {0, }. The Hamiltonian operator is given by any , ∈ {0, }. It is not difficult to show that the time-evolution kernel for a single walker can be put into the following expression irrespective of the values of and :
The term "kernel" is a remnant of continuum theory. In quantum mechanics on continuous spaces, a matrix element of time-evolution operator is given by an integral kernel.
The infinite dihedral group can also be written as the free product ∞ ≅ Z 2 * Z 2 = ⟨ , ′ | 2 = , ′2 = ⟩, where ′ (= ) is another reflection defined by ′ ≔ 2 + 1 − .
, ⋯ , , 1 , ⋯ , , 1 , ⋯ , −1
(Λ/Γ) is the set of square-summable sequences on the orbit space Λ/Γ, and H coin = C is the -
The case = 0 was noted in ref.[38].
dimensional complex vector space that describes internal degrees of freedom of particles. Let {| ⟩} and {| ⟩} be complete orthonormal systems of H position and H coin , respectively. The set {| ⟩ ⊗ | ⟩} then provides a complete orthonormal system of the total Hilbert space H such that the matrix elements of can be defined as ( ,We now define ( , ) as the following × matrix:Since the unitary representation must satisfy the group composition law 1 2 = 1 2 , the unitarity † (= −1 ) = −1 , and the initial condition = , the matrix (44) must also satisfy the following properties:where , ∈ Λ/Γ. Here and stand for the matrix transpose and the × identity matrix, respectively. Now it is a straightforward exercise to show that the matrix (44) can be written aswherẽ ( , ) is a × matrix subject to the conditions ∑ ∈Λ̃ 1 ( , )̃ 2 ( , ) =̃ 1 2 ( , ), ̃ ( , ) = −1 ( , ),̃ ( , ) = , , and̃ ( , ) =̃ ( , ) for any , ∈ Λ and ∈ Γ. It is also straightforward to show that eq. (46) satisfies the following boundary condition:It is now obvious that eq. (46) provides the time-evolution kernel of continuous-time quantum walk with internal degrees of freedom when = R (the additive group of real numbers) and of discretetime quantum walk when = Z (the additive group of integers). It is also obvious that the examples presented in section 3 apply to discrete-time quantum walk as well.
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| {'fraction_non_alphanumeric': 0.10639844476375623, 'fraction_numerical': 0.0447806609754036, 'mean_word_length': 3.5358073915043704, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 14, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 99, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Inspired by the covering-space method in path integral on multiply-connected spaces, we here present a universal formula of time-evolution kernels for continuous-and discrete-time quantum walks on orbit spaces. In this note, we focus on the case in which walkers' configuration space is the orbit space Λ/Γ, where Λ is an arbitrary lattice and Γ is a discrete group whose action on Λ has no fixed points. We show that the time-evolution kernel on Λ/Γ can be written as a weighted sum of time-evolution kernels on Λ, where the summation is over the orbit of initial point in Λ and weight factors are given by a one-dimensional unitary representation of Γ. Focusing on one dimension, we present a number of examples of the formula. We also present universal formulae of resolvent kernels, canonical density matrices, and unitary representations of arbitrary groups in quantum walks on Λ/Γ, all of which are constructed in exactly the same way as for the timeevolution kernel.", 'arxivid': '2301.03193', 'author': ['Satoshi Ohya [email protected] \nInstitute of Quantum Science\nNihon University\nKanda-Surugadai 1-8-14101-8308ChiyodaTokyoJapan\n'], 'authoraffiliation': ['Institute of Quantum Science\nNihon University\nKanda-Surugadai 1-8-14101-8308ChiyodaTokyoJapan'], 'corpusid': 255546615, 'doi': '10.1103/physreva.107.062202', 'github_urls': [], 'n_tokens_mistral': 20417, 'n_tokens_neox': 17924, 'n_words': 9512, 'pdfsha': '3419a3324edab30c29158705043db25b7c4e1f41', 'pdfurls': ['https://export.arxiv.org/pdf/2301.03193v1.pdf'], 'title': ['Quantum Walk on Orbit Spaces', 'Quantum Walk on Orbit Spaces'], 'venue': []} |
arxiv |
RIGIDITY, GRAPHS AND HAUSDORFF DIMENSION
20 Aug 2017
A. IOSEVICHN Chatzikonstantinou
S Mkrtchyan
J Pakianathan
RIGIDITY, GRAPHS AND HAUSDORFF DIMENSION
20 Aug 2017
For a compact set E ⊂ R d and a connected graph G on k + 1 vertices, we define a G-framework to be a collection of k + 1 points in E such that the distance between a pair of points is specified if the corresponding vertices of G are connected by an edge. We regard two such frameworks as equivalent if the specified distances are the same. We show that in a suitable sense the set of equivalences of such frameworks naturally embeds in R m where m is the number of "essential" edges of G. We prove that there exists a threshold s k < d such that if the Hausdorff dimension of E is greater than s k , then the m-dimensional Hausdorff measure of the set of equivalences of G-frameworks is positive. The proof relies on combinatorial, topological and analytic considerations.
Introduction
The Falconer distance conjecture ( [10]) says that if the Hausdorff dimension of a compact subset of R d , d ≥ 2, is greater than d 2 , then the Lebesgue measure of the set of distances determined by pairs of elements of the set is positive. The best current results are due to Wolff ([22]) in dimension 2 and Erdogan ([8]) in R d , d ≥ 3, who established the d 2 + 1 3 threshold. In the context of Ahlfors-David regular sets, the Falconer conjecture was established in the plane by Orponen ([20]). These results build partly on the previous work by Bourgain ([5]), Falconer ([10]) and Mattila (see [19] and the references contained therein). A related conjecture, also pioneered by Falconer ([10]) and studied extensively by Bourgain, says that if the Hausdorff dimension of E is equal to d 2 , then the upper Minkowski dimension of ∆(E) is 1. Bourgain proved that if dim H (E) = d 2 , E ⊂ R d , d ≥ 2, then the upper Minkowski dimension of ∆(E) is > 1 2 + ǫ d for some ǫ d > 0. The Falconer distance conjecture can be viewed as a problem about 2-point frameworks. It is quite interesting to consider (k + 1)-point frameworks with k > 1. For example, one can consider triangles inside sufficiently large sets, properly interpreted. This problem has been extensively studied in a variety of contexts by Bennett, Bourgain, Chan, Furstenberg, Greenleaf, Katznelson, Laba, Pramanik, Weiss, Ziegler, the second and fourth named authors and others (see e.g. [4], [12], [6], [1], [2], [15], [17], [11], [25]). In [3], the authors considered chains, and in [13] necklaces were investigated. More general frameworks were studied in [16].
In this paper we show that in a suitable sense, a nontrivial dimensional threshold can be found for any finite point framework. What we mean by a finite point framework is a finite collection of points in a compact set E of a given Hausdorff dimension, where some, but not all the pairwise distances are specified. We encode these frameworks in a rigorous way using combinatorial graphs. We then define a suitable notion of equivalence and embed the resulting equivalence classes in R m , where m is the number of "essential" edges of the graph which encodes a given framework. We then prove that if the Hausdorff dimension of the ambient set E is larger than a nontrivial threshold s k , then the m-dimensional Hausdorff content of the set of equivalences is positive. The precise formulation can be found in Theorem 2.18 and Theorem 2.20 below.
As the reader shall see below, a rigorous formulation of the Falconer type problem for finite point frameworks naturally leads one to the notions of rigidity and other interesting concepts that combine combinatorial, topological and analytic concepts. The resulting symbiosis makes possible results that were not accessible using the purely analytic methods that were employed in the cases of simplexes, chains and necklaces.
Definitions and statements of results
We shall encode finite point frameworks using combinatorial graphs. Let k ≥ 1 and let K k+1 denote the complete graph with vertex set {1, . . . , k + 1} and edge set ordered lexicographically. Let G k+1,m be a subgraph of K k+1 with k + 1 vertices and m edges inheriting the order. We define ij ∈ G k+1,m to mean that i < j and {i, j} is an edge of G k+1,m , and when ij ∈ G k+1,m ranges over all the edges of G k+1,m , it ascends in the order of the edge set. Let | · | denote the Euclidean distance, · p the L p norm, dim(·) denote the Hausdorff dimension and H s (·) denote the s-dimensional Hausdorff measure. Let L m (·) denote the Lebesgue measure of measurable subsets of R m . Let A B mean that for some constant C > 0, A ≤ C · B, where C is independent of ǫ, δ and the summation index or integration variable (if used to bound the term). Moreover, A ≈ B means that A B and B A.
Definition 2.1. A (k + 1)-tuple x in R d is a tuple x = (x 1 , x 2 , . . . , x k+1 ), x j ∈ R d . Definition 2.2. A framework of G k+1,m in R d is a pair (G k+1,m , x), where x is a (k + 1)-tuple in R d .
A convenient way to specify distances is through the distance function which we now define.
Definition 2.3. Given a graph G k+1,m we define the distance function f G k+1,m (x) on x = (x 1 , . . . , x k+1 ) ∈ R d(k+1) by f G k+1,m (x) = |x i − x j | ij∈G k+1,m .
We also define the distance-squared function F G k+1,m (x) by
F G k+1,m (x) = |x i − x j | 2 ij∈G k+1,m . Definition 2.4 (Graph Distances). The value f G k+1,m (x) is called the G k+1,m -distance of x.
When we restrict our domain to some set X ⊆ R d(k+1) , we call f G k+1,m (x) a G k+1,m -distance on X and we say that x is a realization of this distance in X. The set of G k+1,m -distances on X is f G k+1,m (X) and we denote it by ∆(G k+1,m , X).
Remark 2.5. Equivalence classes of frameworks (equivalent in the sense of the corresponding tuples having equal f G k+1,m -values) can be viewed as subsets of R m since m distances are specified in the sense of Definition 2.4. Given a graph, we ask whether there exists some 0 < s k < d such that any compact subset E of R d , d ≥ 2, of Hausdorff dimension larger than s k contains a positive m-dimensional measure (Hausdorff or Lebesgue, depending on the context) worth of equivalence classes of frameworks of the given graph, in other words, whether the set ∆(G k+1,m , E k+1 ) has positive (Hausdorff or Lebesgue) measure. Complete graphs in R d with at most d + 1 vertices were comprehensively studied in [12]. In fact as we shall see later, when k ≤ d, the only interesting case is the complete graph. Thus in this paper we consider graphs with k > d unless otherwise stated.
Remark 2.6. The distance set ∆(G k+1,m , X) depends on the numbering of the vertices and the order of the edges. Whereas the order of the edges is superficial, inducing only a permutation in the components of the G k+1,m -distances, the numbering of the vertices can significantly change the G k+1,m -distance set. Consider
X = {x 0 } × R d × · · · × R d and a graph G = G ′ ∪ G ′′ ∪ {e} where e is a bridge between G ′ and G ′′ .
Then if we number the vertices of G ′ followed by those of G ′′ , we essentially capture G ′′ -distances only, whereas if we reverse the numbering order of the vertices of G we will capture G ′ -distances only. In the rest of this paper we take X = E k+1 for some E ⊂ R d , so that the numbering of the vertices becomes superficial as well. In particular, the dimension of the G k+1,m -distance set and its Hausdorff (or Lebesgue) measure are independent of the vertex numbering and edge order.
We define the notion of independence for subsets of the edge set of K k+1 and of maximal independence for subsets of the edge set of G k+1,m . We define the set of generic tuples as the complement of the zero set of a certain polynomial. This notion is independent of the graph G k+1,m , depending only on the dimension d and the number of vertices k + 1.
Let us use the following notation for our matrices: If a ij is a matrix, (i, j) ∈ I × J, then for B ⊆ I, C ⊆ J, we defined a B,C to be the submatrix a ij with (i, j) ∈ B × C.
Definition 2.7. We say that x ∈ R d(k+1) is a regular tuple of F G k+1,m if rank DF G k+1,m attains its global maximum at x. A framework (G k+1,m , x) is a regular framework if x is a regular tuple of F G k+1,m . Definition 2.8. A subset H of the edge set of K k+1 is called independent in R d with respect to x 0 ∈ R d(k+1) if the row vectors of DF K k+1 (x 0 ) corresponding to H are linearly independent. We call H independent in R d if there exists some x 0 so that H is independent with respect to x 0 , and x 0 is said to be a witness to the independence of H. We also call H a maximally independent (in R d ) subset of edges of G k+1,m when it is independent and it is not contained in a larger independent edge set of G k+1,m . Definition 2.9. For any nonempty independent in R d set H of edges of K k+1 we define the polynomial P H (x) to be the sum of squares of |H| × |H|-minors of the submatrix of rows of DF K k+1 corresponding to edges of H. Thus,
P H (x) = A⊂{1,...,d(k+1)} |A|=|H| det(DF K k+1 (x) H,A ) 2 .
Let X H denote the zero set of P H . We define the set of generic tuples of R d to be the complement of the zero set X of the polynomial P (x) defined by
P (x) = H independent P H (x) .
We call X the set of critical tuples of R d .
Remark 2.10. We have X = ∪ H X H where the union is taken over all the edge sets H which are independent and the generic tuples are then equal to R d(k+1) \X. Moreover, if a set H of edges is independent then by Definition 2.9 it is generically independent, i.e. independent with respect to any generic x. In fact, the set of generic tuples is precisely the set of tuples that simultaneously witness the independence of every independent edge set.
Remark 2.11. The polynomial P (x) is nontrivial because every P H is nontrivial since there is at least one witness x H for the independence H, which means that P H (x H ) = 0. Thus X is a proper algebraic variety of dimension dim X ≤ d(k + 1) − 1 .
(2.1)
Remark 2.12. It is immediate from the definitions that generic tuples are regular tuples. The other implication does not hold in general.
Definition 2.13. A framework (G k+1,m , x) is called generic in R d if x is a generic tuple in R d and it is called critical in R d if x is a critical tuple in R d .
Our main results concern the dimension of the set ∆(G k+1,m , E k+1 ) and its Hausdorff (or Lebesgue) measure. An important role is played by properties of the graph G k+1,m . In particular it is essential whether the graph is rigid or not.
The key heuristic notion of this paper is that a graph G k+1,m is rigid in R d if once the m quantities t ij in
|x i − x j | = t ij , ij ∈ G k+1,m
are specified, the other distances |x i − x j | for ij ∈ G k+1,m can only take finitely many values as the frameworks (G k+1,m , x) vary over the set of generic frameworks.
For technical reasons, we use a more precise and flexible notion of rigidity described below. A simple example that illustrates the technical obstacles one must contend with is the following. Consider a quadrilateral in the plane with side-lengths 1, 1, 1, 3.
This configuration is perfectly rigid in the heuristic sense, but it is not minimally infinitesimally rigid, as the reader will see, roughly because the rigidity in this case is not stable under small perturbations.
We now turn to precise definition.
Definition 2.
14. An infinitesimal motion u = (u 1 , . . . , u k+1 ) in R d of G k+1,m at x is a (k + 1)-tuple u of vectors u j ∈ R d such that
DF G k+1,m (x) · u = 0 . The set of infinitesimal motions in R d of G k+1,m at x is the kernel of DF G k+1,m (x). Let us denote by V(G k+1,m , x) the set of infinitesimal motions in R d of G k+1,m at x. Let D(x) be the set of infinitesimal motions in R d of K k+1 at x. Remark 2.15. It is evident that D(x) ⊆ V(G k+1,m , x) since the system of equations DF G k+1,m (x) · u = 0 is included in DF K k+1 (x) · u = 0. Definition 2.16. A framework (G k+1,m , x) is called infinitesimally rigid in R d when V(G k+1,m , x) = D(x).
It is unnecessarily restrictive to require of a graph to have all its frameworks be infinitesimally rigid. We shall only require it of generic frameworks. Definition 2.17. A graph G k+1,m is called infinitesimally rigid in R d if all its generic frameworks are infinitesimally rigid. It is called minimally infinitesimally rigid in R d if it is infinitesimally rigid and no proper subgraph (on the same vertex set) is infinitesimally rigid.
Statements of results.
Theorem 2.18. Let G k+1,m be a connected graph that is minimally infinitesimally
rigid in R d , d ≥ 2 and E is a compact subset of R d with dim E > d − 1 k+1 . Then L m (∆(G k+1,m , E k+1 )) > 0 .
Remark 2.19. We shall in the proof of Theorem 2.18 that if G k+1,m is not connected, the proof naturally breaks into consideration of the connected components of the graph.
Theorem 2.20. Let G k+1,m be a graph without isolated vertices and let E be a
compact subset of R d , d ≥ 2 with dim E > d− 1 k+1 . Let H be a maximally independent subset of edges of G k+1,m . Then dim ∆(G k+1,m , E k+1 ) = |H| and H |H| ∆(G k+1,m , E k+1 ) > 0 .
Remark 2.21. We stated Theorem 2.20 using the Hausdorff measure instead of the Lebesgue measure because the edges not in H do not produce any further dimensionality (see Corollary 3.8 and Proposition 3.1).
Remark 2.22. It should be pointed out that for our results we only work out the case where k > d. Theorem 2.18 for k ≤ d was worked out in [12] and the better threshold dim E > dk+1 k+1 was obtained. Theorem 2.20 follows as a consequence, since when k ≤ d, the only minimally infinitesimally rigid graph is the complete graph on k + 1 vertices and the independence condition of Theorem 2.20 is always satisfied (see Theorem 4.6) so H can be taken to be the edge set of G k+1,m and in that case Theorem 2.20 is a consequence of Theorem 2.18 by an application of Fubini's theorem.
Theorem 2.23 (Deforestation). Let G k+1,m , m ≥ 2 be a graph without isolated vertices with a vertex v of degree 1 and let G 1 be the resulting graph when v is removed from G k+1,m . Iterate this process obtaining a sequence G 1 , . . . , G n , until G n has no more such vertices or when G n = K 2 . Then in using Theorem 2.18 or Theorem 2.20 with G k+1,m , the dimensional threshold for E obtained may be taken to be
d − 1 k + 1 − n .
Remark 2.24. Thus trees disjoint from the rest of the graph except for the root can be ignored by applying Theorem 2.23.
We shall now see that our results are fairly sharp in the sense that the critical exponent must in general tend to d as the number of vertices tends to infinity.
Theorem 2.25. Let E be a compact subset of R d , d ≥ 2 of Hausdorff dimension s ∈ (0, dgeneral hold if s < d − ( d 2 ) k .
In dimension d = 2, the difference between the exponents in Theorem 2.18, Theorem 2.20 and Theorem 2.25 is not very large, 2 − 1 k+1 versus 2 − 1 k . In higher dimensions the gap increases, but Theorem 2.25 still shows that the correct critical exponent must in general tend to d as the number of vertices tends to ∞.
Graph distances of subsets of R d
Introduction. Our goal is to prove that
L m (∆(G k+1,m , E k+1 )) > 0 (3.1)
for some dimensional threshold s k with dim E > s k . Here we may assume G k+1,m is connected, since we have
Proposition 3.1. If G 1 , .
. . , G n are the connected components of G k+1,m on k 1 , . . . , k n vertices respectively, then for cartesian products
E k+1 where E ⊆ R d , we have ∆(G k+1,m , E k+1 ) = ∆(G 1 , E k 1 ) × · · · × ∆(G n , E kn ) .
Proof. It is clear that (after reordering the vertices if necessary)
f G k+1,m = (f G 1 , . . . , f Gn ) where f G k+1,m , f G j are the corresponding distance functions of G k+1,m , G j . The result follows.
Thus we only need to consider connected graphs, and requiring of G k+1,m to be connected in Theorem 2.18 is not an essential restriction. If (3.1) does not hold, it may be the case that the dimension of the G k+1,m -distance set is not full. Theorem 2.20 then provides its Hausdorff dimension and positivity of the Hausdorff measure.
We define the notion of congruency for tuples and frameworks.
Definition 3.2. Let x be a (k + 1)-tuple in R d and define the set of tuples congruent to x to be
M x = {(T x 1 , . . . , T x k+1 ) : T ∈ ISO(d)}, (3.2)
where ISO(d) denotes the set of isometries of R d to itself.
Definition 3.3. We say that the framework (G, x) is congruent to the framework (G ′ , x ′ ) if G = G ′ and x is congruent to x ′ in the sense of Definition 3.2.
We now describe some examples.
3.2.
Examples of ∆(G k+1,m , E k+1 ). In the case where G k+1,m is the complete graph K 2 on 2 vertices and E ⊆ R d , we recover the distance set of E
∆(K 2 , E 2 ) = {|x − y| : x, y ∈ E} .
Now consider the complete graph K 4 . Let d = 2 and E = R 2 . We will directly show that L 6 (∆(K 4 , R 8 )) = 0. This is expected, as we will also show that dim ∆(
K 4 , R 8 ) = 5 and H 5 (∆(K 4 , R 8 )) > 0. Split the tuples x ∈ R 8 into two sets A 1 and A 2 , A j ⊂ R 8 , with x ∈ A 1 iff (x 1 , x 2 , x 3 , x 4
) are in convex position and A 2 the rest. We will work with A 1 but A 2 is treated similarly. Let t ij denote the distance from the vertex i to j. By using Euler's theorem for convex quadrilaterals we may obtain the following equation,
t 2 24 = t 2 23 + t 2 14 − t 2 13 + 2t 12 t 34 cos(θ − ψ) . (3.3) Here θ = cos −1 t 2 12 + t 2 13 − t 2 23 2t 2 12 t 2 13 , ψ = cos −1 t 2 13 + t 2 34 − t 2 14 2t 2 13 t 2 34
.
Let t 24 = g(t) wheret = (t 12 , t 13 , t 14 , t 23 , t 34 ) .
Thus,
L 6 (∆(K 4 , A 1 )) ≤ t 24 =g(t) dt 24 dt = 0 .
This happens because equation (3.3) makes us integrate over the zero-dimensional set (it is a singleton) t 24 ∈ {g(t)}. Similarly we may obtain L 6 (∆(K 4 , A 2 )) = 0. Thus K 4 in d = 2 cannot possibly give us a dimensional threshold since even E = R 2 has a K 4 -distance set of zero measure. This happened because the graph had too many edges. Not only ∆(K 4 , R 8 ) is a L 6 -null set, but in fact its dimension is less than 6. By using Corollary 3.8, we find that its dimension is 5. Then using Theorem 2.20 we see that it has positive H 5 -measure. Now consider the following 'double banana' graph G 8,18 on R 3 (dashed edge for emphasis, but it is in the edge set of G 8,18 ),
(∆(G 8,18 , E 8 )) > 0 for E ⊆ R 3 compact with dim E > 3 − 1 9 . 3.3.
A sharp upper bound for the dimension of the distance set. In this section we determine the Hausdorff dimension of ∆(G k+1,m , R d(k+1) ).
If G k+1,m is a minimally infinitesimally rigid graph in R d , then from Corollary 4.10, it must have
m = d(k + 1) − d + 1 2 . (3.4)
We may say that G k+1,m is minimally infinitesimally rigid in R d when its edge set is independent and any edge added to G k+1,m turns the rows of DF G k+1,m into a linearly dependent set of vectors, as the next proposition shows. Proof. If G k+1,m is minimally infinitesimally rigid in R d , then by definition for generic tuples x ∈ R d(k+1) , the kernel of DF G k+1,m (x) has the smallest dimension possible (in view of Theorem 4.7 and the dimension of the rotation group). Thus the edge set can not be enlarged while retaining independence, since a larger independent set of edges would produce an even smaller kernel.
On the other hand, assume G k+1,m has an independent in R d edge set that may not be enlarged while retaining independence. We have just argued that G k+1,m cannot contain a minimally infinitesimally rigid proper subgraph. Assuming then that G k+1,m is not minimally infinitesimally rigid itself, for generic tuples x ∈ R d(k+1) we know that V(G k+1,m , x) properly contains D(x). Thus there must be a set of edges
H ⊂ K k+1 disjoint from those of G k+1,m with V(H ∪ G k+1,m , x) = D(x) . But that implies rank D F H∪G k+1,m (x) > rank D F G k+1,m (x)
, a contradiction to the assumption that G k+1,m has an edge set that may not be enlarged while retaining independence.
Proposition 3.5. If the edge set of G k+1,m is independent in R d , then a minimally infinitesimally rigid (in R d ) graph G k+1,m containing G k+1,m exists.
Proof. If k ≤ d, we just complete G k+1,m to the complete graph K k+1 since that is the only rigid graph on k + 1 vertices in R d (see Theorem 4.6).
Otherwise we pick x at random from a continuous distribution (say, the Gaussian distribution) on R d(k+1) . Since the set of critical frameworks is a proper algebraic variety, it has Lebesgue measure zero and we have almost certainly (that is, with probability 1) picked a generic framework.
As long as the property of independence with respect to x is retained, we keep adding edges to G k+1,m until no more edges may be added. We then end up with a graph that is minimally infinitesimally rigid.
Remark 3.6. The graph G k+1,m need not be unique. For instance, if G k+1,m is a tree, for large enough k there are many different minimally infinitesimally rigid graphs it may complete to.
Theorem 3.7. Let G k+1,m be a connected graph and H a maximally independent in R d subset of edges of G k+1,m . Then
dim ∆(G k+1,m , R d(k+1) ) = |H| .
Proof. Let W ij denote the plane {x : x i = x j }. The map f G k+1,m is smooth away from W = ij∈G k+1,m W ij and the rank of its total derivative does not exceed |H| there.
Thus, since f G k+1,m has regular tuples away from W , we see that f G k+1,m (R d(k+1) \ W ) has dimension |H|. Now consider f G k+1,m restricted to W ij . There the function is smooth away from W ij ∩W kl for kl ∈ G k+1,m with kl = ij. The rank of the derivative is less than or equal to |H|, so inductively f G k+1,m (W ij ) has dimension less than or equal to |H|. Corollary 3.8. If G k+1,m is a connected graph that contains a minimally infinitesimally rigid (in R d ) subgraph G * k+1,m * , then dim ∆(G k+1,m , R d(k+1) ) = m * .
3.4.
Bounds on the number of noncongruent realizations. Let G k+1,m be minimally infinitesimally rigid in R d and let x ∈ R d(k+1) .
We consider the set of preimages of f G k+1,m (x),
N x = {y : f G k+1,m (y) = f G k+1,m (x)} .
Define the equivalence relation y ∼ z by y ∈ M z , where M z is defined in (3.2) to be the set of tuples congruent to z. The set N x is defined by the system of quadratic equations
|y i − y j | 2 = |x i − x j | 2 , ij ∈ G k+1
,m , and results on bounds of the Betti numbers of semi-algebraic varieties by Oleinik and Petrovskii, Thom and Milnor (see [29], [30], [31]) allow us to conclude that N x , hence (since ISO(d) has two connected components), N x /∼, has less than C d,k · 2 dk connected components, for some C d,k > 0. For a better bound see [32]. In particular, when x is regular valued, we may conclude that there are at most C d,k · 2 dk preimages of f G k+1,m (x) up to congruences by Proposition 4.11. If x is not regular valued, it is possible for noncongruent preimages to lie in the same connected component, but in the argument to follow we will avoid critical frameworks. For our purposes we only need the fact that if the critical tuples are removed, then the preimage set N x is finite up to congruences, with the bound independent of x.
3.5. The proof of the dimensional threshold. We prove Theorem 2.18.
Proof. Let G = G k+1,m to ease subscript use. Fix E ⊂ R d compact, and let µ = µ s be a Borel probability measure supported on E, with Frostman exponent s. Thus there exists some constant C µ > 0 with µ(B(x, r)) ≤ C µ r s for all balls B(x, r) of radius r > 0, and we may choose s < dim E arbitrarily close to dim E. (See [23], Chapter 8 for the existence and properties of such measures). In particular, we may choose
s > d − 1 k + 1 . (3.5)
Let us first prove that the set of critical frameworks X is a null set for µ k+1 :
µ k+1 (X) = 0 .
Observe that µ k+1 is a Frostman measure of exponent s(k + 1). This follows easily from the fact that any ball B(x, r) is contained in a concentric cube Q = Q 1 × · · · × Q k+1 of side 2r, where each Q j in turn is contained in a ball B(x j , 2r). Since µ k+1 (Q) = µ(Q 1 ) · · · µ(Q k+1 ) r s(k+1) and s(k +1) > dim(X), which we may assume since the dimension of E is big enough (by (2.1) and (3.5)) and using the fact that sets of positive measure of a Frostman measure have dimension greater or equal to the Frostman exponent (see Lemma 4.15), we conclude that X is a null set for µ k+1 . Let δ > 0 be such that µ k+1 (E k+1 \ X δ ) > 1/2 for X δ the δ-neighborhood of X defined by X δ = {y ∈ R d(k+1) : |y − x| < δ}. Such a δ exists since X is closed and X = δ>0 X δ . We want to avoid getting close to X because our named constants in the arguments to follow blow up near it. Let A = E k+1 \ X δ and let ν(t) be the pushforward of µ k+1 (x) by f G | A , that is, for any measurable function g(t) the integral gdν is defined by
g(t)dν(t) = A g (|x i − x j |) ij∈G dµ(x 1 ) · · · dµ(x k+1 ) .
From now on we shall write dµ k+1 (x) for dµ(x 1 ) · · · dµ(x k+1 ). We shall show that ν(R m ) > 0 and ν ∈ L 2 (R m ) implying L m (supp ν) > 0 which concludes the proof since supp ν ⊂ ∆(G, E k+1 ). For the first claim,
ν(R m ) = µ k+1 (E k+1 \ X δ ) > 1/2 . Now we will prove that ν ∈ L 2 (R m ). Let ν ǫ = φ ǫ * ν, where φ ǫ (t) = ǫ −m φ(ǫ −1 t) and φ ∈ C ∞ c (R m )
is a nonnegative radial function with φ = 1, φ ≤ 1 and supp φ ⊂ B(0, 2). Here we denote by χ the characteristic function of a set. We have,
ν ǫ (t) = A ǫ −m φ ǫ −1 (f G (x) − t) dµ k+1 (x) ≤ A ǫ −m χ f G (x) − t < 2ǫ dµ k+1 (x) .
(3.6) By Lemma 4.13, as ǫ → 0 we have ν ǫ → ν in the weak topology of the dual of C 0 (R m ). We conclude that lim inf ν ǫ 2 ≥ ν 2 from Lemma 4.14, and thus it suffices to bound lim inf ν ǫ 2 . By an application of the triangle inequality on (3.6) we have,
ν ǫ 2 2 ≤ ǫ −2m A A χ f G (x) − t < 2ǫ χ f G (y) − t < 2ǫ dµ k+1 (x)dµ k+1 (y)dt ≤ ǫ −2m |t|<2ǫ dt · A A χ f G (x) − f G (y) < 4ǫ dµ k+1 (x)dµ k+1 (y) ǫ −m A A χ f G (x) − f G (y) < 4ǫ dµ k+1 (x)dµ k+1 (y) .
Consider y to be fixed now. Note that it is a regular tuple since it belongs to A. The condition that the images of x and y are ǫ-close is giving us 2 dk open sets of R d(k+1) where x may lie (as explained in Section 3.4). Let U 1 , . . . , U l be those open sets, and let Z = {z 1 , . . . , z l } be such that each z j ∈ U j ∩ A (potentially picking less than l tuples, if some intersections are empty). Denote by O d (R) the group of rotations of R d . Cover the compact Riemannian manifold O d (R) by ǫ-balls T ǫ 1 , . . . , T ǫ K(ǫ) of finite (uniformly in ǫ) overlap with centers g 1 , . . . , g K(ǫ) .
Then the set {x ∈ A :
|f G (x) − f G (y)| < ǫ} is a subset of the set z∈Z K(ǫ) k=1 {x ∈ A : |(x i − x j ) − g k (z i − z j )| < cǫ, ∀ij ∈ G}
for some c > 0 that depends continuously on y (as Proposition 4.11 shows, f G is biLipschitz in U 1 , . . . , U l , once congruences are identified). Since A is a compact set, c attains a maximum value, so pick such c to lift the dependence on y.
Since |Z| 2 dk , it follows that,
ν ǫ 2 2 ǫ −m A z∈Z K(ǫ) k=1 A χ{|(x i − x j ) − g k (z i − z j )| < cǫ, ∀ij ∈ G}dµ k+1 (x)dµ k+1 (y) 2 dk ǫ −m K(ǫ) k=1 A A χ{|(x i − x j ) − g k (y i − y j )| < cǫ, ∀ij ∈ G}dµ k+1 (x)dµ k+1 (y) . The volume of the ǫ-balls of O d (R) is ≈ ǫ dim O d (R) = ǫ d(d−1)/2 .
In what follows, we estimate the value of a function at a point by twice the average of that function around an ǫ-ball and use the fact that they cover O d (R) with finite overlap to obtain,
ν ǫ 2 2 ǫ −m K(ǫ) k=1 1 ǫ d(d−1)/2 T ǫ k A A χ{|(x i − x j ) − g(y i − y j )| < cǫ, ∀ij ∈ G} dµ k+1 (x)dµ k+1 (y)dg ǫ −dk O d (R) A A χ{|(x i − x j ) − g(y i − y j )| < cǫ, ∀ij ∈ G}dµ k+1 (x)dµ k+1 (y)dg . (3.7)
Here we used (3.4)
to get m + d(d − 1)/2 = dk. For g ∈ O d (R), define ν g by f (x)dν g (x) = f (u − gv)dµ(u)dµ(v) .
Let G ′ ⊂ G be a spanning tree. We continue (3.7) with
ν ǫ 2 2 ǫ −dk O d (R) A A χ{|(x i − x j ) − g(y i − y j )| < cǫ, ∀ij ∈ G ′ }dµ k+1 (x)dµ k+1 (y)dg .
(3.8)
Using Lemma 3.12 (to be proved below) on (3.8) we obtain
ν ǫ 2 2 ν k+1 g (x)dxdg . (3.9)
Theorem 3.10 shows this integral to be finite for s > d − 1 k+1 , concluding the proof that ν ∈ L 2 (R m ) and thus showing that L m (∆(G, E k+1 )) > 0.
We may now go a step further and prove Theorem 2.20.
Proof. If G k+1,m is any connected graph, and H is a maximally independent subset of the edge set of G k+1,m , we may complete H to a minimally infinitesimally rigid graph H k+1,m (see Proposition 3.5). By using Theorem 2.18, we obtain the nontrivial exponent d − 1 k+1 for ∆(H k+1,m , E k+1 ) to have positive Lebesgue measure. We
project ∆(H k+1,m , E k+1 ) → ∆(H, E k+1 ) by (t ij ) ij∈H k+1,m → (t ij ) ij∈H to show that ∆(H, E k+1 ) has positive Lebesgue measure by Fubini. Now, projecting ∆(G k+1,m , E k+1 ) → ∆(H, E k+1 ) by (t ij ) ij∈G k+1,m → (t ij ) ij∈H shows that ∆(G k+1,m , E k+1 ) has positive H |H| -measure, because the projection is Lipschitz. Lastly, Theorem 3.7 shows that dim ∆(G k+1,m , E k+1 ) = |H|.
Moreover, if G k+1,m has connected components G 1 , . . . G n , we will obtain a maximally independent subset H = H 1 ∪ · · · ∪ H n of G k+1,m where each H j is a maximally independent subset of G j for j = 1, . . . , n. Using Proposition 3.1 and what we just argued for connected graphs, we again obtain a positive |H|-Hausdorff measures worth of distances.
We now prove Theorem 2.23.
Proof. As before let G = G k+1,m to avoid notational clutter. Let σ t denote the surface measure of the sphere tS d−1 ⊂ R d of radius t > 0 centered at 0.
Let φ ǫ (x) = ǫ −d φ(ǫ −1 x) where φ ∈ C ∞ c (R d )
is a nonnegative radial function with φ = 1 on B(0, 1), φ ≤ 1 and supp φ ⊂ B(0, 2). Let σ ǫ t = φ ǫ * σ t . We note that c · χ {y:||y|−t|<ǫ} (x) ≤ ǫσ ǫ t (x) ≤ C · χ {y:||y|−t|<2ǫ} (x) (3.10) for some constants c > 0 and C > 0 depending on φ only. Without loss of generality assume v = k + 1 and that {k, k + 1} is an edge of G. Let t = (t ij ) ij∈G , t ij ∈ (0, +∞), and consider the function (on the domain of t just mentioned),
Λ ǫ G,µ (t) = ij∈G σ ǫ t ij (x i − x j )dµ(x 1 ) · · · dµ(x k+1 ) .
Using (3.10) we see that obtaining a bound Λ ǫ G,µ 2 ≤ M with M independent of ǫ is equivalent to obtaining a bound for ν ǫ 2 independent of ǫ, in particular showing that ν ∈ L 2 (R m ). Write then
Λ ǫ µ (t) = ij∈G,ij =k(k+1) σ ǫ t ij (x i − x j )dµ(x 1 ) · · · dµ(x k−1 ) · σ ǫ t k(k+1) (x k − x k+1 )dµ(x k )dµ(x k+1 ) = ij∈G,ij =k(k+1) σ ǫ t ij (x i − x j )dµ(x 1 ) · · · dµ(x k−1 ) (σ ǫ t k(k+1) * µ)(x k )dµ(x k )
From Lemma 4.16 we know that σ ǫ t * µ L 1 (µ) 1 and so using Chebyshev's inequality we may obtain a compact subset E ′ ⊂ E with µ(E ′ ) > 0 and σ ǫ t * µ L ∞ (E ′ ,µ) 1. Denoting by µ ′ the restriction of µ to E ′ , we obtain a Frostman measure of the same exponent. Denote by G ′ the graph G with the vertex v removed. Thus we have now
Λ ǫ G,µ ′ (t) ij∈G ′ σ ǫ t ij (x i − x j )dµ ′ (x 1 ) · · · dµ ′ (x k+1 ) = Λ ǫ G ′ ,µ ′ (t)
The rest of the proof proceeds as in Theorem 2.18, with µ ′ in place of µ and G ′ in place of G.
3.6. The natural measure ν g on E − gE. We denote by S d−1 the (d − 1)dimensional unit sphere centered at 0 in R d . We will need the following result. For s ≤ (d + 2)/2, see Wolff [22] for d = 2, and Erdogan [8] for d ≥ 3. For the case s ≥ (d + 2)/2, see Sjölin [24]). See [23] Chapter 8 for the definition and relevant properties of s-energy, we are only interested in the fact that it will be a finite number.
s > d(4k−1)+2 4k+2 for d 2 < dim E ≤ d+2 2 s > 4kd−1 4k+1 for d+2 2 < dim E .
Let g be a rotation, g ∈ O d (R) and define the measure ν g by
f dν g = f (u − gv)dµ(u)dµ(v) . (3.11)
Then the integral
ν k+1 g (x)dxdg
is a finite quantity and in particular ν g is absolutely continuous for g-a.e.
Remark 3.11. The threshold for s when d 2 < dim E ≤ dim d 2 + 1 is not useful unless d = 2 or d = 3, k = 1, 2 and k = 1, since in that case
d(4k − 1) + 2 4k + 2 ≥ d + 2 2 ≥ dim E .
In particular, below is a Proof. Let ψ ≥ 0 be a smooth radial function supported in {ξ ∈ R d : 1/2 ≤ |ξ| ≤ 4}, identically equal to 1 in {ξ ∈ R d : 1 ≤ |ξ| ≤ 2} with √ ψ also smooth. Let ψ j (ξ) = ψ(2 −j ξ). Moreover, we require +∞ j=−∞ ψ j (ξ) = c, for a suitable constant c > 0, for all ξ = 0 (see [23] §7 for existence of such ψ). Let ν g,j denote the j-th Littlewood-Paley piece of ν g defined byν g,j (ξ) =ν g (ξ)ψ j (ξ). Since ν g is a finite measure, in bounding the pieces, we may assume that j is bounded from below. Using the Littlewood-Paley decomposition of ν g , we may write ν k+1 g (x)dx as j 1 ,...,j k+1 ν g,j 1 (x) · · · ν g,j k+1 (x)dx . This is bounded above by
(k + 1)! j 1 ≥j 2 ≥···≥j k+1 ν g,j 1 (x) · · · ν g,j k+1 (x)dx .
Now using Plancherel, we see that sinceν g,j 2 * · · · * ν g,j k+1 is supported on scale 2 j 2 + · · · + 2 j k+1 ≤ 2 j 2 +1 whileν g,j 1 is supported on an annulus of scale 2 j 1 , the sum vanishes if j 1 − j 2 > 2 for j 2 large. Thus it suffices to consider the case j 1 = j 2 (the case j 1 = j 2 + 1 is similarly treated) and to look at the sum
j 1 =j 2 ≥j 3 ≥···≥j k+1 ν 2 g,j 1 (x)ν g,j 3 (x) · · · ν g,j k+1 (x)dx . (3.12)
From the definition of ν g,j it follows that ν g,j = µ j (−·) * µ j (g·), whereμ j =μ ψ j . By Young's inequality,
ν g,j ∞ ≤ µ j 1 · µ j ∞ .
Trivially µ j 1 ≤ 1 since µ is a probability measure. Also
|µ j (x)| = 2 dj |µ * ψ(−2 j x)| ≤ C N 2 dj (1 + 2 j |x − y|) −N dµ(y) ≤ C ′ N 2 j(d−s)
for any N > 1 since µ is a Frostman measure on E. Using this estimate on the terms corresponding to the indices j 3 , . . . , j k+1 we can bound (3.12) by a constant multiple of
j j≥j 3 ≥···≥j k+1 2 (j 3 +···+j k+1 )(d−s) ν 2 g,j (x)dx j 2 j(k−1)(d−s) ν 2 g,j (x)dx .
It follows that
ν k+1 g (x)dxdg j 2 j(k−1)(d−s) · ν 2 g,j (x)dxdg .
We will show that ν 2 g,j (x)dxdg 2 j(d−s) 2 −jγ(s,d) where the quantity γ(s, d) is defined in Theorem 3.9, which completes the proof.
Since we haveν g,j =μ j (ξ)μ j (gξ), via Plancherel we obtain
ν 2 g,j (x)dxdg = |μ j (ξ)| 2 |μ j (gξ)| 2 dξdg = ∞ 0 S d−1 |μ j (tω)| 2 |μ j (gtω)| 2 dg t d−1 dωdt .
Since O d (R) acts transitively on the sphere, the quantity in the parentheses is constant in ω, and in particular it is a constant multiple of |μ j (tω ′ )| 2 dω ′ . Thus we have that
ν 2 g,j (x)dxdg = C S d−1 |μ j (tω)| 2 dω 2 t d−1 dt = C S d−1 |μ(tω)| 2 ψ(2 −j tω)dω 2 t d−1 dt = C ′ 2 j+2 2 j−1 S d−1 |μ(tω)| 2 dω 2 t d−1 dt .
Since we are summing over j and the intervals [2 j−1 , 2 j+2 ] have finite overlap with each other, we may as well bound ν 2 g,j (x)dxdg by a constant multiple of
2 j+1 2 j S d−1 |μ(tω)| 2 dω 2 t d−1 dt .
The proof is finished by using Theorem 3.9, showing that ν 2 g (x)dxdg < ∞.
The proof is now complete up the proof of Lemma 3.12 and the geometric results in Section 4. We prove the lemma below. The geometric results are established in Section 4. Lemma 3.12. Let G ′ k+1,m be a tree. Then for small enough ǫ
ǫ −dk O d (R) A A χ{|(x i − x j ) − g(y i − y j )| < cǫ : ij ∈ G ′ k+1,m }dµ k+1 (x)dµ k+1 (y)dg (3.13)
is bounded by a constant multiple of
ν k+1 g (x)dxdg .
Proof. First we bound (3.13) by
ǫ −dk χ{|(x 1 − x j ) − g(y 1 − y j )| < (k + 1)cǫ : j > 1}dµ k+1 (x)dµ k+1 (y)dg .
(3.14)
This is accomplished as follows: Fix ij ∈ G ′ k+1,m and let (x 1 , x l 2 , . . . , x lp , x i , x j ) be a path of length p + 1, from x 1 to x j in G ′ k+1,m . Set l 1 = 1 and l p+1 = i, l p+2 = j. Using the triangle inequality we see that the set
{(x, y) : |(x i − x j ) − g(y i − y j )| < cǫ} is contained in the set {(x, y) : p+1 f =1 |(x l f − x l f +1 ) − g(y l f − y l f +1 )| < (p + 1)cǫ} which is contained in {(x, y) : |(x 1 − x j ) − g(y 1 − y j )| < (p + 1)cǫ} .
Since k ≥ p we conclude that (3.13) is bounded by (3.14). Using now the natural measure ν g we write (3.14) as ǫ −dk · · · χ{|z 1 − z j | < cǫ : j > 1}dν g (z 1 ) · · · dν g (z k+1 )dg .
Now it is obvious that taking ǫ → 0 and using the absolute continuity of ν g and the dominated convergence theorem, we may bound (3.14) by a constant multiple of ν k+1 g (z)dzdg finishing the proof.
Geometric results
For d ≥ 2 and each 1 ≤ q ≤ d we show that the edge set of K q+1 is independent in R d . We show that infinitesimal rigidity of a fixed graph G k+1,m is a generic property, either holding for all generic frameworks, or none of them. We count the number of edges a minimally infinitesimally rigid graph must have. The behavior of the distance function near regular tuples is investigated. Our approach follows closely that of [26]. See also [28] for motivation and examples.
Generic Frameworks.
In this section we prove various results for generic frameworks in R d . Some of the statements are for regular frameworks, but as noted in Remark 2.12, generic frameworks are regular.
The following lemma, while technically obvious, serves to remind the reader of the form that DF G k+1,m takes, which will be useful in subsequent proofs in this section.
Lemma 4.1. Fix d ≥ 2.
We have DF G k+1,m (x) · u = 0 if and only if u is a solution to the following system of m equations in d(k + 1) variables:
(x i − x j ) · (u i − u j ) = 0, for ij ∈ G k+1,m . (4.1) Proof. Since F G k+1,m is a function R d(k+1) → R m , DF G k+1,m is a m × d(k + 1) matrix
with rows corresponding to edges ij ∈ G k+1,m and columns corresponding to the scalar components of x. The ij-th row is equal to the following vector (0, . . . , 0,
d(i−1)+1 to di 2(x i − x j ) , 0, . . . , 0, d(j−1)+1 to dj −2(x i − x j ), 0, . . . , 0) .
Here every component x i − x j is also a vector (x 1 , . . . , x k+1 ∈ R d are vectors). Thus we can see that DF G k+1,m (x) · u = 0 is equivalent to (4.1). Proof. Since the rank of the matrix is less than |H|, the |H| row vectors are linearly dependent. Conversely, if the rows are linearly dependent every minor has to be zero since all the submatrices will satisfy the same dependence. Moreover, if H ⊂ H ′ then the matrix corresponding to H ′ will have rank at least that of the one for H. Proof. Note that to each polynomial P H corresponds at least one tuple x 0 for which P H is nonzero, thus the zero sets X H are proper algebraic varieties. Thus the set of generic tuples of G is nonempty, and in particular open dense of full measure (since the complement X is of codimension at least 1, as a proper algebraic variety).
The independence of H for any generic tuple follows from the definition of genericity. In particular if x is generic then x ∈ X, thus x ∈ X H . By Proposition 4.2 it follows that H is independent with respect to x. Proof. If Ax = Bx + b where B is an invertible linear transformation and b a vector, then since the row vectors of DF K k+1 (x) contain the entries ±(x i − x j ), we see that the row vectors of DF K k+1 (Ax) contain the entries ±(Ax i − Ax j ) = ±B(x i − x j ). In particular we see that the rows of DF K k+1 (x) corresponding to H are linearly independent if and only if those of DF K k+1 (Ax) are. Using Proposition 4.2 we conclude that affine transformations preserve genericity. where r ij is the row corresponding to the edge ij ∈ K q+1 . Suppose a < b is such that s ab = 0. Then if we focus on the column corresponding to x a we get the nontrivial equation
j<a s ja (x a − x j ) + j>a s aj (x a − x j ) = 0
which contradicts the fact that x 0 is in general position, i.e. that {x a − x j : j = a} is a linearly independent set of vectors. Proof. Assume x is not in general position. Thus for some 1 ≤ q ≤ d without loss of generality we may assume {x 1 , . . . , x q+1 } are affinely dependent with {x 1 , . . . , x q } affinely independent. As proven in Theorem 4.6, the edge set of K q+1 is independent. Let A be the affine transformation taking each x j , 1 ≤ j ≤ q to e j , the standard j-th basis vector (using Lemma 4.4). Then by affine dependence we must have Ax q+1 = t 1 e 1 + · · · + t q e q with t 1 + · · · + t q = 0. Setting s ij = −t i t j for 1 ≤ i < j ≤ q and s i(q+1) = t i for 1 ≤ i ≤ q we easily check that for any 1 ≤ a ≤ q + 1,
ia∈K q+1 s ia (x a − x i ) + aj∈K q+1 s aj (x a − x j ) = 0 .
Thus the edge set of K q+1 is not independent with respect to (x 1 , . . . , x q+1 ) and so by Theorem 4.3 we conclude that (x 1 , . . . , x q+1 ) is not a generic tuple. We now note that if x were a generic tuple then (x 1 , . . . , x q+1 ) would also be a generic tuple, simply because we are removing d(k + 1) − d(q + 1) column vectors from DF K k+1 to test for the genericity of (x 1 , . . . , x q+1 ). This contradicts what we have found therefore x is not generic.
Lemma 4.8. If x is in general position in R d then dim D(x) = d+1
2 . Proof. First let us show that if γ(t) : (−1, 1) → M x is a smooth curve with γ(0) = x, then γ ′ (0) ∈ D(x). This follows from the fact that the composition F (γ(t)) is constant, so by the chain rule DF (γ(0)) · γ ′ (0) = 0. Note that dim M x = d+1 2 , giving us
dim D(x) ≥ d + 1 2 .
For the reverse inequality, we will show that any infinitesimal motion u ∈ D(x) projects injectively to an infinitesimal motionũ of V(K d+1 ,x) whereũ = (u 1 , . . . , u d+1 ) andx = (x 1 , . . . , x d+1 ). By the rank-nullity theorem and Theorem 4.6,
dim V(K d+1 ,x) = d(d + 1) − d + 1 2 = d + 1 2 ,
establishing the reverse inequality and completing the proof.
Thus it remains to show injectivity. Let u, v ∈ D(x) and assumeũ =ṽ, that is,
u i = v i for 1 ≤ i ≤ d + 1. Since the space D(x) is a vector space, for w = u − v we have (w i − w j ) · (x i − x j ) = 0, for ij ∈ K k+1 . Now if i = 1, .
. . , d we have w i = 0, so that for any j > d we have
w j · (x i − x j ) = 0, for i = 1, . . . , d .
But that means w j is perpendicular to d linearly independent vectors, since x is in general position, thus w j = 0 as well, and u = v. Theorem 4.9. Let (G k+1,m , x 0 ) be an infinitesimally rigid framework in R d with x 0 generic. Then for all generic tuples x, the frameworks (G k+1,m , x) are infinitesimally rigid in R d .
Proof. Since V(G k+1,m , x 0 ) = D(x 0 ), combine Lemma 4.8 and Theorem 4.7 to obtain
dim V(G k+1,m , x 0 ) = d + 1 2 .
Since V(G k+1,m , x 0 ) = ker DF G k+1,m (x 0 ) by the rank-nullity theorem we obtain
dim V(G k+1,m , x 0 ) = d(k + 1) − rank DF G k+1,m (x 0 ) . (4.2)
Combining these two equations we find that
rank DF G k+1,m (x 0 ) = d(k + 1) − d + 1 2 .
Since generic tuples have the same rank (by Theorem 4.3), by using Equation (4.2) with x in place of x 0 we see that dim V(G k+1,m , x) = d+1 2 , implying that V(G k+1,m , x) = D(x), so that (G k+1,m , x) is infinitesimally rigid. Corollary 4.10. A minimally infinitesimally rigid graph G k+1,m in R d satisfies
m = d(k + 1) − d + 1 2 .
Proof. Let G k+1,m be minimally infinitesimally rigid. Then m ≥ rank DF G k+1,m (x 0 ) = d(k + 1) − d+1 2 , as the proof of Theorem 4.9 shows. Let (G k+1,m , x 0 ) be a regular framework. If we assume m > d(k + 1) − d+1 2 , there must be a subset H of edges of G k+1,m such that
rank DF H = d(k + 1) − d + 1 2 = |H| about x 0 , therefore H is infinitesimally rigid about x 0 .
Since that is an open set, by Theorem 4.9 the generic behavior of H is determined, thus H is infinitesimally rigid, which is a contradiction since H has less edges than G k+1,m . The same also hold for f G k+1,m .
x 0 M N x 0 ∩ U U Figure 3. The local behavior of the distance function at a regular tuple.
Proof. Since rank DF G k+1,m (x 0 ) = m is maximal, it stays maximal around an open neighborhood U of x 0 . The Inverse Function Theorem yields local coordinates (p, q) at x 0 such that F G k+1,m (p, q) = p. The manifold M is the image of (p, 0). It clearly is of dimension m. Using Corollary 4.10 and the fact that dim ISO(d) = d+1 2 , the other claims follow.
Remark 4.12. We justly say that the regular frameworks of a minimally infinitesimally rigid graph are locally uniquely realizable, in the sense that modulo isometries the distance function is a local diffeomorphism as Proposition 4.11 shows.
Useful Lemmas.
Here we collect the rest of lemmas that were used, that are not related to graph rigidity. Lemma 4.13. Let φ ∈ C ∞ c (R m ) be a nonnegative radial function with φ = 1, φ ≤ 1 and supp φ ⊂ {t : |t| ≤ 2} and for ǫ > 0 set φ ǫ (t) = ǫ −m φ(ǫ −1 t). Let ν(t) be a Borel probability measure. Then ν ǫ = φ ǫ * ν converges weakly-* to ν.
Proof. Let f ∈ C 0 (R m ) be a nonnegative function that vanishes at infinity. Then
f dν ǫ = (f * φ ǫ )dν .
It is a well known result of mollifiers that f * φ ǫ → f pointwise and by the Dominated Convergence Theorem we may conclude that f dν ǫ → f dν.
Lemma 4.14. Let V be a normed vector space and V * its dual equiped with the operator norm. If ν n → ν weakly-* in V * then lim inf ν n ≥ ν .
Proof.
lim n→∞ inf k≥n sup x =1 x, ν k ≥ lim n→∞ sup x =1 inf k≥n x, ν k ≥ sup x =1
x, ν . Lemma 4.15. If µ is a measure on R n and C µ > 0 a constant with µ(B(x, r)) ≤ C µ r s for some s > 0 and all r > 0 and x ∈ R n , then for any measurable subset A of R n with µ(A) > 0 we have dim A ≥ s.
Proof. Let U j be open balls of radius r j covering A. We then have
0 < µ(A) ≤ ∞ j=1 µ(U j ) ≤ C µ ∞ j=1 r s j .
Taking the infimum over all such collections U j we obtain that H s (A) > 0.
Lemma 4.16. Let σ t denote the surface measure of the sphere tS d−1 ⊂ R d of radius t centered at 0. Let φ ǫ (x) = ǫ −d φ(ǫ −1 x) and φ ∈ C ∞ c (R d ) is a nonnegative radial function with φ = 1, φ ≤ 1 and supp φ ⊂ B(0, 2). Let σ ǫ t = φ ǫ * σ t . Let µ be a Frostman measure on E ⊂ R d , E compact, with Frostman exponent s > d+1 2 . Then there exists a constant C t > 0 independent of ǫ with σ ǫ t * µ L 1 (µ) < C t . Proof. We use Plancherel and the stationary phase of the sphere, (see [23]), that tells us that for ξ of large norm and some c > 0 we have
| σ ǫ t (ξ)| ≤ ct d−1 2 |ξ| − d−1 2
, to obtain that for some C > 0 depending on the diameter of E,
σ ǫ t * µ(x)dµ(x) = σ t (ξ) φ(ǫξ)| µ| 2 (ξ)dξ 1 + |ξ| − d−1 2 | µ| 2 (ξ)dξ 1 + |x − y| − d+1 2 dµ(x)dµ(y) 1 + +∞ C µ x : |x − y| < λ − 2 d+1 dλdµ(y) 1 + +∞ C λ − 2 d+1 s dλ
The last integral is finite by assumption.
Proof of Theorem 2.25
Let q be a positive integer and define E q to be the q − d s -neighborhood of 1 q Z d ∩ [0, q] d with s ∈ d 2 , d to be determined later. It is known (see e.g. [9]) that if we choose q 1 = 2, q i+1 > q i i , then the Hausdorff dimension of E = ∩ i E q i is equal to s. Lemma 5.1. The number of congruence classes of frameworks with k + 1 vertices in Z d ∩ [0, q] d is bounded above by Cq dk .
To prove the lemma, fix one of the vertices at the origin, which we may do since Z d is translation invariant. The number of the remaining k-tuples is ≤ q dk by construction. This proves the lemma.
We now consider an infinitesimally rigid framework on k + 1 vertices in Z d ∩ [0, q] d described by the graph G k+1,m . By Corollary 4.10, the number of edges is m = d(k + 1) − d+1 2 = dk − d 2 . It follows that
H dk−( d 2 ) (∆(G k+1,m , E k+1 q )) ≤ C(q − d s ) dk−( d 2 ) · q dk .
This quantity tends to 0 as q → ∞ if s < d − ( d 2 ) k and Theorem 2.25 is proved.
of ∆(G k+1,m , E k+1 ) 8 3.3. A sharp upper bound for the dimension of the distance set 10 3.4. Bounds on the number of noncongruent realizations 12 3.5. The proof of the dimensional threshold 12 3.6. The natural measure ν g on E − June 19, 2018. The second and fourth listed authors were partially supported by the NSA Grant H98230-15-1-0319.
Figure 1 .
1A framework of K 4 (dashed edge for emphasis).
Figure 2 .
2A framework of the 'double banana' G 8,18 graph. This is not an infinitesimally rigid graph. Each banana may be freely rotated about the line joining the banana ends without altering the edge lengths. Yet it may not be completed into a minimally infinitesimally rigid graph by adding edges because it contains redundant edges (the dashed one, for instance). The solid edges form a maximally independent set H of edges of G 8,18 , thus by an application of Theorem 2.20 we obtain dim ∆(G 8,18 , E 8 ) = 17 and H 17
Proposition 3. 4 .
4Let G k+1,m be a graph. Then the set of edges of G k+1,m is independent in R d and may not be enlarged while retaining independence if and only if G k+1,m is minimally infinitesimally rigid.
Theorem 3. 9 (
9Wolff-Erdogan Theorem). Let µ be a compactly supported Borel measure in R d . Then, for s ≥ d/2 and ǫ > 0,S d−1 |μ(tω)| 2 dω ≤ C ǫ I s (µ)t ǫ−γ(s,d) ,with γ(s, d) = (d+2s−2)/4 if d/2 ≤ s ≤ (d+2)/2 and γ(s, d) = s−1 for s ≥ (d+2)/2 where I s (µ) is the s-energy of µ, I s (µ) = |x − y| −s dµ(x)dµ(y).
Theorem 3 .
310 (Natural measure on E − gE). Let k ≥ 2 and let E ⊂ R d , d ≥ 2 be a compact set with dim E > d 2 .Let µ be a Borel probability measure on E of Frostman exponent s < dim E with s satisfying
Theorem 4 . 3 .
43The set of generic tuples in R d is an open dense set of full Lebesgue measure. Moreover every independent (in R d ) set H is in fact independent in R d with respect to any generic tuple in R d .
Lemma 4 . 4 .
44If A is an invertible affine transformation of R d and we set Ax = (Ax 1 , . . . , Ax k+1 ), then we have that AX = X. Thus invertible affine transformations preserve genericity.
Definition 4. 5 .
5The tuple x = (x 1 , . . . , x k+1 ) is said to be in general position in R d if for every set J ⊆ {1, . . . , k + 1} with |J| ≤ d + 1 we have that {x j : j ∈ J} is affinely independent.
Theorem 4 . 6 .
46Assume q ≤ d. The edge set of K q+1 is then independent in R d , in fact with respect to any tuple in general position.Proof. Let x 0 be in general position and assume that the rows of DF K q+1 (x 0 ) are linearly dependent, say 1≤i<j≤q s ij r ij = 0
Theorem 4 . 7 .
47Let x be a generic tuple in R d . Then x is in general position in R d .
Proposition 4 . 11 .
411Let G k+1,m be a minimally infinitesimally rigid graph in R d and (G k+1,m , x 0 ) be a regular framework. Then there exists some open neighborhood U of x 0 and an embedded m-dimensional submanifold M ⊂ U that contains x 0 , with F G k+1,m restricted on M a diffeomorphism onto its image. Moreover if x ∈ U, letting N x = {y : F G (y) = F G (x)} denote the level curves, we have N x ∩ U = {(T x 1 , . . . , T x k+1 ) : T ∈ ISO(d)} ∩ U .
). Then the conclusion of Theorem 2.18 and Theorem 2.20 does not in
table for the readers convenience. If the values of d, k are not listed on this table then we haveExponents for s when
d
2
< dim E ≤
d + 2
2
d = 2
s >
4k
2k + 1
d = 3, k = 1, 2 s >
12k − 1
4k + 2
d > 3, k = 1
s >
d
2
+
1
3
dim E > s >
4kd − 1
4k + 1
.
Proposition 4.2. If H ⊆ K k+1 is an independent (in R d ) set of edges then x ∈ X H if and only if the rows of DF K k+1 (x) corresponding to edges of H are linearly dependent. Moreover if H ⊂ H ′ (H, H ′ independent), then X H ⊆ X H ′ .
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arxiv |
Assisted coupled quintessence
Luca Amendola
Institut für Theoretische Physic
Departamento de Matemática, ECEO
Universität Heidelberg
Philosophenweg 16D-69120HeidelbergGermany
Tiago Barreiro
Faculty of Sciences and Centre for Astronomy and Astrophysics
Universidade Lusófona de Humanidades e Tecnologias
Campo Grande, 3761749-024LisboaPortugal
Nelson J Nunes
University of Lisbon
1749-016LisbonPortugal
Assisted coupled quintessence
We study models of quintessence consisting of a number of scalar fields coupled to several dark matter components. In the case of exponential potentials the scaling solutions can be described in terms of a single field. The corresponding effective logarithmic slope and effective coupling can be written in a simple form in terms of the individual slopes and couplings of the original fields. We also investigate solutions where the scalar potential is negligible, in particular those leading to transient matter dominated solutions. Finally, we compute the evolution equations for the linear perturbations which will allow these models to be tested against current and future observational data.PACS numbers: 98.80.-k,98.80.Jk
I. INTRODUCTION
In the past decade, our understanding of the evolution of the Universe, its components and respective abundances has increased to an unprecedented level. Results from various independent observations of different scales have provided us with evidence for dark matter. Supernovae combined with other independent observations suggest that the universe is currently undergoing accelerated expansion, possibly due to a negative pressure component usually dubbed dark energy. We have also learned from analysis of the cosmic microwave background radiation that the universe is close to flat and that the large scale structure developed through gravitational instability from a spectrum of adiabatic, nearly Gaussian and nearly scale invariant density perturbations. These conclusions are consistent with the predictions of the simplest inflationary paradigm, a short period of accelerated expansion in the early Universe, introduced to explain the flatness, homogeneity and isotropy of the Universe. Scalar fields are the most popular building blocks to construct candidate models of early Universe inflation and of the present day cosmological acceleration. They are appealing because such fields are ubiquitous in theories of high energy physics beyond the standard model. Models are usually constructed using a single field, however, there is also the interesting possibility that a cosmological behaviour arises from the presence of multiple scalar fields. The idea that our Universe contains a vast number of light scalar fields is also based on expectations from landscape models, see e.g. [1]. Assisted inflation is an example of a model where many fields can cooperate to sustain inflation even if none is able to fuel it if evolving in isolation. Assisted inflation for exponential potentials was proposed in [2] and extended in [3,4] and the inclusion of a background fluid was evaluated in Ref. [5,6]. It was studied for quadratic and quartic potentials in the context of higher dimensional reduction in Ref. [7][8][9], for Bianchi models [10,11], for an ensemble of tachyon fields [12][13][14][15], in braneworld models [16,17], in particular string theory realisations [18][19][20] and taking into account Loop Quantum Cosmology corrections [21]. A multi-field dynamics was also studied in the context of Logamediate inflation [22] and k-inflation [23].
A set of several fields working together could also give rise to the accelerated evolution of the Universe we currently observe. This is usually known as assisted dark energy or assisted quintessence [24][25][26][27].
These fields could, of course, interact with the rest of the world and new forces between matter particles would arise. For common particles, these forces are tightly constrained by solar system and gravitational experiments on Earth. Limits on these forces are not so strong for interactions involving neutrinos or dark matter and current bounds come from cosmological observations. Coupled quintessence, where a scalar field interacts with dark matter was introduced in Refs. [29][30][31] and its dynamics, properties and possible couplings was studied in Refs. [32][33][34][35][36][37][38][39][40].
There might be, however, more than one dark matter species, a suggestion made in Refs. [41][42][43]. The possibility that a scalar field is indeed coupled to more than a single dark matter component has been raised recently in Ref. [44,45] and the phenomenology of such a set up was investigated and compared to observations in Refs. [46][47][48].
A natural extension of these works is, therefore, to investigate the cosmological dynamics considering a group of arXiv:1407.2156v1 [astro-ph.CO] 8 Jul 2014 several scalar fields coupled to various dark matter components. Such a system has been addressed in Ref. [25]. Here we extend that work with an explicit analysis of the background and of linear perturbations. In section II we introduce the general equations for n scalar fields coupled to m matter components. We then apply these to scalar potentials consisting of a sum of exponential terms in section III, and an exponential of a sum of terms in section IV. We follow this in section V with the solutions where the scalar potential is negligible. In section VI we extend our analysis to the linear perturbations. Finally we conclude in section VII.
II. GENERAL EQUATIONS
We consider an ensemble of n scalar fields φ i cross-coupled to an ensemble of m dark matter components ρ α . The cross-couplings are described by the matrix C iα where latin indexes i, j identify scalar field indexes and greek indexes α, β identify the dark matter components. The equation of motion for the fields and the various dark matter components in a spatially flat Friedmann-Robertson-Walker metric with scale factor a(t) is then written as
φ i + 3Hφ i + V ,φi = κ α C iα ρ α ,(1)ρ α + 3Hρ α = −κ i C iαφi ρ α .(2)
The solution for the dark matter component evolution can be given immediately in terms of the values of the fields as
ρ α = ρ α0 exp −3N − κ i C iα (φ i − φ i0 ) .(3)
The rate of change of the Hubble function isḢ
= − κ 2 2 α ρ α + iφ 2 i ,(4)
subject to the Friedmann constraint
H 2 = κ 2 3 α ρ α + i ρ φi .(5)
where ρ φi = i φ 2 i /2 + V (φ 1 , ..., φ n ). In the next two sections we consider two possible forms for V (φ 1 , ..., φ n ), both leading to scaling solutions: a sum of exponential terms and an exponential of a sum of terms.
The dark matter field can in principle include the baryon component. However in this case the coupling will have to satisfy the strong solar system constraints and be in practice negligible.
III. SUM OF EXPONENTIAL TERMS: V (φ1, ..., φn) = M 4 i e −κλ i φ i
With the aim of finding the critical points of the evolution, we will rewrite the equations of motion as a system of first order differential equations. To do this we define the new variables
x i ≡ κφ i √ 6H , y 2 i ≡ κ 2 V i 3H 2 , z 2 α ≡ κ 2 ρ α 3H 2 ,(6)
where
γ 1 = C 11 + C 21 λ 1 λ 2 ,(19)γ 2 = C 22 + C 12 λ 2 λ 1 .(20)
Equations (9) also provide the constraint
x 2 x 1 = λ 1 λ 2 = C 12 − C 11 C 21 − C 22 .(21)
and
γ 1 /λ 1 = γ 2 /λ 2 .
We can now compute the effective equation of state parameter of the Universe knowing that
H H = − 3 2 λ i x i = − 3 2 (1 + w eff )(22)
it is obtained that
w eff = −1 + 1 2 2 3 (λ 1 x 1 + λ 2 x 2 ).(23)
Substituting for x 1 and x 2 as found earlier, we can write
w eff = γ i λ i − γ i ,(24)
where i = 1, 2. Making use of the property that γ 1 /λ 1 = γ 2 /λ 2 we can define and effective coupling of the system such that
C eff ≡ λ eff γ i λ i ,(25)
to write
w eff = C eff λ eff − C eff ,(26)
Moreover, Ω φ = x 2 1 + x 2 2 + y 2 1 + y 2 2 and given the equality w eff = x 2 1 + x 2 2 − y 2 1 − y 2 2 , it results that the total contribution of the fields to the total energy budget is
Ω φ = 1 + 2(x 2 1 + x 2 2 ) − 1 2 2 3 (λ 1 x 1 + λ 2 x 2 ).(27)
Substituting now for x 1 and x 2 we obtain that
Ω φ = 3 − λ eff C eff + C 2 eff (λ eff − C eff ) 2 ,(28)
where the effective λ eff is given by
1 λ 2 eff = 1 λ 2 1 + 1 λ 2 2 .(29)
A typical evolution of the energy densities is illustrated in Fig. 1. The generalization to more than two fields is fairly trivial for this potential. Eqns. (17) and (18) are still valid for n fields × m dark matter components and in general we have, for every i = 1, ..., n. It is also instructive to consider how these quantities simplify when we consider that all the fields are just replicas of, say, the field φ 1 . By this we mean that the matrix C is diagonal with entries C ii = C 11 and λ i = λ 1 for all indexes i. It is easy to verify that in this case λ eff = λ 1 / √ n and C eff = C 11 / √ n and consequently
γ i λ i = n j C j1 λ j = ... = n j C jm λ j(30)w eff = C 11 λ 1 − C 11 ,(31)
takes the same value as for a single field system, however,
Ω φ = 3n − λ 1 C 11 + C 2 11 (λ 1 − C 11 ) 2 ,(32)
has a slight dependence on the number of fields n as illustrated in Fig. 2.
IV. EXPONENTIAL OF A SUM OF TERMS: V (φ1, ..., φn) = M 4 e − i κλ i φ i
For this case we only need to define one single y such that,
x i ≡ κφ i √ 6H , y 2 ≡ κ 2 V 3H 2 , z 2 α ≡ κ 2 ρ α 3H 2 .(33)
FIG. 2: Dependence of w eff (triangles) and Ω φ (circles) with the number of fields in the case when all fields are replicas of the same field and for the sum of exponentials potential. We used λ1 = 10 and C11 = −30.
The evolution is now described by
x i = − 3 + H H x i + 3 2 λ i y 2 + α C iα z 2 α ,(34)y = − 3 2 i λ i x i + 2 3 H H y,(35)z α = − 3 2 i C iα x i + 3 2 + 2 3 H H z α ,(36)H H = − 3 2 1 + i x 2 i − y 2 ,(37)
where the Friedmann equation now reads
i x 2 i + y 2 + α z 2 α = 1.(38)
A. Scalar field dominated solution
When the matter components are negligible (z α = 0), we can use Eqns. (35), (10) and (38) to obtain
x i = λ i √ 6 ,(39)
and then the equation of state parameter reads
w eff = −1 + 1 3 λ eff ,(40)
where the effective slope, λ eff is now
λ 2 eff = i λ 2 i .(41)
For this case, increasing the number of fields increases the value of λ eff and an accelerated expansion becomes more difficult to be attained. Again, this mimics previous results obtained in an assisted inflation setting in [3,4].
B. Scaling solution
In order to find the critical points corresponding to the scaling solution, when all the variables x i , y and z α are non-vanishing, we could proceed as for the previous potential. However, it is considerably easier to first start with a redefinition of variables. Let us observe that the evolution equations are invariant under an orthogonal transformation of x i , λ i and C ij . That is,x
i = Q ij x j ,(42)λ i = Q ij λ j ,(43)C ij = Q il C lj ,(44)
were Q ij is an orthogonal matrix, i.e., Q il Q T lj = Q il Q jl = δ ij . We will now give a working example for two fields and two dark matter components. We will show that, in the case of this potential it is always possible to rotate the fields such thatx 2 = 0. Looking for the case where all the variablesx i ,ŷ andẑ i are non-vanishing, from Eqs. (35) and (36), we obtain the conditions
(λ 1 −Ĉ 11 )x 1 + (λ 2 −Ĉ 21 )x 2 = 3 2 ,(45)(λ 1 −Ĉ 12 )x 1 + (λ 2 −Ĉ 22 )x 2 = 3 2 .(46)
It is easy to show that whenĈ 11 =Ĉ 12 the solution yieldsx 2 = 0. Using the orthogonal transformation (43) with the conditionĈ 11 =Ĉ 12 ,
Q 11 C 11 + Q 12 C 21 = Q 11 C 12 + Q 12 C 22(47)
together with the constraint that the rows of Q are unit vectors, Q 2 11 + Q 2 12 = 1, gives us the result
Q 11 = C 22 − C 21 (C 11 − C 12 ) 2 + (C 22 − C 21 ) 2 ,(48)Q 12 = C 11 − C 12 (C 11 − C 12 ) 2 + (C 22 − C 21 ) 2(49)
where we have chosen the positive roots. Sincex 2 = 0 we do not need to compute the rest of the matrix Q. We can now obtain the effective coupling
C eff =Ĉ 11 = Q 11 C 11 + Q 12 C 21 (50) = C 22 C 11 − C 21 C 12 (C 11 − C 12 ) 2 + (C 22 − C 21 ) 2(51)
and the effective λ eff ,
λ eff =λ 1 = Q 11 λ 1 + Q 12 λ 2(52)= (C 22 − C 21 )λ 1 + (C 11 − C 12 )λ 2 (C 11 − C 12 ) 2 + (C 22 − C 21 ) 2(53)
The scaling solution can then be written asx
1 = 3 2 1 λ eff − C eff ,(54)x 2 = 0.(55)
We can then compute the effective equation of state parameter and the total scalar field contribution from the expressions
w eff = −1 + 2 3 (λ 1x1 +λ 2x2 ) = C eff λ eff − C eff ,(56)Ω φ = 1 + 2(x 2 1 +x 2 2 ) − 2 3 (λ 1x1 +λ 2x2 ) = 3 − λ eff C eff + C 2 eff (λ eff − C eff ) 2 .(57)
It is straightforward to extend these results to the case of n scalar fields and n dark matter components. We can havex i = 0 for i ≥ 2 provided thatĈ
ii =Ĉ ij for j ≥ i.(58)
This can always be achieved through the orthogonal transformation Eqs (42)- (44). Again we only really need to obtain the first row Q 1j , of the Q matrix to obtain the value of the effective coupling C eff . This comes through the solution to the equation
C eff =Ĉ 1i = j Q 1j C j1 ,(59)
for any i, together with the unitarity condition, Q 2 1i = 1. Once we have Q 1i we can obtain the effective slope λ eff
λ eff =λ 1 = j Q 1j λ j .(60)
andx 1 , Ω φ and w eff will be given by the same Eqs. (54)-(57). To see how this would work, let us look at the simple case of a diagonal coupling matrix C. In this case, we get for Q 1i
Q 1i = 1 C ii l 1/C 2 ll ,(61)
such that
1 C 2 eff = i 1 C 2 ii ,(62)λ eff = C eff i λ i C ii .(63)
When all the fields are a copy of field φ 1 , then the expressions become fairly simple and give C eff = C 11 / √ n and λ eff = √ nλ 1 . For this potential both w eff and Ω φ have a strong dependence on the number of fields
w eff = C 11 nλ 1 − C 11 ,(64)Ω φ = 3n − C 11 λ 1 n + C 2 11 (nλ 1 − C 11 ) 2 ,(65)
which is illustrated in Fig. 3 for λ 1 = 10 and C 11 = −30.
V. SCALAR POTENTIAL INDEPENDENT SOLUTIONS
We can find fixed point solutions where the scalar potential energy density is negligible, so these solutions are independent of the type of scalar potential used. These can be divided into three different types.
A. Subdominant potential solution
This critical point corresponds to the case where the scalar potential is negligible and the total energy density is comprised of the scalar fields kinetic energies and the dark matter components. We can use either Eq. (9) or (36) and the Friedmann constraint to obtain
x i = 2 3 C iα ,(66)
for any α. This immediately gives the simple expression for the equation of state parameter and the total scalar field contribution
w eff = Ω φ = i x 2 i = 2 3 i C 2 iα .(67)
B. Kinetic dominated solution
This critical point occurs when only the kinetic energy of the fields is non-vanishing. It is immediately obtained that i x 2 i = 1 and w eff = Ω φ = 1.
(68)
C. Matter dominated solution
The matter dominated solution is the fixed point characterized by x i , y i = 0, (or y = 0) and which essentially means that the partial derivative with respect to the field φ i of the sum of all the dark matter contributions must vanish. This solution was discussed in a number of recent publications for a single field system and two dark matter components with symmetric couplings [45][46][47][48]. For the simple system of two fields and two dark matter components we obtain that the field must settle at the bottom of a valley defined by the flat direction in the φ 1 -φ 2 plane,
α C iα z 2 α = 0,(69)(C 11 − C 12 )(φ 1 − φ 10 ) + (C 21 − C 22 )(φ 2 − φ 20 ) = 1 κ ln − C 11 C 12 ρ 10 ρ 20 = 1 κ ln − C 21 C 22 ρ 10 ρ 20 .(70)
For consistency, we observe that the couplings must satisfy the simple relation
C 11 C 12 = C 21 C 22 ,(71)
otherwise the flat direction is non-existent and the field will keep on evolving. This solution is necessarily unstable. As the dark matter contribution decays with e −3N , eventually the scaling solution or the scalar field dominated solution becomes more important. Such an example is illustrated in Fig. 4. This critical point allows the dark matter to have a substantial contribution before the scaling regime and before the Universe accelerates.
VI. LINEAR DENSITY PERTURBATIONS
We now extend our analysis to linear perturbations. This would allow to test the multi-coupled model with current and future data on galaxy clustering. We perturb the flat-space metric as
ds 2 = −(1 + 2Ψ)dt 2 + a 2 (1 − 2Φ)η ij dx i dx j(72)
where Φ, Ψ are functions of space and time, and expand the fields as φ i → φ i + δφ i . The energy-momentum tensor for the various fields can be written as
T φ µν = i ∂ µ φ i ∂ ν φ i + g µν − 1 2 g ρσ i ∂ ρ φ i ∂ σ φ i − V (φ 1 , ..., φ n ) ,(73)
and for the component α of dark matter as
T dm (α) µν = α ρ α u (α) µ u (α) ν ,(74)
where u (α) µ is the four-velocity of component α of the matter fluid. The conservation equations then read
∇ µ T φ µ ν = −κ i α C iα ρ α ∇ ν φ i (75) ∇ µ T dm(α) µ ν = κ i C iα ∇ ν φ i ρ α .(76)
We want to obtain the equations for the linear perturbations around the background values for the fields,φ i and for the matter componentsρ α . We define the field perturbations as δφ i = φ i −φ i and the matter density contrasts as δ α = (ρ α −ρ α )/ρ α . Going to Fourier space, we obtain the following equations of motion for the φ i field perturbations of wavenumber k
δφ i + 3Hδφ i + j V ,φiφj δφ j + k 2 a 2 δφ i + 2V ,φi Φ − 4φ iΦ − 2κ α C iα ρ α Φ − κ α C iα ρ α δ α = 0,(77)
and for the density contrast of the matter component α,
δ α + θ α a − 3Φ + κ i C iα δφ i = 0,(78)
where θ α = ∇ · v α and v α is the velocity of the α component of the matter fluid. By differentiation we then get,
δ α + 3Φ + κ i C iα δφ i + k 2 a 2 Φ − i C iα δφ i + (2H − κ i C iαφi ) δ α − 3Φ − κ i C iα δφ i = 0.(79)
The ij component of Einstein's equations for i = j yields
Ψ = Φ,(80)
and using this equality from now on, the 00 component gives
3H(Φ + HΦ) + 1 2 κ 2 i φ i δφ i + V ,φi δφ i −φ 2 i Φ + 1 2 κ 2 α ρ α δ α + k 2 a 2 Φ = 0.(81)
Combining Eqs. (79) and (81) and working in the limit of small scales, s.t., (k/a) 2 H 2 , it turns out that the equation of motion for the linear matter perturbations is
δ α + 2H 1 − 1 2 κ i C iαφ i H δ α − 1 2 κ 2 β (1 + i C iα C iβ )ρ β δ β = 0.(82)
Using as time variable, N = ln a, we can rewrite these equations of motion in the form
δ α + 2 − 3 2 β Ω β − i ( 1 2 κ 2 φ 2 i + C iα κφ i ) δ α − 3 2 β (1 + i C iα C iβ )Ω β δ β = 0,(83)
which can also be written, using the definition of x i , as
δ α + 2 − 3 2 β Ω β − i (3x 2 i + √ 6C iα x i ) δ α − 3 2 β (1 + i C iα C iβ )Ω β δ β = 0.(84)
It should be clear that for a given α component a too large positive or negative i C iα C iβ , might imply, respectively, a very strong growth or damping of δ α , a situation that must be avoided.
For the case of two scalar fields coupling with two components of dark matter, the equations for the matter linear perturbations become
δ 1 + 2 − 3 2 (Ω 1 + Ω 2 ) − 3(x 2 1 + x 2 2 ) − √ 6(C 11 x 1 + C 21 x 2 ) δ 1 − 3 2 (1 + C 2 11 + C 2 12 )Ω 1 δ 1 − 3 2 (1 + C 11 C 12 + C 21 C 22 )Ω 2 δ 2 = 0.(85)
and
δ 2 + 2 − 3 2 (Ω 1 + Ω 2 ) − 3(x 2 1 + x 2 2 ) − √ 6(C 12 x 1 + C 22 x 2 ) δ 1 − 3 2 (1 + C 12 C 11 + C 22 C 21 )Ω 1 δ 1 − 3 2 (1 + C 2 12 + C 2 22 )Ω 2 δ 2 = 0.(86)
That is, for a scaling solution they are of the form
δ 1 + A 1 δ 1 − B 1 δ 1 − B 2 δ 2 = 0, (87) δ 2 + A 2 δ 2 − C 1 δ 1 − C 2 δ 2 = 0,(88)
where A j , B j and C j are all constants. The solutions for these coupled equations can be written as δ 1 ∝ e ξN ∝ a ξ and δ 2 = bδ 1 , however, the analytical relations between ξ and b in terms of the A j , B j , C j are too complicated to be of any practical use. In the cases when b B 1 /B 2 and b C 1 /C 2 we can substitute the solutions in the above equations and obtain a set of equations to estimate ξ and b given by
ξ 2 + A 1 ξ − B 1 ≈ 0, (89) ξ 2 + A 2 ξ − C 1 /b ≈ 0,(90)
which give the analytical results
ξ ≈ − 1 2 A 1 ± A 2 1 + 4B 1 ,(91)
b ≈ 1 2
C 1 B 2 1 + B 1 A 2 (A 1 − A 2 ) 2B 1 + (A 1 − A 2 ) A 1 ± A 2 1 + 4B 1 .(92)
Of course this approximation is only reliable when the assumptions b B 1 /B 2 , C 1 /C 2 are indeed verified. This happens for instance when Ω 2 Ω 1 , i.e. when one of the dark matter component is much smaller than the other. The growth rate f ≡ d log δ/d log a = ξ can be then directly compared to observations. This will be performed in future work.
VII. CONCLUSIONS
We have studied a cosmological system composed of a set of scalar fields coupled to an ensemble of dark matter components, thus generalizing previous work. The obtained solution can either be applied to construct early Universe inflationary solutions or provide a mechanism to explain the current accelerated expansion of the Universe and the ratio of abundances between dark energy and dark matter. More specifically, we have investigated two representative types of potential leading to analytical solutions. We have seen that the scalar field dominated solution allows for an inflationary evolution which is easier to attain with the sum of exponentials potential. The effective coupling of the scaling solution has an explicit dependence on the value of the coupling C iα and the potential parameters, λ i , for the sum of exponentials potential. For the exponential of a sum potential, however, the effective coupling can be written solely in terms of the individual couplings. We found a relation between the value of the couplings in order to obtain an early dust like dominated behaviour. Essentially, the fields must settle at the bottom of the effective potential which has a flat direction. Finally, we observed that the equations of motion for the matter density contrasts possess a source or damping term which might lead to an unacceptable growth or damping of the density contrast. It would be interesting to carry out a numerical analysis to test the range of parameter space for which these models are compatible with current and forecasted future large scale structure data.
FIG. 1 :
1Evolution of energy densities for the sum of exponentials potential with λ1 = 10, λ2 = 5.4, C11 = 90, C12 = −8, C21 = −63, C22 = −10.
FIG. 3 :
3Dependence of w eff (triangles) and Ω φ (circles) with the number of fields in the case when all fields are replicas of the same field and for the exponential of sum of fields potential. We used λ1 = 10 and C11 = −30.
FIG. 4 :
4Example of a matter dominated solution followed by a scaling solution by only one of the dark matter components. Evolution of energy densities for the sum of exponentials potential with λ1 = 10, λ2 = 10, C11 = −20, C12 = 40, C21 = −30, C22 = 60. The initial condition for the scalar fields is very near the bottom of the effective potential which causes the initial oscillations before N = −4.
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arxiv |
Estimating Patterns of Classical and Quantum Skyrmion States
Vladimir V Mazurenko
Theoretical Physics and Applied Mathematics Department
Ural Federal University
Mira Str. 19620002EkaterinburgRussia
Ilia A Iakovlev
Theoretical Physics and Applied Mathematics Department
Ural Federal University
Mira Str. 19620002EkaterinburgRussia
Oleg M Sotnikov
Theoretical Physics and Applied Mathematics Department
Ural Federal University
Mira Str. 19620002EkaterinburgRussia
Mikhail I Katsnelson
Institute for Molecules and Materials
Radboud University
NijmegenNetherlands
Estimating Patterns of Classical and Quantum Skyrmion States
(Dated: April 6, 2023)
In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements.
In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements.
I. INTRODUCTION
History of science unambiguously evidences that the development of new theoretical concepts and physical models is impossible without insights from experiments and observations, which is crucial for exploring physical systems of any scale, from planetary objects to atoms and less than atoms. One of the bright examples of this insight is magnetic measurements 1 of a seemingly standard magnet, hematite α-Fe 2 O 3 performed by Smith in 1916, which, at the very end, paved the way to the concept of anisotropic inter-atomic interactions formulated by Igor Dzyaloshinskii 2 and Toru Moriya 3 . In his work Smith has found a ferromagnetic response by applying magnetic field perpendicular to the trigonal axis of α-Fe 2 O 3 . Remarkably, this experimental observation playing the crucial role in the theory of the inter-spin interaction was done about 10 years before the concept of the electron spin itself was introduced. Further experiments 4 confirmed Smith's findings for Fe 2 O 3 and showed robust weak ferromagnetism in other antiferromagnets 5,6 (MnCO 3 , CoCO 3 , NiF 2 ) characterized absence of inversion symmetry center between nearest magnetic ions. This relation with crystallography assumes that we deal with an intrinsic property rather than something related to crystal lattice imperfections (impurities, violation of the stoichiometric composition and others) 7 .
By 1957, a critical mass of experimental data that evidence existence of a symmetry-dependent spontaneous magnetization in a number of antiferromagnets had been accumulated. In this year Igor Dzyaloshinskii proposed an elegant way to explain the magnetic moment by using the symmetry arguments based on the twisting of the spin arrangement due to inversion symmetry breaking 2 . For that anisotropic exchange interaction in the form D ij [S i × S j ] was introduced into the spin Hamiltonian. Such an interaction is antisymmetric with respect to interchange of the spins and favors noncollinear magnetic order. Toru Moriya has proposed a micro-scopic mechanism for this newborn coupling by developing its superexchange theory 3, 8,9 with taking the spinorbit coupling into account. Subsequently, the original Moriya's microscopic theory was consistently improved and refined in papers [10][11][12] . Moreover, different numerical schemes based on the density functional theory calculations were developed to estimate Dzyaloshinskii-Moriya interaction (DMI) from first-principles calculations [13][14][15][16][17] . All these methodological results facilitate modeling magnetic properties of completely different materials and make estimation of DMI to be a routine procedure in the modern computational physics.
Remarkably, even after half century our understanding of the Dzyaloshinskii-Moriya interaction is far from being complete. It can be justified by the example that standard magnetic measurements techniques allow to estimate the magnitude and symmetry of DMI in concrete correlated materials, however, they do not provide the information on the DMI sign, which defines the local twist of the magnetic structure with respect to the atomic rearrangement due to inversion symmetry breaking. Such a problem was solved in 2014 when a new experimental technique, suggested earlier in Ref. 18, was developed in Ref. 19. This technique is based on the interference between two X-ray scattering processes, where one acts as a reference wave allowing to determine the sign of another. Experimentally, the sign of the DMI in correlated materials can be controlled with the occupation of the 3d shell. For instance, as it was shown in Ref. 20 there is a DMI sign change in the series of isostructural weak ferromagnets, MnCO 3 (DMI < 0), FeBO 3 (DMI < 0), CoCO 3 (DMI > 0) and NiCO 3 (DMI > 0). These experimental results agree with magnetic structure features obtained from the DFT calculations and can be explained Moriya's microscopic theory taking into account the occupation change of the correlated states.
Over time, it became clear that the scope of Dzyaloshinskii-Moriya interaction is not limited to antiferromagnetic insulators with weak ferromagnetism. It has been used for prediction of long-range spiral structures in certain magnets 21 without inversion symmetry. Later on such spiral structures were experimentally observed in metallic MnSi and FeGe magnets 22,23 and Fe 1−x Co x Si alloys 24,25 with the B20 crystal group. For these metallic systems, the period of the spiral structures varies in wide ranges from 175Å to 700Å. In turn, magnetic critical temperatures for stabilization of the spin-spiral ground state can be also very different and include technologically important regimes that are close to the room temperature. In addition, it was found that there is a complex interplay between magnetic properties and electronic structure of long-range spin-spiral metallic magnets. For instance, first-principles calculations 26 of the B20 crystal group systems have revealed a strong renormalization of the electronic spectra near the Fermi level due to the dynamical electron-electron correlations, which can also affect values of the magnetic moments as well as isotropic and anisotropic magnetic exchange interactions 27 and, therefore, should be taken into account when one describing these systems theoretically.
The exploration of the materials hosting the DMI spin spirals played an important role in establishing new research field of topologically-protected magnetic structures. It was first shown theoretically 28 in 1989 and then confirmed experimentally 29,30 that the DMI is responsible for forming long-range topologically protected chiral structures, magnetic skyrmions in metallic ferromagnets. The possibility to stabilize and manipulate skyrmions with magnetic and electric fields at the room temperature makes them very promising in numerous technological applications including next-generation memory devices and quantum computing 31 . Undoubtedly, further strides in the field of the magnetic skyrmions as well as creating new technologies that will make of use topological properties of the materials require implementation of the most advanced techniques for generation, detection, exploration and control. In this respect machine learning and quantum computing are of special interest. While the former allows to automatically classify and characterize magnetic structures the latter facilitates imitation of new phases of matter including topological ones.
Keeping in mind the recent progress in developing machine learning and quantum computing techniques for scientific research in this review paper we first focus on the latest activity concerning the implementation of computing methods for exploration of the magnetic skyrmion phases. A special accent will be given on the recently introduced renormalization procedure for calculating structural complexity of an object 32 , which allows straightforward estimation of the phase boundaries in non-collinear magnets in a purely unsupervised manner by using a few magnetic snapshots of the system in question. The second part of the paper is devoted to theoretical analysis of the quantum skyrmions that are ground states of quantum systems with Dzyaloshinskii-Moriya interaction. Our abilities in simulation of such quantum states with classical computers are limited due to the expo-nential growth of the Hilbert space with the number of particles. At the same time quantum computing can be considered as the most promising technology for further exploration of the quantum skyrmion states. In this respect the development of approaches for certification and identification of quantum states of large-scale quantum systems are in demand and attract considerable attention. We discuss the generalization of the procedure for calculating structural complexity of object onto the case of quantum states and report on the classification of the quantum phases in DMI magnet on the basis of the bitstrings obtained after a limited number of projective measurements. The problems concerning imitation of the quantum skyrmions with quantum computers are discussed.
II. CLASSICAL SKYRMIONS
A typical spin Hamiltonian used to simulate skyrmion structures can be written in the following form:
H = ij J ij S i S j + ij D ij [S i × S j ] + B i S z i (1)
where J ij and D ij are the isotropic interaction and Dzyaloshinskii-Moriya vector, respectively, S i is a unit vector along the direction of the ith spin and B denotes the out-of-plane magnetic field. To stabilize a skyrmion state for Hamiltonian (1) the Dzyaloshinskii-Moriya interaction for each bond should be in-plane. In our work we consider D ij that points in the direction perpendicular to the bond between neighboring i and j sites. At low temperatures, different phases realized with Hamiltonian Eq.1 can be identified using conventional techniques, that is, calculating skyrmion number and spin structure factors. The skyrmion number (topological charge) Q is defined as
Q = 1 8π ijk S i · [S j × S k ],(2)
where the summation runs over all nonequivalent elementary triangles that connect neighboring i, j, and k sites.
In the case of skyrmions of a few atoms in size 33 one can use an approach proposed in Refs. 34 and 35. In turn, the spin structure factors are given by
χ q = S z q S z -q ,(3)χ ⊥ q = S x q S x −q + S y q S y −q ,(4)
where q is the reciprocal space vector.
As it was previously shown 30,36 , phase diagram of such system (Eq.1) consists of three clear phases: spin spirals, skyrmion crystal and ferromagnetic state, and two significant intermediate regions, namely, skyrmion-bimeron state and skyrmion gas. Some examples of these spin textures are presented in Fig. 1. By bimeron here we mean a spin texture composed of two half-disk meron domains having Q = ±1/2 divided by neutral rectangular stripe domain 37 . Such quasi-particle can be associated with either elongated skyrmions or broken helix segments, since its length strongly depend on the Hamiltonian parameters. We should note that term bimeron is rather used to describe particles, composed of a pair of merons of different vorticity [38][39][40] . However, to observe them one has to change the symmetry of the DM vector. The detailed description of these quasi-particles is given in Ref. 41.
Unfortunately, the skyrmion number and spin structural factors are very sensitive to temperature and give us inappropriate results even in case when the spin structures still remain visually recognisable 42 (see Fig. 2). This fact aroused significant interest in the development of machine methods for conducting phase classification in this system. Below we will discuss such techniques.
A. Complexity
One of the basic concept which is usually used to analyze various patterns, systems and processes is their structural complexity. Although being intuitively clear since it reflects human's perception of reality, this value is very difficult to describe quantitatively. However, a lot of domains from geology to social sciences require a robust mathematical notion that properly reflects complexity of hierarchical non-random structures. Despite numerous attempts to give a formal definition of this quantity 43-50 , our understanding of these matters is still far from being complete. Recently, some of us have proposed an easy to compute, robust and universal definition of structural (effective) complexity based on inter-scale dissimilarity of patterns 32 . Besides meeting an intuitive perception of what is "complex" and what is "simple", this measure has been shown to be a suitable tool for determining phase transitions in various types of systems, including skyrmion structures.
In its simplest form, the algorithm to compute structural complexity of a given magnetic configuration consisting of L × L atoms can be formulated in the following way 32 : at each iteration, the whole system is divided into blocks of Λ × Λ size, and each block is substituted with a single spin which is calculated as
s ij (k) = 1 Λ 2 l m s Λi+m,Λj+l (k − 1)
, where the lm indices enumerate the spins belonging to the same block, and k is the number of iteration. Then, one can compute overlaps between patterns separated by one step of such an averaging procedure:
O k,k−1 = 1 L 2 L i=1 L j=1 s ij (k) · s ij (k − 1),(5)
with k = 0 corresponding to the original pattern, and O k,k is an overlap of the pattern at scale k with its own self. Defining structural complexity C as an integral characteristic accounting for features emerging at every new scale, we obtain
C = N −1 k=0 C k = N −1 k=0 |O k+1,k − 1 2 (O k,k + O k+1,k+1 ) |,(6)
where N is the total number of averaging steps. Fig.3 shows an example of the implementation of the structural complexity approach to the skyrmion problem. More specifically, it gives the resulting dependence of structural complexity on magnetic field for two-dimensional triangular lattice system described by Hamiltonian Eq.(1). Remarkably, for each value of B the complexity appears to be very robust, fluctuating within 0.01% error range for independent Monte Carlo runs. It means that one can safely use a single magnetization image for each magnetic field value to define the complexity. The extrema of complexity derivatives dC/dB reflect very well both the melting of spin spirals (magnetic labyrinths) into skyrmion crystals, with the transition point being exactly the bimeron phase, as well as the transition between skyrmion crystals and ferromagnets.
Recently, the structural complexity was used to find the phase boundary between self-induced spin-glass and noncollinear magnetically ordered states in elemental Nd at low temperatures 51 .
B. Machine learning methods for phase classification
By the construction the method for calculating structural complexity can be classified as unsupervised one, since it does not use any apriori information on the system in question. At the same time, we would like to stress the advances of various supervised techniques that involve a learning with pre-prepared and labeled data. After the inspiring work of Carrasquilla and Melko 52 , who had demonstrated the ability of neural networks to define phase transitions in magnetic systems, significant efforts have been made in this field. Here we give a brief overview of such approaches aimed to study the properties of the magnetic skyrmions.
In Ref. 42 we have demonstrated the possibility to use machine learning algorithms for exploration phases of non-collinear magnets that can host skyrmionic structures (see Fig. 4). In the work 42 , a square lattice system described by a spin Hamiltonian (1) was considered.
Given the fact, that all presented spin textures have distinct magnetization profiles, we decided to use only z components of spins as an input for machine learning algorithms. We have found out, that a simple singlehidden-layer feed-forward neural network (FFN) with only 64 hidden neurons, trained on a moderate number of clear-phase Monte Carlo configurations, was able to successfully reproduce the entire phase diagram, including intermediate regions. Moreover, it demonstrated good results on unseen data, namely, high-temperature configurations, larger skyrmions and configurations obtained for triangular lattice system. Unfortunately, we found that such a network relies mostly on total magnetization and therefore cannot distinguish spin spirals and paramagnetic state. However, such an inconvenience can be easily overcome by simple sorting of the input vector even in case of 3D systems 53 (see Fig. 5). It was also shown, that standard machine learning techniques like k-nearest neighbours, nearest centroids and supportvector machine work well in case of all clear phases and paramagnetic state.
Later, more complicated convolutional neural networks (CNN) were used to study the effect of uniaxial magnetocrystalline anisotropy pointing in the z direction on the phase diagram of a disk-shaped system 54 , and to construct detailed phase diagrams for skyrmion systems including intermediate regions and paramagnetic state 55 . Moreover, it was shown that such an architecture is able to not only determine phase boundaries but also restore various parameters. The authors of Ref. 56 have demon- strated, that a CNN trained on ground state configurations successfully recovers the chirality and magnetization of a given spin texture, as well as the temperature and external magnetic field at which it was stabilized. It is interesting to note that the accuracy of the algorithm remains remarkably high in presence of disorder caused by the randomly generated site-dependent uniaxial anisotropy. The authors of Ref. 57 have addressed an important problem of finding a topological charge of a given system based on its time-integrated spacedependent magnetization. They demonstrated that, being analytically inaccessible, this quantity can be extracted using a CNN with almost 100% accuracy. It was shown, that such an approach works well on systems of different confined geometries, including random islands, which looks very promising from the point of view of potential application to real experimental data.
Recently, considerable attention has also been paid to the dynamic properties of skyrmion structures. Some of us have shown that the simplest recurrent neural network (RNN) is able to automatically detect different processes occurring with an isolated skyrmion under the influence of picosecond magnetic field pulses 58 (see Fig. 6). Such an approach is promising as a technique which performs an autonomous control of the system's dynamics in case of a prototype of skyrmionic data storage elements 35 . The authors of Ref. 59 have studied a dynamic phase diagram of a particle model for skyrmions in metallic chiral magnets with using CNN-RNN architecture. It was shown, that the network is able to not only draw the correct phase boundaries but also define the exact number of the order parameters of the system in question.
III. QUANTUM SKYRMIONS
The progress in the development of experimental techniques 29,30 for the observation of magnetic skyrmions, topologically protected spin structures, poses new challenges for the theory and numerical simulations of ordered magnetic phases 60 . Nowadays, skyrmions are mostly discussed in the context of spintronics, where these stable magnetic structures are proposed as bits in magnetic memory devices 61 . The need to store more and more information requires the development of ultradense memories. This fact motivates the investigation of skyrmions of a nanoscale size, with recent significant progress. Skyrmions with the characteristic size of a few nanometers have already been observed in real experiments 62 and were theoretically predicted in magnets with DMI 63 , in frustrated magnets 64 , as well as in narrow band Mott insulators under high-frequency light irradiation 65 , and others. On such small characteristic length scales compared to the lattice constant, quantum effects cannot be neglected. Given this, the numerical study of classical spin models can no longer be considered as an comprehensive solution of the problem. Quantum fluctuations play a crucial role, because, strictly speak- ing, the spin itself is a quantum characteristic of an electron.
A common way to approach this problem is to force the quantum system to behave as a classical one. As the result, description of a quantum skyrmionic problem is either done semiclassically assuming that the magnetization dynamics is dominated by classical magnetic excitations that emerge on top of the symmetry-broken ground state of the system 66 , or by means of the Holstein-Primakoff transformation, which only allows to compute quantum corrections to the classical solution 67 . Besides, in paper 68 topological states of small clusters embedded in the ferromagnetic environment were investigated.
Recently, some of us have developed an approach for a characterisation of the quantum skyrmion state 69 in an infinite magnetic systems for which, in contrast to the classical case, the magnetization density is uniform. For that, the following spin model defined on the 19-spin supercell with periodic boundary conditions was considered,Ĥ
= ij J ijŜiŜj + ij D ij [Ŝ i ×Ŝ j ] + B iŜ z i . (7)
HereŜ i is the spin-1/2 operator. For characterization of the quantum ground states obtained at different magnetic field values, the local three-spin correlation function, Q Ψ = N π Ŝ 1 · [Ŝ 2 ×Ŝ 3 ] defined on neighboring lattice sites (here N is the number of non-overlapping triangles in the supercell) was used. Such a correlator gives information about the topology of the entire quan-tum system, for instance, from Fig.7 one can see that the scalar chirality is characterized by non-zero constant value for the magnetic fields 0.3 < B < 0.66, which clear signature of the quantum skyrmion phase.
Theoretically, exponential growth of the Hilbert space is the main factor preventing one from simulating larger systems than 19-site clusters discussed in Ref. 69 and from exploring quantum skyrmions of different kinds and sizes. The topological spin structures such as skyrmions emerge as the result of a competition between different magnetic interactions, leading to a magnetic frustration that restricts the applicability of quantum Monte Carlo methods due to the notorious sign problem 70 . In turn, exact diagonalization (ED) based methods have a severe restriction on the cluster size. For instance, for spin-1/2 Heisenberg-type models, the current limit is 50 lattice sites 71 . Account of the anisotropic terms such as Dzyaloshinskii-Moriya interaction leads to mixing of the sectors of the Hamiltonian with different total spins, which significantly limits our opportunities to use symmetry of the system in question to reduce the size of the Hamiltonian matrix. Thus, this supercell of 19 sites with isotropic and anisotropic exchange interactions between spins defines the current limit on the system size for simulating quantum skyrmions with exact diagonalization approach.
In this tough situation quantum computing provides a promising alternative to the standard approaches aimed at the search for ground and low-lying excited states of quantum Hamiltonian. Over the last decades there has been a fantastic progress in quantum computing on the level of constructing complete operating devices of up to 65 qubits with on-line access as well as in developing numerous algorithms for different fields of research including material science and condensed matter physics 72 . The first attempt 73 of the Google team to demonstrate a quantum supremacy by the example of quantum chaos states have additionally heat up the interest of scientific community to the work on quantum states that are significantly delocalized in Hilbert space. Our preliminary results show that the quantum skyrmion and spin spiral states being solutions of the quantum spin model, Eq.7 are significantly spread over Hilbert space, which is an indication that their further theoretical exploration may be effective with quantum computers. However, it calls for development of the methods for characterization and identification of the quantum states. Below we will discuss such a procedure based on the estimating structural complexity of the bit-string patterns.
Calculating inter-scale dissimilarity of bit-string arrays
Generally, the exploration of the magnetic skyrmions with quantum devices or simulators should include the following steps. First of all, one needs to define a quantum circuit that transforms the initial trivial quantum state |000..0 into desire Ψ Skyrm (Fig.8) being the ground state of the quantum Hamiltonian, Eq.7. There are two alternative solutions of this problem. (i) One can use a variational approach 72,74,75 in which the quantum state is represented with a fixed sequence of one-and two-qubit gates. The parameters of these gates are tuned to get the best approximation of the desire state. Unfortunately, there is no a universal sequence of gates that can approximate the ground state of an arbitrary Hamiltonian. Another problem is that one obtains not a true target state but only its approximation, which is of crucial importance for certain tasks. (ii) On the other hand, in the case of the small-size problems, the exact decomposition of the quantum ground state over basis functions can be found with exact diagonalization. It allows to employ Least Significant Bit (LSB) procedure 76 or similar procedure to find a sequence of gates realizing such a state. In the case of the 19-site quantum skyrmion ground state the LSB circuit contains thousands of gates, which is appropriate for creation and manipulation of such a state on a quantum simulator that imitates quantum logical operations on classical computer, but not on a real quantum device subjected to decoherence. For the latter one needs to develop new approaches aiming at generating quantum circuits that are as compact as possible. In this work we explore the quantum skyrmion state on the quantum simulator by using the LSB procedure.
When quantum is initialized with target state, the measurements in a basis are performed (Fig.8). As it was shown in our previous work 77 to characterize a quantum state with dissimilarity procedure it is necessary to perform measurements in at least two bases because measurements in one fixed basis give the information only about amplitudes of wave function coefficients, but not the local phases. Following Ref. 77 the measurements in σ z and random bases were performed. For each basis the measurement outputs are concatenated together into one string which can be considered as binary array of length L = N × N shots . After sampling of bit-string arrays, we estimate its patterns using the procedure presented in Ref. 77. Below we reproduce the main steps of this procedure.
At every step of coarse-graining k, a vector of the same length L is constructed as
b k i = 1 Λ k Λ k l=1 b k−1 Λ k [(i−1)/Λ k ]+l ,(8)
where b 0 is initial bit-string array containing 0 and 1 elements, square brackets denote taking integer part. According to this expression at each iteration the whole array is divided into blocks of Λ k size, and elements within a block are substituted with the same value resulting from averaging all elements of the block (Fig.9). Index l denotes elements belonging to the same block. In our recent work 77 we have shown that in some cases it is enough to measure only part of the system to extract information about phase transitions. Following this way, for simplicity, we measured only 16 qubits in 19 site system to make the bit-string length an integer power of filter size Λ: log Λ N ∈ N. Dissimilarity between scales k and k + 1 is defined as
D k = |O k+1,k − 1 2 (O k,k + O k+1,k+1 ) |,(9)
where O m,n is the overlap between vectors at scales m and n:
O m,n = 1 L (b m · b n ) .(10)
This expression can be considered as a modification of the structural complexity 32 discussed above in the context of classical magnetic patterns to our quantum problem. There are two quantities of our principal interest: D k that contains scale-resolved information on the pattern structure of the generated bit-string array and overall dissimilarity, D = k D k , where the sum goes over all the renormalization steps. D and {D k } can be used for a unambiguous identification of a quantum state. Fig.10 gives the dissimilarity calculated in σ z and random bases for the ground state of the spin Hamiltonian, Eq.7 taken with |D ij | = 1, J ij = 0.5 and B= 0.4. In these calculations of D we used 16 of 19 bits from each measurement. The calculated D z reveals a smooth transition between spin spiral and skyrmion phases. On the other hand, the transition between skyrmion and ferromagnetic states is abrupt. Figure 11 shows partial dissimilarities {D k } calculated for 19-qubit system in σ z and random bases. Although partial dissimilarities of different non-collinear magnetic phases measured in σ z basis have similar shape, they still have different absolute values. Note, that in σ z basis the value of partial and total dissimilarity is equal to zero for ferromagnetic state. This happens because in σ z basis all measurements result in 0000. . . 0 or 1111. . . 1 depending on the direction of magnetic field which gives exactly zero for Eq. 9. The D r demonstrates different behaviour for ferromagnetic phase, and, peak at k = 4 emerges when filter size Λ k becomes larger than the length of individual measurement bit-string effectively mixing different measurements.
Thus, the dissimilarity metric facilitates the search for phase boundaries of quantum models. Formally, this problem can be solved by choosing an appropriate order parameter, a correlation function characterized by a specific non-zero value for the definite range of model parameters 78,79 . However, in this case one faces the problem that different phases cannot be described with a single order parameter. For instance, by using the scalar chirality we cannot distinguish between the cases of zero and high magnetic fields in the considered DMI magnet, Fig.7. We would like to stress that one can consider the dissimilarity as a worthy alternative to other approaches for detecting phase boundaries in strongly correlated systems. First, the calculation of the dissimilarity is considerably cheaper in computational resources than traditional methods based on the calculation of the correlation functions. Then, the dissimilarity allows the accurate description of the phase boundaries in the cases when the choice of the order parameter is not obvious. It was demonstrated by the examples of the Shastry-Sutherland model of the orthogonal spin dimers and one-dimensional bond-alternating XXZ model hosting topological order 77 .
IV. CONCLUSION
To conclude, implementation of the machine learning methods considerably facilitates and accelerates theoretical characterization of non-collinear magnetic structures originated from the competition between anisotropic Dzyaloshinskii-Moriya and isotropic Heisenberg exchange interactions. As we have shown in this review by the example of the magnetic skyrmions the range of tasks to be solved is very wide, from constructing phase diagrams to estimating parameters of the parent Hamiltonians. In contrast to the traditional approach for description of magnetic systems that assumes accumulation of significant amount of the statistical data and calculations correlation functions of different orders, machine learning approaches do not require a complete information on the system in question. For instance, the temperature-magnetic-field phase diagrams can be constructed on the basis of few snapshots containing only z projection of the spins. In this respect, we would like to emphasize a crucial role of the data preprocessing that can realize in different ways and forms. For instance, it could be simple sorting the magnetization vectors that considerably improves the quality of the supervised neural network classification.
The procedure for estimating structural complexity of an object we described in the main text can be also considered as that performing a kind of preprocessing. It al-lows unsupervised identification of the phase boundaries of physical systems at a smooth varying external parameters such as temperature, magnetic field and others. As a result the complete phase space of the system in question can be divided onto separate phases, whose origin and properties could be further clarified with traditional and machine learning approaches.
The same procedure for estimating pattern complexity can be applied in the case of quantum systems, for which one explores sequences of the bitstrings obtained from the projective measurements. As a prominent example we analyzed the pattern structure of the quantum ferromagnet with Dzyaloshinskii-Moriya interaction. Depending on the value of the external magnetic field one can observe different quantum phases including recently introduced quantum skyrmion one. Our results presented in this paper complete the picture of the quantum skyrmion properties and can be useful for characterization of such a state on the quantum computing devices.
V. ACKNOWLEDGEMENTS
We thank Vladimir Dmitrienko for fruitful discussions.
FIG. 1 .
1(Color online) Monte Carlo solutions of the spin model, Eq. (1) on the rhombic plaquette, corresponding to the spin spiral (a), bimeron (b), skyrmion (c) and ferromagnetic states. This figure is reproduced from Ref. 36. (c) [2018] American Physical Society.
FIG. 2 .
2(Color online) (A) Magnetic labyrinth on a triangular lattice at low temperature (T = 0.02J), (B) spin spirals, (C) mixed skyrmion-bimeron magnetic configuration and (D) pure skyrmions on a square lattice at high temperature (T = 0.4J), and the corresponding spin structure factors. This figure is reproduced from Ref. 32. (c) [2020] National Academy of Sciences.
online) (Top) Structural complexity calculated for the classical two-dimensional triangular lattice magnetic configurations obtained with Hamiltonian Eq.(1) as a function of external magnetic field. The error bars are smaller than the symbol size. (Bottom) Complexity derivative we used for accurate detection of the phases boundaries. Squares and circles correspond to the low (T = 0.02J) and high (T = 0.4J) temperature configurations obtained with using all spin components, meanwhile triangles and diamonds represent results obtained with using only z component. All the results are obtained for |D| = J, only the interaction between the nearest neighbours is taken into account.
FIG. 4 .
4(Color online) (Top) Schematic representation of the machine learning process. Neural network with single hidden layer of sigmoid neurons performs phase classification based on z components of spins of the magnetic configuration. (Bottom) Phase triptych obtained by using the neural network with 64 hidden neurons for |D| = 0.72J. White circles denote the phase boundaries defined with the spin structure factors. This figure is reproduced from Ref. 42. (c) [2018] American Physical Society.
online) Magnetization profiles of configurations belonging to different phases obtained with the spin Hamiltonian Eq. (1) on the two-dimensional triangular lattice (top) and three-dimensional cubic lattice (bottom). This figure is reproduced from Ref. 53. (c) [2019] American Physical Society.
FIG. 6 .
6(Color online) (Top) Illustration of idea of the ultrafast skyrmionic process recognition. Magnetization dynamics is used frame by frame as an input for recurrent neural network providing the process classification. (Bottom) Obtained process diagram with the θ = 40 • magnetic pulses. Phase boundaries determined by means of RNN are indicated by brown dashed lines. This figure is reproduced from Ref. 58. (c) [2019] American Physical Society.
FIG
. 7. (Color online) (a) Calculated scalar chirality and magnetization as functions of the magnetic field. (b) Magnetization density. (c) Calculated spin structural factors. This figure is reproduced from Ref. 69. (c) [2021] American Physical Society.
FIG. 8 .FIG. 9 .
89(Color online) Schematic representation of initialization of the quantum skyrmion state on a quantum simulator(computer). The measurement basis is chosen with onequbit rotational gate U0 that is described in the text. (Color online) Schematic representation of the renormalization procedure for bitstring array. First three steps are shown. At each step the elements belonging to the same block (black rectangle) are averaged.
online) The calculated dissimilarity as a function of the magnetic field. k k FIG. 11. (Color online) Partial dissimilarity of 19-site cluster calculated in (top) σ z and (bottom) random bases for different values of magnetic field B.
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| {'fraction_non_alphanumeric': 0.05826703530252591, 'fraction_numerical': 0.0333276909477327, 'mean_word_length': 4.163639895115082, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this review we discuss the latest results concerning development of the machine learning algorithms for characterization of the magnetic skyrmions that are topologically-protected magnetic textures originated from the Dzyaloshinskii-Moriya interaction that competes Heisenberg isotropic exchange in ferromagnets. We show that for classical spin systems there is a whole pool of machine approaches allowing their accurate phase classification and quantitative description on the basis of few magnetization snapshots. In turn, investigation of the quantum skyrmions is a less explored issue, since there are fundamental limitations on the simulation of such wave functions with classical supercomputers. One needs to find the ways to imitate quantum skyrmions on near-term quantum computers. In this respect, we discuss implementation of the method for estimating structural complexity of classical objects for characterization of the quantum skyrmion state on the basis of limited number of bitstrings obtained from the projective measurements.', 'arxivid': '2304.02201', 'author': ['Vladimir V Mazurenko \nTheoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia\n', 'Ilia A Iakovlev \nTheoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia\n', 'Oleg M Sotnikov \nTheoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia\n', 'Mikhail I Katsnelson \nInstitute for Molecules and Materials\nRadboud University\nNijmegenNetherlands\n'], 'authoraffiliation': ['Theoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia', 'Theoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia', 'Theoretical Physics and Applied Mathematics Department\nUral Federal University\nMira Str. 19620002EkaterinburgRussia', 'Institute for Molecules and Materials\nRadboud University\nNijmegenNetherlands'], 'corpusid': 257952366, 'doi': '10.7566/jpsj.92.081004', 'github_urls': [], 'n_tokens_mistral': 16364, 'n_tokens_neox': 14218, 'n_words': 8453, 'pdfsha': 'b48ac4c389abc8e14dba99da00b74bc94646df38', 'pdfurls': ['https://export.arxiv.org/pdf/2304.02201v1.pdf'], 'title': ['Estimating Patterns of Classical and Quantum Skyrmion States', 'Estimating Patterns of Classical and Quantum Skyrmion States'], 'venue': []} |
arxiv |
QUALITATIVE PROPERTIES OF SOLUTIONS FOR DUAL FRACTIONAL NONLINEAR PARABOLIC EQUATIONS
18 Mar 2023
Wenxiong Chen
Lingwei Ma
QUALITATIVE PROPERTIES OF SOLUTIONS FOR DUAL FRACTIONAL NONLINEAR PARABOLIC EQUATIONS
18 Mar 2023dual nonlocal parabolic equationsfractional time-diffusionnarrow region principle in unbounded domainsaveraging effectsdirect method of moving planesmonotonicity
In this paper, we consider the dual fractional parabolic problemis the right half space. We prove that the positive solutions are strictly increasing in x 1 direction without assuming the solutions be bounded.So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂ α t and the fractional Laplacian (−∆) s . To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂ α t + (−∆) s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or bounded-ness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions.We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems.Mathematics Subject classification (2020): 35R11; 35B50; 35K58; 26A33.
INTRODUCTION
The infancy of fractional calculus dates back to a letter from L'Hôpital in 1695, in which he asked Leibniz how to define the derivative d n f (x) dx n when the order n = 1 2 is not an integer. Since then, this problem has received extensive interest from many mathematicians such as Riemann, Liouville, Riesz, Marchaud, Caputo and so on. They proposed the definitions of fractional derivatives in different forms. The accompanying surprise is that the fractional derivatives can be used Date: March 21, 2023. to model many important physical phenomena, thereby considerable attentions have been paid to the study of qualitative properties of solutions to equations involving fractional derivatives.
In this paper, we investigate the nonlocal parabolic equations involving the fractional time derivative and the fractional Laplace operator as follows
∂ α t u(x, t) + (−∆) s u(x, t) = f (u(x, t)), in R n + × R, u(x, t) = 0, in (R n \R n + ) × R,(1.1)
where R n + := {x ∈ R n | x 1 > 0} is the right half space. The space-time nonlocal equation in (1.1) can be seen as a typical model in the continuous time random walks [32], which is a generalization of the Brownian random walks formulated as the equation involving the local time derivative. The latter describes the particles experience uncorrelated random displacements at fixed time intervals. The time non-locality explains the history dependence introduced in dynamics by the presence of anomalously large waiting time, and the space non-locality accounts for the existence of anomalously large jumps, such as Lévy flights connecting distant regions in space.
The fractional time derivative ∂ α t we consider here is the Marchaud fractional derivative of order α ∈ (0, 1), defined by |u(x, τ )| 1 + |τ | 1+α dτ < +∞ for any t ∈ R}.
∂ α t u(x, t) := C αˆt −∞ u(x, t) − u(x, τ ) (t − τ ) 1+α dτ,(1.
Such fractional time derivative emerges in a variety of physical phenomena, for instance, particle systems with sticking and trapping phenomena, magneto-thermoelastic heat conduction, plasma turbulence and so on (cf. [25,26,27]). The spatial nonlocal pseudo-differential operator in (1.1), the fractional Laplacian (−∆) s is defined as (−∆) s u(x, t) := C n,s P.V.ˆR where 0 < s < 1, C n,s is a normalization positive constant and P.V. stands for the Cauchy principal value. We define L 2s (R n ) := {u(·, t) ∈ L 1 loc (R n ) |ˆR n |u(x, t)| 1 + |x| n+2s dx < +∞}, then u ∈ C 1,1 loc (R n + ) ∩ L 2s (R n ) ensures the integrability of (1.3). This fractional diffusion operator is of great interest due to its applications in physics. To name a few, the fractional Laplacian arises in anomalous diffusion, quasi-geostrophic dynamics, phase transition models, and image reconstruction problems (cf. [1,6,8,28]). Observe that the definition of the fractional time derivative ∂ α t given in (1.2) looks similar to the one-dimensional fractional Laplacian except that such integral only takes account of the interactions in the past, which is related to the fact that the transport is not time reversible and has a memory effect. It is worth mentioning that the nonlocal operator ∂ α t + (−∆) s can be reduced to the local heat operator ∂ t − ∆ as α → 1 and s → 1. During the last decade, considerable attentions have been paid to the investigation of the aforementioned space-time nonlocal equation in order to acquire a clearer understanding of various physical phenomena, and then correspondingly many subtle mathematical problems have appeared. In the context of well-posedness such as the existence, uniqueness and regularity to such nonlocal parabolic equations have been systematically studied in a series of remarkable papers [2,3,4] by Caffarelli and his group. However, so far as we know, there have not been any results on the geometry of solutions for such equations, hence our interest here is to propose a holistic approach to establish the monotonicity of solutions to problem (1.1) in a half space.
The method of moving planes introduced by Alexandroff in [29] is a common technique to study the monotonicity of solutions to local elliptic and parabolic equations. However, this approach cannot be applied directly to psuedo-differential equations involving the fractional Laplace operator due to its nonlocality. To overcome this difficulty, an important progress traces back to the pioneering work of Caffarelli and Silvestre [7], in which they developed an extension method to reduce the nonlocal problem into a local one in a higher dimensional space. Thereby the traditional method of moving planes designed for local equations can be applied for the extended problem to establish the properties of solutions. Another useful approach is to turn the given pseudo-differential equations into their equivalent integral equations, then one can use the method of moving planes in integral forms and the regularity lifting to investigate the properties of solutions (cf. [14,15,16]). These two effective methods have been employed successfully to investigate the elliptic equations involving the fractional Laplacian, and a series of interesting results have been obtained in [5,9,22,23,31,33,39] and the references therein. However, the above two methods can only be applied to equations involving the fractional Laplacian, and sometimes one needs to impose extra conditions on the problems which may not be necessary if dealing with the fractional equations directly. Nearly ten years later, a further progress was made by Chen, Li, and Li [13], who introduced a direct method of moving planes to remove the restrictions and to greatly simplify the proof process. Afterwards, such effective direct method has been widely applied to establish the symmetry, monotonicity, non-existence, and even to obtain estimates in a boundary layer of solutions for various elliptic equations and systems involving the fractional Laplacian, the fully nonlinear nonlocal operators, the fractional p-Laplacians as well as the higher order fractional operators, we refer to [10,11,12,17,19,24,34,35,36,40] and the references therein.
Very recently, many substantial advances have been made in the symmetry, monotonicity, and non-existence of positive solutions for fractional parabolic equations of type (1.1) with the usual local time derivative ∂ t u(x, t), based on the method of moving planes (cf. [18,20,21,30,37,38] and the references therein).
In contrast, the geometric behavior of solutions to nonlocal parabolic equations with simultaneous presence of the fractional time derivative and the fractional Laplacian are still lacking.
Here we present a simple example to illustrate the essential difference between the local time derivative and the fractional time derivative. Let Ω be a bounded domain in R n and [t 1 , t 2 ] be an interval in R. A typical maximum principle for a fractional parabolic problem with the local time derivative ∂ t in the parabolic cylinder Ω × (t 1 , t 2 ] can be formulated as:
If u(x, t) is a solution of ∂ t u(x, t) + (−∆) s u(x, t) ≥ 0, (x, t) ∈ Ω × (t 1 , t 2 ], u(x, t) ≥ 0, (x, t) ∈ Ω c × (t 1 , t 2 ], u(x, t 1 ) ≥ 0, x ∈ Ω, then u(x, t) ≥ 0 in Ω × (t 1 , t 2 ].
Here the initial condition is given at the time moment t = t 1 .
While the maximum principle involving the fractional time derivative ∂ α t established in the subsequent section takes the following form.
Assume that u(x, t) is a solution of
∂ α t u(x, t) + (−∆) s u(x, t) ≥ 0, (x, t) ∈ Ω × (t 1 , t 2 ], u(x, t) ≥ 0, (x, t) ∈ Ω c × (t 1 , t 2 ], u(x, t) ≥ 0, (x, t) ∈ Ω × (−∞, t 1 ], (1.4) then u(x, t) ≥ 0 in Ω × (t 1 , t 2 ].
We can see that due to the nonlocal nature of the fractional time derivative ∂ α t defined in (1.2), in order to guarantee the validity of the classical maximum principle, one must prescribe the initial condition on the whole past time before t 1 instead just on the initial time moment t 1 . If only requiring the initial condition u(x, t 1 ) ≥ 0 for (1.4), then in general the conclusion of the maximum principle may not be valid as will be illustrated by the following counterexample.
For simplicity, we consider functions of t only. Let
u(x, t) := u(t) := sin t, t ∈ (0, 2π], t, t ∈ (−R, 0], −R, t ∈ (−∞, −R].
Through a straightforward calculation, we have, for a sufficiently large R > 0,
∂ α t u(t) ≥ 0, in (0, 2π], u(0) = 0.
However this differential inequality and the initial condition at time moment t = 0 does not guarantee u(t) to be nonnegative in (0, 2π], while apparently, u(t) < 0 in (π, 2π). The main problem here lies in that u(t) < 0 for t < 0.
This example shows that the initial condition on the whole past time before t 1 is necessary to ensure the validity of the maximum principle for parabolic problems with nonlocal time derivative.
Among the literature on qualitative properties of fractional parabolic equations with local time derivative, we would like to mention [20], in which Chen and Wu considered the following
∂ t u(x, t) + (−∆) s u(x, t) = f (u(x, t)), in R n + × R, u(x, t) = 0, in (R n \R n + ) × R,(1.5)
They show that the positive solution u(x, t) is strictly increasing with respect to x 1 in R n + for any t ∈ R by using the method of moving planes.
In the first step, under a weak assumption that the antisymmetric functions is allowed to tend to infinity, they established a narrow region principle and a maximum principle for antisymmetric functions, which are essential ingredients to carry on the direct method of moving planes. However, in the second step, when they employed the limit argument, they still need to assume that the solution u(x, t) be bounded in R n + × R. Inspired by the previous literature, in this paper, we will investigate qualitative properties of solutions of dual-fractional parabolic problem (1.1). Under notably weaker conditions than in [20], that is, allowing the solutions to tend to infinity in some rate, we will prove that they are strictly increasing with respect to x 1 in R n + for any t ∈ R. To this aim, we need to introduce some new ideas and approaches to surmount the difficulties caused by both the dual non-locality in space-time and the weaker assumption on the solutions, as we will deliberate after the introduction of each theorem below.
To illustrate the main results of this paper, we start by presenting the notation that will be used in what follows. Let
T λ := {x = (x 1 , x ′ ) ∈ R n | x 1 = λ for λ ∈ R}
be the moving planes perpendicular to x 1 -axis, Σ λ := {x ∈ R n | x 1 < λ} and Ω λ := {x ∈ R n + | x 1 < λ} be the region to the left of the hyperplane T λ in R n and in R n + respectively. We denote the reflection of x with respect to the hyperplane T λ as
x λ := (2λ − x 1 , x 2 , · · · , x n ).
Let u(x, t) be a solution of (1.1) and u λ (x, t) := u(x λ , t). Define
w λ (x, t) := u λ (x, t) − u(x, t),
which represents the comparison between the values of u(x, t) and u(x λ , t). It is obvious that w λ (x, t) is an antisymmetric function of x with respect to the hyperplane T λ .
We are now in a position to state our main results of this paper. The first one is the narrow region principle for antisymmetric functions in unbounded domains.
{x ∈ Σ λ | λ − 2l < x 1 < λ} with some small l. Suppose that w(x, t) ∈ (C 1,1 loc (Ω) ∩ L 2s (R n )) × (C 1 (R) ∩ L − α (R)) is lower semi-continuous with respect to x on Ω , satisfying w(x, t) ≥ −C(1 + |x| γ ) for some 0 < γ < 2s, (1.6) and ∂ α t w(x, t) + (−∆) s w(x, t) = c(x, t)w(x, t), (x, t) ∈ Ω × R, w(x, t) ≥ 0, (x, t) ∈ (Σ λ \Ω) × R, w(x, t) = −w(x λ , t), (x, t) ∈ Σ λ × R, (1.7)
where c(x, t) is bounded from above.
Then w(x, t) ≥ 0, in Σ λ × R (1.8)
for sufficiently small l. Furthermore, if w(x, t) attains zero at some point (x 0 , t 0 ) ∈ Ω × R, then
w(x, t) ≡ 0, in R n × (−∞, t 0 ]. (1.9)
Note that the condition (1.6) allows w(x, t) to tend to negative infinity as |x| → ∞ as long as its decreasing rate is not greater than |x| γ . The previous research has shown that when studying maximum principles in unbounded domains, one usually need to impose some decay (to zero) assumption on w at infinity, or at least to assume that w is bounded from below. In this respect, Chen and Wu [20] weakened the condition to the type of (1.6) for the first time by constructing a suitable auxiliary function to establish the narrow region principle for the fractional parabolic equation involving the local time derivative ∂ t . However, the auxiliary function given in [20] has failed to work in our situation due to the presence of fractional time derivative ∂ α t . Here we introduce a new approach by applying a perturbation technique in t to construct a sequence of auxiliary functions to estimate the dual-fractional operator ∂ α t + (−∆) s along a sequence of approximate minimum points.
The aforementioned narrow region principle is a crucial ingredient to carry out the method of moving planes as it provides a starting point, then from which we will move the plane T λ along x 1 direction to the right as long as inequality (1.8) (with w(x, t) replaced by w λ (x, t)) holds. Let
λ 0 = sup{λ | w µ (x, t) ≥ 0, ∀ (x, t) ∈ Σ λ × R, µ ≤ λ}.
We will show that λ 0 = +∞. This is usually done by a contradiction argument. Suppose otherwise, then there exists λ k ց λ 0 such that inf w λ k < 0. In [20], the authors took limit along a proper subsequence of such {w λ k }, then applied a Hopf type lemma to the limiting equation to derive a contradiction. In this process, it is necessary to require the solutions be bounded.
In order to remove the boundedness assumption on solutions in [20], we introduce a new idea here. Instead of taking limit along a subsequence of {w λ k }, we apply the averaging effects (see the theorems below) twice, first on the solution u, and then on w λ 0 , to derive a contradiction for sufficiently large k. We believe that this new approach will become a very useful tool in investigating an unbounded sequence of solutions. Theorem 1.2. (Averaging effects) Let D be a domain in R n and t 0 ∈ R be a real number. For any x 0 ∈ R n , if there exists a positive radius r > 0 such that B r (x 0 ) ∩ D = ∅ as shown in Figure 1 and
u(x, t) ≥ C 0 > 0 in D × (t 0 − r 2s α , t 0 + r 2s α ]. (1.10) Assume that u(x, t) ∈ C 1,1 loc (B r (x 0 )) ∩ L 2s (R n ) × C 1 ([t 0 − r 2s α , t 0 + r 2s α ]) ∩ L − α (R) is lower semi-continuous in x on B r (x 0 ), satisfying ∂ α t u(x, t) + (−∆) s u(x, t) ≥ −ε, (x, t) ∈ B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], u(x, t) ≥ 0, (x, t) ∈ B c r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], u(x, t) ≥ 0, (x, t) ∈ B r (x 0 ) × (−∞, t 0 − r 2s α ], (1.11)
for some sufficiently small ε > 0. Then there exists a positive constant
C 1 = C 1 α, n, s, C 0 , diam(D), dist(x 0 , D) such that u(x 0 , t 0 ) ≥ C 1 > 0,
where diam(D) represents the diameter of domain D, and dist(x 0 , D) denotes the distance from point x 0 to domain D.
u C 0 > 0 D x 0 rx 0 ∈ Σ λ , if there exists a ball B r (x 0 ) ⊂ Σ λ such that B r (x 0 )∩D = ∅ and r ≤ dist(x 0 ,T λ ) 2
as shown in Figure 2, and
w(x, t) ≥ C 0 > 0 in D × (t 0 − r 2s α , t 0 + r 2s α ]. (1.12) Assume that w(x, t) ∈ C 1,1 loc (B r (x 0 )) ∩ L 2s (R n ) × C 1 ([t 0 − r 2s α , t 0 + r 2s α ]) ∩ L − α (R) is lower semi-continuous in x on B r (x 0 ), satisfying ∂ α t w(x, t) + (−∆) s w(x, t) ≥ −ε, (x, t) ∈ B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], w(x, t) ≥ 0, (x, t) ∈ (Σ λ \ B r (x 0 )) × (t 0 − r 2s α , t 0 + r 2s α ], w(x, t) ≥ 0, (x, t) ∈ B r (x 0 ) × (−∞, t 0 − r 2s α ], w(x, t) = −w(x λ , t), (x, t) ∈ Σ λ × R, (1.13)
for some sufficiently small ε > 0. Then there exists a positive constant
C 1 = C 1 α, n, s, C 0 , diam(D), dist(x 0 , D), dist(∂D, T λ ), dist(x 0 , T λ ) such that w(x 0 , t 0 ) ≥ C 1 > 0,
where dist(∂D, T λ ) stands for the distance between the boundary ∂D and the hyperplane T λ .
w C 0 > 0 Σ λ D x 0 r x 1 T λ Figure 2.
The positional relationship between the region D and the ball B r (x 0 ) in Σ λ .
Remark 1.1. The so-called averaging effects reveal that the positiveness of the solution to the fractional time-diffusion equations in some region D will be diffused to any other region B disjointed from D. The averaging effects depend on the distance between such two regions, the closer the distance, the greater the effect.
Based on the vital techniques above, we establish the direct method of moving planes suitable for the nonlocal parabolic equation (1.1) to derive the following monotonicity result.
Theorem 1.4. (Monotonicity in a half space) Let u(x, t) ∈ C 1,1 loc (R n + ) ∩ L 2s (R n ) × C 1 (R) ∩ L − α (R)
be a positive solution of (1.1). Assume that u(x, t) is uniformly continuous with respect to x, satisfying u(x, t) ≤ C(1 + |x| γ ) for some 0 < γ < 2s.
If the nonlinear term f ∈ C 1 ([0, +∞)) satisfies f (0) ≥ 0, f ′ (0) ≤ 0 and f ′ is bounded from above, then the solution u(x, t) is strictly increasing with respect to x 1 in R n + for any t ∈ R.
Remark 1.2. To our knowledge, Theorem 1.4 is the first result about the geometric shapes of solutions to the nonlocal parabolic equations involving the fractional time derivative ∂ α t and the fractional Laplacian (−∆) s . Particularly, the holistic approach presented here is also valid for parabolic equations of type (1.1) with the usual local time derivative ∂ t . It is worth mentioning that our results are still novel even if ∂ α t is replaced by ∂ t , since we remove the assumption that u is uniformly bounded in x as compared to the result in [20] .
The remainder part of this paper proceeds as follows. Section 2 is composed by the proofs of various maximum principles used for this study, including the aforementioned narrow region principle. Section 3 is devoted to establishing the averaging effects applicable to the dual fractional operator ∂ α t + (−∆) s . In section 4 , we complete the proof of Theorem 1.4 . The last section presents some useful lemmas.
MAXIMUM PRINCIPLES
Without assuming any decay and boundedness conditions on the antisymmetric function w(x, t) with respect to the space variable x, this section begins by establishing the narrow region principle ( Theorem 1.1 ) in unbounded domains. From now on, C denotes a constant whose value may be different from line to line, and only the relevant dependence is specified in what follows.
Proof of Theorem 1.1 .
We argue by contradiction to derive (1.8).
Note that condition (1.6) allows w(x, t) to go to negative infinity as |x| → ∞; hence, the minimizing sequence of w(x, t) in x may leak to infinity and the minimum may not be attained. To overcome this impediment, we employ the following positive auxiliary function
h(x) := 1 − (x 1 − (λ − l)) 2 l 2 s + + 1 (1 + |x ′ | 2 ) β 2 for some γ < β < 2s to ensure that lim |x|→∞w (x, t) := lim |x|→∞ w(x, t) h(x) ≥ 0. (2.1) It follows that for each fixed t ∈ R, if there is a point x ∈ Ω withw(x, t) < 0, then there must exists x(t) ∈ Ω such thatw (x(t), t) = min x∈Ωw (x, t) < 0. (2.2)
Combining (1.6) with the definition ofw(x, t) and γ < β, one deduces thatw(x(t), t) is bounded. Hence, if (1.8) is not valid, then there exists a positive constant m such that
inf Ω×Rw (x, t) = inf Rw (x(t), t) =: −m < 0. (2.3) This implies that there exists a sequence {(x k , t k )} ⊂ Ω × R such that w(x k , t k ) = −m k → −m as k → ∞.
Let ε k := m − m k , then it is obvious that ε k ≥ 0 and ε k → 0 as k → 0. We further need to introduce the following auxiliary function
v k (x, t) :=w(x, t) − ε k η k (t)
to remedy scenario that the infimum ofw(x(t), t) may not be attained due to t ∈ R, where
η k (t) = η(t − t k ) ∈ C ∞ 0 ((−2 + t k , 2 + t k )) is a smooth cut-off function satisfying η k (t) ≡ 1 in [−1 + t k , 1 + t k ], and 0 ≤ η k (t) ≤ 1. One one hand, we have v k (x k , t k ) =w(x k , t k ) − ε k = −m k − m + m k = −m.
On the other hand, if |t − t k | ≥ 2 and x ∈ Ω, then it follows from (
2.3) that v k (x, t) =w(x, t) ≥ −m. Thus, there exists (x k ,t k ) ∈ Ω × (−2 + t k , 2 + t k ) such that − m − ε k ≤ v k (x k ,t k ) = inf Ω×R v k (x, t) ≤ −m. (2.4) From this, it is not difficult to see that − m ≤w(x k ,t k ) ≤ −m + ε k = −m k < 0. (2.5)
Next, by a direct calculation, we obtain
∂ α t v k (x k ,t k ) = C αˆt k −∞ v k (x k ,t k ) − v k (x k , τ ) (t k − τ ) 1+α dτ ≤ 0.
Then in terms of the definition of v k (x, t) and Lemma 5.1 , we derive
∂ α tw (x k ,t k ) ≤ ε k ∂ α t η k (t k ) ≤ Cε k . (2.6)
To proceed, a combination of the definition ofw(x, t), (1.7), (2.2), (2.5) with |x k − y| < |x k − y λ | and h(y) > h(y λ ) for y ∈ Σ λ yields that
(−∆) s w(x k ,t k ) = (−∆) s w(x k ,t k )h(x k ) =w(x k ,t k )(−∆) s h(x k ) + C n,s P.V.ˆR n h(y) w(x k ,t k ) −w(y,t k ) |x k − y| n+2s dy ≤w(x k ,t k )(−∆) s h(x k ) + C n,sˆΣ λ h(y)w(x k ,t k ) − w(y,t k ) |x k − y λ | n+2s dy +C n,sˆΣ λ h(y λ )w(x k ,t k ) + w(y,t k ) |x k − y λ | n+2s dy ≤w(x k ,t k )(−∆) s h(x k ) + C n,sˆΣ λ 2h(y λ )w(x k ,t k ) |x k − y λ | n+2s dy ≤w(x k ,t k )(−∆) s h(x k ) ≤ C 1 l 2s h(x k )w(x k ,t k ),(2.7)
where on the last line we used the fact established in [20, Lemma 2.1] that there exits a positive constant C 1 such that
(−∆) s h(x) h(x) ≥ C 1 l 2s for all λ − 2l < x 1 < λ with sufficiently small l.
Substituting (2.6) and (2.7) into (1.7), and combining (2.5) with the bounded-ness of c(x, t) from above, we obtain
− C 1 l 2s m k h(x k ) ≥ C 1 l 2s h(x k )w(x k ,t k ) ≥ (−∆) s w(x k ,t k ) = −∂ α t w(x k ,t k ) + c(x k ,t k )w(x k ,t k ) = −h(x k )∂ α tw (x k ,t k ) + c(x k ,t k )h(x k )w(x k ,t k ) ≥ −Cε k h(x k ) − Cmh(x k ).
Now dividing both side of the preceding inequality by −mh(x k ), we deduce that
C 1 l 2s ← C 1 l 2s m k m ≤ Cε k m + C → C,
as k → ∞. Therefore, we derive a contradiction for sufficiently small l, which concludes that (1.8) is true. Finally, we prove (1.9). It follows from (1.8) that
w(x 0 , t 0 ) = min Σ λ ×R w(x, t) = 0. If w(x, t 0 ) ≡ 0 in Σ λ , on one hand, we compute ∂ α t w(x 0 , t 0 ) = −C αˆt 0 −∞ w(x 0 , τ ) (t 0 − τ ) 1+α dτ ≤ 0 and (−∆) s w(x 0 , t 0 ) = C n,s P.V.ˆΣ λ w(y, t 0 ) 1 |x 0 − y λ | n+2s − 1 |x 0 − y| n+2s dy < 0, then ∂ α t w(x 0 , t 0 ) + (−∆) s w(x 0 , t 0 ) < 0.
On the other hand, by virtue of the equation in (1.7), we have
∂ α t w(x 0 , t 0 ) + (−∆) s w(x 0 , t 0 ) = 0. (2.8)
This contradiction implies that w(x, t 0 ) ≡ 0 in Σ λ . In addition, the antisymmetry of w(x, t) with respect to x infers that w(x, t 0 ) ≡ 0 in R n .
Thereby for any fixed x ∈ Σ λ such that w(x, t) ≡ 0 in (−∞, t 0 ), by estimating similarly as above, we have
∂ α t w(x, t 0 ) = −C αˆt 0 −∞ w(x, τ ) (t 0 − τ ) 1+α dτ < 0 and (−∆) s w(x, t 0 ) = 0,
which also means that
∂ α t w(x, t 0 ) + (−∆) s w(x, t 0 ) < 0. It contradicts the equation (2.8), then w(x, t) ≡ 0 in Σ λ × (−∞, t 0 ]
. Applying the antisymmetry of w(x, t) with respect to x again, we deduce that
w(x, t) ≡ 0 in R n × (−∞, t 0 ].
Therefore, the proof of Theorem 1.1 is completed.
In fact, if the region Ω given in Theorem 1.1 is not narrow, then it suffices to assume that the positive part of coefficient c(x, t) is small to guarantee the validity of the similar maximum principle, as pointed out in the following result.
w(x, t) ∈ (C 1,1 loc (Ω) ∩ L 2s (R n )) × (C 1 (R) ∩ L − α (R))
is lower semi-continuous for x on Ω, satisfying
w(x, t) ≥ −C(1 + |x| γ ) for some 0 < γ < 2s,(2.9)
and
∂ α t w(x, t) + (−∆) s w(x, t) = c(x, t)w(x, t), (x, t) ∈ Ω × R, w(x, t) ≥ 0, (x, t) ∈ (Σ λ \Ω) × R, w(x, t) = −w(x λ , t), (x, t) ∈ Σ λ × R,(2.
10)
where c(x, t) ≤ c 0 in Ω × R for some small positive constant c 0 . Then there holds that
w(x, t) ≥ 0 in Σ λ × R. (2.11) Furthermore, if w(x 0 , t 0 ) = 0 for some point (x 0 , t 0 ) ∈ Ω × R, then w(x, t) ≡ 0 in R n × (−∞, t 0 ].
Proof. Since the width of the domain Ω in x 1 direction is finite, then it may well be assumed that Ω is contained in {x ∈ Σ λ | λ − 2a < x 1 < λ} for some a > 0. Thereby all proofs follow almost verbatim from the proof of Theorem 1.1 , the essential difference is that we choose the auxiliary function
h(x) := 1 − (x 1 − (λ − a)) 2 a 2 s + + 1 (1 + |bx ′ | 2 ) β 2
for some γ < β < 2s, where the sufficiently small positive constant b depends on a such that
(−∆) s h(x) h(x) ≥ C 1 a 2s
for any x ∈ Ω and some constant C 1 > 0. In analogy with the notation and calculations in the proof of Theorem 1.1 , if (2.11) is false, then we can finally deduce that
C 1 a 2s ← C 1 a 2s m k m ≤ Cε k m + c 0 → c 0 ,
as k → ∞. It suffices to select the upper bound c 0 of the coefficient c(x, t) to satisfy c 0 < C 1 a 2s , then we derive a contradiction. Hence, (2.11) is valid. Furthermore, by proceeding similarly as the proof of Theorem 1.1 , we can verify that the strong maximum principle holds. This completes the proof of Theorem 2.1.
In the sequel, we will also use the following two simple maximum principles in bounded domains, where the minimum can be attained as compared to unbounded domains.
Theorem 2.2. (Maximum principle in bounded domains)
Let Ω be a bounded domain in R n and t 1 < t 2 be two real numbers. Suppose that
u(x, t) ∈ (C 1,1 loc (Ω) ∩ L 2s (R n )) × (C 1 ([t 1 , t 2 ]) ∩ L − α (R)) is lower semi-continuous in x on Ω, satisfying ∂ α t u(x, t) + (−∆) s u(x, t) ≥ 0, (x, t) ∈ Ω × (t 1 , t 2 ], u(x, t) ≥ 0, (x, t) ∈ Ω c × (t 1 , t 2 ], u(x, t) ≥ 0, (x, t) ∈ Ω × (−∞, t 1 ], (2.12) then u(x, t) ≥ 0 in Ω × (t 1 , t 2 ].
Proof. The proof goes by contradiction. If the conclusion is not valid, then there exists (x 0 , t 0 ) ∈ Ω × (t 1 , t 2 ] such that u(x 0 , t 0 ) = min
Ω×(t 1 ,t 2 ]
u(x, t) < 0.
Combining the exterior condition with the initial condition in (2.12), we directly calculate
∂ α t u(x 0 , t 0 ) + (−∆) s u(x 0 , t 0 ) = C αˆt 0 −∞ u(x 0 , t 0 ) − u(x 0 , τ ) (t 0 − τ ) 1+α dτ + C n,s P.V.ˆR n u(x 0 , t 0 ) − u(y, t 0 ) |x 0 − y| n+2s dy = C αˆt 1 −∞ u(x 0 , t 0 ) − u(x 0 , τ ) (t 0 − τ ) 1+α dτ + C αˆt 0 t 1 u(x 0 , t 0 ) − u(x 0 , τ ) (t 0 − τ ) 1+α dτ +C n,s P.V.ˆΩ u(x 0 , t 0 ) − u(y, t 0 ) |x 0 − y| n+2s dy + C n,sˆΩ c u(x 0 , t 0 ) − u(y, t 0 ) |x 0 − y| n+2s dy < 0,
which contradicts the differential inequality in (2.12). Hence, we verify Theorem 2.2.
Theorem 2.3. (Maximum principle for antisymmetric functions in bounded domains)
Let Ω ⊂ Σ λ be a bounded domain and [t 1 , t 2 ] ⊂ R be a finite interval. Suppose that
w(x, t) ∈ (C 1,1 loc (Ω) ∩ L 2s (R n )) × (C 1 ([t 1 , t 2 ]) ∩ L − α (R)) is lower semi-continuous in x on Ω, satisfying ∂ α t w(x, t) + (−∆) s w(x, t) ≥ 0, (x, t) ∈ Ω × (t 1 , t 2 ], w(x, t) ≥ 0, (x, t) ∈ (Σ λ \ Ω) × (t 1 , t 2 ], w(x, t) ≥ 0, (x, t) ∈ Ω × (−∞, t 1 ], w(x, t) = −w(x λ , t), (x, t) ∈ Σ λ × R, (2.13) then w(x, t) ≥ 0 in Ω × (t 1 , t 2 ].
Proof. We argue by contradiction. If the conclusion is violated, then there exists (x 0 , t 0 ) ∈ Ω × (t 1 , t 2 ] such that w(x 0 , t 0 ) = min
Σ λ ×(t 1 ,t 2 ]
w(x, t) < 0. Now a combination of the exterior condition, the initial condition with the antisymmetry of w(x, t) in x yields that
∂ α t w(x 0 , t 0 ) + (−∆) s w(x 0 , t 0 ) = C αˆt 0 −∞ w(x 0 , t 0 ) − w(x 0 , τ ) (t 0 − τ ) 1+α dτ + C n,s P.V.ˆR n w(x 0 , t 0 ) − w(y, t 0 ) |x 0 − y| n+2s dy = C αˆt 1 −∞ w(x 0 , t 0 ) − w(x 0 , τ ) (t 0 − τ ) 1+α dτ + C αˆt 0 t 1 w(x 0 , t 0 ) − w(x 0 , τ ) (t 0 − τ ) 1+α dτ +C n,s P.V.ˆΣ λ w(x 0 , t 0 ) − w(y, t 0 ) |x 0 − y| n+2s dy + C n,sˆΣ c λ w(x 0 , t 0 ) − w(y, t 0 ) |x 0 − y| n+2s dy < C n,sˆΣ λ w(x 0 , t 0 ) − w(y, t 0 ) |x 0 − y λ | n+2s dy + C n,sˆΣ λ w(x 0 , t 0 ) + w(y, t 0 ) |x 0 − y λ | n+2s dy = 2C n,s w(x 0 , t 0 )ˆΣ λ 1 |x 0 − y λ | n+2s dy < 0,
which is guaranteed by the radial decrease of the kernel resulting from |x 0 − y| < |x 0 − y λ | for y ∈ Σ λ . Obviously, the aforementioned estimate contradicts the differential inequality in (2.13). This completes the proof of Theorem 2.3.
AVERAGING EFFECTS
In this section, we develop two averaging effects ( Theorem 1.2 and Theorem 1.3 ) of the dual nonlocal operator ∂ α t + (−∆) s , which will play crucial roles in the proof of our main results and will become powerful tools in studying fractional problems, particularly in the cases where the solutions are unbounded. We start by presenting the proof of Theorem 1.2 .
Proof of Theorem 1.2 . The lower bound estimate will be obtained by constructing a sub-solution. Denote
ψ(x, t) := φ(x)η(t) := C 1 − x − x 0 r 2 s + η(t),
where η(t) is a smooth cut-off function whose compact support is contained in
(t 0 − r 2s α , t 0 + r 2s α ), satisfying 0 ≤ η(t) ≤ 1, and η(t) ≡ 1 in [t 0 − r 2s α 2 , t 0 + r 2s α 2 ].
In fact, it is well known that
(−∆) s φ(x) = 1 r 2s in B r (x 0 ) (3.1)
for a proper choice of positive constant C, and φ(x) ≡ 0 in B c r (x 0 ). Let
u(x, t) := u(x, t)χ D (x) + δψ(x, t),
where δ is a positive constant to be determined later, and χ D is the characteristic function on D, more precisely, χ D (x) ≡ 1 for x ∈ D and χ D (x) ≡ 0 for x / ∈ D. Next, we attempt to claim that u(x, t) is a sub-solution of u(
x, t) in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ]. For (x, t) ∈ B r (x 0 ) × (t 0 − r∂ α t (u(x, t) − u(x, t)) + (−∆) s (u(x, t) − u(x, t)) ≥ −ε − δφ(x)∂ α t η(t) − (−∆) s (u(x, t)χ D (x)) − δη(t)(−∆) s φ(x) ≥ −ε + C n,s C 0ˆD 1 |x − y| n+2s dy − Cδ r 2s ≥ −ε + C 2 − Cδ r 2s .
Now choosing ǫ = C 2 2 and δ = C 2 r 2s 2C , we deduce that
∂ α t (u(x, t) − u(x, t)) + (−∆) s (u(x, t) − u(x, t)) ≥ 0, (x, t) ∈ B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ].
For
(x, t) ∈ B c r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], we have u(x, t) − u(x, t) = u(x, t) − u(x, t)χ D (x) ≥ 0,
which is ensured by the definition of ψ(x, t) and the exterior condition in (1.11).
While if (x, t) ∈ B r (x 0 ) × (−∞, t 0 − r 2s
α ], then we apply the initial condition in (1.11) to derive
u(x, t) − u(x, t) = u(x, t) ≥ 0.
In summary, we have obtained
∂ α t (u(x, t) − u(x, t)) + (−∆) s (u(x, t) − u(x, t)) ≥ 0, in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], u(x, t) − u(x, t) ≥ 0, in B c r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], u(x, t) − u(x, t) ≥ 0, in B r (x 0 ) × (−∞, t 0 − r 2s α ].
Then the maximum principle established in Theorem 2.2 implies that
u(x, t) ≥ u(x, t) for (x, t) ∈ B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ].
Hence, we verify that u( Proof of Theorem 1.3 . The primary objective of this proof is to construct an antisymmetric subsolution of w(x, t). Let
x, t) is a sub-solution of u(x, t) in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ). Finally, it follows that u(x 0 , t 0 ) ≥ u(x 0 , t 0 ) = δφ(x 0 )η(t 0 ) = Cδ =: C 1 > 0.φ(x) := 1 − x − x 0 r 2 s + and φ λ (x) := 1 − x λ − x 0 r 2 s + , then it is obvious that Φ(x) := φ(x) − φ λ (x)
is an antisymmetric function with respect to the plane T λ . We denote
η(t) ∈ C ∞ 0 ((t 0 − r 2s α , t 0 + r 2s α ))
which is a smooth cut-off function whose value belongs to [0, 1], satisfying
η(t) ≡ 1 in [t 0 − r 2s α 2 , t 0 + r 2s α 2 ].
Let D λ be the reflection domain of D with respect to the plane T λ as illustrated in Figure 3 below, and
w(x, t) = w(x, t)χ D∪D λ (x) + δΦ(x)η(t),
where δ is a positive constant to be determined later, and χ D∪D λ (x) is the characteristic function in the region D ∪ D λ . In the sequel, we aim to show that the antisymmetric function w( , we further select ǫ = C 2 2 and δ = C 2 r 2s 2C to derive
w C 0 > 0 Σ λ D D λ x 0 (x 0 ) λ r r x 1 T λx, t) is a sub-solution of w(x, t) in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ]. For (x, t) ∈ B r (x 0 ) × (t 0 − r∂ α t (w(x, t) − w(x, t)) + (−∆) s (w(x, t) − w(x, t)) ≥ −ε − δΦ(x)∂ α t η(t) − (−∆) s (w(x, t)χ D∪D λ (x)) − δη(t)(−∆) s (φ(x) − φ λ (x)) ≥ −ε + C n,sˆD ∪D λ w(y, t) |x − y| n+2s dy − Cδ r 2s − C n,s δˆB r ((x 0 ) λ ) 1 |x − y| n+2s dy ≥ −ε + C n,s C 0ˆD 1 |x − y| n+2s − 1 |x − y λ | n+2s dy − Cδ r 2s ≥ −ε + C 2 − Cδ r 2s ≥ 0.
For (x, t) ∈ (Σ λ \ B r (x 0 )) × (t 0 − r 2s α , t 0 + r 2s α ], it follows from the exterior condition in (1.13) that
w(x, t) − w(x, t) = w(x, t) − w(x, t)χ D∪D λ (x) ≥ 0. Finally, taking into account (x, t) ∈ B r (x 0 ) × (−∞, t 0 − r 2s
α ], we apply the initial condition in (1.13) to derive
w(x, t) − w(x, t) = w(x, t) ≥ 0.
Through the above arguments, we have deduced that
∂ α t (w(x, t) − w(x, t)) + (−∆) s (w(x, t) − w(x, t)) ≥ 0, in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ], w(x, t) − w(x, t) ≥ 0, in (Σ λ \ B r (x 0 )) × (t 0 − r 2s α , t 0 + r 2s α ], w(x, t) − w(x, t) ≥ 0, in B r (x 0 ) × (−∞, t 0 − r 2s α ], w(x, t) − w(x, t) = − w(x λ , t) − w(x λ , t) , in Σ λ × R.
Then by virtue of the maximum principle for antisymmetric functions established in Theorem 2.3 , we obtain
w(x, t) ≥ w(x, t) in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ]. Hence, we conclude that w(x, t) is a sub-solution of w(x, t) in B r (x 0 ) × (t 0 − r 2s α , t 0 + r 2s α ]. As a consequence, we have w(x 0 , t 0 ) ≥ w(x 0 , t 0 ) = δφ(x 0 )η(t 0 ) = δ =: C 1 > 0.
This completes the proof of Theorem 1.3.
MONOTONICITY IN A HALF SPACE
Based on the maximum principles and the averaging effects introduced in the previous section, we establish the monotonicity of positive solutions for nonlocal parabolic problem (1.1) in a half space by virtue of the direct method of moving planes. For readers' convenience, we first provide a sketch of proof.
We proceed in three steps.
In the first step, we argue that for λ > 0 sufficiently close to 0, i.e. when Ω λ is a narrow region, it holds w λ (x, t) ≥ 0, in Ω λ × R. This provides a starting point for moving the plane T λ to the right along the x 1 -axis.
In the next step, we move the plane T λ continuously to the right along the x 1 -axis as long as inequality (4.1) is valid to its limiting position. We want to show that the plane can be moved all the way to positive infinity. Otherwise, we will apply the maximum principle for antisymmetric functions established in Theorem 2.1 and subtly employ the averaging effects acquired in Section 3 twice to derive a contradiction along a sequence of approximate minimum points.
In the final step, we verify that u(x, t) is strictly increasing with respect to x 1 in R n + for any t ∈ R. To this end, it suffice to derive a strong maximum principle based on (4.1) for the dual nonlocal operator ∂ α t + (−∆) s to arrive at w λ (x, t) > 0, in Ω λ × R for any λ > 0.
Now we show the details.
Proof of Theorem 1.4 . By a direct calculation, we have
∂ α t w λ (x, t) + (−∆) s w λ (x, t) = C λ (x, t)w λ (x, t), (x, t) ∈ Ω λ × R, w λ (x, t) ≥ 0, (x, t) ∈ (Σ λ \Ω λ ) × R, w λ (x, t) = −w λ (x λ , t), (x, t) ∈ Σ λ × R,(4.2)
where the coefficient function
C λ (x, t) =ˆ1 0 f ′ (ςu λ (x, t) + (1 − ς)u(x, t)) dς ≤ C,
which is ensured by f ′ is bounded from above. In order to prove the strict monotonicity of u(x, t) with respect to x 1 , it suffices to show that the antisymmetric function w λ (x, t) > 0 in Ω λ × R for any λ > 0. Now we divide the proof into three steps.
Step 1. Start moving the plane T λ from x 1 = 0 to the right along the x 1 -axis. If the positive λ is small enough, then the assumptions in Theorem 1.4 guarantee that we can apply Theorem 1.1 to the problem (4.2) to derive
w λ (x, t) ≥ 0 in Ω λ × R, (4.3)
where Ω λ is a narrow region for sufficiently small λ > 0. Inequality (4.3) provides a starting point to move the plane T λ .
Step 2. In this step, we continue to move the plane T λ to the right along the x 1 -axis as long as (4.3) is valid to its limiting position. Let
λ 0 := sup {λ | w µ (x, t) ≥ 0, (x, t) ∈ Σ µ × R for any µ ≤ λ} ,
We will show that λ 0 = +∞. Otherwise, if 0 < λ 0 < +∞, then by its definition, there exists a sequence of λ k > λ 0 such that λ k → λ 0 as k → ∞, along which
Σ − λ k × R := {(x, t) ∈ Σ λ k × R | w λ k (x, t) < 0} is nonempty and inf Σ λ k ×R w λ k (x, t) < 0. We first show that inf Σ λ k ×R w λ k (x, t) → 0 as k → ∞.
(4.5)
If not, then there exists a uniformly positive constant M such that
inf Σ λ k ×R w λ k (x, t) < −M < 0. From this, there exists a sequence {(x k , t k )} ⊂ Σ λ k × R such that w λ k (x k , t k ) ≤ −M < 0. (4.6)
If x k is between T λ 0 and T λ k , then by virtue of λ k → λ 0 as k → ∞, we have |x k − (x k ) λ k | → 0 as k → ∞. By u(x, t) is uniformly continuous with respect to x, we derive
w λ k (x k , t k ) = u((x k ) λ k , t k ) − u(x k , t k ) → 0 as k → ∞,
which contradicts with (4.6). In the other case, if x k ∈ Ω λ 0 , then combining the uniform continuity of u(x, t) in x with λ k → λ 0 as k → ∞ again, we have
w λ k (x k , t k ) − w λ 0 (x k , t k ) = u((x k ) λ k , t k ) − u((x k ) λ 0 , t k ) → 0 as k → ∞.
While in terms of (4.6) and w λ 0 (x k , t k ) ≥ 0, we must derive a contradiction that
w λ k (x k , t k ) − w λ 0 (x k , t k ) ≤ −M < 0.
Hence, we deduce that inf
Σ λ k ×R w λ k (x, t) =: −m k → 0 as k → ∞. (4.7)
In the sequel, we denote the sequence
q k := sup Σ − λ k ×R C λ k (x, t),
there are the following two possibilities. Case 1. If q k ≤ ε k → 0 as k → ∞, then it infers that C λ k (x, t) ≤ C in Σ − λ k ×R for sufficiently large k and for any given small positive constant C. Hence, by virtue of Theorem 2.1 to the problem (4.2) with λ = λ k , we conclude that
w λ k (x, t) ≥ 0 in Σ λ k × R
for sufficiently large k, which is a contradiction with the definition of λ k . Case 2. If positive sequence q k 0 as k → ∞, then there exist a positive constant δ 0 and a subsequence of {q k } (still denoted by {q k }) such that q k ≥ δ 0 > 0. In this case, by using the condition f ′ (0) ≤ 0 and (4.7), we can deduce that there exist a positive constant ε 0 and a sequence
{(x k , t k )} ⊂ Σ − λ k × R such that u(x k , t k ) ≥ ε 0 > 0 and w λ k (x k , t k ) = −m k + m 2 k < 0.
Then a combination of u(x, t) = 0 in (R n \ R n + ) × R and the continuity of u(x, t) yields that there exists a small radius r 0 > 0 independent of k, such that
u(x, t) ≥ ε 0 2 > 0 in B r 0 (x k ) × (t k − r 2s α 0 , t k + r 2s α 0 ] ⊂ R n + × R. (4.8)
Now we claim that δ k := dist(x k , T λ k ) = λ k − x k 1 is bounded away from zero for sufficiently large k. If not, then
δ k → 0 as k → ∞. Let v k (x, t) := w λ k (x, t) − m 2 k η k (x, t),
where the sequence of smooth cut-off functions
η k (x, t) := η( x − x k δ k , t − t k δ 2s α k ) ∈ C ∞ 0 B δ k (x k ) × (t k − δ 2s α k , t k + δ 2s α k ) satisfies 0 ≤ η k ≤ 1, and η k (x, t) ≡ 1 in B δ k 2 (x k ) × (t k − δ 2s α k 2 , t k + δ 2s α k 2 ).
We denote the parabolic cylinder
Q δ k (x k , t k ) := B δ k (x k ) × (t k − δ 2s α k , t k + δ 2s α k ), thereby a direct calculation infers that v k (x k , t k ) = w λ k (x k , t k ) − m 2 k η k (x k , t k ) = −m k + m 2 k − m 2 k = −m k , and v k (x, t) = w λ k (x, t) ≥ −m k for (x, t) ∈ Q c δ k (x k , t k ) ∩ (Σ λ k × R). Hence, there exists a point (x k ,t k ) ∈ Q δ k (x k , t k ) such that −m k − m 2 k ≤ v k (x k ,t k ) = inf Σ λ k ×R v k (x, t) ≤ −m k . Moreover, it follows from the definition of v k that −m k ≤ w λ k (x k ,t k ) ≤ −m k + m 2 k < 0. On such minimum point (x k ,t k ) of v k (x, t), we estimate ∂ α t v k (x k ,t k ) = C αˆt k −∞ v k (x k ,t k ) − v k (x k , τ ) (t k − τ ) 1+α dτ ≤ 0, and (−∆) s v k (x k ,t k ) = C n,s P.V.ˆΣ λ k v k (x k ,t k ) − v k (y,t k ) |x k − y| n+2s dy + C n,sˆΣ c λ k v k (x k ,t k ) − v k (y,t k ) |x k − y| n+2s dy ≤ C n,sˆΣ λ k 2v k (x k ,t k ) + m 2 k η k (y,t k ) |x k − y λ k | n+2s dy ≤ Cv k (x k ,t k ) δ 2s k + Cm 2 k δ 2s k ≤ − Cm k δ 2s k + Cm 2 k δ 2s k .
While from the equation in (4.2) and Corollary 5.2 , we derive
∂ α t v k (x k ,t k ) + (−∆) s v k (x k ,t k ) = ∂ α t w λ k (x k ,t k ) + (−∆) s w λ k (x k ,t k ) − m 2 k ∂ α t η k (x k ,t k ) − m 2 k (−∆) s η k (x k ,t k ) ≥ C λ k (x k ,t k )w λ k (x k ,t k ) − Cm 2 k δ 2s k ≥ −Cm k − Cm 2 k δ 2s k ,
where the last line is ensured by f ′ with an upper bound. Then it follows from the above estimates that
− Cm k δ 2s k ≥ −Cm k − Cm 2 k δ 2s k .
Multiplying both sides by δ 2s k −m k and combining (4.7) with the assumption lim k→∞ δ k = 0, we obtain
0 < C ≤ Cδ 2s k + Cm k → 0 as k → ∞.
This contradiction indicates that δ k is bounded away from zero for sufficiently large k. Furthermore, since λ k → λ 0 as k → ∞, then we deduce that there exists a subsequence of
{x k , t k } (still denoted by {x k , t k }) such that {(x k , t k )} ⊂ Σ λ 0 × R and dist{x k , T λ 0 } ≥ δ 0 > 0.
As a consequence of (4.8), we further select a radius r 1 := min{r 0 , δ 0 } such that
u(x, t) ≥ ε 0 2 > 0 in B r 1 (x k ) × (t k − r 2s α 1 , t k + r 2s α 1 ] ⊂ Ω λ 0 × R. (4.9)
Before continuing, we illustrate the ideas of the proof based on Figure 4 below for ease of understanding. Our purpose is to show that w λ k (x k , t k ) > 0, which will contradict (x k , t k ) ∈ Σ − λ k × R. With this aim in mind, combining (4.9) with the averaging effects established in Theorem 1.2 , we first claim that the solution u(x, t) has a positive lower bound in
B r 1 (x k ) × (t k − r 2s α 1 , t k + r 2s α 1 ].
We next apply the exterior condition in (1.1) to derive that u(x, t) has a smaller upper bound in
B r 2 (x k ) ∩ R n + × (t k − r 2s α 1 , t k + r 2s α 1 ]
. It follows that the antisymmetric function w λ 0 (x, t) is positively bounded away from zero in B r 2 (x k ) ∩ R n + × (t k − r 2s α 1 , t k + r 2s α 1 ]. We further use the averaging effects demonstrated in Theorem 1.3 to deduce that
w λ 0 (x, t) ≥ ε 2 > 0 in B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s
α . Finally, a combination of the continuity of w λ (x, t) with respect to λ and λ k → λ 0 as k → ∞ yields that w λ k (x k , t k ) ≥ ε 2 2 > 0 for sufficiently large k.
x ′ T 0 We now carry out the details of the proof.
x 1 T λ0 T λ k T 2λ0 x k r1x k r1 x k r2Letx k = (2λ 0 , (x k ) ′ ), since dist(T λ 0 , T 2λ 0 ) = λ 0 > 2r 1 , then we have B r 1 (x k )∩B 2r 1 (x k ) = ∅.
Here the main purpose is to claim that there exists a positive constant ε 1 = ε 1 (α, n, s, ε 0 , λ 0 , r 1 ) such that
u(x, t) ≥ ε 1 > 0 in B r 1 (x k ) × (t k − r 2s α 1 , t k + r 2s α 1 ].
(4.10)
If not, that is to say
u(x, t) < ε in B r 1 (x k ) × (t k − r 2s α 1 , t k + r 2s α 1 ]
for any ε > 0, (4.11) then applying the condition that f is a C 1 function, we have
|f (u(x, t)) − f (0)| ≤ C|(u(x, t)) − 0| < Cε in B r 1 (x k ) × (t k − r 2s α 1 , t k + r 2s α 1 ].
Furthermore, it follows from the assumption f (0) ≥ 0 that
f (u(x, t)) > −Cε in B r 1 (x k ) × (t k − r 2s α 1 , t k + r 2s α 1 ] for any ε > 0.
Then by virtue of (1.1), (4.9) and the averaging effects for the nonlocal operators established in Theorem 1.2 , and combining with the continuity of u(x, t), we derive
u(x, t) ≥ ε 1 > 0 in B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s α ,
where the positive constant ε 1 = ε 1 (α, n, s, ε 0 , λ 0 , r 1 ). It apparently contradicts (4.11), which verifies (4.10). Next, letx k = (0, (x k ) ′ ), using the exterior condition u(x, t) ≡ 0 in (R n \ R n + ) × R and the continuity of u(x, t) again, we deduce that there exists some positive small radius r 2 < r 1 2 independent of k, such that
u(x, t) ≤ ε 1 2 in B r 2 (x k ) ∩ R n + × (t k − r 2s α 1 , t k + r 2s α 1 ],(4.12)
where ε 1 is given in (4.10). Note that for any point x ∈ B r 2 (x k ) ∩ R n + , its reflection point x λ 0 with respect to the plane T λ 0 belongs to B r 2 (x k ) ∩ Σ 2λ 0 ⊂ B r 1 2 (x k ) by r 2 < r 1 2 . Then a combination of (4.10) and (4.12) yields that
w λ 0 (x, t) = u(x λ 0 , t) − u(x, t) ≥ ε 1 − ε 1 2 = ε 1 2 > 0 in B r 2 (x k ) ∩ R n + × (t k − r 2s α 1 , t k + r 2s α 1 ] . (4.13)
The ultimate aim is to demonstrate that there exists a positive constant ε 2 = ε 2 (α, n, s, ε 0 , λ 0 , r 1 ), such that
w λ 0 (x, t) ≥ ε 2 > 0 for (x, t) ∈ B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s α .
(4.14)
Otherwise, there holds that
w λ 0 (x, t) < ε in B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s α for any ε > 0. (4.15)
Considering (4.2) and the definition of λ 0 , we obtain
∂ α t w λ 0 (x, t) + (−∆) s w λ 0 (x, t) = C λ 0 (x, t)w λ 0 (x, t), in B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s α , w λ 0 (x, t) ≥ 0, in Σ λ 0 \ B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s α , w λ 0 (x, t) ≥ 0, in B r 1 2 (x k ) × −∞, t k − ( r 2 2 ) 2s α .
Meanwhile, if follows from (4.15) and f ∈ C 1 that
C λ 0 (x, t)w λ 0 (x, t) = f (u λ 0 (x, t)) − f (u(x, t)) > −Cε in B r 1 2 (x k ) × t k − ( r 1 2 ) 2s α , t k + ( r 1 2 ) 2s α ,
for any ε > 0. Thereby using (4.13) and the averaging effects established in Theorem 1.3 for the antisymmetric function w λ 0 (x, t), and combining with the continuity of u(x, t), we conclude that
w λ 0 (x, t) ≥ ε 2 > 0 in B r 1 4 (x k ) × t k − ( r 1 4 ) 2s α , t k + ( r 1 4 )
2s α for some positive constant ε 2 = ε 2 (α, n, s, ε 0 , λ 0 , r 1 ), which contradicts the assumption (4.15). Hence, we verify that (4.14) is valid. Moreover, applying (4.14), and combining the continuity of w λ (x, t) in λ with λ k → λ 0 as k → ∞, we finally derive
w λ k (x, t) ≥ ε 2 2 > 0 in B r 1 2 (x k ) × t k − ( r 1 2 )
2s α , t k + ( r 1 2 )
2s α for sufficiently large k, which means that w λ k (x k , t k ) ≥ ε 2 2 > 0 for sufficiently large k. Hence, it contradicts the assumption that the sequence {(x k , t k )} ⊂ Σ − λ k × R, therefore we must have λ 0 = +∞.
Step 3. In this final step, we prove that u(x, t) is strictly increasing with respect to x 1 in R n + for any t ∈ R. By virtue of Step 1 and Step 2, we have deduced that w λ (x, t) ≥ 0 in Σ λ × R for any λ > 0.
In fact, it suffices to show that w λ (x, t) > 0 in Ω λ × R for any λ > 0, (4.16) then we can conclude that u(x, t) is strictly increasing with respect to x 1 in R n + for any t ∈ R. Suppose (4.16) is violated, then there exist a fixed λ 0 > 0 and a point (x 0 , t 0 ) ∈ Ω λ 0 × R such that w λ 0 (x 0 , t 0 ) = min Σ λ 0 ×R w λ 0 (x, t) = 0.
Since w λ 0 (x, t 0 ) is not identically zero in Σ λ 0 due to the exterior condition u(x, t) ≡ 0 in (R n \R n + )× R and the interior positivity of the solution u(x, t) in R n + × R, then through a straightforward calculation, we obtain ∂ α t w λ 0 (x 0 , t 0 ) + (−∆) s w λ 0 (x 0 , t 0 )
= C αˆt 0 −∞ w λ 0 (x 0 , t 0 ) − w λ 0 (x 0 , τ ) (t 0 − τ ) 1+α
dτ + C n,s P.V.ˆR n w λ 0 (x 0 , t 0 ) − w λ 0 (y, t 0 ) |x 0 − y| n+2s dy ≤ C n,s P.V.ˆΣ λ 0 w λ 0 (y, t 0 ) 1 |x 0 − y λ 0 | n+2s − 1 |x 0 − y| n+2s dy < 0, which contradicts the equation ∂ α t w λ 0 (x 0 , t 0 ) + (−∆) s w λ 0 (x 0 , t 0 ) = f (u λ 0 (x 0 , t 0 )) − f (u(x 0 , t 0 )) = 0. This verifies (4.16).
Finally, based on (4.16), for each fixed t ∈ R, for any pointsx = (x 1 , x ′ ) andx = (x 1 , x ′ ) in R n + withx 1 <x 1 , if choosing λ =x 1 +x 1 2 as illustrated in Figure 5 below, then we must have
0 < w λ (x, t) = u(x λ , t) − u(x, t) = u(x, t) − u(x, t).
It implies that u(x, t) is strictly increasing with respect to x 1 in R n + for any t ∈ R.
x 1
x ′ 0 T λ xx Figure 5. The positions of the pointsx,x and the plane T λ .
This completes the proof of Theorem 1.4.
APPENDIX
In this section, we provide some facts on the fractional time derivative that are used repeatedly in establishing our main results.
Lemma 5.1. Let η(t) ∈ C ∞ 0 ((−2, 2)) be a smooth cut-off function, satisfying η(t) ≡ 1 in [−1, 1] and 0 ≤ η(t) ≤ 1. Then there exists a positive constant C 0 that depends only on α such that |∂ α t η(t)| ≤ C 0 for t ∈ (−2, 2).
Proof. By using the definitions of the fractional time derivative ∂ α t and the smooth cut-off function η(t), we directly compute
|∂ α t η(t)| = C α ˆt −∞ η(t) − η(τ ) (t − τ ) 1+α dτ , ≤ C α ˆ− 3 −∞ η(t) (t − τ ) 1+α dτ + C α ˆt −3 η(t) − η(τ ) (t − τ ) 1+α dτ ≤ C α α + CC α ˆt −3 (t − τ ) (t − τ ) 1+α dτ ≤ C α α + CC α 5 1−α 1 − α =: C 0 .
Hence, we complete the proof of Lemma 5.1 .
As a byproduct, we derive the following result by rescaling and translation.
first introduced by Marchaud in 1927. The normalization positive constant C α = α Γ(1−α) , and Γ denotes the Gamma function. In order to guarantee the singular integral in (1.2) is well defined, we may assume that u(x, ·) ∈ C 1 (R) × L − α (R), where L − α (R) is a class of slowly increasing functions given by L − α (R) := {u(x, ·) ∈ L 1 loc (R) |ˆt −∞
Theorem 1.1. (Narrow region principle for antisymmetric functions) Let Ω be an unbounded narrow region containing in the narrow slab
Figure 1 .
1The positional relationship between the region D and the ball B r (x 0 ) in R n . Theorem 1.3. (Averaging effects for antisymmetric functions) Let D ⊂ Σ λ be a domain and t 0 ∈ R be a real number. For any
Theorem 2.1. (Maximum principle for antisymmetric functions) Let Ω be an unbounded domain in Σ λ with a finite width in x 1 direction. Suppose that
α
], combining (1.10), (1.11), (3.1) and Corollary 5.2 with B r (x 0 ) ∩ D = ∅, we directly calculate
Now the proof of Theorem 1.2 is completed. Now we turn our attention to the proof of Theorem 1.3 regarding the averaging effects for the antisymmetric functions.
Figure 3 .
3The positions of B r (x 0 ), B r ((x 0 ) λ ), D and D λ .
5.2 and r ≤ dist(x 0 ,T λ ) 2
Figure 4 .
4The positional relationship between the balls.
Corollary 5. 2 .α
2For any t 0 ∈ R and r > 0, + t 0 ), where the smooth cut-off function η(·) and the positive constant C 0 are defined in Lemma 5.1 .
ACKNOWLEDGMENTSThe work of the first author is partially supported by MPS Simons foundation 847690. This work of the second author is partially supported by the National Natural Science Foundation of China (NSFC Grant No.12101452).
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| {'fraction_non_alphanumeric': 0.10891057068427099, 'fraction_numerical': 0.038299111693607106, 'mean_word_length': 3.0122046221760583, 'pattern_counts': {'":': 0, '<': 63, '<?xml version=': 0, '>': 52, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 95, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper, we consider the dual fractional parabolic problemis the right half space. We prove that the positive solutions are strictly increasing in x 1 direction without assuming the solutions be bounded.So far as we know, this is the first paper to explore the monotonicity of possibly unbounded solutions for the nonlocal parabolic problem involving both the fractional time derivative ∂ α t and the fractional Laplacian (−∆) s . To overcome the difficulties caused by the dual nonlocality in space-time and by the remarkably weak assumptions on solutions, we introduced several new ideas and our approaches are quite different from those in the previous literature. We first establish an unbounded narrow region principle without imposing any decay and boundedness assumptions on the antisymmetric functions at infinity by estimating the nonlocal operator ∂ α t + (−∆) s along a sequence of suitable auxiliary functions at their minimum points, which is an essential ingredient to carry out the method of moving planes at the starting point. Then in order to remove the decay or bounded-ness assumption on the solutions, we develop a new novel approach lies in establishing the averaging effects for such nonlocal operator and apply these averaging effects twice to guarantee that the plane can be moved all the way to infinity to derive the monotonicity of solutions.We believe that the new ideas and techniques developed here will become very useful tools in studying the qualitative properties of solutions, in particular of those unbounded solutions, for a wide range of fractional elliptic and parabolic problems.Mathematics Subject classification (2020): 35R11; 35B50; 35K58; 26A33.', 'arxivid': '2303.10304', 'author': ['Wenxiong Chen ', 'Lingwei Ma '], 'authoraffiliation': [], 'corpusid': 257631573, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 25300, 'n_tokens_neox': 21592, 'n_words': 13087, 'pdfsha': 'bab63db7cbd9586cd44c1ab95955e9adc6729dcb', 'pdfurls': ['https://export.arxiv.org/pdf/2303.10304v1.pdf'], 'title': ['QUALITATIVE PROPERTIES OF SOLUTIONS FOR DUAL FRACTIONAL NONLINEAR PARABOLIC EQUATIONS', 'QUALITATIVE PROPERTIES OF SOLUTIONS FOR DUAL FRACTIONAL NONLINEAR PARABOLIC EQUATIONS'], 'venue': []} |
arxiv |
EvIcon: Designing High-Usability Icon with Human-in-the-loop Exploration and IconCLIP
200y
I-Chao Shen [email protected]
Fu-Yin Cherng [email protected]
Takeo Igarashi
[email protected]
The University of Tokyo
Japan
Wen-Chieh Lin
Bing-Yu Chen
The University of Tokyo
Japan
National Chung Cheng University
Taiwan
EvIcon: Designing High-Usability Icon with Human-in-the-loop Exploration and IconCLIP
Number z
xx200y4 [email protected], National Yang Ming Chiao Tung University, Taiwan 5 [email protected], National Taiwan University, Taiwan
Interface icons are prevalent in various digital applications. Due to limited time and budgets, many designers rely on informal evaluation, which often results in poor usability icons. In this paper, we propose a unique human-in-the-loop framework that allows our target users, i.e., novice and professional UI designers, to improve the usability of interface icons efficiently. We formulate several usability criteria into a perceptual usability function and enable users to iteratively revise an icon set with an interactive design tool, EvIcon. We take a large-scale pre-trained joint image-text embedding (CLIP) and fine-tune it to embed icon visuals with icon tags in the same embedding space (IconCLIP). During the revision process, our design tool provides two types of instant perceptual usability feedback. First, we provide perceptual usability feedback modeled by deep learning models trained on IconCLIP embeddings and crowdsourced perceptual ratings. Second, we use the embedding space of IconCLIP to assist users in improving icons' visual distinguishability among icons within the user-prepared icon set. To provide the perceptual prediction, we compiled IconCEPT10K, the first large-scale dataset of perceptual usability ratings over 10, 000 interface icons, by conducting a crowdsourcing study. We demonstrated that our framework could benefit UI designers' interface icon revision process with a wide range of professional experience. Moreover, the interface icons designed using our framework achieved better semantic distance and familiarity, verified by an additional online user study. † Corresponding author uating icons designed for specific users (e.g., elders or users with lower computer literacy), conducting adequate usability testing is even more laborious.Zhao et al. [ZKH * 20] reported that designers often consult other UI designers' feedback on icons. These informal evaluations often failed to provide comprehensive and objective information about how target users would perceive and use the icons [BAR92, RDF11], thus leading to low usability icons. As shown inFigure 2, even for icons of standard tags (e.g., "Search" and "Calendar"), accidentally omitting the critical visual features when adjusting icons' style could lead to poor usability[Lin94,CUG20]. These examples further illustrate the importance of getting objective and instant feedback from users. These findings underscore the need for an objective and comprehensive usability testing approach that is cost-and time-efficient. Although several automatic graphical icon synthesis methods have been proposed [LRFN04, KPL08], involving humans (i.e., "human-in-the-loop") in the design process has several advantages (e.g., solving computationally complex problems [Hol16], building users' trustworthiness to interactive systems [LGM20]).Hence, instead of proposing an automatic icon synthesis method,
Introduction
Amid the ubiquity of digital technologies, including computers, intelligent appliances, and wearable devices, interface icons play an increasingly important role in representing various functions with benefits including improving interface scannability (i.e., the ease of reading and understanding the content of the interface), saving space on small screens, and conveying information universally [SABAG * 05, SM14]. The usability of icons is determined by several characteristics, including visual complexity, style, familiarity, etc. [MCdB01,IMC07] Existing design guidelines (e.g., Google's Material Design) provide designers with implications of icon design regarding visual characteristics. However, collecting users' perceptual feedback on the icons is still an irreplaceable step to assess the icon's usability [BAR92]. Yet, conducting formal usability tests (e.g., inviting real users to perform usability tasks) can be time-consuming and requires extra effort [RDF11, DHF * 17b, SL19], which could significantly lengthen the iterative process of interface icon design. Moreover, when eval- Figure 1: EvIcon provides two types of instant perceptual usability feedback. (a) An UI designer can improve a single icon's usability and target different demographic users (e.g., elder people or non-tech workers) with the "semantic distance" and "familiarity" feedback. (b) Moreover, an UI designer can improve the usability of an icon set by (i) identifying poor visual distinguishability and (ii) revise the "archive" icon and (iii) the "calendar" icon using the visual distinguishability graph.
(b) (c) (a) (b) (i) (ii) (iii)
we propose EvIcon, an interactive framework to reduce the workload of performing usability tests for a user-prepared interface icon set. EvIcon comprises two main parts: (i) a novel human-in-theloop formulation of icon and icon set design and (ii) an interactive tool with instant perceptual usability feedback. Our main idea is to formulate the common icon usability criteria into perceptual usability functions. Among all icon-related features, we select semantic distance and familiarity as the usability criteria since they are the most critical indications of icons' effectiveness at conveying information [MCdB99,SABAG * 05,WMLB13, SM14,CLKL16] and are commonly used by professional artists. Semantic distance stands for the degree of closeness between an icon and the tag it represents and familiarity referred to the frequency with which icons are encountered. In Figure 1(a), we show examples of icons with different semantic distances and familiarity levels. Moreover, prior research has found that using icons with close semantic distance and high familiarity can significantly increase user performance on interfaces [MCdB99, SABAG * 05, WMLB13, SM14,CLKL16]. Hence, due to the importance of these two indications for icon's usability [SABAG * 05, MI09,SM14], we focus on providing icon designers with semantic distance and familiarity predictions on icon designs in this paper. Moreover, as an icon is usually designed and displayed within an icon set [Kur00], we also use visual distinguishability as a critical usability criterion for designing an icon set [Kur00, LRFN04, SABAG * 05]. The goal is to prevent users from confusing icons of different tags.
To reach the goal of this study mentioned above, we gathered the first large-scale dataset of single-colored interface icon usability ratings coined as IconCEPT10K. The reason we focus on the single-colored icons is that single-colored icons have been recommended by popular online resources (e.g., Font Awesome and Noun Project) and major software providers (e.g., Google and Ap-
Low Usability Icon
High Usability Icon Filter Search Calendar tags Figure 2: Example icons in IconCEPT10K with usability rated by crowdworkers. Icons with low usability have three common shortcomings. First, these icons make users misunderstand their tag with others (e.g., Filter). Second, they use unconventional metaphors to transmit the meanings of a concept (e.g., Search). Last, these icons omit the critical features, so users fail to recognize the target concept (e.g., Calendar).
ple) due to their scalability in various screen sizes and applications as the prevalence of flat UI design [SRS18,LC20]. Also, icons are usually designed in single-colored in the first place and then edited their color later, tailoring to the configuration of display devices [GSF01, ZKH * 20]. Moreover, prior works found that icons' coloring is more critical to icons' visual attractiveness than effectiveness in the conveyance of meaning [Hsi17, CUG20, SZL * 21]. Accordingly, we consider devising icons in single-colored is common in the design process. Hence, we focus on the single-colored icons in the present study.
Our perceptual usability function comprises two components. First, we took a large-scale pre-trained joint image-text embedding (CLIP [RKH * 21]) and fine-tuned it to embed icon visuals with icon tags in the same embedding space (IconCLIP). Second, we collected usability ratings for a curated icon dataset of 50 base tags. We expanded the base tags by using the tags associated with each icon; thus, our usability prediction model can recognize unseen tags and is scalable for future use. After building the perceptual usability function, we present an interactive user interface with two types of instant feedback (as shown in Figure 1) to support refining icons' usability efficiently. Users can iteratively revise the initial icon in the prepared icon set and query for predicted usability results. The first feedback is the predicted perceptual usability of the revised icon (Figure 1(a)). The second feedback is the icon's visual distinguishability to other icons in (i) the user-prepared icon set and (ii) our icon dataset. This feedback is realized by providing an interactive two-dimensional visualization of the IconCLIP embedding (see Figure 1(b)).
To understand the benefits of EvIcon for designers, we conducted a user study with six UI designers and asked them to revise icon sets with and without using EvIcon. We further conducted an online user study on the revised icons to verify whether EvIcon can assist UI designers to improve icons' usability. The result shows that EvIcon can assist UI designers with a wide range of professional experiences to improve the usability of their icon designs. The major contributions and novelties of this paper include:
• We propose a novel human-in-the-loop formulation, EvIcon, for submitted to COMPUTER GRAPHICS Forum (5/2023). refining the usability of an icon set, while previous works focus on providing supports for designing a single icon ignoring the icon usability.
• We gathered IconCEPT10K, the first icon dataset with high-level perceptual usability ratings, instead of low-level visual perceptual properties such as visual saliency.
We implemented EvIcon as a web application so anyone can test EvIcon on their own icon set. We will also release the source code, pretrained models, and the collected dataset (IconCEPT10K).
Related Works
Icon Design and Analysis
Icon plays an essential role in visual communication, including graphic design and user interface design. Prior studies [Git86,Hor94,Hor96] provide a thorough introduction on how to design icons and recommended practices. Icon's usability is mainly associated with the ability to convey the information it represents. Previous research identified several features that heavily influence an icon's usability, including visual complexity, semantic distance, and familiarity [MCdB01,MI09,SM14,KFZI20]. Some studies reported that the users' age [LMG11] and experience [IMC07,AMW21] also influence the effect of these features on icons' usability.
Researchers have proposed various methods to support icon design and generation due to the complex relationship between icons' features and usability. Zhao et al. [ZKH * 20] developed a system to generate icons containing compound meanings automatically. Some works focus on generating icons based on filenames [LRFN04], data content [KPL08], and man-made object category [SC21]. Other prior works focus on learning icons' appearance similarity [LGG18], creating scale variations of icons [BL15], and selecting an icon set based on crowdsourced ratings [LKC * 16]. Compared to previous works [LKC * 16, LGG18], our system lets designers devise the final icon set on their own with our perceptual usability feedback instead of directly obtaining an icon set from an optimization process.
Assistive Authoring Tool for Visual Design
Assistive visual content authoring has gained increasing interest in the past few years since the surge of the need for novel visual content. Many works utilized personal editing histories to assist 2D sketch [XCW14], 3D shape sculpturing [PXW18], and viewpoint selection [CGW * 14]. On the other hand, various prior works have incorporated real-time physical simulation into their interactive tools for designing physically valid furnitures [UIM12] and model airplanes [UKSI14]. Among them, many recent works leveraged collected visual content data to assist 2D sketch [LZC11], multiview clipart design [SLS * 21], and mobile apps user interface design [LCS * 18, DHF * 17a, DHF * 17c]. Other studies crowdsourced and modeled large-scale users' perception about tappability for the mobile interfaces [SL19] and visual importance on graphic designs [BKO * 17] to assist designers in diagnosing the perceptual issues in their designs. Additionally, Rosenholtz et al. [RDF11] conducted a thorough qualitative study with professional design teams and showed that designers benefited from tools with low-level perceptual prediction in the agile assessment of usability. Unlike previous works that only focused on providing low-level visual perceptions feedback, we provide high-level usability feedback such as semantic distance and familiarity. Moreover, we provide visual distinguishability feedback to support revising an icon set's usability, which is rarely addressed in prior related research.
Human-in-the-loop Exploration
Prior studies have demonstrated the feasibility of conducting usability evaluation on crowdsourcing platforms via performing benchmark user testings [KRG13] and collecting human visual importance [BKO * 17]. As exploring various huge design spaces with usability evaluations is a ubiquitous task in visual design, this task is realized by various interactive optimization techniques, including interactive evolutionary computation [Tak01] and human-inthe-loop Bayesian optimization [KSI14, KSSI17, KSG20, CSSI21, BBDF10]. Unlike previous methods, our human-in-the-loop framework focuses on providing instant perceptual usability feedback to support users' exploration instead of providing the final design using the optimization-based method due to the following reasons. First, the state-of-the-art human-in-the-loop optimization methods work best in relatively lower-dimensional parameter spaces (e.g., 6 − 15) [KSI14,KSSI17,KSG20,CSSI21,BBDF10], whereas reducing the design dimensions of interface icons into such low dimensions would omit the nuanced features that are crucial for the high-level usability perceived by users and designers. Hence, our framework makes designers finalize the icons and the icon sets iteratively and manually. Second, previous human-in-the-loop optimization methods use "selection" as the main interaction approach, whereas the task of icon design requires more complicated design interactions than selections [ZKH * 20]. Therefore, the current human-in-the-loop optimization methods are not suitable for the inputs of our framework to design high-usability icons.
Problem Overview
Given an interface icon set I provided by a designer. The goal of our framework is to assist this designer in revising the usability of prepared icons into a new interface icon setÎ efficiently. We expect that each icon I in I is associated with n text tags (T I = t 0 ,t 1 , ...,t n−1 ) that represent the semantic and visual concepts of the icon such as "search", "next", "television", and "map". We characterize the usability of an icon using common perceptual usability metrics including semantic distance, familiarity, and visual distinuguishability, which are commonly used by professional icon designers [MCdB01,Kur00,SM14]. However, these metrics of an icon are usually hard to evaluate mathematically from the icon image since the assessments of these metrics require extensive user testing to collect users' self-reports and feedback. Hence, we collected a large-scale icon dataset and the crowdsourced perceptual ratings of these icons on Amazon Mechanical Turk (AMT). We used the collected ratings to train usability classifiers. For each tag A, we trained a separate classifier f sd A and f fam A for classifying the semantic distance and familiarity of an icon belongs to tag A. For each classifier, it predicts "Very Good", "Good", "Neutral", "Bad", and "Very Bad" as the different levels for the semantic distance and submitted to COMPUTER GRAPHICS Forum (5/2023). Human-in-the-loop exploration with user interface Figure 3: Overview of EvIcon. We collect a large-scale icon and tag collection. And we compiled a dataset IconCEPT10K, comprises 10, 000 icons across 50 base tags, their associated tags, and crowdsourced semantic distance and familiarity ratings. We also fine-tune a pre-trained joint text-image embedding (CLIP) into IconCLIP using this collection. EvIcon computes and presents designers with instant perceptual usability feedback to assist revising high-usability icon sets.
familiarity. The goal of our framework is to enable users to revise an icon I ∈ I that maximizes the following perceptual usability function:
i * = arg max(w sd φ sd (I, T I ) + w f am φ f am (I, T I ) + w vd φ vd (I)), (1)
where φ is a semantic perceptual function. In our work, the semantic perceptual function comprises three parts:
• semantic distance: φ sd (I, T I ) = P( f sd A == Very Close|I, T I ) • familiarity: φ f am (I, T I ) = P( f fam A == Very Good|I, T I ) • visual distinguishability: φ vd (I) = ∑ J∈I ∥ρ I − ρ J ∥ 2 2 ,
where P( f sd A == Very Close) stands for the probability of an icon being classified as having the "Very Close" semantic distance.
To measure visual distinguishability, it is important to measure the distance with respect to the semantic concept difference instead of just pixel-level difference. To achieve this, we obtain an embedding space where icons of the same tags stay closer to each other than those of different tags. We describe how to obtain this embedding space in Section 5.2. The embedded coordinates of icon i in this space are represented by ρ i . The goal of φ sd (i) is to encourage the revised icon to be classified as "Very Close," while the aim of φ vd (i) is to separate the refined icon from other icons in I. To optimize Equation (1) and iteratively refine the icon set I, designers need to be involved in the process to specify their design requirements. Instead of providing designers with automatic synthesis results, we have developed an interactive interface that guides them in designing highly usable icons.
EvIcon User Interface
We propose an interactive and exploratory design tool, EvIcon, to present perceptual usability feedback of an individual icon and visual distinguishability between icons. Our interface augments existing vector graphics design tools with additional usability feedback panels. As shown in Figure 3, our interface contains three main panels: (i) the main canvas panel which includes a vector graphics editor for icon revision and a list to present the uploaded icon set, (ii) the perceptual feedback panel (box with blue borderline), (a) Initial icon.
(b) Modified icon. (c) Two types of inplace warnings. Figure 4: A warning will be displayed in place to draw attention to the poor adjustment compared to the last usability inspection. Blue highlights will indicate the paths suggested to be added back, while light-blue highlights will mark those suggested to be removed.
and (iii) the distinguishability visualization panel (box with orange borderline).
User Workflow
To use EvIcon, a designer first prepares a set of icons and corresponding tags under designing. Next, the designer can select an icon from the icon set, and EvIcon would infer its predicted usability. Then, the designer can check the predicted usability for general users or users with particular demographics in the perceptual usability feedback panel. Furthermore, the designer can revise the selected icon to improve its usability and inspect the visual distinguishability of the revised icon using the interactive distinguishability graph. The designer can repeat these steps until the perceptual usability of the selected icon or the visual distinguishability between icons is satisfied.
Interface Components
Main Canvas Panel
The designer can revise the icons using the vector graphics editor in this panel. During the iterative revision process, EvIcon also submitted to COMPUTER GRAPHICS Forum (5/2023). provides in-place warnings when the predicted perception usability drops. This in-place visual warning is helpful for building the connection between the revised icon and the perceptual prediction. We highlighted the paths of an icon that we encourage the designers to add and remove in two different colors as shown in Figure 4.
Perceptual Usability Feedback Panel
EvIcon shows the predicted level of perceptual usabilities (semantic distance and familiarity) of the icon under revision ( Figure 5). The designers can switch between tabs to check the predicted usabilities for target audiences with particular demographics. To present the levels of semantic distance and familiarity in a way designers can easily understand, instead of showing rating scores directly, we use "Very Bad", "Bad", "Neutral", "Good", and "Very Good" to represent five different levels of user perceptions, and semantic distance is presented as "Semantics" on the interface of EvIcon. We highlighted "Very Bad" and "Bad" in red, "Neutral" in black, and "Good" and "Very Good" in green to enhance readability.
Distinguishability Visualization Panel
EvIcon presents an interactive distinguishability graph to help designers compare the relative visual distance between icons in the prepared icon set I. After the designers revise an icon, they can check the updated embedded coordinate of the icon. We connected the icons in the prepared icon set using grey links (as shown in Figure 6(b)) and changed the color of the links into red if the connected icons were too close to each other (see Figure 6(c)). This interactive design aims to warn designers of the inadequate visual distinguishability in the prepared icon set, and prevent them from refining icons that fall into the wrong tag.
EvIcon Implementation
Icon and Crowdsourced Perceptual Rating Dataset
Our goal of data collection is to gather an icon dataset covering a comprehensive range of tags that UI designers are likely to design. Since unlimited tags exist for interface icons, it is impractical to enumerate them all and collect them at once. To address this issue, we expanded the tags we can cover by adopting the following data collection procedure. First, we collected icons of 212 base tags reported in prior work [LCS * 18], including "Search", "Crop", "Message", "Pause", "Filter", "Calendar", and "Archive". We collected However, it is tedious and repetitive to collect users' perceptual usability ratings for all icons; thus we selected the top 50 base tags that are semantically independent based on the icon numbers of each tag. For each selected base tag, we further selected 200 representative icons with respect to the uniqueness of icon shapes using the following process. After normalizing the size of icons from different resources into 28 × 28 pixels, we applied the principal component analysis (PCA) on icons' pixel values after removing the duplicated icons. Then, we set the projection to preserve 90% of the variances to generate the final principal components and utilize them to represent each icon. Next, we performed K-Means clustering [AV07] on these projected icon representations and set K = 10 based on the results of the Elbow method (i.e., ten clusters in a subset) [KS96]. We obtained 200 icons from each base tag by randomly sampling 20 icons from each cluster. After repeating the same process to all base tags, we acquired the curated dataset with in total 10, 000 icons in which the variety of icons of each function increased compared to the raw dataset.
After obtaining the curated dataset, we used Amazon Mechanical Turk (AMT) to collect users' perceived semantic distance and familiarity with the selected 10, 000 icons. We recruited 5, 559 workers participating in the crowdsourcing task (3,498 males and 2,061 females; mean age = 33.1 with a standard deviation of 8.90). The workers' self-report ages and occupations were divided into three age levels (elder: age > 50 yrs; adult: 50 > age > 20 yrs; teenager: age < 20 yrs) and occupational categories (technology, business, and others) which are used as the demographic information of their ratings when building perceptual usability prediction. Each worker finished five assignments and rated icons of five tags in each assignment (i.e., 25 icons in total) with an average completion time of 8 minutes. The workers were asked to rate each icon on a 5point Likert scale to specify their assessment of the icon's semantic distance and familiarity [MCdB99,IMC07]. The workers also rated their perceived familiarity with each tag on the same 5-point Likert scale. We described the rating distribution of the 50 base tags and the content of the questions in ?? of the supplementary material. The order of the icons was randomized. In general, we spent two days collecting all the rating data in parallel using MTurk API. In the final rated dataset, we collected 138, 964 unique ratings. We describe the details of the distribution of the collected ratings and the AMT crowdsource task in ?? of the supplemental material. We will include the selected 10, 000 icons and the collected perceptual usability ratings as our IconCEPT10K dataset.
Perceptual Usability and Visual Distinguishability Feedback
Given an input icon I and its associated tags T = t 0 ,t 1 , ...,t n−1 , we want to build a classifier that can predict its perceptual usability ratings (semantic distance and familiarity). However, there are unlimited possible tags designers want to design; and it is tedious to collect icons of all possible tags and their perceptual usability ratings. Thus, it is vital to design a classification method to predict the perceptual usability ratings for icons of unseen tags. To address this need, we designed our classification method based on the pretrained joint embedding (CLIP) [RKH * 21] which is learned from loose image-text pairing information.
Introduction to CLIP Embedding Space
CLIP [RKH * 21] is a joint image-text embedding trained on 400 million text-image pairs. The representations learned by CLIP have been shown to be effective for various downstream tasks such as zero-shot image classification. CLIP jointly trains an image encoder g and a text encoder h, that map images and text into a shared embedding space. Unlike previous works on natural image editing using CLIP embedding space [PWS * 21, AZF * 21], the target image domain of our application (single-colored icon image) is different from the training images used in the pre-trained CLIP model. Thus, instead of using the pre-trained CLIP model to extract image and text representations directly, we finetune the original CLIP model using our icon dataset to obtain IconCLIP.
Finetuning CLIP on icon image We let S icon = {(I i , T i )|i = 0, ..., N} denote the icon dataset used for finetuning the original CLIP model. For each icon I i , we converted the associated tags T i into a sentence s i using the prompt template "A icon looks like a {tag 0 , tag 1 , ..., tag n−1 }". We use a pre-trained CLIP ViT-B/32 model as the base model, which uses ViT-B/32 [DBK * 21] as the image encoder and Transformer [VSP * 17] as the text encoder. We follow the training procedure described in the original CLIP [RKH * 21]. Given a training pair (an icon image I i and a sentence s i ), CLIP produces a scalar score: g(I) T h(s i ) that is high when the image and text are mismatched. We finetune the pre-trained model by minimizing a symmetric InfoNCE loss [VdOLV18].
Perceptual Usability Prediction
We designed our classifier F Θ using a deep fully-connected neural network without convolutional layers (i.e., a MLP). As illustrated in Figure 7(c), F Θ takes three different inputs: the input image embedding, the input sentence embedding, and the discrete demographics We fine-tune the general-purpose CLIP into IconCLIP using "icon tags"-"icon image" pairs. (c) We predicted the perceptual usability ratings of an input icon and its associated tags using the IconCLIP embedding space and the target demographic information. : text-to-image retrieval by IconCLIP. Among the four tags we used as queries, only "attachment" and "map" are in the 212 base tags. The IconCLIP embedding space recognizes the meaning of "envelop" and "alarm" because we used the tags associated with icons (we use the green box for positive results and the red box for negative results).
vector (age: three levels and occupation: three categories) and the output are five usability ratings of semantic distance and familiarity. We obtained the image and sentence embedding using the image encoder and the text decoder of IconCLIP. And F Θ process these inputs with four fully-connected layers (using ReLU activations and 256 channels per layer.
Visual Distinguishability
As discussed in Section 4.2.3, EvIcon provides a distinguishability graph to help users compare the relative visual distance between icons in the prepared icon set. We directly use the embedding space of the finetuned IconCLIP as our similarity measurement space. We used Uniform Manifold Approximation and Projection (UMAP) [MHSG18] to project the 512d feature vector to 2d.
6. Evaluation
Evaluation of IconCLIP
To evaluate the IconCLIP embedding space, we performed a top-k image retrieval evaluation. We split the overall collected icons into a training set and a testing set. We used the training set to finetune IconCLIP, and we performed the retrieval test as follows: we used each icon image in the testing set as a query and used the rest of the testing set as a retrieval set. And we consider a retrieved icon image as a positive result if it shares a common tag with the query image. The MAP@5 (mean average precision at rank 5) of the retrieval test is 74.3. On the other hand, we also performed a text-to-image retrieval test and showed the qualitative image retrieval results in Figure 8. We can observe that the top-5 nearest neighbors match the tags in the sentence even when the concepts are not in the 212 base tags we used for collecting the icons. This suggests that the tags associated with the icons expand the embedding space of IconCLIP.
Evaluation of perceptual usability feedback models
As mentioned in Section 5.2.2, we trained a unified network to predict usability ratings based on the icon's image embeddings, tag embeddings, and demographic information. To demonstrate the ability to predict usability ratings of unseen tags, we split the icons of 50 base tags into 45 tags as seen and 5 tags as unseen tags. It should be noted that we only use the base tags as the selection criteria, but we used all the associated tags within the base tags as training signals, so it is not restricted to these base tags. We performed two types of evaluation of our prediction model.
In-domain Evaluation
First, we evaluated the prediction precision and recall on the 45 base functions we used for training. Among the icons of these 45 base functions, we randomly split the data into 90/10 as training/testing data. For semantic distance, our models achieved 83.6% for precision and 84.1% for recall. For familiarity, our models achieved 76.3% for precision and 77.6% for recall.
Out-of-domain Evaluation
Second, we also evaluated the prediction precision and recall on all icons belonging to the 5 base tags we held out during training. For semantic distance, our models achieved 66.4% for precision and 69.5% for recall. For familiarity, our models achieved 67.1% for precision and 68.4% for recall.
Evaluation with UI Designers
To evaluate how EvIcon's interaction design can support designers' revision process, we conducted a user study with six professional UI designers (five females and one male; ages ranging from 22 to 34 years old). We recruited a similar number of domain experts with prior similar works [PXW18, XKG * 16, SSII18]. The self-reported professional experience of the designers ranges from one to ten years. All of them used Adobe Illustrator † for initial icon design. † https://www.adobe.com/products/illustrator.html
Procedure and Tasks
After introducing EvIcon and the meaning of two types of feedback, the designers practiced using EvIcon for ten minutes. We then asked them to complete the practice tasks (e.g., reporting the perceptual usability of an icon in different age groups of users) to ensure they understand how to use EvIcon. In the formal sessions, the designers were asked to improve the usability of two icon sets. For the design brief to guide the designers when revising icons, we informed the designers the scenario of the evaluation is that a client asked them to improve the icon sets so that these icons can be used in a wide range of applications and users (e.g., elders). Each icon set contains three icons of tags "Archive", "Print", and "Filter". We selected these tags based on their average familiarity level collected via the crowdsourced study in Section 5.1 ("Archive": 3.8; "Filter": 3.9; "Print": 4.2) to ensure we included established and uncommon tags in the evaluation. Moreover, "Archive" and "Print" are in the 45 seen tag set, and "Filter" is in the 5 unseen tag set used in Section 6.1. We denoted these icons as the original icons in the following sections.
As shown in Figure 9, the icons in the two sets are different, and we instructed each designer to improve the usability of one icon set with EvIcon and another set without EvIcon, both in fifteen minutes. The combination of the icon set and two conditions were randomly assigned, and the order of conditions was counterbalanced to avoid the learning effect. Under both conditions, the designers can freely edit icons using the design tool of their choice and search online for the information. However, under the without EvIcon condition, the designers can not access the icons and perceptual ratings we collected. We recorded the revision process and the revised icons. In the end, we obtained 36 revised icons from six designers in total, and we found that all designers spent the entire time budget (15 minutes) for each condition.
Result
Revised icons In Figure 9, we show the revised icons of all three tags with and without using EvIcon. To further verify that EvIcon can help designers improve icons' usability, we launched an additional crowdsourced evaluation on AMT to collect 213 (140 males and 73 females; 19 to 64 years old) crowdworkers' usability ratings of all original and revised icons pair in an assignment. We collected averaged 57.8 unique ratings for each original/revised icon pair. Each crowdworker would only rate an icon pair revised by the same designers to eliminate the influence of individual designers' abilities. We used the majority vote of all received ratings as the final rating of each revised icon to reduce the effects of spammers as shown in Figure 9. To reduce the mutual influence of icons in different pairs of revised and original icons, we calculated the final ratings of the original icons by averaging the majority vote ratings across the different pairs of revised icons provided by the designers. We can see from Figure 9 that most of the revised icons with EvIcon (blue bars) obtained higher AMT ratings of semantic distance and familiarity than those without EvIcon. Moreover, to demonstrate the usefulness of EvIcon in improving the visual distinguishability within the icon set, we computed the mutual distances between the 512-dim embedded vector of each revised icon. In the We plot the crowdsourced evaluation results ("AMT rating") of each icon. The gray/blue/red bar denotes the AMT rating of original icon/icon revised with EvIcon/icon revised without EvIcon. The ratings ranged from 1 ("Very Bad") to 5 ("Very Good") for the bar chart of semantic distance (seman distant) and familiarity (fam). We can see that most of the icons revised with EvIcon received higher AMT ratings than icons revised without EvIcon. We also show the visual distinguishability score between each icon in the embedding space. The visual distinguishability between icons revised with EvIcon is the furthest.
rightmost panel of Figure 9, the mutual distance between icons revised with EvIcon is farther than the original icons and icons revised without EvIcon, which suggests better visual distinguishability. Figure 10 illustrates example revision processes for the icons "Archive" and "Print." Throughout each design step, EvIcon provided feedback on "Semantics" (semantic distance) and "Familiarity." The crowdsourced evaluation results, collected via Amazon Mechanical Turk (AMT), are displayed next to the finalized icon (the right-most icon of each block in Figure 10) for each revision process. The evaluation outcomes demonstrate that icons revised with EvIcon generally outperformed those revised without it, as shown in Figure 10, with higher ratings given for both "Semantics" and "Familiarity." Revised icons for diverse demographics We investigated whether icons revised with EvIconresult in higher usability ratings from older users (> 50 years old) to demonstrate the tool's ability to create more inclusive designs for users with diverse demographics We found that the revised icons obtained higher AMT mean semantic distance (with: 3.49; without: 3.35) and familiarity (with: 2.83; without: 2.81) ratings across tags. We show the two examples of the revised icons using EvIcon in Figure 11. Crowdsourced evaluation on revised icons As shown in Figure 12, we compared the averaged mode ratings of the revised icons by their tag and whether they were revised with EvIcon using Cohen's d. We can see that for the icons revised by all designers, the "Archive" and "Filter" icons revised with EvIcon received a higher level of semantic distance (Archive: d = 1.48; Filter: d = 0.63; Figure 12(a)) and familiarity (Archive: d = 1.02; Filter: d = 0.4; Figure 12(b)) than those without using EvIcon with the moderate to the large magnitude of the mean difference. Yet, the "Print" icons revised with EvIcon obtained the same level of semantic dis-tance and familiarity as those without EvIcon. These results suggest that the designers benefit most from using EvIcon in improving the usability of icons of unestablished tags (i.e., "Archive" and "Filter"). Since most of the designers and users have not formed the common visual metaphors for the unestablished tags, EvIcon's feedback helps designers navigate the vast variations of "Archive" and "Filter" icons and find the best way to revise the icons.
Revised icon retrieval test The retrieval test aims to demonstrate that the existing icons in our dataset are used primarily as inspiration rather than copied directly. In Figure 13, we present the closest example from our dataset for each revised icon and provide the PSNR/SSIM scores. We observe that for the semantic concept with simpler shapes, such as "Filter," the revised icons are generally closer to the existing icons in our dataset. However, for the semantic concept with more complex shapes, designers tend to make more significant revisions (e.g., "Print", "Archive"), resulting in greater distances between the revised icons and their closest examples.
Post-study interview In the post-study interviews, all six designers gave positive attitudes towards EvIcon. The designers mentioned that when revising icons with EvIcon, they got the idea of how to revise an icon to meet public understanding more easily by checking the perception feedback constantly. They found the perception feedback convincing as it was generated based on data labeled by over two thousand crowdworkers:
• "EvIcon keeps me on the right track and ensures that my design can be understood by others while I modify the icon design based on my creativity. "(P3) • "The good or bad rating provided by the system is promising and helpful in designing high-usability interface icons, compared to designing the icons on my own."(P5) Some designers were amazed by the perception feedback for spesubmitted to COMPUTER GRAPHICS Forum (5/2023). Figure 11: We show two examples of revised icons with and without EvIcon. We can see that the icons revised by EvIconwith better predicted semantic distance and familiarity levels also achieved higher AMT ratings from the elder crowdworkers (age > 50).
Bad
cific demographics since they have experienced struggling to de- sign interface icons targeting a specific category of users while having limited knowledge or access to the users: Figure 13: For each icon revised by the designers, we show its closest example in our dataset and its corresponding PSNR/SSIM at the bottom.
• "
Although icons play an important role in interface design, there is not much information about which icons are friendly or recognizable to elders. "(P6)
Designers also found the distinguishability visualization panel helpful. Both P2 and P6 said they would check the related distance between the uploaded icon and the icons in the suggestion panel to see how they could improve their design. P2, P3, P5, and P6 mentioned they could derive some graphical design features from the icon suggestion panel that can be added to their own designs:
• "It is interesting that the system provides designs from other designers based on current target function."(P3) • "I can see those good icons in the suggestion panel, and think about how to start my design based on the recommendations. It will help save my time to grasp users' thoughts at the beginning of the design flow."(P5)
Designers also discussed possible benefits EvIcon could bring if applied in their current workflow. P5 said it would save lots of time to notice the perception gap between designers, engineers, and average users earlier with EvIcon, instead of finding out in usability testing after several design iterations and discussions. As designers, participants usually care a lot about aesthetics while designing icons, EvIcon could also provide assistances to balance between aesthetics and usability.
• "It was nice that I could see the perception differences between public users and my personal thoughts and styles."(P2) • "Designers often want to design an aesthetic and unique icon, but sometimes they go too far that the icon becomes unrecognizable to users. With EvIcon, it would be easier to take both aesthetic and usability into consideration at the same time."(P3) • "Designers often add more styling details in the later phase of the iteration and worsen the icons' distinguishability. With EvIcon , we can check the perception feedback in each iteration to ensure the quality of our designed icons."(P4)
The designers also mentioned that the perception feedback could improve communication with their colleagues or clients if EvIcon is included in their design process.
• "I could convince the clients that my design is good with EvIcon."(P3) • "The results from EvIcon would be a promising report to defend our design against clients."(P4)
The designers confirmed that EvIcon could generally be useful and mentioned possibilities of how EvIcon can assist in different design phases. Moreover, they are willing to use EvIcon in their design process if it becomes a mature product in the future.
Limitations and Future Work
Single icon style As flat design continues to be a popular trend in digital design, our framework currently focuses on improving the usability of single-colored icons. However, we recognize the importance of expanding EvIcon's capabilities to include a wider range of icon styles. To achieve this, we plan to build a diverse dataset of icons and use metrics proposed in [GAHG17] to explore and compare icons in different styles. By doing so, we aim to improve the generalizability of EvIcon and make it more adaptable to the changing trends and preferences in digital design. Expanding the dataset and incorporating new metrics will enable EvIcon to produce icons in a wider variety of styles, ensuring that it remains a valuable tool for UI designers across different industries and contexts. User interface with limited icon editing functions While EvIcon's vector graphics editor offers a basic set of design tools, it may not be sufficient for the needs of some UI designers. The absence of advanced features could limit creativity and lead to a less efficient workflow. Our plan to integrate EvIcon as a plugin for professional design tools such as Adobe Illustrator and Sketch ‡ is aimed at addressing these limitations. By providing access to a more comprehensive suite of design tools, UI designers can expand their creative options and improve their efficiency. The integration will enable designers to access EvIcon's icon creation and editing features within their preferred design software, eliminating the need to switch between multiple tools. Ultimately, this will enable designers to produce higher quality icons more efficiently, resulting in better user experiences for their products. Supporting validations for general use of icons While the proposed framework is primarily focused on supporting designers in validating and revising icons for user interface design, it is important to note that icons have many other applications beyond UI design. Icons are widely used in presentation slides, infographics, and other forms of visual communication, where their design requirements may differ from those in UI design. For example, the icons used in infographics may require better abilities to convey information rather than better familiarity with viewers. Given the versatile nature of icons, we plan to extend the usage scenario and target audience of EvIcon to support icon improvement and selection for more general purposes. By doing so, we aim to make EvIcon a more versatile tool that can assist designers across various fields and contexts, not just limited to UI design. Expanding EvIcon's capabilities to accommodate different design requirements and user needs will enhance its value and relevance, making it an even more valuable tool for designers working on a wide range of projects.
Ultimately, this will enable designers to create more effective and engaging visual content across various domains, resulting in better user experiences for their audiences.
Conclusion
In this paper, we propose a human-in-the-loop framework called EvIconthat aims to enhance the usability of interface icon sets. Our framework includes a novel perceptual usability formulation and an interactive design tool that enable users to modify icons' effectiveness in conveying information. We also introduce the first icon dataset, IconCEPT10K, which features high-level perceptual usability ratings, such as semantic distance and familiarity, from over 5,000 crowdworkers. To demonstrate the effectiveness of EvIcon, we conducted a user study with six UI designers. Our quantitative and qualitative results show that using EvIconresulted in an icon set with improved usability, as rated by over 200 crowdworkers. These findings suggest that EvIconis an effective tool for facilitating the design process.
Figure 5 :
5EvIcon provides predicted perceptual usability feedback. Apart from viewing perception feedback for general people (a), users can inspect the feedback from different demographics categories including (b) age and (c) occupation.
Figure 6 :
6(a) Different color-codings indicate different semantic concept clusters. The icons are linked (b) in grey but will change (c) to red in order to notify poor visual distinguishability.these single-colored icons from multiple online resources, including Google Material Icons, Icon8, and The Noun Project. Although these icons are collected from different websites, they share similar visual styles due to the prevalence of flat UI design[Arl14,SRS18]. Overall, we collected 2, 613, 438 single-colored icons and their associated tags provided by the original designers. There are 191, 472 unique tags representing a wide range of concepts, and they provided us with a rich resource to model the relationship between icons and tags. We then used this icon and tag collection to train a joint image-text embedding.
Figure 7 :
7(a) The general-purpose CLIP [RKH * 21] is a joint image-text embedding trained on 400 million text-image pairs. (b)
Figure 8
8Figure 8: text-to-image retrieval by IconCLIP. Among the four tags we used as queries, only "attachment" and "map" are in the 212 base tags. The IconCLIP embedding space recognizes the meaning of "envelop" and "alarm" because we used the tags associated with icons (we use the green box for positive results and the red box for negative results).
Figure 9 :
9The six original icons and their examples of revised icons with and without EvIcon.
Figure 10 :
10Icon revision process starting from left to right by designers with and without EvIcon. (a) presents the revision processes for "Archive" icons of two designers, and (b) presents the revision processes for "Print" icons of two designers. Eight groups of revision process with the prediction of perception feedback and the crowdsourced evaluation results ("AMT rating") of the finalized icons are presented.
Figure 12 :
12The AMT ratings of the icons revised by all designers. (a) The ratings of semantic distance rating. (b) The ratings of familiarity rating. The error bars represent the standard deviation.
The feedback from a specific demographic is very useful. I can adjust the icons according to the feedback from my target user's category provided by the system. This tool definitely helps this."(P4) • "I am touched to see how this tool supports elders' feedback!designers
revised
result
most similar
icon in dataset
filter
print
archive
PSNR/SSIM
(40.62/0.248)
(42.13/0.83)
(38.77/0.29)
(37.01/0.17)
(38.46/0.20)
(38.95/0.73)
submitted to COMPUTER GRAPHICS Forum (5/2023).
‡ https://www.sketch.com/ submitted to COMPUTER GRAPHICS Forum (5/2023).
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Energy-brushes: Interactive tools for illustrating stylized elemental dynamics. J Xing, H Kazi R, Grossman T, Wei L.-Y, J Stam, G Fitzmaurice, 10.1145/2984511.2984585doi: 10.1145/2984511.2984585. 7Proceedings of the 29th Annual Symposium on User Interface Software and Technology. the 29th Annual Symposium on User Interface Software and TechnologyNew York, NY, USAAssociation for Computing MachineryUIST '16[XKG * 16] XING J., KAZI R. H., GROSSMAN T., WEI L.-Y., STAM J., FITZMAURICE G.: Energy-brushes: Interactive tools for illustrating stylized elemental dynamics. In Proceedings of the 29th Annual Sympo- sium on User Interface Software and Technology (New York, NY, USA, 2016), UIST '16, Association for Computing Machinery, p. 755-766. URL: https://doi.org/10.1145/2984511.2984585, doi: 10.1145/2984511.2984585. 7
Iconate: Automatic compound icon generation and ideation. Zhao N, W Kim N, M Herman L, H Pfister, W Lau R, J Echevarria, Bylinskii Z, Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems (2020). the 2020 CHI Conference on Human Factors in Computing Systems (2020)1ZKH * 20. submitted to COMPUTER GRAPHICS Forum (5/2023[ZKH * 20] ZHAO N., KIM N. W., HERMAN L. M., PFISTER H., LAU R. W., ECHEVARRIA J., BYLINSKII Z.: Iconate: Automatic compound icon generation and ideation. In Proceedings of the 2020 CHI Confer- ence on Human Factors in Computing Systems (2020), pp. 1-13. 1, 2, 3 submitted to COMPUTER GRAPHICS Forum (5/2023).
| {'fraction_non_alphanumeric': 0.0532468541245922, 'fraction_numerical': 0.036934907565636166, 'mean_word_length': 4.650281618023554, 'pattern_counts': {'":': 3, '<': 1, '<?xml version=': 0, '>': 5, 'https://': 13, 'lorem ipsum': 0, 'www.': 4, 'xml': 0}, 'pii_count': 5, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Interface icons are prevalent in various digital applications. Due to limited time and budgets, many designers rely on informal evaluation, which often results in poor usability icons. In this paper, we propose a unique human-in-the-loop framework that allows our target users, i.e., novice and professional UI designers, to improve the usability of interface icons efficiently. We formulate several usability criteria into a perceptual usability function and enable users to iteratively revise an icon set with an interactive design tool, EvIcon. We take a large-scale pre-trained joint image-text embedding (CLIP) and fine-tune it to embed icon visuals with icon tags in the same embedding space (IconCLIP). During the revision process, our design tool provides two types of instant perceptual usability feedback. First, we provide perceptual usability feedback modeled by deep learning models trained on IconCLIP embeddings and crowdsourced perceptual ratings. Second, we use the embedding space of IconCLIP to assist users in improving icons\' visual distinguishability among icons within the user-prepared icon set. To provide the perceptual prediction, we compiled IconCEPT10K, the first large-scale dataset of perceptual usability ratings over 10, 000 interface icons, by conducting a crowdsourcing study. We demonstrated that our framework could benefit UI designers\' interface icon revision process with a wide range of professional experience. Moreover, the interface icons designed using our framework achieved better semantic distance and familiarity, verified by an additional online user study. † Corresponding author uating icons designed for specific users (e.g., elders or users with lower computer literacy), conducting adequate usability testing is even more laborious.Zhao et al. [ZKH * 20] reported that designers often consult other UI designers\' feedback on icons. These informal evaluations often failed to provide comprehensive and objective information about how target users would perceive and use the icons [BAR92, RDF11], thus leading to low usability icons. As shown inFigure 2, even for icons of standard tags (e.g., "Search" and "Calendar"), accidentally omitting the critical visual features when adjusting icons\' style could lead to poor usability[Lin94,CUG20]. These examples further illustrate the importance of getting objective and instant feedback from users. These findings underscore the need for an objective and comprehensive usability testing approach that is cost-and time-efficient. Although several automatic graphical icon synthesis methods have been proposed [LRFN04, KPL08], involving humans (i.e., "human-in-the-loop") in the design process has several advantages (e.g., solving computationally complex problems [Hol16], building users\' trustworthiness to interactive systems [LGM20]).Hence, instead of proposing an automatic icon synthesis method,', 'arxivid': '2305.17609', 'author': ['I-Chao Shen [email protected] ', 'Fu-Yin Cherng [email protected] ', 'Takeo Igarashi \[email protected]\nThe University of Tokyo\nJapan\n', 'Wen-Chieh Lin ', 'Bing-Yu Chen ', '\nThe University of Tokyo\nJapan\n', '\nNational Chung Cheng University\nTaiwan\n'], 'authoraffiliation': ['[email protected]\nThe University of Tokyo\nJapan', 'The University of Tokyo\nJapan', 'National Chung Cheng University\nTaiwan'], 'corpusid': 258960103, 'doi': '10.48550/arxiv.2305.17609', 'github_urls': [], 'n_tokens_mistral': 22495, 'n_tokens_neox': 18820, 'n_words': 11531, 'pdfsha': '1d2b5108ad00e31d67c1560acd3232234823e3d7', 'pdfurls': ['https://export.arxiv.org/pdf/2305.17609v1.pdf'], 'title': ['EvIcon: Designing High-Usability Icon with Human-in-the-loop Exploration and IconCLIP', 'EvIcon: Designing High-Usability Icon with Human-in-the-loop Exploration and IconCLIP'], 'venue': ['Number z']} |
arxiv |
On the independence of Robinson's set of axioms for propositional calculus
18 Aug 2022
Benoît Jubin
On the independence of Robinson's set of axioms for propositional calculus
18 Aug 2022
We give a normal five-valued truth table proving independence of one of the axioms in Robinson's set of axioms for propositional calculus from 1968, answering a question raised in his article, where he uses a non-normal table. We also give a normal four-valued table proving independence of one of the other axioms, where he uses a normal five-valued table.
In his article [2], Thacher Robinson introduces a set of axioms for propositional calculus and proves their independence using many-valued truth tables. In one of his independence proofs (for the axiom (S) below), he uses a non-normal four-valued table, and he then asks in the last paragraph for a normal table showing independence. 1 He writes that Paul Bernays constructed for him such a normal six-valued table that was subsequently lost, and that apparently no normal five-valued tables exist. In this note, I give a normal five-valued table proving the required independence. The fact that it is normal implies the existence of a five-valued logic satisfying modus ponens and all of Robinson's axioms except that one.
The question is that of proving the independence of (S) in Robinson's set of axioms: 23 1 All these concepts are explained in his article. Normality is a property of a truth table for implication which is equivalent to the associated logic satisfying modus ponens. We may use the singular "table" to denote a set of tables, one for each connective, and say that this set is normal when its table for implication is normal. 2 Robinson gives two sets of axioms, one containing negation and the other containing falsum, but we can join these axioms in one set and consider both negation and falsum as primitive, instead of defining one of these connectives from the other and implication. 3 Robinson uses a commuted form of orelim but the tables given here satisfy both forms.
(K) p → q → p (1) (S) (p → q → r) → (p → q) → p → r (2) peirce ((p → q) → p) → p (3) andelimr p ∧ q → p (4) andeliml p ∧ q → q (5) andintro p → q → p ∧ q (6) orintror p → p ∨ q (7) orintrol p → q ∨ p (8) orelim p ∨ q → (p → r) → (q → r) → r (9) contrap (p → ¬q) → q → ¬p (10) notelim ¬p → p → q (11) falseelim ⊥ → p(12)
where the sole inference rule is modus ponens
mp p & p → q =⇒ q.(13)
An answer is given by the following normal five-valued table, where the only designated value (the value considered true) is 0: which validates modus ponens and all axioms except (S), which is for instance false for the assignment
[p ← 3, q ← 0, r ← 2].(14)
To prove independence of (K), Robinson uses a normal five-valued table. I found a normal four-valued table, where the designated values are 0, 1, 2: Testing my program on other examples, I found a shorter normal truth-table for
B' (p → q) → (q → r) → p → r,(16)W (p → p → q) → p → q,(17)pon p → (p → q) → q,(18)X ((((p → q) → q) → p) → r) → ((((q → p) → p) → q) → r) → r.(19)
Here is a normal three-valued table where the designated values are 0, 1:
modus ponens and all axioms except (K), which is for instance false for the assignment[p ← 2, q ← 1].(15)Addendum: another truth table
modus ponens and {W, pon, X} but falsifies B' with the assignment[p ← 1, q ← 0, r ← 1].(20)
implication for the following independence problem. In [1, §3], Meyer and Parks use a normal four-valued table (with one designated value), found by Sobociński, to prove independence of B' from {W, pon, X} where
Acknowledgments I would like to thank Norman Megill for bringing this problem to my attention, Mario Carneiro for independently confirming the truth tables, and Jean-Baptiste Bianquis for useful advice on OCaml matters.
Independent axioms for the implicational fragment of Sobocinski's three-valued logic. K Robert, Zane Meyer, Parks, Zeitschr. f. math. Logik und Grundlagen d. Math., Bd. 18Robert K. Meyer and Zane Parks, Independent axioms for the implicational fragment of Sobocinski's three-valued logic, Zeitschr. f. math. Logik und Grundlagen d. Math., Bd. 18, S. 291-295 (1972).
Independence of two nice sets of axioms for the propositional calculus. T , Thacher Robinson, The Journal of Symbolic Logic. 332T. Thacher Robinson, Independence of two nice sets of axioms for the propositional cal- culus, The Journal of Symbolic Logic, Vol. 33, No. 2 (June 1968), 265-270.
| {'fraction_non_alphanumeric': 0.07291433683739655, 'fraction_numerical': 0.02236636099306643, 'mean_word_length': 3.9910714285714284, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "We give a normal five-valued truth table proving independence of one of the axioms in Robinson's set of axioms for propositional calculus from 1968, answering a question raised in his article, where he uses a non-normal table. We also give a normal four-valued table proving independence of one of the other axioms, where he uses a normal five-valued table.", 'arxivid': '2109.14745', 'author': ['Benoît Jubin '], 'authoraffiliation': [], 'corpusid': 238226823, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 1422, 'n_tokens_neox': 1256, 'n_words': 813, 'pdfsha': 'b132d544efabde9725f05fcc82a17ef1fc7aedb0', 'pdfurls': ['https://export.arxiv.org/pdf/2109.14745v4.pdf'], 'title': ["On the independence of Robinson's set of axioms for propositional calculus", "On the independence of Robinson's set of axioms for propositional calculus"], 'venue': []} |
arxiv |
arXiv:quant-ph/0007095v1 25 Jul 2000 Note on reversibility of quantum jumps *
1996
Michael B Mensky †e-mail:[email protected]
P.N.Lebedev Physical Institute
117924MoscowRussia
arXiv:quant-ph/0007095v1 25 Jul 2000 Note on reversibility of quantum jumps *
Physics Letters A
22231996
It has been recently proved that a quantum jump may be reversed by a unitary process provided the initial state is restricted by some conditions. The application of such processes for preventing decoherence, for example in quantum computers, was suggested. We shall show that in the situation when the quantum jump is reversible it supplies no information about the initial state additional to the information known beforehand. Therefore the reversibility of this type does not contradict the general statement of quantum measurement theory: a measurement cannot be reversed. As a consequence of this, the coherence of a state (say, in a quantum computer) cannot be restored after it is destroyed by dissipative processes having a character of measurement.PACS number: 03.65.BzThe problem of preventing quantum decoherence in real systems became recently important in connection with the question about realizability of quantum computers[1,2]. Decoherence as a physical phenomenon may be considered in a more general framework of theory of quantum noise[3]. However, decoherence as a physical process arising in the course of a quantum measurement, has interesting specific aspects. Some of them will be discussed here, with important conclusions about possibility to prevent decoherence.
Mabuchi and Zoller has considered recently [4] a specific type of dissipation processes that may be characterized as a quantum jump i.e. disappearing of a photon, for example its absorption by a detector. The quantum jump may be described by an annihilation operator c (from the pair c, c † ) as the transition |ψ → c|ψ . It has been proved in [4] that the quantum jump can be reverted with the help of an unitary evolution provided the system has been before the jump in a state from a certain subspace. As a result, the initial state may be restored coherently, with the same phase relations as before the jump. This process was suggested as a possible mechanism for preventing decoherence, with possible application in theory of quantum computers.
A quantum jump is an example of a dissipative process. Therefore, the result of Mabuchi and Zoller [4] proves that some of dissipative processes may be reverted. Then the procedure providing the inversion of quantum jumps may serve as a method of preventing dissipation. We shall show however that the dissipation prevented in this way is not accompanied by obtaining new information and therefore cannot be identified with the decoherence arising in the process of a quantum measurement. It seems plausible that this is a general situation: decoherence cannot be inversed if any information is supplied by the process leading to this decoherence. This essentially restricts applicability of the procedure of Mabuchi and Zoller.
Our goal is therefore to show that the inversion of a quantum jump is possible only in the case when the jump supplies no information (additional to the information we had already before the jump), therefore it cannot be considered to be a measurement. The reversible dissipation is not a decoherence arising in the course of a quantum measurement.
1. Let the quantum jump be described by the annihilation operator c (from the pair of creation-annihilation operators c † , c). As it has been proved in [4], the quantum jump c may be reverted (i.e. the initial state of the system recovered) with the help of a unitary evolution, provided that the initial state belongs to some subspace of the state space H.
This means that the action of the operator c on an arbitrary state from the specified subspace is identical with the action of some unitary operator U . Recovering of the initial state is then possible with the help of the evolution described by the operator U † = U −1 .
For the goal of the general theoretical analysis of this situation, we shall denote by H 1 the subset of all vectors with this property, so that
c| H 1 = U | H 1 .(1)
If the quantum jump c could give some information about the initial state, then it might be interpreted as a measurement. The contradiction with quantum measurement theory could arise in this case: the effect of the measurement on the system might be completely discharged by a certain unitary evolution. Our task is to show that this is not the case. We shall prove that, as a consequence of Eq. (1), the event of the quantum jump c gives no information about the initial state (other than the information following from the fact that the initial state belongs to H 1 ). Therefore this event cannot be interpreted as a measurement.
For this end, we shall derive some properties of the states belonging to H 1 and prove that the quantum jump gives no information about the initial state besides that this state had these properties.
First of all, it follows directly from Eq. (1) that for an arbitrary vector |ψ ∈ H 1 the following equations are satisfied:
c|ψ = U |ψ , ψ|c † = ψ|U † .(2)
This means (because of unitarity of U ) that the mean photon number for an arbitrary state from the specified subset, |ψ ∈ H 1 , is equal to unity:
ψ|N |ψ = 1(3)
where N = c † c is an operator of the photon number. Let us expand the state |ψ in a series of terms corresponding to definite photon numbers:
|ψ = c 0 |ψ 0 + c 1 |ψ 1 + c 2 |ψ 2 + . . . + c n |ψ n + . . .(4)
where |ψ n is a (normalized) state with n photons. Then Eq. (3) reads as follows:
p 1 + 2p 2 + 3p 3 + . . . + np n + . . . = 1(5)
with positive numbers p n = |c n | 2 . For this equation being fulfilled, at least one of the numbers p 1 , p 2 , . . . p n , . . . must be non-zero. Therefore, the state |ψ ∈ H 1 cannot be vacuum. The expansion (4) must contain at least one non-vacuum component.
The quantum jump c diminishes the number of photons by unity. Therefore, the fact that the jump (the click of the detector) occurred, gives the information that the initial state has contained not less than one photon (could not be vacuum). However we know this already from Eq. (5). The event of the jump gives no new information and cannot be considered to be a measurement (provided we know already that the system has been in the subset H 1 before the jump).
Of course, if two quantum jumps occur, this will supply some new information: that the number of photons was not less than 2. This could be a measurement. This however again leads to no contradiction, because the action of the operator c 2 (describing a double jump) is not equivalent to the action of a unitary operator even in the subset H 1 . The double jump is a measurement, and its effect cannot be discharged by a unitary evolution. It is irreversible, in complete correspondence with general principles of quantum theory of measurements.
2. Let us suppose now that not only a single jump, but also a double jump may be unitarily reversed provided the system has been in the subset H 2 before the jumps. This means that two unitary operators U 1 , U 2 exist such that
c| H 2 = U 1 | H 2 , c 2 H 2 = U 2 | H 2 .(6)
Then the following relations may be readily derived for an arbitrary state from the subset, |ψ ∈ H 2 :
ψ|N |ψ = 1, ψ|N (N − 1)|ψ = 1.(7)
Using the expansion (4), we may rewrite the same in the form
p 1 + 2p 2 + 3p 3 + . . . + np n + . . . = 1 2p 2 + 6p 3 + . . . + n(n − 1)p n + . . . = 1.(8)
It is seen from Eqs. (8), that there is at least 2 photons in an arbitrary state of the subset H 2 (i.e. such a state cannot be a superposition of the vacuum and the 1-photon state). Therefore, neither a single, nor a double jump give no additional information in the case when the effects of both a single jump and a double jump can be unitarily discharged. the single and double jumps are not in this case measurements.
It is evident that the same consideration is applicable also to the case of a multiple jump with an arbitrary multiplicity.
3. Quantum jump that means for example a click of a detector is considered usually as a sort of measurement in the sense that it supplies a new information. It has been shown above that the quantum jump may give no information if something is known about the initial state. In this case a measurement supplying nontrivial information might take place in the preceding step when the initial state had been prepared. This step should contain projection from the complete space of states H onto some subspace belonging to H ∞ (or H ∈ if the situation with double jumps is considered) . After this preliminary measurement the quantum jump (or double jump) gives no new information. One can formally say that such a jump is a measurement, but it should be clearly understood that this measurement gives no additional information.
This is not at all astonishing and is in fact common in quantum theory of measurement. Indeed, the text-book example of a quantum measurement is the measurement of an observable A with a discrete spectrum. If we have a series of repeated measurements of this type (beginning from an unspecified state), then only the first measurement supplies a non-zero information giving the measurement output a i . All subsequent measurements of A will give with certainty the same result. The situation is quite analogous to the above discussion of quantum jumps if the stage of the preparation of an initial state is taken into account.
4. The above arguments support the general statement about irreversibility of quantum measurements (in the case when the initial state is not specified). This is important in the context of quantum computers and other devices depending on coherent character of their evolution.
One may hope to prevent decoherence resulting from dissipative processes, applying some or another correcting procedures, for example those proposed in [4]. It is shown in [4] that the initial state may be coherently restored after a certain dissipative processes. However the arguments of the present paper (apparently applicable to a more general situation) demonstrate that the restoration of the coherence is not always possible.
The class of dissipative processes leading to the irreversible decoherence may be specified by the concept of measurement or information. The restoration of the coherence turns out to be impossible if the dissipation is accompanied by obtaining information about the state of a quantum system. 1 This essentially restricts the circle of situations in which recoherence is in principle feasible. This resulting restriction should be taken into account together with other principal difficulties in creating quantum computers [2].
Of course, we do not necessarily mean a measurement arranged on purpose. Instead, it may be an interaction with the environment (reservoir) that results in recording the information in the state of the environment even if nobody is interested in this information.
ACKNOWLEDGEMENT The author is indebted to I.Bialynicki-Birula and P.Zoller for stimulating discussions and W.Schleich for his kind hospitality in Ulm University during the workshop of the Ulm and Innsbruck quantum-optics groups.
A Ekert, ; D Wineland, Proceedings of the 14th ICAP. the 14th ICAPNew YorkAIP Press450A.Ekert, in Proceedings of the 14th ICAP, edited by D.Wineland et al. (AIP Press, New York, 1995), p.450.
. Rolf Landauer, Phys. Lett. A. 217188Rolf Landauer, Phys. Lett. A 217, 188 (1996).
C W Gardiner, Quantum Noise. BerlinSpringerC.W.Gardiner, Quantum Noise (Springer, Berlin, 1991).
Inversion of quantum jumps in quantum-optical systems under continuous observation. H Mabuchi, P Zoller, Phys. Rev. Lett. 763108H. Mabuchi and P.Zoller, Inversion of quantum jumps in quantum-optical systems under continuous observation, Phys. Rev. Lett. 76, 3108 (1996).
| {'fraction_non_alphanumeric': 0.03745600804424334, 'fraction_numerical': 0.01592089827383945, 'mean_word_length': 4.232354230600614, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'It has been recently proved that a quantum jump may be reversed by a unitary process provided the initial state is restricted by some conditions. The application of such processes for preventing decoherence, for example in quantum computers, was suggested. We shall show that in the situation when the quantum jump is reversible it supplies no information about the initial state additional to the information known beforehand. Therefore the reversibility of this type does not contradict the general statement of quantum measurement theory: a measurement cannot be reversed. As a consequence of this, the coherence of a state (say, in a quantum computer) cannot be restored after it is destroyed by dissipative processes having a character of measurement.PACS number: 03.65.BzThe problem of preventing quantum decoherence in real systems became recently important in connection with the question about realizability of quantum computers[1,2]. Decoherence as a physical phenomenon may be considered in a more general framework of theory of quantum noise[3]. However, decoherence as a physical process arising in the course of a quantum measurement, has interesting specific aspects. Some of them will be discussed here, with important conclusions about possibility to prevent decoherence.', 'arxivid': 'quant-ph/0007095', 'author': ['Michael B Mensky †e-mail:[email protected] \nP.N.Lebedev Physical Institute\n117924MoscowRussia\n'], 'authoraffiliation': ['P.N.Lebedev Physical Institute\n117924MoscowRussia'], 'corpusid': 119487339, 'doi': '10.1016/0375-9601(96)00636-6', 'github_urls': [], 'n_tokens_mistral': 3002, 'n_tokens_neox': 2688, 'n_words': 1998, 'pdfsha': '4831b35f2c63f43bb06cc82dabfea1b12bbdd46a', 'pdfurls': ['https://export.arxiv.org/pdf/quant-ph/0007095v1.pdf'], 'title': ['arXiv:quant-ph/0007095v1 25 Jul 2000 Note on reversibility of quantum jumps *', 'arXiv:quant-ph/0007095v1 25 Jul 2000 Note on reversibility of quantum jumps *'], 'venue': ['Physics Letters A']} |
arxiv |
AdS/CFT and strongly coupled quark matter
Sep. 12-16 2006. Sep. 20-22 2006. Nov. 21-22 2006. Nov. 20-28 2006. Dec. 26-27 2006
Makoto Natsuume
Institute of Particle and Nuclear Studies
Theory Division
KEK, High Energy Accelerator Research Organization
305-0801TsukubaIbarakiJapan
AdS/CFT and strongly coupled quark matter
CCAST)
2006Sep. 12-16 2006. Sep. 20-22 2006. Nov. 21-22 2006. Nov. 20-28 2006. Dec. 26-27 2006
We review the AdS/CFT description of gauge theory plasmas for non-experts. We discuss the low shear viscosity, jet quenching, and J/ψ-suppression, which are three major signatures for the quarkgluon plasma observed at RHIC experiments. Based on invited talks presented at "Frontiers in the physics of quark-gluon plasma"(July 7-8 2006, RIKEN), "String theory and quantum field theory"
I. INTRODUCTION
In this note, we review the connection between AdS/CFT duality and quark-gluon plasma (QGP) experiments at RHIC (see Ref. [1] for a review of QGP physics). RHIC stands for Relativistic Heavy Ion Collider at Brookhaven National Laboratory. The name "heavy ion" comes from the fact that it collides heavy ions such as gold nuclei 197 Au instead of usual e + e − , pp or pp. The goal of the experiment is to realize the deconfinement transition and form the quark-gluon plasma. In principle, it should be possible to form QGP if one has high enough temperature or high enough density. However, it is not an easy job to confirm QGP formation because of the following problems: First, what one observes is not QGP itself but only the by-products after hadoronization, and one has to infer what had happened from the by-products. Second, those secondary particles are mostly strongly-interacting, and the perturbative QCD is not very reliable for the current and near-future experimental temperatures.
To resolve these problems, many attempts are made to identify the generic signatures of QGP. Some of the generic signatures discussed to date are as follows:
1. The elliptic flow which may be the consequence of very low viscosity of QGP 2. The jet quenching
J/Ψ-suppression
All of these signatures have been discussed in the AdS/CFT duality, so I review recent developments focusing on these phenomena.
II. A SHORT COURSE ON STRING THEORY
Since this review is aimed at non-experts, I first give a short course on string theory, in particular emphasizing * Electronic address: [email protected] on how gauge theories and black holes are described in string theory. Figure 1 shows main ingredients of string theory. There are two kinds of string: open strings with endpoints and closed strings with no endpoints. As I will show you shortly, an open string represents a gauge theory. A closed string represents a graviton. An open string propagates through spacetime just like a closed string, but an open string can also have its endpoints on an object, the so-called D-brane. The open string I consider in this lecture is always of this type.
The easiest way to see that an open string represents a gauge theory is look at how the string oscillates. Figure 2 shows the simplest open string oscillation (in 4 dimensions). 1 As you can see, the string can oscillate in Since these open strings are constrained to have their endpoints on the D-brane, the gauge theory described by the D-brane is localized on the D-brane. The Dbranes arise with various dimensionalities. A D-brane with a p-dimensional spatial extension is called the Dpbrane. Namely, the D0-brane is point-like, the D1-brane is string-like, the D2-brane is membrane-like, and so on. Thus, the Dp-brane describes a (p+1)-dimensional Yang-Mills theory. We are interested in 4-dimensional gauge theories, so consider the D3-brane in order to mimic QCD.
On the other hand, a closed string represents a graviton. Again, the easiest way to see this is to look at how the string oscillates (Fig. 4). In general, the oscillations on a string have two modes: the left-moving modes and right-moving modes. For an open string, these modes mix each other at endpoints, but these modes become independent for a closed string. So, one can oscillate the right-moving mode in one direction and the left-moving mode in the other direction. In a sense, a closed string oscillates in two directions simultaneously. This property explains the spin-2 nature of the graviton. In fact, a graviton also oscillates in two directions simultaneously. Strictly speaking, a closed string represents a graviton and two undiscovered scalar particles, the dilaton and the axion. Since each mode has two degrees of freedom, a closed string has 4 degrees of freedom at this level (in 4 dimensions). The graviton has only 2 degrees of freedom, and two scalar fields cover the remaining degrees of freedom.
We have seen that we can get a gauge theory from the D-brane, but it is not clear if the D-brane is simply described by a gauge theory. This is because string theory is more than a gauge theory, namely it has gravitons. At this point, it is not clear if the effect of gravity can be neglected. According to general relativity, any energymomentum tensor curves spacetime. The D-brane of course has some energy, so how the D-branes curve spacetime? Since gravity is described by the Newton potential
φ Newton ∼ GM r , (2.1)
one can measure the effect of curvature by GM . According to string theory, the Newton constant G and the mass M of the D-brane are given by
G ∼ g 2 s , (2.2) M ∼ N/g s , (2.3)
where g s is the string coupling constant which governs the strength of the interactions between strings. 2 So, one gets GM ∼ g s N . This means that as long as g s N ≪ 1, one can neglect the effects of gravity and spacetime remains flat. In this case, the D-brane is simply described by a gauge theory. On the other hand, when g s N ≫ 1, the D-brane starts to curve spacetime, and eventually it should become a black hole. So, in this case, the D-brane can be described by a black hole. The black hole here is not the usual Schwarzschild-like black hole. We consider a D-brane, an object with infinite spatial extension. Thus, the black hole formed from the D-brane has an horizon which extends indefinitely and is called the "black brane."
To summarize, one can describe the D-brane both by a gauge theory and by a black hole; which description is better depends on the value of g s N . The string coupling and the Yang-Mills coupling g YM are related by g s ∼ g 2 YM (see Sec. III B), so g s N is nothing but the standard 't Hooft coupling, λ := g 2 YM N . This suggests that the black hole is the large 't Hooft coupling limit of a gauge theory. Since 't Hooft coupling is the effective coupling of a gauge theory, the black hole is the strong coupling limit of the gauge theory (Fig. 5). Thus, the strategy here is to use black holes in order to compute gauge theory observables in the strong coupling regime. 3 Our argument here is very rough, but more refined version of the argument is known as the AdS/CFT duality [3,4,5,6] (see Ref. [7] for a review). The precise correspondence is as follows: Here, N = 4 means that the theory has 4 supercharges which are the maximum number of supercharges for a 4-dimensional theory. Also, AdS 5 stands for the fivedimensional anti-deSitter space. DeSitter was a Dutch astronomer who found a solution of Einstein equation with a constant positive curvature in 1917. The space AdS 5 instead has a constant negative curvature; this explains the prefix "anti." One can reach this correspondence by studying the D3-brane more carefully, but I will skip the argument. Instead, I explain the correspondence from the symmetry point of view in Appendix. We will use the finite temperature version of the duality and its cousins; in this case, one needs to replace AdS 5 by a black hole in AdS 5 , which is known as the Schwarzschild-AdS 5 black hole: N = 4 SYM at finite temperature ↔ type IIB string theory on (Schwarzschild-AdS 5 black holes) × S 5 .
III. BLACK HOLES AND HYDRODYNAMICS
According to the RHIC experiment, QGP behaves like a liquid. The AdS/CFT then implies that a black hole also behaves like a liquid. Then, plasma quantities should be calculable from black holes. In fact, black holes and hydrodynamic systems behave similarly. Consider adding a perturbation to a black hole, e.g., drop some object (Fig. 6). Then, the shape of the black hole horizon becomes irregular, but such a perturbation decays quickly, and the black hole returns to the original symmetric shape. The no-hair theorem is one way to see this. According to the theorem, the stationary black hole is unique and symmetric. Thus, the perturbed black hole cannot be stable. If you regard this as a diffusion, the diffusion occurs since the perturbation is absorbed by the black hole.
This behavior is very similar to a liquid. Suppose that one drops a ball in a water pond. Then, you generate surface waves, but they decay quickly, and the water pond returns to a state of stable equilibrium. In hydrodynamics, this is a consequence of viscosity. Thus, one can consider the notion of viscosity for black holes as well. And the "viscosity" for black holes should be calculable by considering the above process.
Let me remind you of freshman physics of viscosity. As a simple example, consider a fluid between two plates and move the upper plate with velocity v (Fig. 7). As the fluid is dragged, the lower plate experiences a force. This force is the manifestation of the viscosity. In this case, the force F the lower plate experiences per unit area is given by
F A = η v L . (3.1)
The proportionality constant η is called the (shear) viscosity. Microscopically, the viscosity arises due to the momentum transfer between molecules. Figure 7 shows a closeup view of the fluid and I put an artificial boundary to divide the fluid into two parts. The molecules collide with each other and are exchanged randomly through the boundary. But in the situation where you move the upper plate, the molecules in the upper-half part, on average, have more momentum in the x-direction than the ones in the lower-half part. These molecules are exchanged, which means that momentum in the x-direction is transported through the boundary.
Going back to the black hole, how can one calculate the plasma viscosity? I will first give a brief explanation and its implications and then I will justify the claim more in detail.
A. Quick argument
There are many ways to compute the viscosity and I will explain one simple method. In the gravity side, the diffusion occurs by black hole absorption. So, it is natural to associate the shear viscosity with the absorption cross section by black holes. (I explain this point more in detail later. The detailed argument suggests that this is the cross section for the graviton of particular polarization.) Now, there is a general theorem on black holes [8] that states that the cross section σ BH is equal to the horizon FIG. 6: When one adds a perturbation to a black hole, the black hole behavior is similar to a hydrodynamic system. In hydrodynamics, this is a consequence of viscosity. area A for a broad range of black holes 4 :
η ∝ lim ω→0 σ BH = A . (3.2)
But the horizon area is the famous quantity, namely it represents the black hole entropy, so it must be the 4 As a simple example, this is true for the usual Schwarzschild black hole as well. Precisely speaking, the theorem applies only to the low energy limit ω → 0.
plasma entropy 5
S BH = A 4Gh k B (3.3) (k B : Boltzmann constant)
. Then, the shear viscosity divided by the entropy becomes constant. 6 The constant can be determined from the argument later and the result is
η s =h 4πk B . (3.4)
This value is very small. In comparison, η/s for water is about 3 × 10 3 under normal circumstances. Now, the point is that all the relations we used (cross section versus horizon area, black hole entropy versus horizon area) are generic, so the result must be universal as well. Namely, it does not depend on the details of black holes nor the details of gauge theories. So, the claim is [9] Gauge theory plasmas which have gravity duals have a universal low value of η/s at large 't Hooft coupling. This is a rather indirect argument, but this claim has indeed been checked for many known gravity duals.
This result is very important, so let me rephrase in a different way (Fig. 8). Gauge theories of which we can actually compute the shear viscosity are supersymmetric gauge theories, not the real QCD. We compute the shear viscosity from black holes, but the gravity dual of QCD is not known. So, one cannot use AdS/CFT directly to compute QCD properties. However, as we saw, the quantity corresponding to the shear viscosity is universal FIG. 8: Gauge theories described by D-branes are mostly supersymmetric gauge theories, and not the real QCD. However, the black hole quantity corresponding to the "shear viscosity" is universal, so probably the results for supersymmetric gauge theories are directly applicable to the real QCD.
FIG. 9: The bulk graviton produces the back-to-back scattering of gluons on the boundary. In black hole picture, it is natural to regard the graviton decay rate as the absorption cross section by the black hole.
on the black hole side, so one can immediately apply the N = 4 result to the real QCD even though these two theories are completely different.
In fact, RHIC suggests that QGP has a very low viscosity and the estimated value [10,11]
η s ∼ O(0.1) ×h k B ? (3.5)
is very close to the above AdS/CFT value. 7 One important point is that the temperature in question is still order of Λ QCD . At this range of temperature, QCD is still strongly coupled and pQCD is not very reliable. In fact, the naive extrapolation of the weak coupling result gives a larger value for η/s. Thus, the AdS/CFT duality which predicts the strong coupling behavior may be useful to analyze QGP. 7 See Refs. [12,13] for recent estimates.
B. More in detail I now describe the relation between the shear viscosity and the absorption cross section. To do so, one first has to understand the interactions of bulk and boundary fields. These fields can interact at the boundary. For example, the graviton produces the back-to-back scattering of gluons (Fig. 9). The relevant interactions are
S int ∼ d 4 x δφF 2 µν + δh µν T YM µν + · · · (3.6)
(φ: dilaton, h µν : graviton, F µν and T YM µν : the field strength and the energy-momentum tensor of the gauge theory). Such interactions can be obtained by expanding the D-brane action around the expectation values of the bulk fields. The D-brane is described by the so-called Dirac-Born-Infeld (DBI) action. The DBI-action with at most two derivatives contains the following term:
L ∼ e −φ F 2 µν + · · · . (3.7)
If φ is constant, one gets the standard gauge theory Lagrangian with g 2 YM ∼ g s , where g s := e φ . If φ fluctuates, the action is expanded as
L ∼ e − φ F 2 µν + δφF 2 µν , (3.8) where φ = φ − δφ.
The second term is nothing but the first term in Eq. (3.6). This fact could be stated as
Bulk field fluctuations act as sources of boundary fields I just rephrase the same statement, but this leads to the so-called "GKP-Witten relation," which is the definition of the AdS/CFT. Given the interaction term, one can easily calculate the graviton decay rate (with polarization h xy , where x and y are directions along the brane) from the standard field theory formula:
σ QFT = 1 S f 1 2ω final states d 3 p 1 (2π) 3 2ω 1 d 3 p 2 (2π) 3 2ω 2 × (2π) 4 δ 4 (p f − p i )|M| 2 = 8πḠ hω d 4 x e iωt [T YM xy (t, x), T YM xy (0, 0)] .
The first equality is just the Fermi's golden rule with matrix element M. The matrix element is proportional to T YM xy , so the formula is written by a correlator of the energy-momentum tensor. (This is an optical theorem.)
The factor S f is a statistical factor for identical particles in the final state.
In black hole description, it is natural to regard this decay rate as the absorption cross section of the graviton. This has been checked for the D3-brane at zero temperature [14], so 8
σ BH = 8πḠ hω d 4 x e iωt [T YM xy (t, x), T YM xy (0, 0)] . (3.9)
We can use this relation to compute the shear viscosity since the viscosity is given by a Kubo formula microscopically:
η = lim ω→0 1 2hω d 4 x e iωt [T YM xy (t, x), T YM xy (0, 0)] .
(3.10) This formula has the same form as the absorption cross section, so one can immediately write the viscosity in terms of the cross section:
η = lim ω→0 σ BH 16πG . (3.11)
We have written plasma quantities in terms of black hole quantities. Using Eqs. (3.2) and (3.3), one obtains
η s = A 16πG A 4Gh k B =h 4πk B . (3.12)
This is the previous formula (3.4).
C. Further checks of universality
We saw that in gauge theories at strong coupling the ratio of shear viscosity to entropy density is universal. How far has the universality been shown? Recently, the universality has been extended to a variety of situations which are not covered in the original proofs.
The shear viscosity was first computed for the D3brane [15]. This corresponds to the N = 4 SYM which is a scale-invariant theory. (Actually, the theory has a larger symmetry, a conformal symmetry. See Appendix.) But the universality has been shown even for theories with scales, i.e., nonconformal theories. Examples include Dp-branes for p = 3, the Klebanov-Tseytlin geometry, the Maldacena-Nunez geometry, and the N = 2 * 8 As a cautionary remark, one should note that Eq. (3.9) is known to hold for the D3-brane at zero temperature but it is unclear if it also holds for generic AdS black holes. The problem is that a black hole is the strong coupling limit of a gauge theory, so Eq. (3.9) is the strong coupling statement. But, of course, it is not easy to compute the right-hand side in gauge theory. One can compute it in the weak coupling, but weak coupling results differ from strong coupling results in general as we will see in Sec. III D.
In the special case of the zero-temperature D3-brane, such a comparison actually makes sense due to the nonrenormalization theorem, and this is the case where Eq. (3.9) has been checked. But it is an open question if this is also true for generic AdS black holes. Here, we simply assume that the relation holds at strong coupling.
system (These theories also have a reduced number of supersymmetries). In fact, various universality arguments were demonstrated for these cases [9,16,17,18]. However, these proofs did not cover the cases with a chemical potential. So, the natural question is what happens to the universality at finite chemical potential. Actually, it is not easy to realize the realistic finite density in AdS/CFT, i.e., baryon number density. But there is a simple alternative. Namely, consider charged AdS black holes instead of neutral black holes. A black hole is known to obey thermodynamic-like laws and its first law is written as dM = T dS BH + ΦdQ (3.13) (T : black hole temperature, Φ: electromagnetic potential, Q: black hole charge). As one can see, the electromagnetic field Φ plays a role of a chemical potential. In AdS/CFT, such a charge arises as follows. As in Eq. (2.4), the full geometry involves S 5 . One can add an angular momentum along S 5 , which is known as the "spinning" D3-brane solutions [19,20]. The angular momentum becomes a Kaluza-Klein charge after the S 5 reduction.
What is the gauge theory interpretation of the charge? The symmetry of S 5 corresponds to an internal symmetry of the SYM, R-symmetry SO(6). This SO(6) rotates adjoint scalars in the N = 4 supermultiplet. (See Appendix.) The R-symmetry group SO(6) is rank 3, so one can add at most three independent charges. The threecharge solution is known as the STU solution [21]. When all charges are equal, the STU solution is the well-known Reissner-Nordström-AdS 5 black hole. Thus, the charge in question is a U (1) R charge.
Because the charge corresponds to the U (1) R charge, this is by no means realistic. However, the theory has interesting features which are common to the real QCD. For instance, the phase diagram is qualitatively similar to the QCD diagram [22,23]. 9 This system does not represent a realistic chemical potential, but it may mimic the realistic case and one may learn an interesting lesson for gauge theory plasmas at finite density.
The shear viscosity for charged AdS black holes was computed by 4 groups, and the result turns out to be η/s = 1/(4π) again [24]- [27]. So, the universality seems to hold even at nonzero chemical potential. If this is true for generic chemical potential, η/s = 1/(4π) may be true even at finite baryon number density. In fact, when there is a chemical potential associated with a global symmetry, the universality has been proven in more general settings as well [28]- [30].
The theories described so far all have matter in the adjoint representation and not in the fundamental repre-sentation such as quarks. However, the universality has been proven even in the presence of fundamental matter [31]. One way to include fundamental matter is to include D7-branes in addition to D3-branes. I will describe one simple way to realize fundamental matter in the next section, and the D3-D7 system is an extension of the method. This D3-D7 system is known as Karch-Katz model [32].
Gauge theory plasmas described so far are stationary ones. The real plasma at RHIC is of course a rapidly changing system, and it is desirable to study such a plasma as well. In the gravity side, this corresponds to a time-dependent black hole. The universality has been claimed even for such a case [33]. 10
D. Other issues
We saw that η/s = 1/(4π), but this is in the strong coupling limit or in the large 't Hooft coupling limit λ → ∞. Real QCD of course has finite coupling, so how about η/s at finite coupling? In general, the shear viscosity is given by η ∼ ρvl mfp (3.14)
(ρ: mass density,v: mean velocity, l mfp : mean free path). At weak coupling, the interaction becomes weaker, so the mean free path becomes larger. The viscosity arises due to the momentum transfer, and the transfer is more effective at weak coupling. So, at weak coupling, one expects η/s ≫ 1/(4π). 11 For the N = 4 SYM, the finite-λ correction can be computed, and the correction indeed raises the value of η/s [44]. The correction is about 9% if one uses λ = 6π (or α SYM = 1/2). Thus, the strong coupling result seems a good approximation. The universality does not hold for finite-λ corrections though, so it is not clear how realistic the result is. Figure 10 shows both the weak coupling result and the strong coupling result. They both increase at weak coupling. However, there is a difference. The viscosity is proportional to the mean free path, so is inversely proportional to the coupling constant. Thus, the naive 10 In this subsection, we included relatively recent discussions only. See Refs. [34]- [43] for early computations and computations of the other transport coefficients. 11 Actually, we have two parameters g YM and N (or g YM and λ), so there are two kinds of finite-coupling corrections. First is the finite-λ correction described here. This corresponds to the socalled α ′ -corrections in the gravity side. The effective actions of string theory contain higher curvature terms in addition to the Einstein-Hilbert action; e.g., S ∼ √ −g(R + α ′3 O(R 4 ) + · · ·). The finite-λ correction comes from such higher curvature terms. On the other hand, the finite-g YM correction comes from the socalled string loop corrections since the string coupling constant is proportional to g 2 YM . Such a correction to η has never been evaluated. extrapolation of the weak coupling result suggests the perfect-fluid-like behavior in the strong coupling limit. Instead, AdS/CFT tells that η/s cannot be small indefinitely and is saturated.
Incidentally, our problem is the strong coupling problem, so one might wonder how useful lattice computations are. A lattice computation of the viscosity for the pure Yang-Mills theory has been done in Ref. [46]. Unfortunately, the result has very large error bars, so one cannot make a definite statement, but it would be very interesting to make the errors smaller.
IV. HEAVY QUARKS IN MEDIUM
Recently, there has been much discussion on heavy quark dynamics in plasma medium. Two applications have been discussed: issues related to J/ψ suppression and issues related to jet quenching. In this section, I quickly summarize the discussion.
A. J/Ψ-suppression
Since J/Ψ is heavy, charm pair production occurs only at the early stages of the nuclear collision. However, if the production occurs in the plasma medium, charmonium formation is suppressed due to the Debye screening. One technical difficulty is that the cc pair is not produced at rest relative to the plasma. Therefore, the screening length is expected to be velocity-dependent. Such a computation has been done only for the Abelian plasma [47].
First, one has to understand how to realize a heavy quark in AdS/CFT. To do so, let us go back to string can have endpoints in various ways; there are N 2 possibilities. This means that the string transforms as the adjoint representation of SU (N ) gauge theory. Now, consider an infinitely long string (Fig. 11). In this case, the string can have endpoints in N different ways. This means that the string transforms as the fundamental representation of SU (N ) gauge theory. In this sense, such a long string represents a "quark." Such a string has an extension and tension, so the string has a large mass, which means that the long string represents a heavy quark.
This kind of string has been widely studied in the past to measure the heavy quark potential [48,49]. For a qq pair, two individual strings extending to the boundary is not the lowest energy configuration. Instead, it is energetically favorable to have a single string that connects the pair (Fig. 12). The energy difference is interpreted as a qq potential, and one gets a Coulomb-like potential for N = 4.
At finite temperature [50,51], it is no longer true that a string connecting the qq pair is always the lowest energy configuration; for large enough separation of the pair (L s ), isolated strings are favorable energetically. This phenomenon is the AdS/CFT description of the Debye screening and L s may be interpreted as the screening length.
References [52,53,54] proposed how to compute the velocity dependence of the screening length. 12 They computed the screening length in the qq rest frame, i.e., they considered the plasma flowing at a velocity v. Such a "plasma wind" is obtained by boosting a black hole.
At the leading order in v, the screening length is obtained as (screening length)
∝ 1 ǫ 1/4 0 (1 − v 2 ) 1/4 (4.1)
∼ (boosted plasma energy density) −1/4 for the N = 4 SYM, where ǫ 0 is the unboosted energy density. Thus, the screening effect becomes stronger than the v = 0 case. One can also compute the screening length for the other gauge theories [55]. The leading behavior in v seems universal. Namely, if one writes the screening length as
(screening length) ∝ (1 − v 2 ) ν , (4.2)
the exponent ν is determined by the speed of sound c s :
4ν = 1 − 3 4 (1 − 3c 2 s ) + · · · (4.3)
when the theory is nearly conformal, i.e., c 2 s ∼ 1/3. One can make a simple estimate of the exponent for QCD. According to the lattice results cited in Ref. [60], all groups roughly predict 1/3 − c 2 s ∼ 0.05 around 2T c . Bearing in mind that our results are valid to large-N theories and not to QCD, Eq. (4.3) gives ν ∼ 0.22. It would be interesting to compare this number with lattice calculations and experimental results.
One can also study the screening length at finite chemical potential [55,56]. At the leading order, the screening length at finite chemical potential is the same as the one at zero potential for a given energy density.
B. Jet quenching
Another interesting QGP phenomenon is jet quenching. In the parton hard-scattering, jets are formed, but the jets have to travel in the QGP medium, so the jets are strongly suppressed. This phenomenon is known as jet quenching. So, the interesting quantity is the energy loss rate of partons. The AdS/CFT descriptions of jet quenching have been proposed recently [61]- [64]. The proposals have been quickly extended to the other gauge theories [65]- [73]. 13 To discuss jet quenching, one now moves the fundamental string with a velocity v along a brane direction. Then, the momentum carried by the string flows towards the horizon and one interprets the flow as the energy loss rate. For example, the energy loss rate for the N = 4 SYM becomes
dp dt = − π 2 √ λT 2 v √ 1 − v 2 . (4.4)
Unfortunately, the result obtained in this way has some drawbacks. First, the result is not universal and is modeldependent. Second, the black hole results become exact only in the λ → ∞ limit, but the result does not have a finite large-λ limit. These two drawbacks are in contrast to the η/s case. This may suggest that one has to be careful to apply AdS/CFT to QGP. Namely, one should not always take AdS/CFT results at face value.
V. TOWARDS "ADS/QGP"
Hydrodynamic description of gauge theory plasmas using AdS/CFT is very powerful due to the universality. The AdS/CFT may be useful to analyze experiments. Conversely, experiments or the other theoretical approaches (such as lattice gauge theory) may be useful to confirm AdS/CFT. This approach is also important since currently there are many loose ends on the AdS/CFT derivation. One has to clear up these loose ends, but it may be hard to make further progress within string theory alone. So, the inputs from the other areas may be useful to increase confidence in the AdS/CFT derivation. On the other hand, one has to be careful to apply AdS/CFT to QGP if the universality does not hold (e.g., jet quenching).
In this talk, I emphasized the universality approach, but of course finding the gravity dual of QCD is desirable. The biggest assumption of the universality argument is that such a dual indeed exists, so finding the dual is necessary for the universality approach as well. One well-known model of QCD is the Sakai-Sugimoto model [82]. This was a successful model in the confining phase, but unfortunately it is not a good model in the plasma phase. The Sakai-Sugimoto model involves the D4-brane compactified on a circle. Above the deconfinement temperature, the model looks as a true five-dimensional theory.
Another problem of the current approach concerns the large-λ limit; AdS/CFT and QGP are actually different limits. The large-λ limit of the AdS/CFT is g YM → 0 and N → ∞. But of course QGP has a large-λ since 13 See Refs. [74]- [80] for the other issues. See also Ref. [81] for an early attempt.
One can reach this correspondence by studying the D3brane more carefully, but instead I explain the correspondence from the symmetry point of view in this Appendix. 3) representations), the former represent a gauge field. The latter represent scalars. The spatial dimension is 9 and the brane dimension is 3, so there are 6 scalar fields. Thus, the gauge theory represented by the D3-brane inevitably comes with scalar fields. The N = 4 SYM is such a theory.
The field contents of the N = 4 SYM include the gauge field A µ and the scalars φ i . (The color indices are suppressed for simplicity. We will also suppress the spinor indices.) In addition, there are 4 fermions λ I due to supersymmetry (which comes from the supersymmetry in superstring.) The Lorentz transformation properties are different, but they come from similar string oscillations, which means that all these fields transform as the adjoint representation. Namely, the theory has no fundamental representation such as quarks.
Also, if we have only D3-branes, as in the present case, the directions transverse to the brane are all isotropic. These directions correspond to the scalar fields, so the isotropy means that there is a global SO(6) symmetry for φ i . Such a global symmetry is known as R-symmetry.
The action for the N = 4 SYM is given by
L = 1 g 2 Y M {− 1 4 F 2 µν − 1 2 (D µ φ i ) 2 − i 2λ I γ µ D µ λ I + O(φ 4 ) + O(λλφ)} .(A2)
First 3 terms are standard kinetic terms for the gauge field, the scalar fields, and the fermions. In addition, the action contains interaction terms which are written only schematically; φ 4 interactions and Yukawa-like interactions.
The gauge field and the scalars have mass dimension 1 and the fermions have mass dimension 3/2, so all terms in the action have mass dimension 4, which means that the theory has no dimensionful parameter and the theory is scale invariant. Actually, it is often the case in a relativistic field theory that the scale invariance and Poincaré symmetry SO(1, 3) combine into a larger symmetry, the conformal symmetry SO(2, 4). Thus, the N = 4 SYM has the global SO(2, 4) × SO(6) R symmetry.
Why AdS5 × S 5
The AdS 5 space is a spacetime with constant negative curvature. We are familiar with spaces with positive curvature, like the sphere, and have no problem visualizing them, but it is more difficult to visualize a space with negative curvature. A space with constant negative curvature is known as a hyperbolic space. One way to visualize a hyperbolic space is illustrated in famous woodcuts by Escher (e.g., "Circle Limit III"). It is not easy to visualize AdS 5 , but one can represent AdS 5 as a hypersurface in a flat spacetime, which is useful for our purpose.
The AdS 5 space is described by
X 2 0 + X 2 5 − X 2 1 − · · · − X 2 4 = l 2 (A3)
in a six-dimensional flat space
ds 2 6 = −dX 2 0 − dX 2 5 + dX 2 1 + · · · + dX 2 4 .(A4)
[Even though the space (A4) has "two times," the AdS itself has the signature (−, +, · · · , +).] Here, l gives the characteristic length scale of the AdS 5 space and it is related to the 't Hooft coupling from the discussion in the text. Note that if all signs are positive in Eq. (A3), Eq. (A3) defines a sphere. Also, Eq. (A3) is a higher dimensional generalization of the standard hyperbola x 2 − y 2 = 1. In this form, it is clear that AdS 5 has a SO(2, 4) symmetry just like the N = 4 conformal symmetry; to say that theories are dual to each other, they should have the same symmetries.
In addition, the full geometry in Eq. (A1) involves S 5 which has a SO(6) symmetry. This is the same symmetry as the N = 4 R-symmetry. So, the gravity side also has the global SO(2, 4) × SO(6) symmetry.
FIG. 1 :
1Main ingredients of string theory. An open string can have endpoints on the D-brane (left) and describes a gauge theory. A closed string represents a graviton (right). FIG. 2: The simplest oscillations of an open string.
FIG. 3 :
3N D-branes represent a SU (N ) gauge theory. two directions. So, the open string has two degrees of freedom at this level. These degrees of freedom represent the two polarizations of the photon. In this sense, the open string represents a gauge theory. Of course, our interest here is not QED, but rather QCD, so how can one describe a Yang-Mills theory? An open string has endpoints on a D-brane, but if there are N coincident D-branes, open strings can have endpoints in various ways(Fig. 3). These new degrees of freedom precisely correspond to SU (N ) degrees of freedom.
FIG. 4 :
4A closed string has two independent modes, which corresponds to the spin-2 nature of the graviton.FIG. 5: A gauge theory is described by a black hole in the strong coupling.
N
= 4 super-Yang-Mills theory (SYM) ↔ type IIB string theory on AdS 5 × S 5 . (2.4)
FIG. 7 :
7When one moves the upper plate, the fluid is dragged due to the viscosity and the lower plate experiences a force. The bottom figure is the close-up view of the fluid.
FIG. 10 :
10The coupling constant dependence of η/s both from the weak coupling result and the strong coupling result. The red line represents the weak coupling result, i.e., a naive extrapolation of the perturbative N = 4 SYM result (Based on Ref.[45]). The blue line represents the strong coupling result, i.e., the N = 4 AdS/CFT result with the first order correction in λ (Based on Ref.[44]).
Fig. 3 .
3When there are multiple number of D-branes, an open FIG. 11: An infinitely long string transforms as the fundamental representation. FIG. 12: For a qq pair, the left configuration is not the lowest energy configuration. Instead, the right configuration is energetically favorable. The energy difference is interpreted as a qq potential. At finite temperature, this is no longer true. For large enough separation of the pair, the left configuration becomes favorable. This phenomenon is the AdS/CFT description of the Debye screening.
APPENDIX A: THE ADS/CFT DUALITYThe precise correspondence between gauge theory and gravity is as follows: N = 4 super-Yang-Mills theory (SYM) ↔ type IIB string theory on AdS 5 × S 5 .
1 .
1Why N = 4 SYM To understand why the N = 4 SYM appears, let us go back to the open string oscillations on the D3-brane. Superstring theory actually requires 10-dimensional spacetime for consistency. The open strings on a D-brane are bounded on the D-brane, so the D3-brane represents a 4-dimensional gauge theory, but these open strings still oscillate in the full 10-dimensional spacetime. Thus, the simplest open string oscillations have 8 degrees of freedom instead of 2 discussed in the text. What are these degrees of freedom? In other words, what kind of gauge theory the D3-brane represents? To see this, note that there are two types of string oscillations. First are the oscillations in the brane and the other are the oscillations out of the brane. From the 4-dimensional point of view (in terms of SO(1,
g YM ∼ O(1) and N = 3. We are not taking the same limits, so it is not clear why both give such close results.I would like to thank Elena Cáceres, Kengo Maeda, and Takashi Okamura for collaboration and for their comments on the manuscript. It is a pleasure to thank Testuo Hatsuda, Tetsufumi Hirano, Kazunori Itakura, Tetsuo Matsui, Osamu Morimatsu, Berndt Müller, and Tadakatsu Sakai for useful conversations. I would also like to thank the organizers of various workshops for the opportunity to give this lecture. The research of M.N. was supported in part by the Grant-in-Aid for Scientific Research (13135224) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.Acknowledgments
String theory is actually a 10-dimensional theory, but we discuss in 4-dimensions for illustrating purpose.
Note that the D-brane is infinitely heavy in the weak coupling limit gs ≪ 1, which is consistent with the hypersurface picture of the D-brane, and the open strings attached represent the fluctuations of the D-brane.
Conversely, such a correspondence has also been used for the microscopic derivation of black hole entropy[2].
We use the unit c = 1 in this lecture.6 Precisely speaking, one divides by the entropy density s. We consider black holes with infinite extension, so the entropy itself diverges. The area for the absorption cross section should be understood in a similar way.
This is the case of the RN-AdS black hole with compact horizon or compact SYM. In this review, we consider the RN-AdS black hole with noncompact horizon, which is always in plasma phase.
See also Refs.[55]-[59] for extensions and the other issues.
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arxiv |
NaRe2(PO4)3 phosphate-based ceramic with kosnarite structure as a matrix for technetium immobilization. Production. Properties
L S Alekseeva
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
A V Nokhrin
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
A I Orlova
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
M.SBoldin
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
E A Lantcev
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
А А Murashov
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
K K Korchenkin
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
D V Ryabkov
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
V N Chuvil'deev
Lobachevsky State University of Nizhny Novgorod
Russian Federation
23 Gagarina ave603022Nizhny Novgorod
NaRe2(PO4)3 phosphate-based ceramic with kosnarite structure as a matrix for technetium immobilization. Production. Properties
2 V.G. Khlopin Radium Institute, JSC, Russian Federation, 194021 St. Petersburg, 28 2nd Murinsky ave.mineral-like matriceskosnariteceramicshydrolytic testingleaching mechanism
NaRe2(PO4)3 phosphate-based ceramic with the structure of kosnarite mineral was obtained by spark plasma sintering. Rhenium (Re) served as a chemical and structural analog of technetium. The ceramic relative density was 85%. The mechanism of Re static leaching from NaRe2(PO4)3 ceramic at room temperature was investigated. The leaching rate of rhenium was 1.3×10 -5 g/(cm 2 ×day).
Introduction
Technetium-99 ( 99 Tc) is one of the most widespread and long-lived radiotoxic isotopes in spent nuclear fuel (SNF). It is one of the key elements in SNF separation strategies such as UREX + for isolation and encapsulation in solid waste forms [1]. Immobilizing 99 Tc is a complex scientific and practical challenge. Thus far, immobilization of 99 Tc as metal alloys [2][3][4], cements [5][6][7], and glasses [2,[8][9] has been researched. During glass transition, which is the most widely used process for high-level radioactive waste (RAW), technetium is partly oxidized or disproportionated and therefore evaporates as heptaoxide, causing environmental pollution. An alternative approach to immobilizing 99 Tc consists in introducing a cation into a mineral-like matrix [10][11][12][13][14][15][16], which has the potential to help avoid formation of highly mobile pertechnetate TcO4ˉ ions during RAW processing.
Tc-containing compounds with the structures of pyrochlore (Nd2Tc2O7), perovskite (SrTcO3) and lamellar perovksite (Sr2TcO4) have been studied in [13,14]. 28 days of hydrolytic tests of Nd2Tc2O7 pyrochlore-based ceramic helped establish the Tc leaching rate to be 1.4810 -7 g/(mm 2 ·day), which is approximately four times slower than the rate of leaching from borosilicate glass (~6.4310 -7 g/(mm 2 ·day)). A disadvantage of the compounds studied in [13,14] is the long (2-10 days) duration of high-temperature annealing during solid-phase synthesis.
Including technetium in compounds in Mg2Ti1-xTcxO4 spinels has been studied in [11,16]. Synthesis of such compounds is demanding and time consuming and their leaching rate was (3-7)10 -3 g/(m 2 ·day) after testing for 40 days at room temperature.
In [15] describes compounds with NZP structure of ARe2(PO4)3 (where А = AM [alkali metal]). These compounds were produced by fusing concentrated phosphoric acid, ammonium perrhenate, and AM chlorides, further annealed at 400 °C in air and at 550 °C under an inert atmosphere in a vacuum-sealed ampoule. There have been no studies of chemical stability of the synthesized compounds. This paper researches NaRe2(PO4)3 phosphate with kosnarite structure as a potential ceramic matrix for immobilizing 99 Tc. Rhenium (Re) served as a chemical and structural analog of technetium. Solution chemistry was used to obtain NaRe2(PO4)3 while the ceramic was produced through Spark Plasma Sintering (SPS) [17].
Materials and methods
A solution of 1M orthophosphoric acid was gradually added to a mixture of sodium chloride and ammonium perrhenate solutions taken in stoichiometric amounts with constant vigorous stirring in order to obtain NaRe2(PO4)3 powders with the NZP structure. Their synthesis occurred as per the following formula: NaCl + 2NH4ReO4 + 3H3PO4 = NaRe2(PO4)3 + HCl + 2NH4OH + 4H2O. The resulting clear solution was evaporated at 80 °C with constant stirring, dried at 200 and 300 °C until moisture was completely removed and the intermediate reaction products partially decomposed. The load turned black after annealing. The resulting powder was annealed in a vacuum-sealed quartz ampoule at 600, 700, or 800 °C for 12 hours.
A Shimadzu LabX XRD-6000 X-ray diffractometer (CuКα filtered radiation) was used to determine the phase composition of the powders and ceramics. A DSC-204 F1 Phoenix thermal analyzer was used to perform a differential thermal analysis within 25-1200 °С range.
The heating rate was 10 °C/min. A Shimadzu FTIR-8400S IR Fourier spectrophotometer was used to study the functional composition of the compounds at room temperature within a frequency range of 400-4000 cm -1 .
A Dr. Sinter model SPS-625 was used to produce the ceramics. The powders were placed in a graphite mold with an inner diameter of 10 mm and heated by millisecond pulses of direct electric current of high power (up to 3 kA). A Chino IR-AH pyrometer focused on the graphite mold surface was used to measure the sintering temperature. Sintering was performed in vacuum (6 Pa). The accuracy of temperature measurements was ± 10 °C, with a pressure maintenance accuracy of 1 MPa. The dilatometer in the Dr. Sinter model SPS-625 was used to monitor the shrinkage and shrinkage rate of the powders. During the experiment, the normalized weight loss was calculated using the following formula:
NLi = aki / (Moi × S),(1)
where NLi is the normalized weight loss of element i, g/cm 2 ; aki is the mass of component i dissolved during leaching, g; M0i is the mass concentration of the element in the sample at the beginning of testing, g/g; S is the sample surface area, cm 2 .
The leaching rate Ri was calculated using the following formula:
Ri = NLi / tn,(2)
where tn is the time interval, days.
The DeGroot -van der Sloot model [18] was used to determine the mechanism of cation leaching from the ceramic, which can be represented as the following equation:
lgBi = Algt + const,(3)
where Bi is the total yield of Re from the sample during contact with water, mg/m 2 ; t is the contact time, days. The Bi value was calculated using the following formula:
Bi = Ci(L/S)√tn / (√tn-√tn-1),(4)
where Ci is the concentration of Re in the solution by the end of the n th period, mg/l; L/S is the ratio of solution volume to sample surface area, l/m 2 ; tn and tn-1 is the time of the n-and n-1 experiment stages, respectively, in days.
The following leaching mechanisms correspond to the values of the coefficient A in equation (3): <0.35 -leaching from the surface of the compound; 0.35-0.65 -diffusion from inner layers; >0.65 -dissolution of the compound surface layer [19,20].
Results and discussion
According to the XRD data (Fig. 1), a monophase product is obtained after annealing at 700 °C. Once annealed at 600 and 800 °C, the powder contains insignificant traces of ReP2O7 rhenium pyrophosphate and ReO2 rhenium (IV) oxide, respectively. The resulting monophasic compounds crystallized in the expected NZP structure and belonged to the R-3c space group.
Further studies were performed with the powder synthesized at 700 °C.
The particle size distribution of the powder was heterogeneous. The electron microscopy data suggested that the synthesized powder contained two types of particles: large faceted particles approx. 10 μm in size, which are likely to be single crystals, and agglomerates consisting of particles up to 1 μm in size (Fig. 2a, b). Energy dispersive microanalysis confirmed the presence of Re in the structure of the synthesized powder (Fig. 2c).
Differential scanning calorimetry (DSC) (Fig. 3) revealed an endothermic effect in the temperature range of 970-1030 °C, which corresponded to the decomposition of NaRe2(PO4)3
and was confirmed by the thermal gravimetric analysis (TGA).
A wide band in the range of 980-1120 cm -1 and a high-frequency band at 1247 cm -1 of the IR spectrum under study (Fig. 4) can be attributed to asymmetric stretching ν3 vibrations of the PO4 3ion. We believe that a low-intensity high-frequency band in the range of 1247 Preliminary sintering of NaRe2(PO4)3 powders was performed by heating them to 1100 °С at a rate of 50 °С/min (Seq. # 1) to define the optimal SPS modes. Fig. 5a shows the sintering modes as dependences between temperature (Т, °С), applied uniaxial pressure (Р, kN), vacuum levels (Vac, Pa), and SPS time. Fig. 5b shows the dependence between shrinkage (L, mm), shrinkage rate (S, mm/s), and the heating temperature. Fig. 5b demonstrates that the powder shrinkage starts at Т = 600 °С; with the shrinkage rate being small and not exceeding Smax ~ 4.510 -3 mm/s (Fig. 5b). A shrinkage peak was observed at 1000 °С, which is likely to be associated with a phase transition in NaRe2(PO4)3 powder. It should also be noted that at temperatures above 850 °C powder decomposition was observed, which manifested itself in a decrease in the vacuum level in the sintering chamber of the Dr.
Sinter model SPS-625 (Fig. 5a). Sample decomposition during SPS upon heating to 1100 °С prevented its density from being reliably determined through the Archimedes' principle.
As a way to minimize or exclude decomposition of the powder sample, further sintering of the ceramics was performed at Тs = 800 °С with different heating rates (Seq. # 2-4). It is of interest to note that the maximum shrinkage rates Smax increased slightly from ~0.210 -3 to ~1.510 -3 mm/s with an increase in the heating rate from 50 to 200 °С/min. The sintered samples were a powder compressed to a density ρrel ≈ 85% with a particle size similar to the size of the initial powder particles d = 1-10 μm (Fig. 6). After sintering, many particles retained their faceted shape, indicating a low intensity of diffusion during SPS. The post-sintering density was insufficient to produce durable ceramics. The phase composition of the ceramic did not change after sintering (Fig. 7). Table 1 shows the minimal rates of Re leaching from NaRe2(PO4)3 samples. Fig. 8 shows graphs of dependence between the normalized weight loss NLi, leaching rate Ri, and testing time t. The resulting data suggest that the rhenium leaching rate on day 28 was R = 1.310 -5 g/(cm 2 ·day). The calculated leaching rate is slightly lower, yet comparable to the rate of technetium leaching from other mineral-like compounds [11,13,14,16]. It should be noted that the real surface area S of the ceramics under study is greater than the value calculated based on geometric dimensions because of their increased porosity (Fig.6). We therefore believe that these results characterize the terminal value R for this compound.
Using formula (3) to determine the mechanism of rhenium leaching from NaRe2(PO4)3 ceramic, the dependence between coefficient B and the experiment time t was plotted in logarithmic coordinates (Fig. 9). The data in Fig. 9 show that the value of coefficient A is -
Conclusions
NaRe2(PO4)3 monophasic phosphate with the structure of the kosnarite mineral was obtained by synthesis at 700 °C. An increased or decreased synthesis temperature does not allow a monophasic product to be obtained. The synthesized NaRe2(PO4)3 compound manifests a non-uniform particle size distribution, with large single-crystal particles of ~ 10 μm and agglomerated μm particles present in the powder.
In terms of achieving the maximum possible density of the ceramic (ρrel= 85.3%) while maintaining the phase composition, the optimal SPS mode consists in heating the sample to Тs = 800 °С at a rate of Vh = 100 °С/min under uniaxial pressure of Р = 70 MPa.
The terminal rate of rhenium leaching from NaRe2(PO4)3 ceramic on Day 28 was 1.3·10 -5 g/(cm 2 ·day), which makes it possible to class the phosphate under study as highly hydrolytically stable. It was established that the dominant mechanism of rhenium leaching in the static mode at room temperature is the leaching of cations from the surface of the ceramic. Table 1). lg t, day lgB, mg/m 2
A
Sartorius CPA balance was used to measure the density of the sintered samples by hydrostatic weighing in distilled water. Microstructural parameters of the samples were analyzed using a Jeol JSM-6490 scanning electron microscope (SEM) with an Oxford Instruments INCA 350 X-ray microanalyzer. The chemical stability of the ceramics was studied by leaching in a static mode during 28 days. The tests were performed at room temperature in distilled water. The Re concentration in water samples was determined using an ELEMENT 2 high-resolution inductively coupled plasma mass spectrometer using external calibration. Calibration was performed using the ICP-MS-68A-A High-Purity Standards solutions and iDplus Performance time-of-flight mass spectrometer.
0.38-0.3. It allows us to conclude that rhenium leaching occurs through washing out of the open surface of NaRe2(PO4)3 ceramic. This result is qualitatively in good agreement with the previous conclusion that open porosity has a significant effect on the high rate of Re leaching from NaRe2(PO4)3 ceramic.
Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .
23456Microstructure of initial NaRe2(PO4)3 at various magnifications (a, b) and the results of energy dispersive microanalysis (b) of synthesized powder. SEM DSC data for NaRe2(PO4)3 powder after annealing at 700 °C IR spectrum of NaRe2(PO4)3 powder after annealing at 700 °C Typical representation of the temperature dependence between shrinkage (L) and shrinkage rate (S) of NaRe2(PO4)3 powder under SPS (Sample # 1) Macrostructure (a, c, e) and microstructure (b, d, f) of NaRe2(PO4)3 samples produced through SPS at heating rate Vh = 50 (a, b), 100 (c, d), 200 (e, f) °С/min (See
Fig. 7 .
7X-ray diffraction patterns of NaRe2(PO4)3 ceramic
Fig. 8 .
8Hydrolytic tests results for NaRe2(PO4)3 ceramics. Dependence of the normalized weight loss (NL), rhenium leaching rate (R), and testing time t
Fig. 9 .
9Logarithmic dependence between the rhenium yield and time of contact with
Figure 3
Figure 6
Figure 7
Figure 8
8Figure 8
Figure 9
9Figure 9
cm -1 , which is not typical for phosphates, may be associated with the influence of the Re 4+ highly charged ion that polarizes the P-O-Re bond. The band in the range of 883 cm -1 powder was performed under constant heating rate. There was no isothermal holding at the sintering temperature.Table 1contains the main sintering parameters of the ceramic samples -heating rate Vh, average applied uniaxial pressure P, sintering temperature Ts and isothermal time ts at sintering temperature.corresponded to symmetric stretching ν1 vibrations. The bands in the 640-438 cm -1 range
belong to bending vibrations: the bands within 650-540 cm -1 corresponded to asymmetric ν4
vibrations while the 445 см -1 band corresponded to symmetric ν2 vibrations.
SPS of NaRe2(PO4)3 Table 1. Main parameters of sintering NaRe2(PO4)3 powders, density and hydrolytic stability
of the resulting samples
Sequence Ts, °С
Vh,
°С/min
ts,
min
Р, MPa
ρ,
g/cm 3
ρrel, %
NLRe,
g/cm 2
RRe,
g/(cm 2 ·day)
1
1100
50
0
70
-
-
-
-
2
800
50
0
70
4.154
84.9
4.4·10 -4
1.3·10 -5
3
800
100
0
70
4.174
85.3
-
-
4
800
200
0
70
4.168
85.2
-
-
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| {'fraction_non_alphanumeric': 0.06162973725632643, 'fraction_numerical': 0.05622149574201881, 'mean_word_length': 4.304645686148576, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'NaRe2(PO4)3 phosphate-based ceramic with the structure of kosnarite mineral was obtained by spark plasma sintering. Rhenium (Re) served as a chemical and structural analog of technetium. The ceramic relative density was 85%. The mechanism of Re static leaching from NaRe2(PO4)3 ceramic at room temperature was investigated. The leaching rate of rhenium was 1.3×10 -5 g/(cm 2 ×day).', 'arxivid': '2111.12973', 'author': ['L S Alekseeva \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'A V Nokhrin \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'A I Orlova \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'M.SBoldin \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'E A Lantcev \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'А А Murashov \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'K K Korchenkin \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'D V Ryabkov \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n', 'V N Chuvil'deev \nLobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod\n'], 'authoraffiliation': ['Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod', 'Lobachevsky State University of Nizhny Novgorod\nRussian Federation\n23 Gagarina ave603022Nizhny Novgorod'], 'corpusid': 244709288, 'doi': '10.1134/s0020168522030013', 'github_urls': [], 'n_tokens_mistral': 8616, 'n_tokens_neox': 7160, 'n_words': 3588, 'pdfsha': '95589c9fb6204d38f26bf735b11dafbe61df9189', 'pdfurls': ['https://arxiv.org/pdf/2111.12973v1.pdf'], 'title': ['NaRe2(PO4)3 phosphate-based ceramic with kosnarite structure as a matrix for technetium immobilization. Production. Properties', 'NaRe2(PO4)3 phosphate-based ceramic with kosnarite structure as a matrix for technetium immobilization. Production. Properties'], 'venue': []} |
arxiv |
Note on the closed-form MLEs of k-component load-sharing systems
16 Mar 2012
Chanseok Park
Department of Mathematical Sciences
Clemson University Clemson
29634SC
Note on the closed-form MLEs of k-component load-sharing systems
16 Mar 2012Reliabilityload-sharingmaximum likelihood estimate (MLE)closed-form solution
Consider a multiple component system connected in parallel. In this system, as components fail one by one, the total load or traffic applied to the system is redistributed among the remaining surviving components, which is commonly referred to as load-sharing.(2004)andSingh et al. (2008)proposed different load-sharing models and developed parametric inference for the these models.Recently Kim and KvamHowever, their parametric estimates are calculated using iterative numerical methods. In this note, we provide the general closed-form MLEs for the two load-sharing models provided by them.
Introduction
Most research work involving load sharing models has mainly focused on the characterization of system reliability under a known load-sharing rule and parameters.
The parameter estimation of the load-sharing rule has not yet been fully developed.
Recently, parametric inference for reliability under the equal load-sharing rule has been considered by Kim and Kvam (2004) and Singh et al. (2008). They solved the likelihood estimating equations to find the maximum likelihood estimators (MLEs) of the load-sharing parameters.
However, they provide no general closed form solutions for the MLEs, but instead use iterative numerical methods to calculate their estimates. It is well known that there are some problematic issues associated with iterative numerical methods such as stability and convergence.
In this note, we provide the general closed-form MLEs for the two load-sharing models provided by Kim and Kvam (2004) and Singh et al. (2008).
Kim-Kvam load-sharing model
Consider k-component system connected in parallel. Following Kim and Kvam (2004), we assume the following:
(i) A system is made up of k components whose lifetimes are independent and have identical exponential distributions with initial failure rate θ.
(ii) After the first component fails, the failure rates of the remaining k − 1 components change to λ 1 θ where λ 1 > 0. After the next component failure, the failure rates of the surviving k − 2 components change to λ 2 θ. Then, after the next component failure, the the failure rates of the surviving k − 3 components change to λ 3 θ and so on and so forth.
(iii) There are n repeated measurements of independent systems. That is, we have a random sample of independent systems of size n.
Let X im denote the lifetime of the m-th component in the i-th parallel system where i = 1, 2, . . . , n and m = 1, 2, . . . , k. For notational convenience, we can re-index the lifetimes such that X i1 < X i2 < · · · < X ik . Then the time spacing between the (j −1)-th failure and j-th failure for the i-th system is T ij = X ij −X i,j−1 with X i0 = 0.
As shown in Kim and Kvam (2004), the likelihood function for the i-th system is
L i (θ, Λ) = (k!)θ k · k j=1 λ j−1 · exp − θ k j=1 (k − j + 1)λ j−1 t ij ,
where λ 0 = 1 and Λ = (λ 1 , . . . , λ k−1 ). It is immediate that the likelihood function for a random sample of size n is given by
L(θ, Λ) = (k!) n θ nk · k j=1 λ n j−1 · exp − θ n i=1 k j=1 (k − j + 1)λ j−1 t ij .(1)
Taking the logarithm of (1), differentiating with respect to θ, λ 1 , . . . , λ k−1 , denoting partial derivative of log L with respect to θ as ℓ θ = ∂ log L/∂θ and partial derivative of log L with respect to λ j−1 as ℓ j−1 = ∂ log L/∂λ j−1 , we obtain the log-likelihood estimating equations shown below:
ℓ θ = nk θ − n i=1 k j=1 (k − j + 1)λ j−1 t ij = 0(2)
and (2) and (3) can be rewritten as
ℓ j−1 = n λ j−1 − θ n i=1 (k − j + 1)t ij = 0 (3) for j = 2, 3, . . . , k. For convenience, we denote t •j = n i=1 t ij . Then,ℓ θ = nk θ − k j=1 (k − j + 1)λ j−1 t •j = 0(4)
and
ℓ j−1 = n λ j−1 − θ(k − j + 1)t •j = 0.(5)
It is immediate from solving (5) for λ j−1 that we have
λ j−1 = n θ(k − j + 1)t •j , j = 2, . . . , k.(6)
Since λ 0 = 1, we rewrite (4) as
nk θ − kt •1 − k j=2 (k − j + 1)λ j−1 t •j = 0.(7)
Substituting (6) into (7) gives
nk θ − kt •1 − n(k − 1) θ = 0.(8)
Solving (8) for θ, we obtain the MLE of θ, denoted byθ,
θ = n kt •1 = n k n i=1 t i1 .(9)
The MLEs of λ j−1 , denoted byλ j−1 , are also easily obtained by substituting (9) into
(6)λ j−1 = kt •1 (k − j + 1)t •j = k n i=1 t i1 (k − j + 1) n i=1 t ij , j = 2, 3, . . . , k.
Singh-Sharma-Kumar load-sharing model
Consider k-component system connected in parallel. Following Singh et al. (2008), we assume the following:
(i) A system is made up of k components whose lifetimes are independent and have exponential distributions with initial failure rate θ.
(ii) After the first component fails, the failure rates of the remaining k − 1 components change to λ 1 θ where λ 1 > 0. After the next component failure, the failure rates of the surviving k − 2 components change to λ 2 θ. Then, after the next component failure, the the failure rates of the surviving k − 3 components change to λ 3 θ and so on and so forth.
After the failure of a certain number of components, say, after the s-th component failure (s ≥ 2), the failure rates of the k − s remaining components change to λ s tθ (linearly increasing failure rate). In a similar manner, after the (s + 1)th component failure, the failure rates of the k − s − 1 remaining components change to λ s+1 tθ, and so on. Finally, after the last failure, the failure rate of the last component becomes λ k−1 tθ.
(iii) There are n repeated measurements of independent systems. That is, we have a random sample of independent systems of size n.
Again, let X im denote the lifetime of the m-th component in the i-th parallel system where i = 1, 2, . . . , n and m = 1, 2, . . . , k. For notational convenience, we can re-index the lifetimes such that X i1 < X i2 < · · · < X ik . Then the time spacing between the (j − 1)-th failure and j-th failure for the i-th system is
T ij = X ij − X i,j−1 with X i0 = 0.
As is given in Singh et al. (2008), the likelihood function for the i-th system is
L i (θ, Λ) =(k!)θ k · k j=1 λ j−1 · k j=s+1 t ij × exp − θ s j=1 (k − j + 1)λ j−1 t ij + 1 2 k j=s+1 (k − j + 1)λ j−1 t 2 ij ,
where λ 0 = 1 and Λ = (λ 1 , . . . , λ k−1 ). It is immediate that the likelihood function for a random sample of size n is given by
L(θ, Λ) =(k!) n θ nk · k j=1 λ n j−1 · n i=1 k j=s+1 t ij × exp − θ n i=1 s j=1 (k − j + 1)λ j−1 t ij + 1 2 k j=s+1 (k − j + 1)λ j−1 t 2 ij .(10)
Taking the logarithm of (10), differentiating with respect to θ, λ 1 , . . . , λ k−1 , denoting partial derivative of log L with respect to θ as ℓ θ = ∂ log L/∂θ and partial derivative of log L with respect to λ j−1 as ℓ j−1 = ∂ log L/∂λ j−1 , we obtain the loglikelihood estimating equations shown below:
ℓ θ = nk θ − n i=1 s j=1 (k − j + 1)λ j−1 t ij + 1 2 k j=s+1 (k − j + 1)λ j−1 t 2 ij = 0,(11)ℓ j−1 = n λ j−1 − θ n i=1 (k − j + 1)t ij = 0, j = 2, 3, . . . , s,(12)
and
ℓ j−1 = n λ j−1 − θ 2 n i=1 (k − j + 1)t 2 ij = 0, j = s + 1, . . . , k.(13)
Let us y ij define as
y ij = (k − j + 1)t ij I [j≤s] + 1 2 (k − j + 1)t 2 ij I [j>s] ,(14)
where I A is an indicator function whose value is one if A is satisfied and is zero if A is not satisfied. Then (11), (12) and (13) can be expressed as
ℓ θ = nk θ − n i=1 k j=1 λ j−1 y ij = 0,(15)
and
ℓ j−1 = n λ j−1 − θ n i=1 y ij = 0, j = 2, 3, . . . , k.(16)
Notice that, by using (14), we have combined the two equations in (12) and (13) which resulted in the equation (16).
For convenience, let y •j = n i=1 y ij . Using this and considering λ 0 = 1, we can rewrite (15) and (16) as follows:
ℓ θ = nk θ − y •1 − k j=2 λ j−1 y •j = 0,(17)
and
ℓ j−1 = n λ j−1 − θy •j = 0, j = 2, 3, . . . , k.(18)
Solving (18) for λ j−1 , we have
λ j−1 = n θy •j , j = 2, 3, . . . , k.(19)
Substituting (19) into (17) gives
nk θ − y •1 − n(k − 1) θ = 0.(20)
Solving (20) for θ, we obtain the MLE of θ, denoted byθ,
θ = n y •1 = n n i=1 y i1 .(21)
The MLEs of λ j−1 are also easily obtained by substituting (21) into (19)
λ j−1 = y •1 y •j = n i=1 y i1 n i=1 y ij , j = 2, 3, . . . , k.
Notice that given the definition of y ij , we can rewrite y •j in the following manner:
y •j = n i=1 (k − j + 1)t ij = (k − j + 1) n i=1
t ij , j = 2, 3, . . . , s 1 2 n i=1 (k − j + 1)t 2 ij = 1 2 (k − j + 1) n i=1 t 2 ij , j = s + 1, . . . , k .
Therefore we can write the closed form MLEs for θ and λ j−1 aŝ
θ = n k n i=1 t i1 andλ j−1 = k n i=1 t i1 (k − j + 1) n i=1 t ij , j = 2, 3, . . . , s k n i=1 t i1 1 2 (k − j + 1) n i=1
t 2 ij , j = s + 1, . . . , k .
Reliability estimation based on system data with an unknown load share rule. H Kim, P H Kvam, Lifetime Data Analysis. 10Kim, H. and Kvam, P. H. (2004). Reliability estimation based on system data with an unknown load share rule. Lifetime Data Analysis, 10:83-94.
. 10.1023/B:LIDA.0000019257.74138.b6http://dx.doi.org/10.1023/B:LIDA.0000019257.74138.b6.
A classical and Bayesian estimation of a k-components load-sharing parallel system. B Singh, K Sharma, A Kumar, Computational Statistics & Data Analysis. 52Singh, B., Sharma, K., and Kumar, A. (2008). A classical and Bayesian estimation of a k-components load-sharing parallel system. Computational Statistics & Data Analysis, 52:5175-5185.
. 10.1016/j.csda.2008.05.026http://dx.doi.org/10.1016/j.csda.2008.05.026.
| {'fraction_non_alphanumeric': 0.10498742665549037, 'fraction_numerical': 0.05081726739312657, 'mean_word_length': 3.007136859781696, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 9, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Consider a multiple component system connected in parallel. In this system, as components fail one by one, the total load or traffic applied to the system is redistributed among the remaining surviving components, which is commonly referred to as load-sharing.(2004)andSingh et al. (2008)proposed different load-sharing models and developed parametric inference for the these models.Recently Kim and KvamHowever, their parametric estimates are calculated using iterative numerical methods. In this note, we provide the general closed-form MLEs for the two load-sharing models provided by them.', 'arxivid': '1203.3747', 'author': ['Chanseok Park \nDepartment of Mathematical Sciences\nClemson University Clemson\n29634SC\n'], 'authoraffiliation': ['Department of Mathematical Sciences\nClemson University Clemson\n29634SC'], 'corpusid': 88518867, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 3843, 'n_tokens_neox': 3368, 'n_words': 1883, 'pdfsha': '8398dc73e913059bc21cedba4591dff0c3970787', 'pdfurls': ['https://arxiv.org/pdf/1203.3747v1.pdf'], 'title': ['Note on the closed-form MLEs of k-component load-sharing systems', 'Note on the closed-form MLEs of k-component load-sharing systems'], 'venue': []} |
arxiv |
3 Mar 2017
Zdeněk Dušek [email protected]
Faculty of Science Rokitanského 62
University of Hradec Králové
500 03Hradec KrálovéCzech Republic
3 Mar 2017The affine approach to homogeneous geodesics in homogeneous Finsler spaces Zdeněk Dušek Dedicated to Professor Oldřich Kowalski on the occasion of his 80th birthday Address of the author:Homogeneous Finsler space, homogeneous geodesic
In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present paper, homogeneous geodesics in Finsler homogeneous spaces are studied using the affine method, which was developed in earlier papers by the author. The mentioned statement is proved correctly and it is further proved that any homogeneous Berwald space or homogeneous reversible Finsler space admits a homogeneous geodesic through any point.MSClassification: 53C22, 53C60, 53C30
Introduction
Let M be either a pseudo-Riemannian manifold (M, g), or a Finsler space (M, F ), or an affine manifold (M, ∇). If there is a connected Lie group G which acts transitively on M as a group of isometries or of affine diffeomorphisms, then M is called a homogeneous manifold . It can be naturally identified with the homogeneous space (G/H, g), where H is the isotropy group of the origin p ∈ M .
A geodesic γ(s) through the point p is homogeneous if it is an orbit of a one-parameter group of isometries. More explicitly, if s is an affine parameter and γ(s) is defined in an open interval J, there exists a diffeomorphism s = ϕ(t) between the real line and the open interval J and a nonzero vector X ∈ g such that γ(ϕ(t)) = exp(tX)(p) for all t ∈ R. The vector X is called a geodesic vector. The diffeomorphism ϕ(t) may be nontrivial only for null geodesics in a properly pseudo-Riemannian manifold or for geodesics in affine manifolds.
A homogeneous Riemannian manifold (M, g) or a homogeneous Finsler space (M, F ) is always a reductive homogeneous space: We denote by g and h the Lie algebras of G and H respectively and consider the adjoint representation Ad: H × g → g of H on g. There exists a reductive decomposition of the form g = m + h where m ⊂ g is a vector subspace such that Ad(H)(m) ⊂ m. For a fixed reductive decomposition g = m + h there is the natural identification of m ⊂ g = T e G with the tangent space T p M via the projection π: G → G/H = M .
Using this natural identification and the scalar product or the Finsler metric on T p M , we obtain the invariant scalar product , or the invariant Minkowski norm F and its fundamental tensor g on m. In the pseudo-Riemannian reductive case, geodesic vectors are characterized by the following geodesic lemma: Lemma 1 ( [12], [10], [8]) Let (G/H, g) be a reductive homogeneous pseudo-Riemannian manifold and X ∈ g. Then the curve γ(t) = exp(tX)(p) is geodesic with respect to some parameter s if and only if
[X, Z] m , X m = k X m , Z(1)
for all Z ∈ m and for some constant k ∈ R. If k = 0, then t is an affine parameter for this geodesic. If k = 0, then s = e −kt is an affine parameter for the geodesic. The second case can occur only if the curve γ(t) is a null curve in a properly pseudo-Riemannian space.
The Finslerian version of this lemma was proved in [13]:
Lemmag Xm ([X, Z] m , X m ) = 0 (2)
for all Z ∈ m.
Another possible approach is to study the manifold M using a more fundamental affine method, which was proposed in [7] and [9]. It is based on the well known fact that a homogeneous manifold M with the origin p admits n = dimM fundamental vector fields (Killing vector fields) which are linearly independent at each point of some neighbourhood of p. Recall that a parametrized curve in a manifold M is regular if γ ′ (t) = 0 for all values of t. It is well known that, in a homogeneous space M = G/H with an invariant affine connection ∇, each regular orbit of a 1-parameter subgroup g t ⊂ G on M is an integral curve of an affine Killing vector field on M .
∇ Z γ(t) Z = k γ · Z γ(t)(3)
holds along γ, where k γ ∈ R is a constant. If k γ = 0, then t is the affine parameter of geodesic γ. If k γ = 0, then the affine parameter of this geodesic is s = e kγ t .
In the paper [11], it was proved that any homogeneous Riemannian manifold admits a homogeneous geodesic through the origin. The generalization to the pseudo-Riemannian (reductive and nonreductive) case was obtained in [5] in the framework of a more general result, which says that any homogeneous affine manifold (M, ∇) admits a homogeneous geodesic through the origin. Here the affine method from [9] and [7], based on the study of integral curves of Killing vector fields, was used. The proof is also using differential topology, namely smooth mappings S n → S n .
In pseudo-Riemannian geometry, null homogeneous geodesics are of particular interest, see [10] for instance. In [2], an example of a 3-dimensional Lie group with an invariant Lorentzian metric which does not admit a light-like homogeneous geodesic was described. Here the standard geodesic lemma was used, because the example is reductive. In the paper [6], the affine method used in [5], [7] and [9] for the study of homogeneous affine manifolds was adapted to the pseudo-Riemannian case and it was shown that any Lorentzian homogeneous manifold of even dimension admits a light-like homogeneous geodesic through the origin.
Recently, in the paper [14], the existence of a homogeneous geodesic in homogeneous Finsler space of odd dimension was claimed. The algebraic method developed in [11] and based on the reductive decomposition was generalized to the Finserian situation and also differential topology and mappings S n → S n were used. Surprisingly, neither the affine method nor the affine result from [5] was referred. Moreover, the proof contains a serious gap.
In the present paper, the original result is reproved and the gap in the proof is indicated. It is shown how the affine method can be adapted to the Finslerian setting and the mentioned result is proved correctly. Further, it is proved that in homogeneous Berwald spaces and in homogeneous reversible Finsler spaces a homogeneous geodesic always exists.
Basic settings
Recall that a Minkowski norm on the vector space V is a nonnegative function F : V → R which is smooth on V \ {0}, positively homogeneous (F (λy) = λF (y) for any λ > 0) and whose Hessian g ij = ( 1 2 F 2 ) y i y j is positively definite on V\{0}. Here (y i ) are the components of a vector y ∈ V with respect to a fixed basis B of V and putting y i to a subscript means the patrial derivative. Then the pair (V, F ) is called the Minkowski space. The tensor g y with components g ij (y) is the fundamental tensor. The Cartan tensor C y has components C ijk (y) = ( 1 4 F 2 ) y i y j y k . A Finsler metric on the smooth manifold M is a function F on T M which is smooth on T M \ {0} and whose restriction to any T x M is a Minkowski norm. Then the pair (M, F ) is called the Finsler space. On a Finsler space, functions g ij and C ijk depend smoothly on x ∈ M and on o = y ∈ T x M . Further, we recall that the slit tangent bundle T M 0 is defined as T M 0 = T M \ {0}. Using the restriction of the natural projection π: T M → M to T M 0 , we naturally construct the pullback vector bundle π * T M over T M 0 , as indicated in the following diagram:
π * T M T M T M 0 ∨ π > M. π ∨
For a given local coordinate system (x 1 , . . . , x n ) on U ⊂ M , at any x ∈ M , one has a natural basis
{ ∂ ∂x 1 , . . . , ∂ ∂x n } of T x M .
It is natural to express tangent vectors y ∈ T x M with respect to this basis. Then (x i , y i ) is the natural coordinate system on T U 0 . We define further functions on T U 0 , namely the formal Christoffel symbols γ i jk and the nonlinear connection N i j , by the formulas
γ i jk = g is 1 2 ∂g sj ∂x k − ∂g jk ∂x s + ∂g ks ∂x j , N i j = γ i jk y k − C i jk γ k rs y r y s .(4)
The Chern connection is the unique linear connection on the vector bundle π * T M which is torsion free and almost g-compatible, hence its connection forms satisfy
dx j ∧ ω i j = 0, dg ij − g kj ω k i − g ik ω k j = 2C ijs (dy s + N s k dx k ).(5)
It follows that it holds
ω i j = Γ i jk dx k , Γ i jk = Γ i kj , Γ l jk = γ l jk − g li (C ijs N s k − C jks N s i + C kis N s j ),(6)
see some monograph, for example [1] or [3] for details. If we fix a nowhere vanishing vector field V on M , we obtain an affine connection ∇ V on M . In the local chart, it is expressed with respect to arbitrary vector fields
W 1 = W i 1 ∂ ∂x i and W 2 = W i 2 ∂ ∂x i by the formula ∇ V W1 W 2 | x = W 1 (W i 2 ) + W j 2 W 1 k Γ i jk (x, V ) ∂ ∂x i .(7)
The affine connection ∇ V on M is torsion free and almost metric compatible, which means
∇ V W1 W 2 − ∇ V W2 W 1 = [W 1 , W 2 ], W g V (W 1 , W 2 ) = g V (∇ V W W 1 , W 2 ) + g V (W 1 , ∇ V W W 2 )+ +2C V (∇ V W V, W 1 , W 2 ),(8)
for arbitrary vector fields W, W 1 , W 2 . Using the affine connection ∇ V , we define the derivative along a curve γ(t) with velocity vector field T . Let W 1 , W 2 be vector fields along γ, we define
D W1 W 2 = ∇ T ′ W ′ 1 W ′ 2 ,(9)
where the vector fields T ′ , W ′ 1 and W ′ 2 on the right-hand side are smooth extensions of T , W 1 and W 2 to the neighbourhood of γ(t). The definition above does not depend on the particular extension. A regular smooth curve γ with tangent vector field T is a geodesic if D T ( T F (T ) ) = 0. In particular, a geodesic of constant speed satisfies D T T = 0.
We now reprove with minor technical modifications the result from [14]. It is using the algebraic method developed in [11] for the Riemannian metric. Proof. Let G be a group of isometries acting transitively on M and H the isotropy subgroup of the origin p ∈ M . We can write M = G/H. Denote by g and h the corresponding Lie algebras, by K the Killing form of g and by rad(K) the null space of K. Because K is nondegenerate on h, we can put m = h ⊥ and fix the reductive decomposition g = h + m. It holds rad(K) ⊆ m and there are the two possible cases:
If rad(K) = m, then [g, g] m is a proper subset of m. We choose arbitrary vector X ∈ [g, g] ⊥ m and for any Z ∈ g it holds [X, Z] m ∈ [g, g] m , hence
g X (X, [X, Z] m ) = 0 ∀Z ∈ g
and X is a geodesic vector. If rad(K) m, we fix an invariant scalar product , on m. For each unit vector X ∈ S n−1 ⊂ m, we define the operator α X : m → m by the formula
g X (α X U, V ) = K(U, V ) ∀U, V ∈ m.
If the Finsler space is Riemannian, there is just one operator α and we can continue as in [11]: There always exists a nonzero eigenvector Y of α with nonzero eigenvalue λ, because rad(K) m. We obtain
g(Y, [Y, Z] m ) = 1 λ g(α(Y ), [Y, Z] m ) = 1 λ K(Y, [Y, Z] m ) = = 1 λ K(Y, [Y, Z]) = 1 λ K([Y, Y ], Z]) = 0 ∀Z ∈ m (10)
and Y is a geodesic vector. We have used here also the invariance of the Killing form K. In a general Finsler space, if there is a vectorX ∈ m such that the eigenvector YX of αX satisfies YX =X, we can use similar steps as in the formula (10) above, write
gX(X, [X, Z] m ) = 1 λ(X) gX(αX (X), [X, Z] m ) = 1 λ(X) K(X, [X, Z] m ) = = 1 λ(X) K(X, [X, Z]) = 1 λ(X) K([X,X], Z]) = 0 ∀Z ∈ m
andX is a geodesic vector. In the paper [14], the mapping v: S n−1 → S n−1 was constructed by the assignment X → Y X , where Y X is the eigenvector of the operator α X with maximal absolute value of the eigenvalue λ(X). The mapping v was claimed to be continuous and the fixed point theorem was used. However, this part of the proof is not well justified and probably it is wrong. For the family of operators α X , the assignment of the eigenvector Y X with maximal absolute value of the eigenvalue λ(X) is not a continuous mapping in general. This seems to be a serious gap in the proof, because there is not an obvious way how to correct it.
Affine method for Finsler spaces
First, let us formulate simple observations which follow from homogeneity of the Finsler metric F .
Proposition 5 Let (M, F ) be a homogeneous Finsler space, G a group of isometries acting transitively on M , X * a Killing vector field generated by the vector X ∈ g, φ(t) = exp(tX) and γ(t) the integral curve of X * through p ∈ M . Along the curve γ(t), it holds
φ(t)(p) = γ(t), φ(t) * (X * (p)) = X * (γ(t))(11)
and
F (φ(t)(p), φ(t) * V ) = F (p, V ), g (γ(t),X * (γ(t))) (φ(t) * U, φ(t) * V ) = g (p,X * (p)) (U, V ),(12)
for all t ∈ R and for all U, V ∈ T p M .
Proposition 6
With the same assumptions as in Proposition 5, along the curve γ(t), it holds
g (γ(t),X * (γ(t)) (D X * X * γ(t) , φ(t) * U ) = g (p,X * (p)) (D X * X * p , U ),(13)
for all t ∈ R and for all U ∈ T p M . Consequently, if
D X * X * p = 0,(14)
then the curve γ(t) is a homogeneous geodesic.
We shall now give a correct proof of Theorem 4.
Theorem 7 Let (M, F ) be a homogeneous Finsler space of odd dimension and p ∈ M . Then M admits a homogeneous geodesic through p.
Proof. Let us consider the Killing vector fields K 1 , . . . , K n which are linearly independent at each point of some neighbourhood U of p and denote by B the basis {K 1 (p), . . . , K n (p)} of T p M . Any tangent vector X ∈ T p M has coordinates (x 1 , . . . x n ) with respect to the basis B. These coordinates determine the Killing vector field X * = x 1 K 1 + . . . + x n K n and an integral curve γ of X * through p.
We are going to show that there exists a vectorX ∈ T p M such that the integral curve γ ofX * through p is geodesic. Let us consider the sphere S n−1 of vectors X ∈ T p M whose coordinates (x 1 , . . . , x n ) with respect to B have the norm equal to 1 with respect to the standard Euclidean scalar product , on R n . In other words, the scalar product , is chosen in a way that the above basis B is orthonormal. We stress that this scalar product does not come from any Finslerian product g used so far. For each X ∈ S n−1 , denote by v(X) the derivative D X * γ(t) X * | t=0 . Further, denote by t(X) the vector v(X) − v(X), X X. Then, for each X ∈ S n−1 , t(X) ⊥ X with respect to the above Euclidean scalar product. Clearly, the map X → t(X) defines a smooth tangent vector field on the sphere S n−1 . If n is odd, according to a well known fact from differential topology, there is a vectorX such that t(X) = 0.
To finish the proof, we use the formula (8) and the standard fact that C X * (X * , X * , X * ) = 0. We observe that, for each X ∈ S n−1 ⊂ T p M , it holds
g (p,X) (v(X), X) = g (p,X) (D X * γ(t) X * t=0 , X * p ) = 0(15)
and hence v(X) lies in the orthogonal complement of X in T p M with respect to the scalar product g (p,X) . The vector t(X) is the projection of v(X) to another complementary subspace of X in T p M and hence v(X) = 0 if and only if t(X) = 0. If follows, using also Proposition 6 and formula (14), that the integral curve of the vector fieldX * through p is a homogeneous geodesic.
Let us now recall that the Finsler metric F is called a Berwald metric if the Christoffel symbols Γ i jk (x, y) of the Chern connection in natural coordinates do not depend on the direction y, hence Γ i jk (x, y) = Γ i jk (x). We further recall that the Finsler metric F is reversible if, for any point x ∈ M and for any vector y ∈ T x M , it holds F (x, y) = F (x, −y). Proof. If the Finsler metric F is Berwald, using formulas (7) and (9) we easily deduce that for any Killing vector field X * it holds
D X * X * = ∇ X * X * X * = ∇ −X * −X * −X * = D −X * −X * .(16)
For a reversible Finsler metric, one can check by the straightforward calculations and using formula (4), that it holds g ij (x, y) = g ij (x, −y), C ijk (x, y) = −C ijk (x, −y), γ i jk (x, y) = γ i jk (x, −y), N i j (x, y) = −N i j (x, −y).
Further, using formula (6), we obtain Γ i jk (x, y) = Γ i jk (x, −y)
and using formula (7), we obtain again that formula (16) is valid also in this situation. Formula (16) is essential for the next step. Let us use the same notation and setting as in the proof of Theorem 7 and let us consider the mappings v: S n−1 → T p M and t: S n−1 → T p M . For n even, let us assume that t(X) = 0 everywhere. Putting f (X) = t(X)/ t(X) , where the norm comes from the Euclidean scalar product , , we obtain a smooth map f : S n−1 → S n−1 without fixed points. According to a well known statement from differential topology, the degree of f is deg(f ) = (−1) n , because it is homotopic to the antipodal mapping. On the other hand, according to formula (16), we have v(X) = v(−X) and hence f (X) = f (−X) for each X. If Y is a regular value of f , then the inverse image f −1 (Y ) consists of even number of elements. Hence deg(f ) is an even number, which is a contradiction. Hence, the assumption t(X) = 0 was wrong.
It follows that there is again a vectorX ∈ T p M such that t(X) = 0 and also v(X) = 0. Consequently, the integral curve of the vector fieldX * through p is a homogeneous geodesic.
The integral curve γ of a nonvanishing Killing vector field Z on M = (G/H, ∇) is geodesic if and only if
Let (M, F ) be a homogeneous Finsler space of odd dimension and p ∈ M . Then M admits a homogeneous geodesic through p.
Theorem 8
8Let (M, F ) be a homogeneous Berwald space or a homogeneous reversible Finsler space and let p ∈ M . Then M admits a homogeneous geodesic through p.
2 ([13]) Let (G/H, g) be a homogeneous Finsler space. The vector X ∈ g is a geodesic vector if and only if it holds
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arxiv |
Solvable delay model for epidemic spreading: the case of Covid-19 in italy
0123456789
Luca Dell'anna [email protected]
Dipartimento di Fisica e Astronomia "G. Galilei"
Università degli Studi di Padova
via F. Marzolo 835131PadovaItaly
Solvable delay model for epidemic spreading: the case of Covid-19 in italy
012345678910.1038/s41598-020-72529-y
We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.In a very few months a viral infection called Covid-19 (Coronavirus disease 19) originated in China, breaking through the borders of all the countries, rapidly spread all over the globalized world. Italy is one of the hardest hit countries suffering from the very dramatic consequences of this disease. The outbreak of the virus, the new coronavirus which caused the infection, seems out of our control. In the absence of a therapy and a vaccine, social distancing measures and a strict lockdown appear to be the most effective means to contain the growth of the infection. We should remind that there are places in the world where often infectious diseases, also those already defeated in the so-called more developed countries, can still cause very severe consequences among the local populations.Even if we cannot answer the question why a virus starts spreading and which is its origin, we can still wonder how it diffuses. The aim of this work is, therefore, to provide a simple handy model for epidemic spreading, which could depend only on the couple of parameters which generally characterize an infectious disease: the infection rate and the infectiousness (or recovery) time. Both these quantities can be taken from the experience, therefore, we do not need further parameters to fit the data which could cause artificial predictions. We will show that the model we are presenting have the same, or even higher, predictive power than that of one of the most widely used technique in epidemiology, the SIR model 1-3 . This latter model requires the presence of a recovery rate related to the number of recovered persons, without considering that the new cases of recovery (and fatality) come from infected cases occurred previously. The model we are proposing, instead, is based on the fact that the closed cases come from the infected ones after an average delay recovery time, therefore, contrary to the SIR model, formulated in terms of a set ordinary differential equations, it is described by just a functional retarded differential equation, bringing predictions more under control. In this work we derive the exact analytical solution of this model in the limit of a large population, also in the presence of a time-dependent infection rate, which is the case when containment measures are implemented in order to reduce the spreading of the infection. Moreover, the definition of the so-called basic reproduction number R 0 (a parameter determining whether a infectious disease can spread or not) comes out naturally in our delay model. Actually delay models in epidemiology have been already implemented in many cases[5][6][7][8][9][10]. We consider the case where the infection period is constant and provide for the first time an analytical result for the spreading of the disease in the early stage of the infection.We finally apply this technique to give a quantitative description of the diffusion of Covid-19 in Italy, showing the current scenario based on the actual situation and what would have happened without the containment measures. Generally it is quite difficult to give a reliable forecast on the fate of the epidemic spreading because it heavily depends on individual and social behaviors, on the effectiveness of the containment measures already implemented, or that will be taken, by the government and on the future political decisions. At the time being, even if the situation in Italy is improving, it seems that more efforts are needed in order to change course and open
Solvable delay model for epidemic spreading: the case of Covid-19 in italy Luca Dell'Anna
We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.
In a very few months a viral infection called Covid-19 (Coronavirus disease 19) originated in China, breaking through the borders of all the countries, rapidly spread all over the globalized world. Italy is one of the hardest hit countries suffering from the very dramatic consequences of this disease. The outbreak of the virus, the new coronavirus which caused the infection, seems out of our control. In the absence of a therapy and a vaccine, social distancing measures and a strict lockdown appear to be the most effective means to contain the growth of the infection. We should remind that there are places in the world where often infectious diseases, also those already defeated in the so-called more developed countries, can still cause very severe consequences among the local populations.
Even if we cannot answer the question why a virus starts spreading and which is its origin, we can still wonder how it diffuses. The aim of this work is, therefore, to provide a simple handy model for epidemic spreading, which could depend only on the couple of parameters which generally characterize an infectious disease: the infection rate and the infectiousness (or recovery) time. Both these quantities can be taken from the experience, therefore, we do not need further parameters to fit the data which could cause artificial predictions. We will show that the model we are presenting have the same, or even higher, predictive power than that of one of the most widely used technique in epidemiology, the SIR model [1][2][3] . This latter model requires the presence of a recovery rate related to the number of recovered persons, without considering that the new cases of recovery (and fatality) come from infected cases occurred previously. The model we are proposing, instead, is based on the fact that the closed cases come from the infected ones after an average delay recovery time, therefore, contrary to the SIR model, formulated in terms of a set ordinary differential equations, it is described by just a functional retarded differential equation, bringing predictions more under control. In this work we derive the exact analytical solution of this model in the limit of a large population, also in the presence of a time-dependent infection rate, which is the case when containment measures are implemented in order to reduce the spreading of the infection. Moreover, the definition of the so-called basic reproduction number R 0 (a parameter determining whether a infectious disease can spread or not) comes out naturally in our delay model. Actually delay models in epidemiology have been already implemented in many cases [5][6][7][8][9][10] . We consider the case where the infection period is constant and provide for the first time an analytical result for the spreading of the disease in the early stage of the infection.
We finally apply this technique to give a quantitative description of the diffusion of Covid-19 in Italy, showing the current scenario based on the actual situation and what would have happened without the containment measures. Generally it is quite difficult to give a reliable forecast on the fate of the epidemic spreading because it heavily depends on individual and social behaviors, on the effectiveness of the containment measures already implemented, or that will be taken, by the government and on the future political decisions. At the time being, even if the situation in Italy is improving, it seems that more efforts are needed in order to change course and Scientific RepoRtS | (2020) 10:15763 | https://doi.org/10.1038/s41598-020-72529-y www.nature.com/scientificreports/ rapidly stop the spreading of the disease. Further measures might be useful, like, for instance, (i) running more diagnostic tests, at least, on all the doctors and medical workers who are in contact with many patients, (ii) improving the food distribution to avoid the crowding in the food shops and to ensure subsistence goods also to those who need, (iii) providing medical devices like surgical masks to all the population. As last remark, we remind that the outbreak of Covid-19 has been declared a pandemic by the World Health Organization. Many countries are already heavily overwhelmed by this infection and by the risk for the public health, therefore, in a networked world we all have to behave and operate with an improved spirit of cooperation. The bitter lesson imparted by this tough situation is that we cannot save ourselves alone.
Results
the model. Let us introduce the model, assuming that the full population is constant, uniform, homogeneously mixed, and counts N persons who can be divided in three parts, susceptible, infected and recovered persons, whose numbers, at a given time t, are S(t), I(t) and R(t), respectively.
Let us define I o the initial infected persons at time t = 0 , and introduce P(t) , the probability of remaining infectious at later time t after becoming infectious. P(t) is a monotonic decreasing function with P(0) = 1 and lim t→∞ P(t) = 0 . The initial number of the first infectious persons, therefore, decreases according to I o P(t) , meanwhile other susceptible persons become infected after coming in contact with those already infected, with a rate of infection α , which counts the number of contacts per unit of time, times the probability for a infected person to transmit the infection. The probability of new infections at a given time x is, therefore, proportional to the ratio S(x)/N of persons who are still susceptible and the number of infected persons who are still infectious, I(x)P(t − x) . At a later time t, the total number of infections are, therefore, given by Equivalently, writing P ′ (t) = dP(t) dt , Eq. (1) can be written as
Since P(t) is a non-increasing function, P ′ (t) is negative, therefore the last two terms of Eq. (2) reduces the increase of infection due to the first term. For that reason we can identify those terms as minus the variation of the removed cases It is convenient, for the benefit of future discussion, to introduce the total number of infected persons, either those who are still infected at time t, I(t), and those who recovered or died, R(t),
From Eqs. (2) and (3), since S(t) + F(t) = N , we have that F(t) fulfills the following equation which is valid for any choice of P(t).
Standard SIR model. If we now choose P(t) = e −βt , inserting it in Eqs. (2)-(3) we recover the well-celebrated SIR model [1][2][3] where β is the recovery rate. In order to make a comparison with what follows let us solve these equations when the population N is very large, and as long as F(t) ≪ N , such that S(t) ≃ N . In this situation we have and solving dF(t) dt = αI(t) , with the initial condition F(t = 0) ≡ F o = I o , we get that the growth of the total number of infections, at the early stage, has the following form
I(t) = I o P(t) + α N t 0 S(x)I(x)P(t − x)dx (2) dI(t) dt = α N S(t)I(t) + I o P ′ (t) + α N t 0 S(x)I(x)P ′ (t − x)dx (3) dR(t) dt = −I o P ′ (t) − α N t 0 S(x)I(x)P ′ (t − x)dx (4) F(t) = I(t) + R(t) (5) dF(t) dt = α F(t) − R(t) 1 − F(t) N (6) dS(t) dt = − α N S(t)I(t) (7) dI(t) dt = α N S(t)I(t) − βI(t) (8) dR(t) dt =βI(t) (9) dI(t) dt = (α − β) I(t)= �(t − T) , a step function, namely P(t) = 1 for 0 ≤ t ≤ T and P(t) = 0 for t > T , inserting it in the Eqs. (2), (3), being P ′ (t − x) = −δ(t − x − T) , we get From the equations above it is easy to see that dR(t) dt = dF(t−T) dt , therefore R(t) = F(t − T) + C , with C a constant value.
We remind that, contrary to I(t), either F(t) and R(t) are both cumulant quantities, namely they are monotonic increasing functions. Requiring that F(t) saturates at t → ∞ , the constant value has to be C = 0 , therefore This equation describes the realistic fact that the total number of cases at some time t becomes that of removed cases at later time t + T , namely after an infectious period T. This seems to be the case also for the new coronavirus spreading, by looking at some reported data for Covid-19 in Italy, shown in Fig. 3 (see also Ref. 4 ). Equation (14) allows us to write Eq. (5) in terms of only the function F(t). Eq. (5), for the delay model, therefore, reads where F(t − T) = 0 for t < T . This delay differential equation is known to be linked to non-Markovian dynamics 11 . If we consider the case where the population N is very large, and as long as F(t) ≪ N , we can neglect the logistic term, 1 − F(t) N ≃ 1 , so to have We expect that this functional retarded differential equation, Eq. (16), at least, at the early stage of the infection, could describe accurately the spreading of the epidemic disease.
Basic reproduction number. Let us rewrite Eq. (16), for t > T , in the following form where we introduce and naturally identify R 0 as the so-called basic reproduction number which is a widely used parameter for predicting whether the infectious disease will spread into a population or turns off, and represents the average number of cases originated by a single infectious case during the infectiousness period. Eq. (17) implies that the first derivative of F(t) is equal to its increment in a time interval T, divided by T, namely F(t) is linear in t if the rate is equal to the critical value α = T −1 ( R 0 = 1 ). For α > T −1 ( R 0 > 1 ), the function F(t) increases more than linearly, while for α < T −1 ( R 0 < 1 ), F(t) goes slower than linearly (see Fig. 1). If we let α vary in time, when α = T −1 ( R 0 = 1 ) the function F(t) has an inflection point, where it changes from being concave to convex or vice versa. Making a comparison with the SIR model, where www.nature.com/scientificreports/ R 0 = α/β , one can identify β , the recovery rate with the inverse of the recovery time β ∼ 1/T . Notice that R 0 is well defined as long as F(t) ≪ N , namely in the early stage of the infection. In general terms one has to define the generalized reproduction number R t = α(1 − F(t)/N)T so that Eq. (15) can be written in the same form of Eq. (17) with R t instead of R 0 .
(10) F(t) = F o β − α e (α−β)t β − α . (11) dS(t) dt = − α N S(t)I(t) (12) dI(t) dt = α N S(t)I(t) − α N S(t − T)I(t − T) (13) dR(t) dt = α N S(t − T)I(t − T) (14) R(t) = F(t − T) (15) dF(t) dt = α F(t) − F(t − T) 1 − F(t) N (16) dF(t) dt = α F(t) − F(t − T) (17) dF(t) dt = R 0 F(t) − F(t − T) T (18) R 0 = α T
Analytical solution. In this section we will provide the exact solution of Eq. (16). Writing the time t as t = nT + t ′ , where n = ⌊ t T ⌋ is the integer part of t/T, the solution of Eq. (16) is given by where the functions A ℓ fulfill the following iterative equation
with A 0 (t) = 0 for any t < T and A 0 (T) = 1 , so that, for ℓ = 1 , we recover A 1 (t) = e αt . The full exact solution is, therefore, obtained by solving a cascade of n local integrals. The proof of Eqs. (19) and (20) is given in Methods.
At time t = nT , from Eq. (20), performing the chain of integrals, and putting the results in Eq. (19), we get the following exact result For instance, for n = 1 and n = 2 , namely up to twice the infectiousness period, the total number of cases is simply F(nT) = F o e nαT − (n − 1) αT e (n−1)αT . Surprisingly we find that Eq. (21) depends only on (αT) , which is the basic reproduction number R 0 . It is easy to check from Eq. (21) that, while for large R 0 = αT , F(nT) is dominated by an exponential behavior, for R 0 = 1 , F(nT) becomes linear in n. From Eqs. (19) and (20) we can also write the following equation
By iteration one gets simply where I m fulfills the following recursive equation, with inital value I 0 = 1,
The final exact result for any time is, therefore, where t ′ = mod(t, T) . For practical reasons, in order to avoid indeterminate forms, for t ′ = 0 and m = 0 , in Eq. (25) one can add an infinitesimal term ǫ → 0 , so to have (αt ′ + ǫ) m . Once we have the total number of infections F(t) at any time, we get also the number of removed cases, R(t) = F(t − T) , and we can easily calculate, from Eq. (25), the number of persons who are still infected, at a given time t, which, by definition and from Eq. (16), is given by
Comparison between the delay model and the standard SIR model. As we have seen, one assumption the standard SIR model is based on is that the time in which individuals remain infectious is described by an exponential distribution, which is however biologically rather unrealistic. In reality, infectious periods are fairly closely centered about the mean duration of an infection. A constant infectious period is therefore a more realistic assumption. The conventional SIR model being formulated in terms of ordinary differential equations, requires the presence of an effective recovery (and fatality) rate which might not correspond to the actual rate since the new cases of recovery (and fatality) come from infected cases occurring a few days earlier. For that reason, instead of writing the problem in terms of ordinary differential equations one has to do it in terms of functional (11)(12)(13). As shown in Fig. 2, even with the same initial conditions and the same R 0 , the growth and the expected peak of the spreading of the infectious disease are quite different between the two models, even if the asymptotic final values are the same. For R 0 ≃ 2.5 the SIR model predicts a much lower peak of I(t) with respect to that expected from the delay model, which is much sharper and occurs much earlier. In other words, the outbreak of an epidemic disease might be underestimated by the standard SIR model. We notice also that the analytic expression for F(t) in Eq. (25) describes fairly well the increase of the infection, at least in its early stage.
(19) F(t) = F(nT + t ′ ) = F o n ℓ=1 A ℓ (T) A n+1 (t ′ ) (20) A ℓ (t) = e αt 1 − α A ℓ−1 (T) −1 t 0 dt ′ e −αt ′ A ℓ−1 (t ′ ) (21) F(nT) = F o n ℓ=0 (−1) ℓ ℓ! (n − ℓ) αT ℓ e (n−ℓ)αT (22) F(t) = F(nT + t ′ ) = e αt ′ F(nT) − α t ′ 0 ds e −αs F (n − 1)T + s (23) F(nT + t ′ ) = e αt ′ n m=0 (−α) m I m (t ′ ) F (n − m)T (24) I m (t ′ ) = t ′ 0 ds I m−1 (s) = t ′ m m!(25)F(t) = F(nT + t ′ ) =e αt ′ n m=0 (−1) m m! (αt ′ ) m F (n − m)T =F o e αt ′ n m=0 n−m ℓ=0 (−1) ℓ+m ℓ! m! (αt ′ ) m (n − m − ℓ) R 0 ℓ e (n−m−ℓ)R 0 (26) I(t) = F(t) − R(t) = F(t) − F(t − T) = 1 α dF(t) dt .= F(nT + t ′ ) = F o n ℓ=1 A ℓ (T) A n+1 (t ′ )
, where now the functions A ℓ are given by F o r i n s t a n c e , where t 1 and t 2 are the times where the steps are located, τ 1 and τ 2 make the function to be smooth, α 1 is the initial observed infection rate which causes the starting exponential growth of the epidemic disease, α 2 the intermediate rate, which fits with the data, supposed to be reached after the first decree of lockdown, and α 3 the supposed asymptotic infection rate after the second decree of lockdown. Fixing the average of recovery and fatality time T, the reproduction number is also a function of time, therefore we define with a profile shown in Fig. 4. More precisely R t = α(t)(1 − F(t)/N)T , but as we will see, because of the containment measures, F(t) ≪ N at any time. Solving Eq. (30), or, analogously, using the recursive relation in Eq. (29), with the time-dependent rate α(t) given by Eq. (31), with the parameters reported in Fig. 4, we obtain the solution F(t) which slowly goes to saturation over time, in perfect agreement with the data for the total number of confirmed infected cases, as shown by Fig. 5, where the blue line is the expected curve, while the red points are the official data. The dotted gray lines in Fig. 5 represent F(t) if the containment measures had not been taken. As one can see from Figs. 4-5, only when R t becomes smaller than 1, the curve flattens allowing for a stop of the epidemic spreading, avoiding that a large part of the population gets infected. For R t ≃ 1 , F(t) would increase linearly, and I(t) would become almost constant, meaning that the number of new infections would be equal to the number of closed cases. A reliable forecast has to take into account the fact that the official data of infectious cases are made by counting mostly the symptomatic cases, probably discarding other infectious cases which could transfer the virus even without or with mild symptoms. Moreover, the data of both the total number of infected persons and that of the recovered ones could be affected by the procedure, the realization times and the number of the diagnostic tests. However, since our model relies on the infectiousness time, it does not need a fitting of the data for recovered persons which may be affected by systematic errors. This uncertainty on the data for closed cases would compromise the result for the SIR model. On the contrary, our theoretical prediction based on the delay model agrees fairly well with the data-set for total infected cases, as shown in Fig. 5.
A 1 (t) = e t 0 α(t ′ )dt ′ , A 2 (t) = e T+t T α(t ′ )dt ′ 1 − e − T 0 α(t ′ )dt ′ t 0 dt ′ α T + t ′ e − T+t ′ T α(t ′′ )dt ′′ e27) dF(t) dt = α(t) F(t) − F(t − T) . (28) A ℓ+1 (t) = e ℓT+t ℓT α(t ′ )dt ′ 1 − A ℓ (T) −1 t 0 dt ′ α ℓ T + t ′ e − ℓT+t ′ ℓT α(t ′′ )dt ′′ A ℓ (t ′ ) . (29) F(t) = F(nT + t ′ ) = e nT+t ′ nT α(s)ds F(nT) − t ′ 0 ds α(nT + s)e − nT+s nT α(t ′′ )dt ′′ F (n − 1)T + s (30) dF(t) dt = α(t) F(t) − F(t − T) 1 − F(t) N
As a final remark we remind that most of the confirmed infected cases in Italy are counted after the appearance of the symptoms and the persons who exhibit severe ones are mostly hospitalized, and afterwards counted as infected persons. Some of them, unfortunately, die approximately 4 days after (therefore after approximately 9 days from the onset of the first symptoms, as reported by the Istituto Superiore di Sanità 14 ). We observe that, splitting the closed cases between real recovered persons, R R , and dead persons, R D , www.nature.com/scientificreports/ and since the confirmation of recovery needs extra diagnostic tests which are not widely performed yet, the most reliable data are those related to dead persons R D (t) , which are found to be linked to the total number of confirmed infected cases, F(t), in the following way with γ = 1 7 and a delay time of T d = 4 days, as show in Fig. 6. The number of victims follows the number of total confirmed cases and it is equal to 1/7 of its value four days before.
(31) α(t) = α 1 − α 2 1 + e (t−t 1 )/τ 1 + α 2 − α 3 1 1 + e (t−t 2 )/τ 2 + α 3 (32) R t = α(t)T (33) R(t) = R R (t) + R D (t)
The fatality of the sick persons, those who exhibit some symptoms, is therefore quite high, lim t→∞ R D (t)
F(t) = γ ≃ 14%.
Discussion
We present a simple but realistic model for describing epidemic spreading, based on the fact that the closed cases come from infected ones at an early time. This observation allows us to formulate the problem in terms of a single functional differential equation depending on two well defined clinically relevant parameters: the infection rate and the infectiousness time. We provide the exact analytical solution for such an equation, in the limit of a large population, finding how it depends on the basic reproduction number R 0 = αT , see Eqs. (21) and (25). Contrary to the result of the conventional SIR model, the total number of cases has a combined polynomial and exponential growth. We derive the analytic solution also in the presence of a generic time-dependent infection rate, which is the case when some measures are taken to weaken the spreading of the epidemic disease. We apply, therefore, our model to study the spreading of Covid-19 in Italy, allowing the infection rate to vary in time, as a result of some containment measures implemented by the government in order to mitigate the consequences of the infection www.nature.com/scientificreports/ on the population. We find perfect agreement between the official data and the expected theoretical results. In general terms, the reproduction number should be suppressed well below 1 in order to rapidly recover the initial condition. By a rough estimation, in order to have a decline of the infection as fast as its growth, containment measures or possible therapies should be so effective to reduce the basic reproduction number and reach the final value R f such that R f ≃ R 0 2R 0 −1 , starting from an initial value R 0 . In the case of Covid-19 in Italy, the initial value for the basic reproduction number was R 0 ≃ 2.6 , while the current one (April) seems to settle at R f ≃ 0.8 , implying a rather slow decline of the infection. Finally we discussed the fatality rate, showing that the number of victims is exactly a fraction of the total number of cases few days before. Before we conclude a final comment is in order. The confirmed cases are mostly symptomatic or mild symptomatic. There are also asymptomatic cases which may contribute to the spreading of the infection. However, by scaling arguments, the infection rates of the symptomatic and asymptomatic are expected to be equal, otherwise either symptomatic or asymptomatic cases might become irrelevant. Under the hypothesis that the infectiousness time does not depend on the strength of the symptoms, the ratio between the total number of asymptomatic and symptomatic cases should be constant, although it could be very large. As a result, the total number of infected persons should be equal to the number of symptomatic cases times an overall pre-factor greater than one. The conclusion is, therefore, that, as far as the time evolution of the infection is concerned, which is the aim of this work, the study of only symptomatic cases is still relevant and greatly meaningful.
(34) R D (t) ≃ γ F(t − T d )
Methods
Solution of the retarded differential equation. For t ≤ T , the solution of Eq. (16) is F(t) = F o e αt . Let us consider t = T + dt with infinitesimal dt, from Eq. (16)
Using this result we can calculate Analogously, from that, we can proceed calculating and going on by adding infinitesimal time steps, we find iteratively that with A 1 (T) = e αT and defining We can notice that at any step T we can perform the same calculation since we can factorize the function F as where, therefore, F(n T) = F o n ℓ=1 A ℓ (T) and
(35) F(T + dt) = F(T) + dt α(F(T) + F(0)) = F(T)(1 + α dt) − F o α dt = F o e αT (1 + α dt) − F o α dt (36) F(T + 2dt) = F(T + dt) + dt α(F(T + dt) + F(dt)) = F o e αT (1 + α dt) 2 − F o α dt (1 + α dt) + e αdt (37) F(T + 3dt) =F(T + 2dt) + dt α(F(T + 2dt) + F(2dt)) = F o e αT (1 + α dt) 3 − F o α dt (1 + α dt) 2 + e αdt (1 + α dt) + e 2αdt
In the continuum limit, dt → 0 and m → ∞ , keeping finite the time interval m dt = t , reminding that we finally obtain the result reported Eq. (20).
In the presence of time dependent infection rate, splitting again the time in n intervals T and the residual time in m infinitesimal intervals dt, we define Proceeding iteratively as done for the constant rate case, but now taking trace of the different values of α, after several steps, similar to those done previously, we find that Eq. (46) can be generalized in the following way whose continuum limit is given in Eq. (28). (49) F n T + m dt = F n T + (m − 1)dt 1 + α (n) m dt − α (n) m dt F (n − 1) T + (m − 1)dt
(50) A ℓ (m dt) = m � i=1 � 1 + α (ℓ−1) i dt � − A ℓ−1 (T) −1 dt m−1 � j=0 α (ℓ−1) j A ℓ−1 (j dt) j � i=1 � 1 + α (ℓ−1) i dt � m−j−1 ,
Figure 1 .
1R 0 for different slops of the epidemic curve as compared with its increment in a time interval T. Scientific RepoRtS | (2020) 10:15763 | https://doi.org/10.1038/s41598-020-72529-y
Time-dependent infection rate: analytical solution. Let us now consider the possibility of having a time-dependent infection rate α(t) in the dynamical equation for the total number of infected persons Also in this more general case the exact solution, valid for any profile of α(t) , can be written in the same form of Eq. (19), namely, F(t)
(t ′′ )dt ′′ and so on. For constant α, Eq. (28) reduces to Eq. (20). See "Methods" for more details about the derivation. The solution F(t) has therefore to fulfill the following recursive equation, after splitting the time in n intervals T with the residual time t ′ = mod(t, T)This general result implies that if we knew the time dependence of the infection rate or if we could tailor its evolution by, for instance, containment measures, we can know the exact analytical expression of F(t), the total number of infected persons, as a function of time, as long as F(t) is much smaller than N.Covid-19 in Italy.Let us consider the delay model in its general form, Eq. (15) where the infection rate α varies in time as the effect of some containment measures taken in order to reduce the impact of an infection on the population. As an example, let us suppose that α(t) is modified by social distancing measures, lockdown and the shutdown of
(
Figure 2 .
2(a) Number of susceptible S(t) (dotted lines), infected I(t) (solid lines), and recovered R(t) (dashed lines) persons as functions of time, for the SIR model, Eqs. (6-8) (blue lines) and for the delay model, Eqs. (11-13) (red lines), with initial conditions I o = I(t = 0) = 150 and R(0) = 0 , and S(t) + I(t) + R(t) = N = 6 × 10 7 , with parameters α = 0.23 per unit of time and T = β −1 = 11 units of time (e.g. days), therefore the basic reproduction number in both the models is R 0 ≃ 2.5 . (b) Total number of infected persons F(t) = I(t) + R(t) , in log-scale, as a function of time from the standard SIR model (blue solid line) and from the delay model (red solid line). The gray dotted lines are the analytical results from Eq. (10) for the SIR model and Eq. (25) for the delay model, valid in the first stage of the infection. activities, as it is happening in Italy (and in many other countries) to mitigate and reduce the spreading of the new coronavirus, Covid-19, after two main decrees imposed by the Italian Prime Minister ordering the lockdown of the whole national territory, taken on March 11th (lockdown and shutdown of many stores) andMarch 22th 2020 (shutdown of many factories and strengthening of social distancing measures), and after some other measures taken right before for local regions (e.g. the decree of March 8-th for the lockdown of Lombardy and other areas). As a result, we can imagine that α(t) decreases smoothly after those dates taking into account the adaptation time for the individuals to the new social behaviors and the period needed to complete the last activities before the blockade of the factories. Let us suppose, therefore, that α(t) can change in time according to a smooth step function as in Eq. (31),
Figure 3 .
3Total number of confirmed cases of Covid-19 in Italy, F(t) (red dots), reported in Ref. 12 , since 21th February to 22th March 2020, compared with the closed cases, R(t) (blue dots), in the same period of time. If the numbers of closed cases are shifted in time by T ≃ 11 days (blue circles) they fairly overlap with the total numbers of cases. Scientific RepoRtS | (2020) 10:15763 | https://doi.org/10.1038/s41598-020-72529-y
Figure 4 .
4Time-dependent reproduction number R t = α(t)T , as a function of time, based on the profile for the infection rate described by Eq. (31). We take T about 11 or 12 days 4 , t 1 between 13th and 14th March 2020, t 2 on 26th March, τ 1 ∼ 2 days, τ 2 ∼ 1 day. The initial value is R 0 = α 1 T ≃ 2.65 (in agreement with other estimates, see e.g.13 ), the intermediate value is R t = α 2 T ≃ 1.45 , and the final value, R t = α 3 T ≃ 0.85 . The vertical dotted blue lines point the dates of the main laws for the containment measures (8th-11th March and 22th March 2020). The gray dotted lines correspond to R t in the absence of the first and the second containment measures.
Figure 5 .
5(a) Total number of infected persons over time, F(t) (red points), from official data for Covid-19 in Italy 12 , where N = 6 · 10 7 , up to 25th April. The blue line is the theoretical prediction F(t) as solution of Eq. (30), with initial conditions, fixed at t = 0 the 21th February, F o ≃ 150 , and α = α 1 ≃ 0.23 ( R 0 ≃ 2.65 ), and using the profile for the infection rate given by Eq. (31), with the parameters reported in Fig. 4. The gray dotted lines are the expected curves for F(t) if the first and the second containment measures (8-11th March and 22th March) had not been taken. (b) Daily number of infected persons, �F(t) , compared with the theoretical result obtained performing dF(t) dt from the solution of Eq. (30). Scientific RepoRtS | (2020) 10:15763 | https://doi.org/10.1038/s41598-020-72529-y
( 38 )Figure 6 .
386F(T + m dt) =F o e αT (1 + α dt) m − F o α dt m−1 j=0 e jαdt (1 + α dt) m−j−1 ≡ F o A 1 (T)A 2 (m dt) (39) =F(T) A 2 (m dt) (a) Total number of confirmed cases of Covid-19 in Italy, F(t) (red dots), reported in Ref. 12 , up to 25th April 2020, compared with the deceased cases, R D (t) (blue dots), in the same period of time. If the numbers of dead persons are shifted in time by T d ≃ 4 days and rescaled by γ −1 = 7 (blue circles) they perfectly overlap with the total numbers of cases. The blue solid line is theoretical prediction for F(t), as solution of Eq. (30). (b) Daily number of infected persons, �F(t) (red dots), and daily number of victims �R D (t) (blue dots). The blue circles are the daily number of victims after rescaling according to Eq. (34), with T d = 4 days and γ = 1/7 . The solid blue line is dF(t) dt from the solution of Eq. (30). , for m dt = T , we have an expression for F(2T) in terms of the function at early time, F(2T) = F(T)A 2 (T) . We can now start again with the iteration One can proceed in the same way as before getting which can be written as where
+ dt) = F(2T)(1 + α dt) − α dtF(T) = F(T)A 2 (T)(1 + α dt) − α dtF(2T + m dt) = F(T) A 2 (T) A 3 (m dt) = F(2T) A 3 (m dt) (44) A 3 (m dt) = (1 + α dt) m − A 2 (T)
t) = α(n T + m dt) ≡ α (n) m .
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| {'fraction_non_alphanumeric': 0.06551490344775632, 'fraction_numerical': 0.024834894407754313, 'mean_word_length': 3.9823595646190415, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 4, 'https://': 5, 'lorem ipsum': 0, 'www.': 7, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We study a simple realistic model for describing the diffusion of an infectious disease on a population of individuals. The dynamics is governed by a single functional delay differential equation, which, in the case of a large population, can be solved exactly, even in the presence of a time-dependent infection rate. This delay model has a higher degree of accuracy than that of the so-called SIR model, commonly used in epidemiology, which, instead, is formulated in terms of ordinary differential equations. We apply this model to describe the outbreak of the new infectious disease, Covid-19, in Italy, taking into account the containment measures implemented by the government in order to mitigate the spreading of the virus and the social costs for the population.In a very few months a viral infection called Covid-19 (Coronavirus disease 19) originated in China, breaking through the borders of all the countries, rapidly spread all over the globalized world. Italy is one of the hardest hit countries suffering from the very dramatic consequences of this disease. The outbreak of the virus, the new coronavirus which caused the infection, seems out of our control. In the absence of a therapy and a vaccine, social distancing measures and a strict lockdown appear to be the most effective means to contain the growth of the infection. We should remind that there are places in the world where often infectious diseases, also those already defeated in the so-called more developed countries, can still cause very severe consequences among the local populations.Even if we cannot answer the question why a virus starts spreading and which is its origin, we can still wonder how it diffuses. The aim of this work is, therefore, to provide a simple handy model for epidemic spreading, which could depend only on the couple of parameters which generally characterize an infectious disease: the infection rate and the infectiousness (or recovery) time. Both these quantities can be taken from the experience, therefore, we do not need further parameters to fit the data which could cause artificial predictions. We will show that the model we are presenting have the same, or even higher, predictive power than that of one of the most widely used technique in epidemiology, the SIR model 1-3 . This latter model requires the presence of a recovery rate related to the number of recovered persons, without considering that the new cases of recovery (and fatality) come from infected cases occurred previously. The model we are proposing, instead, is based on the fact that the closed cases come from the infected ones after an average delay recovery time, therefore, contrary to the SIR model, formulated in terms of a set ordinary differential equations, it is described by just a functional retarded differential equation, bringing predictions more under control. In this work we derive the exact analytical solution of this model in the limit of a large population, also in the presence of a time-dependent infection rate, which is the case when containment measures are implemented in order to reduce the spreading of the infection. Moreover, the definition of the so-called basic reproduction number R 0 (a parameter determining whether a infectious disease can spread or not) comes out naturally in our delay model. Actually delay models in epidemiology have been already implemented in many cases[5][6][7][8][9][10]. We consider the case where the infection period is constant and provide for the first time an analytical result for the spreading of the disease in the early stage of the infection.We finally apply this technique to give a quantitative description of the diffusion of Covid-19 in Italy, showing the current scenario based on the actual situation and what would have happened without the containment measures. Generally it is quite difficult to give a reliable forecast on the fate of the epidemic spreading because it heavily depends on individual and social behaviors, on the effectiveness of the containment measures already implemented, or that will be taken, by the government and on the future political decisions. At the time being, even if the situation in Italy is improving, it seems that more efforts are needed in order to change course and open', 'arxivid': '2003.13571', 'author': ['Luca Dell'anna [email protected] \nDipartimento di Fisica e Astronomia "G. Galilei"\nUniversità degli Studi di Padova\nvia F. Marzolo 835131PadovaItaly\n', 'Luca Dell'anna [email protected] \nDipartimento di Fisica e Astronomia "G. Galilei"\nUniversità degli Studi di Padova\nvia F. Marzolo 835131PadovaItaly\n'], 'authoraffiliation': ['Dipartimento di Fisica e Astronomia "G. Galilei"\nUniversità degli Studi di Padova\nvia F. Marzolo 835131PadovaItaly', 'Dipartimento di Fisica e Astronomia "G. Galilei"\nUniversità degli Studi di Padova\nvia F. Marzolo 835131PadovaItaly'], 'corpusid': 214713628, 'doi': '10.1038/s41598-020-72529-y', 'github_urls': [], 'n_tokens_mistral': 11899, 'n_tokens_neox': 10656, 'n_words': 7031, 'pdfsha': 'a1f1f89af0fa6ef1ab5f8a15c0042b6f89126a84', 'pdfurls': ['https://arxiv.org/pdf/2003.13571v1.pdf'], 'title': ['Solvable delay model for epidemic spreading: the case of Covid-19 in italy', 'Solvable delay model for epidemic spreading: the case of Covid-19 in italy', 'Solvable delay model for epidemic spreading: the case of Covid-19 in italy', 'Solvable delay model for epidemic spreading: the case of Covid-19 in italy'], 'venue': []} |
arxiv |
Large Deviations for Heavy-Tailed Factor Models
4 Dec 2007 February 2, 2008
Boualem Djehiche
Department of Mathematics Royal Institute of Technology SE
100 44Stockholm
Jens Svensson
Department of Mathematics Royal Institute of Technology SE
100 44Stockholm
Large Deviations for Heavy-Tailed Factor Models
4 Dec 2007 February 2, 2008Large deviationsheavy tailsregular variationfactor models AMS 2000 Subject Classification: Primary: 60F10 Secondary: 60G50
We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms contribute to the behaviour of the tail-probability of the sum. A simple conditional Monte Carlo algorithm is also provided together with a comparison between the simulations and the large deviation approximation. We also study large deviation probabilities for stochastic processes with factor structure. The processes involved are assumed to be Lévy processes with regularly varying jump measures. Based on the results of the first part of the paper, we show that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process.
Introduction
This paper is devoted to the study of large deviations of sums of dependent random variables and processes, where the dependence is generated through a factor model. Factor models are important in both financial theory and practice, because this form of structural dependence is both realistic and tractable. From a theoretical point of view, different types of factor models give intuition to economic phenomena: the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) are examples where factor structure is a fundamental property (see e.g. Cochrane (2001)). From an applied point of view, factor models are useful as approximations of other models and for dimension reduction. In many cases, reducing the number of dimensions of a model can make it tractable in practice.
Often, the random variables or vectors involved are assumed to be normally distributed, or at least light-tailed. A random variable X is called light-tailed if its tail-distribution P (X > λ) tends to zero faster than e −cλ for some c > 0.
Empirical studies of financial time series often conclude that data are heavytailed, i.e. the previous condition is not satisfied (see e.g. Cont (2001) for a review of some of these empirical findings). Consequently, light-tailed factor models may not be suited for describing the tail-properties of financial data. Therefore, it is of interest to incorporate the assumption of heavy tails into a factor model. As we will see, heavy-tailed factor models display qualitatively different behaviour from standard light-tailed models.
In the first part of the paper, we restrict ourselves to the class of regularly varying random variables and vectors. This class is fairly rich and includes popular distributions such as Pareto and student's t. See e.g. Embrechts et al. (1997) and Resnick (2004) for treatments of the univariate and multivariate case, respectively. A random variable X is regularly varying if there exist α ≥ 0 and p ∈ [0, 1] such that lim x→∞ P (X > tx) P (|X| > x) = pt −α and lim x→∞ P (X ≤ −tx)
P (|X| > x) = (1 − p)t −α ,(1)
for t > 0. We refer to p as the tail balance parameter. The definition can also be formulated in terms of sequences instead of a continuous parameter x.
Clearly, regularly varying random variables are heavy-tailed according to the above definition.
Since we will allow for dependence between factors, we also need the corresponding class of random vectors. For random vectors, regular variation is defined through convergence of measures. Specifically, an R d -valued random vector X is said to be regularly varying if there exist a sequence a n → ∞ and a measure µ on R d such that
lim n→∞ nP (a −1 n X ∈ B) = µ(B)(2)
and µ(B) < ∞ for every Borel set B ⊂ R d satisfying 0 / ∈ B and µ(∂B) = 0, where B and ∂B denote the closure and boundary of B, respectively. We write X ∈ RV(α, µ). See Hult and Lindskog (2006) for details about equivalent definitions of regular variation.
Using this class of distributions, we define a factor model for the vector (R 1 , . . . , R n ) by letting
R i = d j=1 L ij F j + ε i , i = 1, . . . , n,(3)
where F d = (F 1 , . . . , F d ) T is a regularly varying random vector, ε i are i.i.d. regularly varying random variables and L i = (L i1 , . . . , L id ) are i.i.d. random vectors.All the random variables and vectors involved are assumed to be independent. The components of F d are referred to as factors, L ij as factor loadings and ε i as idiosyncratic components.
A sum of variables from this model can be expressed as
S n = n i=1 R i = n i=1 d j=1 L ij F j + n i=1 ε i .(4)
The tail probability P (S n > λ) exhibits different asymptotic behaviour depending on the relation between the tail indices of the independent sums n i=1 ε i and
n i=1 d j=1 L ij F j .
Recall that (see e.g. Embrechts et al. (1997)) if two independent regularly varying random variables X and Y have different tail indices, 0 < α X < α Y , then
P (X + Y > λ) ∼ P (X > λ), as λ → ∞,
which means that the random variable with heaviest tail, or smallest tail index, dominates the tail probability of the sum. On the other hand (see e.g. Embrechts et al. (1997)), if X 1 , X 2 , . . . are i.i.d. regularly varying random variables with tail balance parameter p, we have with n fixed,
P ( n i=1 X i > λ) ∼ npP (|X 1 | > λ), as λ → ∞,(5)
where
a(x) ∼ b(x) as x → ∞ denotes lim x→∞ a(x)/b(x) = 1.
In fact, Relation (5) is still valid when n → ∞ if λ = λ n increases sufficiently fast. Asymptotic probabilities of this kind are called large deviation probabilities.
For an appropriate choice of λ n we have
P ( n i=1 X i > λ n ) ∼ npP (|X 1 | > λ n ), as n → ∞.(6)
We refer to Mikosch and Nagaev (1998) for details about the choice of sequence λ n under different distributional assumptions.
In this paper we consider regularly varying random variables with tail indices larger than 2, for which it was shown in Nagaev (1970) that if λ n is such that √ n log n/λ n → 0 as n → ∞, then Relation (6) holds. Similarly, for tail probabilities of the sum S n given by (4), we have two different situations. As n → ∞ with λ n ∼ n, the tail behaviour of S n is determined by the tail probability of the sum n i=1 d j=1 L ij F j , whereas, when λ → ∞ with n fixed, it is determined by the sum with the heaviest tail.
To obtain an expression where both sums contribute to the tail behaviour of S n , we study the influence of the choice of λ n on the behaviour of large deviation probabilities of the form P (S n > λ n ), when n → ∞. In the main result of the paper, Theorem 1, we identify conditions under which there exists a sequence λ n such that both sums contribute to the large deviation probability of S n . In particular, ε i should have heavier tail than F d . We also show that the i.i.d. random vectors L i only contribute through their expectations.
Using the obtained results, we also study sums of heavy-tailed processes with factor structure. We adapt results from Hult and Lindskog (2005) to our case and derive a large deviation principle for our processes on D([0, 1], R), the space of real-valued càdlàg functions on [0, 1]. Here we note that extreme events during a finite time interval occur due to one large jump. Moreover, using 1, we conclude that this large jump can come from either the factor or the idiosyncratic part of the process.
The paper is organised as follows. In Section 2, we derive a large deviation result for sums of dependent random variables from a heavy-tailed factor model. Section 3 contains a numerical example where, under some further assumptions on the factor model, we derive a conditional Monte Carlo algorithm. Moreover, we compare the simulation results with the analytical approximations. Section 4 deals with large deviation results for heavy-tailed Lévy processes with factor structure. Some proofs and technical results are collected in Section 5.
Large Deviations for Heavy-Tailed Factor Models
In this section we investigate under which conditions both the factors and the idiosyncratic components in (4) contribute to the large deviation probability P (S n > λ n ) as n → ∞.
Consider the model given by (3), which in matrix notation reads
R n = Λ n F d + ε n ,(7)
where Λ n denotes the matrix (L i ) n i=1 . We assume that the vector of risk-factors F d is regularly varying i.e.
lim n→∞ nP (a −1 n F d ∈ B) = µ(B),
for Borel sets B ⊂ R d satisfying 0 / ∈ B and µ(∂B) = 0, where µ is given and has tail index α F > 2. Furthermore, the rows of the matrix of factor loadings Λ n , L i , are independent copies of a random vector L = (L 1 , . . . , L d ) with E|L j | αF +δ < ∞ for j = 1, . . . , d and some δ > 0. The elements ε 1 , . . . , ε n of the idiosyncratic term are i.i.d. and regularly varying random variables with tail index α ε > 2.
Denoting S L n,j = n i=1 L ij , we get S n = d j=1 S L n,j F j + n i=1 ε i .(8)
By the law of large numbers,
lim n→∞ S L n,j n = lim n→∞ 1 n n i=1 L ij = EL j a.s.,
as n → ∞, which suggests that
P ( d j=1 S L n,j F j > λ n x) ∼ P ( d j=1 (EL j )F j > λ n n x), as n → ∞.(9)
To verify this, we use Lemmas 1 and 2, below.
Lemma 1. Let X be a d × 1 regularly varying random vector, X ∈ RV(α, µ) and let A n = 0 be a sequence of 1 × d random vectors independent of X such that A n → A = 0 a.s., as n → ∞ and E(sup n |A n | ∞ ) α+δ < ∞, where, |A| ∞ = sup |x|=1 |Ax|.
Then, for 0 < λ n ↑ ∞ and x > 0, we have
lim n→∞ P (A n X > λ n x) P (|X| > λ n ) = x −α µ A −1 (1, ∞) .
Proof. See Section 5.
Lemma 2. Let X i , i = 1, 2, . . . be a sequence of i.i.d. random variables E|X 1 | r < ∞, r > 1. Then E(sup k | k i=1 X i |/k) r < ∞. Proof.
The result follows directly from the L p maximum inequality for martingales, see eg. Durrett (1996).
Lemma 2 is needed to verify the conditions of Lemma 1. Indeed, under the integrability assumptions E|L j | αF +δ < ∞ on L, it follows that E|S L n,j /n| αF +δ < ∞ and that E(sup k |S L k,j /k|) αF +δ < ∞. Now, applying Lemma 1, we conclude that for fixed
x > 0 lim n→∞ P ( d j=1 S L n,j F j > λ n x) P (|F d | > λ n /n) = x −αF µ (EL) −1 (1, ∞) .
We now consider the tail-behaviour of the sum S n . If F d and ε 1 have the same tail indices, we expect F d to dominate the extremal behaviour, i.e. we expect the idiosyncratic components to become less relevant as n grows due to the law of large numbers. Thus, the variation of the sum is mainly due to variation of the factors. If we want to use large deviation probabilities as approximations for finite n, we should try to avoid this behaviour. In the following Theorem, which is the main result of the paper, we state the behaviour of the tail probability of our sum under different assumptions.
Theorem 1. Let F d = (F 1 , . . . , F d ) be a regularly varying random vector, F d ∈ RV(α F , µ)
, and ε i be a sequence of i.i.d. regularly varying random variables, ε i ∈ RV(α ε ), with tail balance parameter p. Consider the factor model given in (7) and the sum S n in Equation (8). Let γ n ≫ ρ n denote lim n→∞ γ n /ρ n = ∞.
(1) If α F ≤ α ε , then for any λ n ≫ n,
lim n→∞ P (S n > λ n x) P (|F d | > λ n /n) = x −αF µ (EL) −1 (1, ∞) . (2) Assume that P (|F d | > x) = L |F | (x)x −αF and P (|ε| > x) = L |ε| (x)x −αε , where α F > α ε > 2. Define θ F = (α F − 1)/(α F − α ε ), θ ε = θ F − 1. If α F > α ε , we have three different possibilities: (a) If λ n ≫ n θF , then lim n→∞ P (S n > λ n x) nP (|ε| > λ n ) = px −αε . (b) If λ n ≪ n θF , then lim n→∞ P (S n > λ n x) P (|F d | > λ n /n) = x −αF µ (EL) −1 (1, ∞) .
(c) If λ n ∼ n θF , and (11) and for C = ∞,
lim n→∞ L |ε| (n θF ) L |F | (n θε ) = C ∈ [0, ∞],(10)then for 0 ≤ C < ∞, lim n→∞ P (S n > λ n x) P (|F d | > λ n /n) = x −αF µ (EL) −1 (1, ∞) + x −αε pClim n→∞ P (S n > λ n x) nP (|ε| > λ n ) = px −αε .
Remark 1. Theorem 1 (c) provides a choice for λ n that, given the tail indices of F and ε, yields the asymptotic behaviour (11). Qualitatively, it also shows that for both parts to contribute to the large deviation behaviour, the idiosyncratic part must have heavier tail than the factors. (10) can be difficult to verify. The slowly varying functions of the norms are often not known, and are not easy to calculate explicitly. Examples where Condition (10) is satisfied include:
Remark 2. Condition
(a). L |F | (x) = c 1 , L |ε| (x) = c 2 (b). L |F | (x) → c 1 , L |ε| (x) → c 2 (c). L |F | (x) = a 1 log x + b 1 , L |ε| (x) = a 2 log x + b 2 .
Example 1. As an illustration of the application of Theorem 1, we consider the case of independent Pareto-distributed factors and idiosyncratic components. Assume that d = 10, i.e. F 10 = (F 1 , . . . , F 10 ). We have L |F | = L |ε| = 1 so that C = 1. Let α F = 5 and α ε = 3. With λ n = n (5−1)/(5−3) = n 2 we obtain
P ( n i=1 R i > λ n x) = P ( 10 j=1 S L n,j F j + n i=1 ε i > λ n x) ∼ P ( 10 j=1 S L n,j F j > λ n x) + P ( n i=1 ε i > λ n x) ∼ 10 j=1 P ( S L n,j F j n > nx) + npP (ε 1 > n 2 x) ∼ n −5 10 j=1 (EL j ) −5 x −5 + px −3 .(12)
Before proving Theorem 1, we state a partial result.
Lemma 3.
Assume that X is a regularly varying d-dimensional random vector, X ∈ RV (µ, α X ), and Y i is a sequence of i.i.d. regularly varying random variables, Y 1 ∈ RV(α Y ), with tail balance parameter p. Let A n be a sequence of d-dimensional random vectors satisfying E(sup n |A n | ∞ ) αX +δ < ∞, for some δ > 0, and A n a.s.
− − → A = 0. Furthermore assume that A n , Y i and X are independent for all i and n.
Consider the tail probabilities
F |X| (x) = P (|X| > x), F * (x) = P (nA n X + n i=1 Y i > x), F 1 (x) = P (nA n X > x), F 2 (x) = P ( n i=1 Y i > x),
where x > 0. Assume that there exists a sequence λ n ≫ n such that
lim n→∞ F 2 (λ n x) F |X| (λ n /n) = Qx −αY ,(13)
where Q ∈ [0, ∞]. Then,
lim n→∞ F 1 (λ n ) F * (λ n x) = 1 x −αX + x −αY Q/µ A −1(14)
and
lim n→∞ F 2 (λ n ) F * (λ n x) = 1 x −αY + x −αX µ A −1 /Q ,(15)
where, µ A −1 = µ(A −1 (1, ∞)). If Q is zero or infinite, we interpret the right hand side of relations (14)-(15) as limits.
Proof. See Section 5.
Proof of Theorem 1. We only derive Relation (11), the other relations are proved in a similar fashion. First, we compute Q in (13). This gives us the sequence λ n via the tail indices. We then apply Lemma 3 to obtain the results. We have, with
F 2 (λ n x) = P ( n i=1 ε i > λ n x), lim n→∞ F 2 (λ n x) F |F | (λ n /n) = lim n→∞ F 2 (λ n x) nF |ε| (λ n ) nF |ε| (λ n ) F |F | (λ n /n) = lim n→∞ F 2 (λ n x) nF |ε| (λ n ) I1 L |ε| (n θF ) L |F | (n θε ) I2 nλ −αε n (λ n /n) −αF
I3
.
From (6) we get I 1 → px −αε and, by assumption, I 2 → C. For simplicity, we restrict ourselves to the case I 3 = 1. This condition gives the expression for λ n . We then have Q = pC. Applying Lemma 3 we obtain, with
µ L −1 = µ L −1 (1, ∞) , lim n→∞ F * (λ n x) F |F | (λ n /n) = lim n→∞ F * (λ n x) F 1 (λ n ) F 1 (λ n ) F |F | (λ n /n) = (x −αF + Qx −αε /µ L −1 )µ L −1 = µ L −1 x −αF + px −αε C,
and we arrive at relation (11).
The above results rely on the regular variation of the components involved. In the case of light-tailed random variables, the decomposition in Theorem 1 is no longer valid. We illustrate this in the following corollary by assuming light-tailed factors.
Corollary 1. Let X > 0 be a light-tailed random variable with tail distribution
F X (x) ∼ e −g(x) , where g(x) − cx → ∞, as x → ∞ for some c > 0. Let Y i , i = 1, 2, . .
. be a sequence of i.i.d. regularly varying random variables with tail-index α > 0, Y i ∈ RV (α). Then, for any sequence λ n such that λ n /n → ∞,
lim n→∞ P (nX + n i=1 Y i > λ n ) P ( n i=1 Y i > λ n ) = 1.
Proof. Considering Equation (13), we have
F 2 (λ n ) F |X| (λ n /n) = P ( n i=1 Y i > λ n ) P (X > λ n /n) ∼ e −g(λn/n) nλ −α n ,
so that log Q = lim n→∞ g(λ n /n) + log n − α log λ n = ∞.
Hence, using Equation (15) we obtain the result.
Simulation
To see how the approximations derived in the previous section behave, we will present a short simulation study. Since tail probabilities are rare events, naive Monte Carlo Simulation can be very slow. To achieve a given relative error, a huge number of simulations are often needed. Methods of variance reduction are therefore crucial for obtaining a satisfactory estimation. We present a method for estimating the tail probability of a sum of variables from our factor model, under certain restrictive conditions.
Variance reduction algorithms for sums of heavy-tailed random variables are often based on the observation that, asymptotically, a sum is determined by its largest term. This is then used for conditioning or change of measure, Denoting M n,d = max(ε 1 , . . . , ε n , S L n,1 F 1 , . . . , S L n,d F d ) and assuming that the all variables are continuous, we have
P (S n > x) = P ( d j=1 S L n,j F j + n i=1 ε i > x) = P (S F d + S ε n > x) = nP (S n > x, M n,d = ε n ) + dP (S n > x, M n,d = S L n,d F d ).
Conditioning yields If the distributions of ε and S L n,d F d are known, these probabilities can be calculated explicitly. Alternatively, conditioning on Λ n and calculating the last probability by simulation only requires knowledge of the marginal distribution of F d .
P (S n > x, M n,d = ε n ) = EP (S n > x, M n,d = ε n |ε 1 , . . . , ε n−1 , S L n,1 F 1 , . . . , S L n,d F d ) = EP (ε > (x − S n−1 ) ∨ M n−1,d |ε 1 , . . . , ε n−1 , S L n,1 F 1 , . . . , S L n,d F d ).
In Table 1, we compare the analytical approximation of the tail probability in Example 1 to simulations using the above algorithm. Since it is a large deviation result, the approximation performs best when we consider regions far out in the tail, i.e. when λ n x = n 2 x is large. The resulting probabilities in these regions range from small to extremely small. As expected, we obtain the worst results for x = 0.1 and n = 10 3 . In Theorem 2 we establish a large deviation result for the process
S n (t) = d j=1 S L n,j F j (t) + n i=1 ε i (t).
Theorem 2. Assume that F d (t) is a d-dimensional Lévy process and that ε i (t), i = 1, . . . , n are i.i.d. Lévy processes. Furthermore, assume that their Lévy measures are regularly varying with tail indices satisfying α F > α ε > 2. Let
P (|F d (1)| > x) = L |F | (x)x −αF , P (|ε(1)| > x) = L |ε| (x)x −αε ,
and assume that L |F | (x) and L |ε| (x) satisfy condition (10) in Theorem 1. Then,
lim n→∞ γ n P (λ −1 n S n ∈ B) =m(B),(16)
for all Borel sets B ∈ D([0, 1], R) with 0 / ∈ B andm(∂B) = 0. We denote this property by S n ∈ LD((γ n , λ n ),m, D([0, 1], R)).
Moreover,m puts all mass on step functions with one step, i.e. 1] , v ∈ [0, 1], y ∈ R}. That is, any extreme event during the interval is due to one large jump of either the factor or the idiosyncratic part of the process.
m(V c 0 ) = 0, where V 0 = {x ∈ D([0, 1], R) : x = y1 [v,
The proof of Theorem 2 is given in Section 5, below. We end this section with an example.
Example 2. Let F d (t) and ε(t) be compound Poisson processes
F d (t) = N F t i=1 Z i ε i (t) = N ε t i=j W ij ,
where Z i = (Z 1 i , . . . , Z d i ) are random vectors with i.i.d. components such that P (|Z 1 1 | > x) = x −αF with tail balance parameter p F and W ij are i.i.d. random variables such that P (|W 11 | > x) = x −αε with tail balance parameter p ε . N F t and N ε t are Poisson processes with intensities λ F and λ ε , respectively. Assume that the tail-indices satisfy α F > α ε > 2. Both F d (1) and ε i (1) are regularly varying, and with | · | = | · | 1 , we have
P (|F d (1)| > x) ∼ dλ F P (|Z 11 | > x) and P (|ε 1 (1)| > x) ∼ λ ε P (|W 11 | > x).
The conditions of Theorem 2 being satisfied, we get γ n P (S n ∈ λ n B) →m(B), wherem puts all its mass on step functions with one step. Moreover,
m t (x, ∞) := lim n→∞ γ n P (λ −1 n S n (t) ∈ (x, ∞))
is explicitly given by (see (11), above)
m t (x, ∞) = tp F d j=1 (EL j ) −αF x −αF + tp ε λ ε dλ F x −αε .
Proofs and Technical Results
To prove Lemma 1, we use the following multivariate version of Breiman's Lemma proved by Basrak, Davis and Mikosch (2002).
Lemma 4 (Breiman's lemma). Let X be a d×1 regularly varying random vector and let A be a k × d random matrix, independent of X. If 0 < E|A| α+δ ∞ < ∞ for some δ > 0, then lim n→∞ P (AX ∈ a n B) P (|X| > a n ) = E(µ • A −1 (B)).
for any Borel set B ⊂ R d satisfying 0 / ∈ B and µ(∂B) = 0.
Proof of Lemma 1. We split X into positive and negative parts,
X = X + − X − , where X + = (X + 1 , . . . , X + d ), X − = (X − 1 , . . . , X − d )
. The infimum and supremum of the vector A k is interpreted component-wise, i.e. sup k>M A k = (sup k>M A 1 k , . . . , sup k>M A d k ) and analogously for the infimum. Fix M > 0. For n > M we have,
P (A n X > λ n x) = P (A n (X + − X − ) > λ n x) ≤ P ( sup k>M A k X + − inf k>M A k X − > λ n x) = P (( sup k>M A k , inf k>M A k )(X + , −X − ) T > λ n x).(17)
The same argument also provides a lower bound,
P (A n X > λ n x) ≥ P (( inf k>M A k , sup k>M A k )(X + , −X − ) T > λ n x).(18)
The probability P (A n X > λ n x)/P (|X| > λ n ) is thus bounded from above and below. To determine these bounds, we need to show regular variation of the vector (X + , −X − ) T .
Let E 1 = R d \{0} and E 2 = {z ′ ∈ R 2d \{0} : z ′ = (z + , −z − ) T , z ∈ R d \{0}}
and define the continuous transformation
T : E 1 −→ E 2 x −→ (x + , −x − ) T .
Any relatively compact set K 2 of E 2 is of the form
K 2 = {z ′ = (z + , −z − ) ∈ R 2d \{0} : z ∈ R d \{0}},
bounded away from 0, i.e. 0 / ∈ K 2 . Since z ′ = 0 ⇒ z = 0, it is obvious that the inverse images of these sets in R d \{0} are bounded away from 0 as well.
Hence, if K 2 is compact in R 2d \{0} then K 1 = T −1 (K 2 ) is compact in R d \{0}.
Therefore, vague convergence of a sequence of measures µ n on E 1 implies vague convergence of the induced measuresμ n = µ n • T −1 on E 2 . Specifically, since |T (x)| = |x| and T (ax) = aT (x) for any a > 0,
P (T (X) ∈ λ n B) P (|T (X)| > λ n ) = P (X ∈ T −1 (λ n B)) P (|X| > λ n ) = P (X ∈ λ n T −1 (B)) P (|X| > λ n ) v − → µ(T −1 (B)),
Therefore, the vector T (X) = (X + , −X − ) T is regularly varying.
Since, E sup n |A n | ∞ < ∞ it follows that E|(sup k>M A k , inf k>M A k )| ∞ < ∞, so we can use the multivariate version of Breiman's lemma to determine the bounds (17) and (18). This yields
E µ • ( inf k>M A k , sup k>M A k ) −1 (1, ∞) x −α ≤ lim inf n→∞ P (A n X > λ n x) P (|X| > λ n ) ≤ lim sup n→∞ P (A n X > λ n x) P (|X| > λ n ) (19) ≤ E µ • ( sup k>M A k , inf k>M A k ) −1 (1, ∞) x −α . Since A n a.s. − −−− → n→∞ A we have inf k>M A k a.s. −−−−→ M→∞ A and sup k>M A k a.s. −−−−→ M→∞ A. It
remains to verify that we can evaluate these limits inside the expectations. We have
µ • ( inf k>M A k , sup k>M A k ) −1 (1, ∞) ≤ µ • ( sup k>M A k , inf k>M A k ) −1 (1, ∞) ≤ µ • (sup k A k , inf k A k ) −1 (1, ∞)
and
Eµ • (sup k A k , inf k A k ) −1 (1, ∞) = Eµ(z ∈ R d : (sup k A k , inf k A k )(z + , −z − ) T > 1) ≤ Eµ(z ∈ R d : (sup k |A k | ∞ )1 T 2d (z + , z − ) T > 1) = E(sup k |A k | ∞ ) α µ(z ∈ R d : 1 T 2d (z + , z − ) T > 1) = E(sup k |A k | ∞ ) α µ(z ∈ R d : 1 T d |z| > 1) < ∞,
with |z| = (|z 1 |, . . . , |z d |). Hence, by Dominated Convergence,
lim M→∞ Eµ(z ∈ R d : ( sup k>M A k , inf k>M A k )(z + , −z − ) T > 1) = Eµ(z ∈ R d : (A, A)(z + , −z − ) T > 1) = Eµ(z ∈ R d : Az > 1).
A similar calculation applies to the lower bound in equation (19), with the same limit. Letting M → ∞ in that equation yields the conclusion.
Proof of Lemma 3. We first note that if U and V are independent random variables, we have
P (U + V > x) ≥ P (U > (1 + δ)x)P (|V | < δx) + P (|U | < δx)P (V > (1 + δ)x). Therefore, setting U = nA n X and V = n i=1 Y i , we get F * (x) ≥ F 1 ((1 + δ)x)P (| n i=1 Y i | < δx) + F 2 ((1 + δ)x)P (|nA n X| < δx) .(20)
Furthermore, since for δ ∈ (0, 1/2) we have
{U + V > x} ⊂ {U > (1 − δ)x} ∪ {V > (1 − δ)x} ∪ {U > δx, V > δx},
it follows that
F * (x) ≤ F 1 ((1 − δ)x) + F 2 ((1 − δ)x) + F 1 (δx)F 2 (x).(21)
Relation (14) is then obtained by dividing both sides in (20) and (21) by F 1 (λ n ), and inverting.
The lower bound consists of two parts. The first part is
lim n→∞ F 1 ((1 + δ)xλ n ) F 1 (λ n ) P (| n i=1 Y i )| < δλ n x) = lim n→∞ F 1 ((1 + δ)λ n x) F |X| (λ n /n) F |X| (λ n /n) F 1 (λ n ) P (| n i=1 Y i )| < δλ n x) = x −αX (1 + δ) −αX ,
where we have used Lemma 1 and the fact that n/λ n → 0, as n → ∞, i.e. λ n is in the large deviation region which imlpies that (cf. Proposition 3.1 in Mikosch and Nagaev (1998))
lim n→∞ P (| n i=1 Y i )| < δλ n ) = 1
and lim n→∞ P (|nA n X| < δλ n ) = 1.
The second part is
lim n→∞ F 2 ((1 + δ)λ n x) F 1 (λ n ) P (|nA n X| < δλ n x) = lim n→∞ F 2 ((1 + δ)λ n x) F |X| (λ n /n) F |X| (λ n /n) F 1 (λ n ) P (|nA n X| < δλ n x) = Q (1 + δ)x −αY µ(A −1 (1, ∞)) −1 ,
using Assumption (13) and Lemma 1.
The upper bound is treated similarly, although it consists of three parts. The first part is treated using Lemma 1 as above. The second part is The third and last part is lim n→∞ F 1 (δλ n z) F 1 (λ n z)
lim n→∞ F 2 ((1 − δ)λ n x) F 1 (λ n ) = lim→(zδ) −α X F 2 (δλ n ) →0 = 0.
Hence, with µ A −1 = µ(A −1 (1, ∞)), it follows that
1 (1 − δ)z −αX + Q (1 − δ)x −αY /µ A −1 ≤ lim inf n→∞ F 1 (λ n ) F * (λ n x) ≤ lim sup n→∞ F 1 (λ n ) F * (λ n x) ≤ 1 (1 + δ)x −αX + Q (1 + δ)x −αY /µ A −1 .
Letting δ → 0 proves the first relation. The second relation is shown analogously.
The following proof of Theorem 2 relies on several results from the work by Hult and Lindskog (2005), adapted to our conditions. All the arguments in their proofs apply, with obvious modifications.
Proof of Theorem 2. By Theorem 1, we have that lim n→∞ γ n P (λ −1 n S n (1) > x) =μ(x, ∞), whereμ is given by (11) and γ −1 n = P (|F d (1)| > λ n /n). Since both F d and ε are Lévy-processes, we also have lim n→∞ γ n P (λ −1 n S n (t) > x) = tμ(x, ∞)
for every t ∈
importance sampling. Examples of such algorithms includeJuneja et al. (2002), where measures for importance sampling are chosen by the so-called hazard rate twisting method.Dupuis et al. (2006) use a dynamic algorithm to change measure for each term in the sum, making sure that the rare event in question occurs. In the setting of a portfolio loss depending on multivariate t-distributed risk factors,Glasserman et al. (2002) derive an importance sampling algorithm using a quadratic approximation of the portfolio loss.Using the conditioning approach suggested inAsmussen and Kroese (2006) we can state a simulation algorithm for our factor model with i.i.d. factors, i.i.d. loadings and i.i.d. idiosyncratic components.
P
(S n > x, M n,d = S L n,d D d ) = EP (S n > x, M n,d = S L n,d F d |ε 1 , . . . , ε n , S L n,1 F 1 , . . . , S L n,d−1 F d−1 ) = EP (S L n,d F d > (x − S n−1 ) ∨ M n,d−1 |ε 1 , . . . , ε n , S L n,1 F 1 , . . . , S L n,d−1 F d−1 ).
F 2 (( 1 F
21− δ)λ n x) F |X| (λ n /n) |X| (λ n /n) F 1 (λ n ) = Q (1 − δ)x −αY µ(A −1 (1, ∞)) −1 .
0 .
0ε ) −m 0 (B c 0,ε ) = δμ(y ∈ R : |y| > x) m 1 (B c 0,ε ) −m 1−δ (B c 0,ε ) = δμ(y ∈ R : |y| > x).Finally, we have α n λn,1 (1) = sup{P n s,t (x, B c x,λn ) : x ∈ R; s, t ∈ [0, 1]; t − s ∈ [0, 1]} = P (|S n (1) − 0| > λ n ) → 0,as n → ∞, since λ n is in the large deviation region. The conditions of Theorem 13 in Hult and Lindskog (2005) are hence satisfied. This proves the first part of Proposition 2. It remains to show thatm puts all its mass on step functions with one step. Let B(p, ǫ, [0, 1]) = {x ∈ D([0, 1], R d ) : x has ǫ-oscillation p times in [0, 1]}, where, for ǫ > 0 and p a positive integer, the process x ∈ D([0, 1], R d ) is said to have ǫ-oscillation p times in [0, 1] if there exist t 0 , . . . , t p ∈ [0, 1] with t 0 < . . . < t p such that |x ti − x ti−1 | > ǫ for 1 = 1, . . . , p. Using Lemma 21 in Hult and Lindskog (2005), we get lim inf n→∞ γ n P (S n ∈ B(2, λ n ǫ, [0, 1])) = 0. Since the convergence of γ n P (λ −1 n S n ∈ B) tom(B) is equivalent to lim inf n→∞ γ n P (λ −1 n S n ∈ G) ≥m(G) for all open and bounded G, and G = B(2, ǫ, [0, 1]) is open, we have that m(B(2, ǫ, [0, 1])) = 0 for all ǫ > It thatm(V c 0 ) ≤m( ǫ∈Q,ǫ>0 B(2, ǫ, [0, 1])) = 0.
Large Deviations for Factor ProcessesIn this section, we study the large deviation behaviour of sums of heavy-tailed processes with factor structure. We assume that, in Equation (4), F d = {F d (t) : t ∈ [0, 1]} and ε i = {ε i (t) : t ∈ [0, 1]} are Lévy processes, whose increments are regularly varying, or equivalently, whose Lévy measures are regularly varying.
AcknowledgementsThe authors would like to thank Allan Gut and Filip Lindskog for valuable discussions as well as comments on the paper. Financial support from the Göran Collert Foundation is gratefully acknowledged.
Improved algorithms for rare event simulation with heavy tails. S Asmussen, D P Kroese, Adv. Appl. Probab. 382Asmussen, S. and Kroese, D.P. (2006) Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Probab. 38 (2), 545-558.
Regular variation of GARCH processes. B Basrak, R A Davis, T Mikosch, Stoch. Proc. Appl. 99Basrak, B., Davis, R.A. and Mikosch, T. (2002) Regular variation of GARCH pro- cesses, Stoch. Proc. Appl. 99, 95-116.
On some limit theorems similar to the arc-sine law. L Breiman, Theory Probab. Appl. 10Breiman, L. (1965) On some limit theorems similar to the arc-sine law, Theory Probab. Appl. 10, 323-331.
. J H Cochrane, Princeton University PressAsset PricingCochrane, J. H. (2001) Asset Pricing, Princeton University Press.
Empirical properties of asset returns: stylized facts and statistical issues. R Cont, Quantitative Finance. 1Cont, R. (2001) Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance 1, 223-236.
Importance sampling for sums of random variables with regularly varying tails. P Dupuis, K Leder, H Wang, Working paper. Brown UniversityDupuis, P., Leder, K. and Wang, H. (2006) Importance sampling for sums of random variables with regularly varying tails. Working paper. Brown University.
R Durrett, Probability: Theory and Examples. Duxbury pressDurrett, R. (1996) Probability: Theory and Examples, Duxbury press.
Modelling Extremal Events for Insurance and Finance. P Embrechts, C Klüppelberg, T Mikosch, Springer-VerlagBerlinEmbrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin.
Portfolio value-atrisk with heavy-tailed risk factors. P Glasserman, P Heidelberger, P Shahabuddin, Mathematical Finance. 12Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2002) Portfolio value-at- risk with heavy-tailed risk factors, Mathematical Finance 12, 239-269.
Extremal behavior of regularly varying stochastic processes. H Hult, F Lindskog, Stoch. Proc. Appl. 115Hult, H. and Lindskog, F. (2005) Extremal behavior of regularly varying stochastic processes. Stoch. Proc. Appl. 115, 249-274.
Regular variation for measures on metric spaces. H Hult, F Lindskog, 80Nouvelle SériePublications de l'Institut MathématiqueHult, H. and Lindskog, F. (2006) Regular variation for measures on metric spaces. Publications de l'Institut Mathématique, Nouvelle Série, 80, 121-140.
Simulating heavy-tailed processes using delayed hazard rate twisting. S Juneja, P Shahabuddin, ACM Transactions on Modeling and Computer Simulation. 12Juneja, S. and Shahabuddin, P. (2002) Simulating heavy-tailed processes using delayed hazard rate twisting. ACM Transactions on Modeling and Computer Simulation 12, 94- 118.
Large deviations of heavy-tailed sums with applications in insurance. T Mikosch, A Nagaev, 1Mikosch, T. and Nagaev, A. (1998) Large deviations of heavy-tailed sums with appli- cations in insurance, Extremes 1, 81-110.
On large deviation probabilities for sums of independent random variables. A Nagaev, Mathematical Institute, TashkentDoctor of Science ThesisNagaev, A. (1970) On large deviation probabilities for sums of independent random variables. Doctor of Science Thesis, Mathematical Institute, Tashkent.
On the foundations of multivariate heavy-tail analysis. S Resnick, J. Appl. Probab. 41Resnick, S. (2004) On the foundations of multivariate heavy-tail analysis. J. Appl. Probab. 41, 191-212.
| {'fraction_non_alphanumeric': 0.10850412686636372, 'fraction_numerical': 0.027110575288262388, 'mean_word_length': 3.0022268959544722, 'pattern_counts': {'":': 0, '<': 30, '<?xml version=': 0, '>': 153, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 41, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms contribute to the behaviour of the tail-probability of the sum. A simple conditional Monte Carlo algorithm is also provided together with a comparison between the simulations and the large deviation approximation. We also study large deviation probabilities for stochastic processes with factor structure. The processes involved are assumed to be Lévy processes with regularly varying jump measures. Based on the results of the first part of the paper, we show that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process.', 'arxivid': '0712.0459', 'author': ['Boualem Djehiche \nDepartment of Mathematics Royal Institute of Technology SE\n100 44Stockholm\n', 'Jens Svensson \nDepartment of Mathematics Royal Institute of Technology SE\n100 44Stockholm\n'], 'authoraffiliation': ['Department of Mathematics Royal Institute of Technology SE\n100 44Stockholm', 'Department of Mathematics Royal Institute of Technology SE\n100 44Stockholm'], 'corpusid': 115174588, 'doi': '10.1016/j.spl.2008.08.011', 'github_urls': [], 'n_tokens_mistral': 12208, 'n_tokens_neox': 11031, 'n_words': 6521, 'pdfsha': '21a8f5c57eea23fcfa330cad859b8fd71e3618f1', 'pdfurls': ['https://arxiv.org/pdf/0712.0459v1.pdf'], 'title': ['Large Deviations for Heavy-Tailed Factor Models', 'Large Deviations for Heavy-Tailed Factor Models'], 'venue': []} |
arxiv |
A Dependency Look at the Reality of Constituency
2018
Xinying Chen
Foreign Languages Research Center
School of Foreign Studies
Xi'an Jiaotong University
No.28 Xianning West Road710049Xi'an, ShaanxiP.R. China
Carlos Gómez-Rodríguez
Departamento de Computación. Facultade de Informática
LyS Research Group
Universidade da Coruña. FASTPARSE Lab
15071 AElviña, CoruñaSpain
Ramon Ferrer-I-Cancho
Complexity & Quantitative Linguistics Lab
Departament de Ciències de la Computació
LARCA Research Group
Universitat Politècnica de Catalunya
Campus Nord, Edifici Omega, Jordi Girona Salgado 1-308034BarcelonaCataloniaSpain
A Dependency Look at the Reality of Constituency
Glottometrics
402018104
A comment on "Neurophysiological dynamics of phrase-structure building during sentence processing" by Nelson et al (2017), Proceedings of the National Academy of Sciences USA 114(18), E3669-E3678. Recently, Nelson et al. (2017) have addressed the fundamental problem of the neurophysicological support for complex syntactic operations of theoretical computational models. They interpret their compelling results as supporting the neural reality of phrase structure. Such a conclusion opens various questions.First, constituency is not the only possible reality for the syntactic structure of sentences. An alternative is dependency, where the structure of a sentence is defined by word pairwise dependencies (Fig.
Cancho, 2004)
, thus a bigram model misses 50% of the dependencies. Bigrams are a weak baseline, as the common practice in computational linguistics is using at least smoothed trigram models, and often 5-gram models, to obtain meaningful predictions (Jozefowicz, Vinyals, Schuster, Shazeer, & Wu, 2016). A higher-order lexical n-gram model would strengthen the current results. The authors also employ more sophisticated n-gram models. One is an unbounded model based on part-of-speech categories, implying a dramatic loss of information with respect to the original words which might explain its poor performance. The other is a syntactic n-gram, but not enough information is provided about its definition and implementation. Regardless, since the model is obtained from a corpus derived from a toy grammar and lexicon, its probabilities are likely to be unrealistic and thus it is problematic.
In sum, dependency offers a better approach to the syntactic complexity of languages and merge. n-gram models of higher complexity should be the subject of future research involving realistic sentences.
Figure 1 :
1Syntactic dependency structure of the sentence in Fig 2 A of Nelson et al. (2017) according to Universal Dependencies (McDonald et al., 2013).
1 Foreign Languages Research Center, School of Foreign Studies, Xi'an Jiaotong University, No.28 Xianning West Road, 710049 Xi'an, Shaanxi, P.R. China. 2 Universidade da Coruña. FASTPARSE Lab, LyS Research Group. Departamento de Computación. Facultade de Informática, Elviña 15071 A Coruña, Spain 3 Complexity & Quantitative Linguistics Lab, LARCA Research Group, Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Campus Nord, Edifici Omega, Jordi Girona Salgado 1-3, 08034 Barcelona, Catalonia (Spain). Corresponding author, [email protected].
Acknowledgements X.C. is supported by the Social Science Fund of Shaanxi State (2015K001). C.G.R is funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714150 FASTPARSE), and by the TELEPARES-UDC project (FFI2014-51978-C2-2-R) from MINECO (Ministerio de Economía y Competitividad). R.F.C is funded by the grants 2014SGR 890 (MACDA) from AGAUR (Generalitat de Catalunya) and the grant TIN2014-57226-P from MINECO.
The myth of language universals: language diversity and its importance for cognitive science. N Evans, S C Levinson, Behavioral and Brain Sciences. 32Evans, N., & Levinson, S. C. (2009). The myth of language universals: language diversity and its importance for cognitive science. Behavioral and Brain Sciences 32, 429-492.
Euclidean distance between syntactically linked words. R Ferrer-I-Cancho, Physical Review E. 7056135Ferrer-i-Cancho, R. (2004). Euclidean distance between syntactically linked words. Physical Review E 70, 056135.
Natural language processing and the Now-or-Never bottleneck. C Gómez-Rodríguez, Behavioral and Brain Sciences. 3974Gómez-Rodríguez, C. (2016). Natural language processing and the Now-or-Never bottleneck. Behavioral and Brain Sciences 39, e74.
Exploring the limits of language modeling. R Jozefowicz, O Vinyals, M Schuster, N Shazeer, Y Wu, arXiv:1602.02410arXiv preprintJozefowicz, R., Vinyals, O., Schuster, M., Shazeer, N., & Wu, Y. (2016). Exploring the limits of language modeling. arXiv preprint arXiv:1602.02410.
Dependency parsing. S Kübler, R Mcdonald, J Nivre, Morgan and Claypool PublishersKübler, S., McDonald, R. & Nivre, J. (2009). Dependency parsing. Morgan and Claypool Publishers.
Syntactic flexibility in the noun: Evidence from picture naming. N A Lester, F Moscoso Del Prado Martín, Proceedings of the 38th Annual Conference of the Cognitive Science Society. Papafragou, A., Grodner, D., Mirman, D., & Trueswell, J.C.the 38th Annual Conference of the Cognitive Science SocietyAustin, TXCognitive Science SocietyLester, N. A. & Moscoso del Prado Martín, F. (2016). Syntactic flexibility in the noun: Evidence from picture naming. In: Papafragou, A., Grodner, D., Mirman, D., & Trueswell, J.C. (Eds.), Proceedings of the 38th Annual Conference of the Cognitive Science Society (pp. 2585-2590). Austin, TX: Cognitive Science Society.
Dependency distance as a metric of language comprehension difficulty. H Liu, Journal of Cognitive Science. 9Liu, H. (2008). Dependency distance as a metric of language comprehension difficulty. Journal of Cognitive Science 9, 159-191.
Universal dependency annotation for multilingual parsing. R Mcdonald, J Nivre, Y Quirmbach-Brundage, Y Goldberg, D Das, K Ganchev, K B Hall, S Petrov, H Zhang, O Täckström, C Bedini, N Bertomeu, J Lee, Proceedings of ACL. ACLMcDonald, R., Nivre, J., Quirmbach-Brundage, Y., Goldberg, Y., Das, D., Ganchev, K. Hall, K.B., Petrov, S., Zhang, H., Täckström, O., Bedini, C., Bertomeu, N. & Lee, J. (2013). Universal dependency annotation for multilingual parsing. Proceedings of ACL (pp. 92-97).
Dependency in language-2011. I Mel'čuk, Proceedings of the international conference on dependency linguistics. K. Gerdes, E. Hajicova, & L. Wannerthe international conference on dependency linguisticsDepLing; BarcelonaMel'čuk, I. (2011). Dependency in language-2011. In: K. Gerdes, E. Hajicova, & L. Wanner (Eds.), Proceedings of the international conference on dependency linguistics, DepLing 2011, Barcelona, September 5-7, 2011.
Neurophysiological dynamics of phrase-structure building during sentence processing. M J Nelson, I El Karoui, K Giber, X Yang, L Cohen, H Koopman, S S Cash, L Naccache, J T Hale, C P Pallier, S Dehaene, Proceedings of the National Academy of Sciences. 11418Nelson, M. J., El Karoui, I., Giber, K., Yang, X., Cohen, L., Koopman, H., Cash, S. S., Naccache, L., Hale, J. T., Pallier, C.P. & Dehaene, S. (2017). Neurophysiological dynamics of phrase-structure building during sentence processing. Proceedings of the National Academy of Sciences, 114 (18), E3669-E3678.
Bare phrase structure, label-less trees, and specifier-less syntax: Is minimalism becoming a dependency grammar?. T Osborne, M Putnam, T Gross, The Linguistic Review. 28Osborne, T., Putnam, M., & Gross, T. (2011). Bare phrase structure, label-less trees, and specifier-less syntax: Is minimalism becoming a dependency grammar? The Linguistic Review 28, 315-364.
| {'fraction_non_alphanumeric': 0.06652027899767501, 'fraction_numerical': 0.04004133298889176, 'mean_word_length': 4.852607709750567, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'A comment on "Neurophysiological dynamics of phrase-structure building during sentence processing" by Nelson et al (2017), Proceedings of the National Academy of Sciences USA 114(18), E3669-E3678. Recently, Nelson et al. (2017) have addressed the fundamental problem of the neurophysicological support for complex syntactic operations of theoretical computational models. They interpret their compelling results as supporting the neural reality of phrase structure. Such a conclusion opens various questions.First, constituency is not the only possible reality for the syntactic structure of sentences. An alternative is dependency, where the structure of a sentence is defined by word pairwise dependencies (Fig.', 'arxivid': '1708.07722', 'author': ["Xinying Chen \nForeign Languages Research Center\nSchool of Foreign Studies\nXi'an Jiaotong University\nNo.28 Xianning West Road710049Xi'an, ShaanxiP.R. China\n", 'Carlos Gómez-Rodríguez \nDepartamento de Computación. Facultade de Informática\nLyS Research Group\nUniversidade da Coruña. FASTPARSE Lab\n15071 AElviña, CoruñaSpain\n', 'Ramon Ferrer-I-Cancho \nComplexity & Quantitative Linguistics Lab\nDepartament de Ciències de la Computació\nLARCA Research Group\nUniversitat Politècnica de Catalunya\nCampus Nord, Edifici Omega, Jordi Girona Salgado 1-308034BarcelonaCataloniaSpain\n'], 'authoraffiliation': ["Foreign Languages Research Center\nSchool of Foreign Studies\nXi'an Jiaotong University\nNo.28 Xianning West Road710049Xi'an, ShaanxiP.R. China", 'Departamento de Computación. Facultade de Informática\nLyS Research Group\nUniversidade da Coruña. FASTPARSE Lab\n15071 AElviña, CoruñaSpain', 'Complexity & Quantitative Linguistics Lab\nDepartament de Ciències de la Computació\nLARCA Research Group\nUniversitat Politècnica de Catalunya\nCampus Nord, Edifici Omega, Jordi Girona Salgado 1-308034BarcelonaCataloniaSpain'], 'corpusid': 3947009, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2513, 'n_tokens_neox': 2124, 'n_words': 1063, 'pdfsha': '2d0cd9adf7c0f1bb5f52062c64435cabeb9eeeeb', 'pdfurls': ['https://arxiv.org/pdf/1708.07722v3.pdf'], 'title': ['A Dependency Look at the Reality of Constituency', 'A Dependency Look at the Reality of Constituency'], 'venue': ['Glottometrics']} |
arxiv |
L3Cube-IndicSBERT: A simple approach for learning cross-lingual sentence representations using multilingual BERT
Samruddhi Deode
Mksss ' Cummins
College of Engineering for Women
MKSSS' Cummins College of Engineering for Women
JANHAVI GADRE *
L3Cube Pune, L3Cube PuneIndia, India
MKSSS' Cummins College of Engineering for Women
ADITI KAJALE *
L3Cube PuneIndia
MKSSS' Cummins College of Engineering for Women
ANANYA JOSHI *
L3Cube PuneIndia RAVIRAJ
JOSHI
Indian Institute of Technology Madras
L3Cube PuneIndia
L3Cube-IndicSBERT: A simple approach for learning cross-lingual sentence representations using multilingual BERT
CCS Concepts: • Computing methodologies → Lexical semanticsNat- ural language processingLanguage resources Additional Key Words and Phrases: Natural Language Processing, Sentence BERT, Sentence Transformers, Semantic Textual Similarity, Indian Regional Languages, Low Resource Languages, Text Classification, IndicNLP, BERT, Natural Language Inference
The multilingual Sentence-BERT (SBERT) models map different languages to common representation space and are useful for cross-language similarity and mining tasks. We propose a simple yet effective approach to convert vanilla multilingual BERT models into multilingual sentence BERT models using synthetic corpus. We simply aggregate translated NLI or STS datasets of the low-resource target languages together and perform SBERT-like finetuning of the vanilla multilingual BERT model. We show that multilingual BERT models are inherent cross-lingual learners and this simple baseline fine-tuning approach without explicit cross-lingual training yields exceptional cross-lingual properties. We show the efficacy of our approach on 10 major Indic languages and also show the applicability of our approach to non-Indic languages German and French. Using this approach, we further present L3Cube-IndicSBERT, the first multilingual sentence representation model specifically for Indian languages Hindi, Marathi, Kannada, Telugu, Malayalam, Tamil, Gujarati, Odia, Bengali, and Punjabi. The IndicSBERT exhibits strong cross-lingual capabilities and performs significantly better than alternatives like LaBSE, LASER, and paraphrase-multilingual-mpnetbase-v2 on Indic cross-lingual and monolingual sentence similarity tasks. We also release monolingual SBERT models for each of the languages and show that IndicSBERT performs competitively with its monolingual counterparts. These models have been evaluated using embedding similarity scores and classification accuracy.
INTRODUCTION
Natural Language Processing (NLP) is an interdisciplinary field that focuses on developing techniques to process and understand human language [27]. Semantic Textual Similarity (STS) is a crucial task in NLP, which measures the equivalence between the meaning of two or more text segments [2,5]. The aim of STS is to identify the semantic similarity between text inputs, taking into account their meaning rather than just surface features like word frequency and length [1]. The concept is widely used in various NLP applications, including question-answering [15], information retrieval [25], text generation [16], etc.
One common tool used for this purpose is BERT (Bidirectional Encoder Representations from Transformers) [10], a pre-trained transformer-based language model that has achieved state-of-theart performance on a wide range of NLP tasks. However, BERT is not well-suited for semantic similarity tasks as it is trained to predict masked words in a sentence, which does not directly optimize for semantic similarity [42]. To address this limitation, Sentence-BERT (SBERT) [32] was proposed, a modified version of the BERT architecture designed to generate sentence representations for the improved semantic similarity between sentences. The SBERT makes use of a siamese network [23] and is trained using specific datasets like STS, resulting in representations specifically geared for semantic similarity.
Recent works are focused on multilingual SBERT models capable of encoding sentences from different languages to the same representation space [36,41]. With these models, it is possible to extend NLP tasks to different languages without training a language-specific model. These multilingual models often employ teacher-student training [14,33] or are based on translation ranking tasks [13]. These methods make use of parallel translation corpus in target languages for training a cross-lingual model [3,9,38]. Even vanilla multilingual BERT models have been shown to have surprisingly good cross-lingual properties [31,40]. However, their performance is not good as the multilingual sentence BERT models.
In this work, we propose a simple approach to learning crosslingual sentence representations without using any parallel corpus. We leverage pre-trained multilingual transformer models and finetune them using our mixed training strategy, as depicted in Figure 1. We mix the monolingual translated NLI / STSb corpus for target languages and fine-tune the multilingual BERT model in an SBERT setup. We show that this simple mixed data training is sufficient to train a multilingual SBERT model with strong cross-lingual properties. This strategy is capable of significantly amplifying the cross-lingual properties of the existing vanilla multilingual BERT model. Our approach is inspired by a recent work [17] that shows that translated STSb and NLI can be used to train high-quality monolingual SBERT models.
We present L3Cube-IndicSBERT a multilingual SBERT model for 10 Indian regional languages Hindi, Marathi, Kannada, Telugu, Malayalam, Tamil, Gujarati, Odia, Bengali, Punjabi, and English. The IndicSBERT uses MuRIL [22] as the base model and performs better than existing multilingual/cross-lingual models like LASER, LaBSE, and paraphrase-multilingual-mpnet-base-v2. These models are compared on monolingual and cross-lingual sentence similarity tasks. We also evaluate these models on real text classification datasets to show that the synthetic data training generalizes well to real datasets. Further, we also release monolingual SBERT models for individual languages to show that IndicSBERT performs competitively with the monolingual variants.
Our main contributions are as follows:
• We propose a simple strategy to train cross-lingual sentence representations using a pre-trained multilingual BERT model and synthetic NLI/STS data. Unlike previous approaches, it does not use any cross-lingual data or any complex training strategy. • We present IndicSBERT 12 , the first multilingual SBERT model trained specifically for Indic languages. The model performs better than state-of-the-art LaBSE and paraphrase-multilingualmpnet-base-v2 models. • We also release monolingual SBERT models for 10 Indic languages. To the best of our knowledge, this work is first to introduce the majority of these models.
The subsequent sections of the paper are organized as follows: Section 2 examines prior research on improving BERT performance and surveys previous work on sentence-BERT models. In Section 3.1, the datasets utilized in this study are outlined, while Section 3.2 provides details on the various models used and Section 3.3 delves into the specifics of the SBERT training procedure. Section 4 outlines the evaluation strategy used for the models and presents the key findings from our experiment. Finally, the paper concludes with a summary of all observations. This work is released as a part of the MahaNLP project.
RELATED WORK
BERT [10] (Bidirectional Encoder Representations from Transformers) is a pre-trained transformer network that is widely regarded as one of the best language models for natural language processing (NLP) tasks, such as text categorization and named entity recognition. Vanilla BERT models serve as a starting point for many NLP 1 https://huggingface.co/l3cube-pune/indic-sentence-bert-nli 2 https://huggingface.co/l3cube-pune/indic-sentence-similarity-sbert tasks, and researchers and practitioners often use their pre-trained weights to fine-tune models on specific tasks. mBERT [10] (Multilingual BERT) is a BERT-based language model, pretrained using MLM (Masked Language Modeling) objective on 104 different languages. XLM-RoBERTa [7] is another large-scale, cross-lingual language model developed by Facebook, trained on 100 different languages, making it a highly effective model for multilingual NLP tasks. A study [34] reveals that pretraining data size and a designated monolingual tokenizer are important factors that affect performance, and replacing the original multilingual tokenizer with a specialized monolingual tokenizer improves the downstream performance of the multilingual model for most languages and tasks. Despite numerous attempts at building better and bigger multilingual language models (MLLMs), as shown in [11], there has been limited research focused on creating models specifically for lowresource languages. In [30], the authors show the effectiveness of novel data-efficient methods using matrix factorization and lexically overlapping tokens for the adaptation of pre-trained multilingual models to low-resource languages and unseen scripts.
For the Indian languages, the available multilingual models include IndicBERT [21]. It follows the architecture of the original BERT model but is trained on a large corpus of text from several Indian languages. Another multilingual model, MuRIL [22] (Multilingual Representations for Indian Languages) has been pre-trained on 17 Indic languages.
Sentence embedding models [6,8,26,41] are superior to word embedding models [4,12,28,29] as they capture the meaning of the entire sentence rather than individual words. While BERT is trained to generate word embeddings, Sentence-BERT [32] modifies the architecture and fine-tunes the pre-trained BERT model for generating sentence embeddings. SBERT also includes additional training methods, such as the Siamese and triplet network architectures, that allow for more effective training of sentence embeddings. The SBERT model is trained using supervised datasets like NLI and STS that help in understanding the sentence semantics.
Numerous unsupervised methods have been proposed that learn meaningful sentence embeddings directly from text without the need for labeled training data. These include TSDAE which rebuilds noisy versions of input data while maintaining the data's original semantics. SimCSE is a contrastive learning technique that learns to encode the semantic similarity of phrase pairs into their embeddings. However, in this work, we focus solely on the supervised approaches for learning sentence embeddings.
LaBSE [13], a sentence-BERT model is designed to generate languageagnostic sentence embeddings that can be used for cross-lingual NLP tasks, while LASER [3] is a multilingual sentence embedding model that generates high-quality sentence embeddings for multiple lowresource languages. These models have been explicitly trained using parallel translation corpus. Similarly, by aligning the embeddings of parallel sentences in many languages, Cross-Lingual Transfer (CT) [3] technique learns a shared space for sentence embeddings across multiple languages. Thus, in the multilingual category, several BERT, as well as Sentence-BERT models, have been developed to date.
However, monolingual models are typically found to be performing better than multilingual ones. In a previous study [35], a German RoBERTa-based BERT model, with slight adjustments to its hyperparameters, was found to yield superior results than all other German and multilingual BERT models. Similarly, in [37] a Czech RoBERTa language model has been shown to perform better than other Czech and multilingual models. In [39] and [19], monolingual BERT models for the Marathi language were studied and found to perform better than their multilingual counterparts. Hence, to obtain an improved performance with rich sentence embeddings, monolingual Sentence-BERT models were proposed. Similarly, in this study, we propose monolingual SBERT models for the ten most prominent Indic languages. Additionally, we also propose a multilingual model tailored specifically to these languages. Considering that other multilingual models are trained to support a greater number of languages, our model is better suited for Indian languages, as it is specifically optimized for them.
EXPERIMENTAL SETUP 3.1 Datasets
The results shown in [17], indicate the efficacy of using synthetic datasets in creating MahaSBERT-STS and HindSBERT-STS. Thus, we utilize the machine-translated IndicXNLI and STSb datasets for training our models. Our models are evaluated on the synthetic STSb dataset, as well as on real-world classification datasets. The 3 datasets are described below.
The IndicXNLI 3 dataset comprises of English XNLI data translated into eleven Indian languages including Hindi and Marathi. To train the monolingual Sentence-BERT models, we use the training samples of the corresponding language from IndicXNLI. To ensure balanced training data for the multilingual IndicSBERT, we combine and randomly shuffle the IndicXNLI training samples of ten languages.
The STS benchmark (STSb) 4 dataset is commonly utilized for evaluating supervised Semantic Textual Similarity (STS) systems. The dataset includes 8628 sentence pairs from captions, news, and forums and is divided into 5749 for training, 1500 for development and 1379 for testing. To make the dataset accessible for all ten Indian languages used in this study, we translate it using Google Translate and use the resulting train samples of the corresponding language for each monolingual model and a combined dataset of ten languages for the multilingual model. We use the testing samples from the corresponding translated STSb dataset to evaluate each model based on the embedding similarity metric. For evaluating the cross-lingual property, we construct a dataset of STSb sentence pairs with each pair comprising two sentences from different languages.
We also evaluate the models on real text classification datasets. We perform this evaluation to show that the sentence representations from the models trained using synthetic datasets also generalize well to real datasets. We choose the IndicNLP news article classification datasets [24] for the purpose of evaluation. The classification datasets consist of train, validation, and test sets in an 8:1:1 ratio.
Models
BERT is a deep, bi-directional model based on the Transformer architecture, which has been trained on a large, unlabeled corpus. Multiple pre-trained BERT models, both monolingual and multilingual, are publicly available for use. In our experiment, we use different BERT models, including both monolingual and multilingual ones which are described below. The training procedure is applied over some of these models which serve as a base for creating Sentence-BERT.
Multilingual BERT models:
• mBERT 5 : A pre-trained multilingual BERT-base model that has been trained on 104 languages using a combination of the next sentence prediction (NSP) and Masked Language Modeling (MLM) [10] objectives.
• MuRIL 6 (Multilingual Representations for Indian Languages): a BERT-based model which supports 17 Indian languages [22]. It is pre-trained using masked language modeling and next-sentence prediction objectives on parallel data, which includes the translations as well as transliterations on each of the 17 monolingual corpora. It has shown state-of-theart performance on a variety of language understanding tasks.
• LaBSE 7 [13] (Language-agnostic BERT sentence embedding): It is a transformer-based model that learns languageagnostic sentence representations through a cross-lingual sentence retrieval task. It was trained on parallel sentence pairs from 109 languages using a Siamese network based on the BERT architecture. The model's ability to support 109 languages makes it a powerful tool for multilingual applications and cross-lingual natural language processing tasks.
• paraphrase-multilingual-mpnet-base-v2 8 : It is based on a Multilingual Pretrained Transformer (MPT) architecture. This model supports 50 languages and is trained on a paraphrase identification task. It has achieved state-of-the-art performance on the paraphrase identification task on several benchmark datasets. tasks including machine translation, sentiment analysis, and cross-lingual information retrieval.
Monolingual BERT models:
We also use the monolingual BERT models for the 10 Indic languages, released by L3cube-Pune 9 as the base models. These models are termed as HindBERT, MahaBERT [19], KannadaBERT, Telugu-BERT, MalayalamBERT, TamilBERT, GujaratiBERT, OdiaBERT, Ben-galiBERT, and PunjabiBERT. Further details about these models can be found in [18].
SBERT Training
In order to achieve competitive performance, sentence embedding models typically require significant amounts of training data and fine-tuning over the target task. Unfortunately, in many scenarios, only limited amounts of training data are available. Several unsupervised and semi-supervised approaches have been proposed to overcome the lack of a large training dataset. However, the models trained using unsupervised techniques give inferior performance than those trained using supervised learning.
In this work, we, therefore, use a supervised training approach, wherein we address the scarcity of specialized datasets, such as NLI and STS, in Indian languages by machine translating the English versions of these datasets into the respective Indian languages. 9 https://huggingface.co/l3cube-pune We follow a two-step procedure to train the monolingual SBERT models and the multilingual IndicSBERT model. The monolingual BERT model serves as the base for monolingual SBERT while MuRIL serves as the base model for IndicSBERT.
In the first step of the training procedure, natural language inference, or textual entailment, is performed. This task involves determining the logical relationship between a premise and hypothesis, represented as text sequences. The aim is to classify the relationship into three categories: entailment (hypothesis can be inferred from the premise), contradiction (negation of the hypothesis can be inferred from the premise), or neutral (no clear relationship between the two). In this step, the base model is trained on the IndicXNLI dataset, which consists of 392702 sentence pairs, each labeled as entailment, contradiction, or neutral.
To improve the effectiveness of the model, we utilize the Multiple Negatives Ranking Loss function instead of the Softmax-Classification-Loss used in [32]. This is because the Multiple Negatives Ranking Loss, which considers multiple negative samples simultaneously, is better suited for similarity-based problems where the goal is to learn similarities and dissimilarities between examples. This results in a more complex decision boundary and improves the model's robustness to outliers and variations in data.
The training data consists of triplets: [(a1, b1, c1), . . . , (an, bn, cn)], where (ai, bi) are considered similar sentences and (ai, ci) are dissimilar. An entry for bi is randomly picked from the set of sentences labeled as 'entailment' for ai, and an entry for ci is picked from the set of sentences labeled as 'contradiction' for ai, referred to as hard-negatives. Although they are similar to ai and bi on a lexical level, they mean different things on a semantic level. The model is trained using 1 epoch, with a batch size of 4, AdamW optimizer, and a learning rate of 2e-05. The AdamW optimizer extends the Adam optimizer and adds weight decay regularization to prevent overfitting and improve the model's generalization.
The models obtained after applying the first step (NLI only) of training are named as MahaSBERT 10 , HindSBERT 11 , Kan-nadaSBERT 12 , TeluguSBERT 13 , MalayalamSBERT 14 , TamilS-BERT 15 , GujaratiSBERT 16 , OdiaSBERT 17 , BengaliSBERT 18 , and PunjabiSBERT 19 that are made publicly available.
In the second step, the model from step one is fine-tuned using the translated STSb dataset. The STS benchmark is a commonly used dataset for evaluating the performance of NLP models in determining the similarity between two pieces of text. It comprises sentence pairs with human-annotated similarity scores on a scale of 0-5. The fine-tuning process uses the Cosine Similarity Loss as the loss function, which measures the similarity between two vectors 10 https://huggingface.co/l3cube-pune/marathi-sentence-bert-nli 11 https://huggingface.co/l3cube-pune/hindi-sentence-bert-nli 12 https://huggingface.co/l3cube-pune/kannada-sentence-bert-nli 13 https://huggingface.co/l3cube-pune/telugu-sentence-bert-nli 14 https://huggingface.co/l3cube-pune/malayalam-sentence-bert-nli 15 https://huggingface.co/l3cube-pune/tamil-sentence-bert-nli 16 https://huggingface.co/l3cube-pune/gujarati-sentence-bert-nli 17 https://huggingface.co/l3cube-pune/odia-sentence-bert-nli 18 https://huggingface.co/l3cube-pune/bengali-sentence-bert-nli 19 https://huggingface.co/l3cube-pune/punjabi-sentence-bert-nli in a multi-dimensional space. Cosine similarity loss considers the angle between vectors rather than their magnitudes, making it a robust measure of similarity. It is derived by computing the vectors for the two input texts, taking the dot product of the two vectors and dividing it by the product of the magnitudes of the two vectors. The result is a value between -1 and 1, where -1 indicates complete dissimilarity and 1 indicates complete similarity. The training process involves 4 epochs with Cosine Similarity Loss as the loss function and uses an AdamW optimizer with a learning rate of 2e-05.
The final models obtained after applying the two-step procedure (NLI + STS) are named as MahaSBERT-STS 20 , HindSBERT-STS 21 , KannadaSBERT-STS 22 , TeluguSBERT-STS 23 , MalayalamSBERT-STS 24 , TamilSBERT-STS 25 , GujaratiSBERT-STS 26 , OdiaSBERT-STS 27 , BengaliSBERT-STS 28 , and PunjabiSBERT-STS 29 and are made publicly available. In addition to the models mentioned above, we also release the multilingual SBERT models named IndicSBERT and IndicSBERT-STS. These multilingual models support the 11 languages of English, Hindi, Marathi, Kannada, Telugu, Malayalam, Tamil, Gujarati, Odia, Bengali, and Punjabi. Fig. 6. Cross-lingual performance of models for English with Indian languages
EVALUATION 4.1 Evaluation Methodology
We evaluate the SBERT models on the basis of the Embedding Similarity score as well as classification accuracy. The Embedding Similarity evaluation is performed by calculating the Pearson and Spearman rank correlation of the embeddings for different similarity metrics with the gold-standard scores. A high score in embedding similarity indicates that the embeddings being compared are of high quality in relation to the benchmark embeddings.
In our experiment, we use the cosine similarity metric and the value of Spearman correlation to evaluate sentence similarity models. The choice of cosine similarity is based on its superiority compared to traditional distance metrics such as Euclidean or Manhattan distance. Unlike these distance metrics, cosine similarity measures the cosine of the angle between the vectors representing the sentences and considers only their direction, making it less affected by scaling and more computationally efficient. Additionally, cosine similarity takes into account shared terms and contexts, providing a more accurate representation of semantic relationships between sentences. Spearman correlation, on the other hand, is used in preference to Pearson correlation because it is more robust to non-linear relationships and handles ties in data. Unlike Pearson correlation, which assumes a linear relationship, Spearman correlation measures the rank relationship between two variables, making it better equipped to accurately assess a model's performance in cases where the relationship is non-linear.
In this study, the text classification datasets were used to evaluate the performance of BERT and SBERT models in generating sentence embeddings. The K Nearest Neighbors (KNN) algorithm was applied to classify the texts based on proximity. The Minkowski distance, a generalized form of both the Euclidean and Manhattan distances, is employed. The optimal value of k was determined using a validation dataset and then used to calculate the accuracy of the test dataset, with results reported in Tables 2, 3. Table 1 presents the Embedding Similarity scores of monolingual SBERT models, while the classification accuracies are displayed in Table 2. Table 3 presents the similarity and accuracy results of the IndicSBERT. Table 4 compares the zero-shot performance of various multilingual models with that of IndicSBERT, while the superior cross-lingual performance of IndicSBERT is shown in Table 6. Our observations from these tables are discussed below.
Evaluation Results & Discussion
1. AVG pooling shows better performance than CLS We find that monolingual SBERT models generate superior embedding similarity scores when AVG pooling is utilized instead of CLS, across all 10 Indic languages. The same trend is observed for In-dicSBERT, where AVG pooling produces better results than CLS for embedding similarity scores. Hence, the AVG pooling values are reported in this work.
NLI + STS training works better
Fine-tuning the pre-trained models using NLI followed by STSb gives an upper hand over single-step training using the NLI dataset alone. Figure 4 compares the embedding similarities for the Vanilla, One-step trained, and Two-step trained monolingual models. We observe that the Two-step trained models surpass the one-step and Vanilla models in terms of performance across all 10 Indic languages. Fine-tuning with the STSb dataset results in a significant increase in embedding similarity for the monolingual SBERT models as well as for IndicSBERT, as demonstrated by Tables 1, 3, and Figures 4,5. Figure 5 demonstrates that the two-step training on IndicSBERT, which employs MuRIL as its base model, increases the embedding similarity scores nearly two-fold in comparison to the vanilla MuRIL model. While we mainly focus on cross-lingual performance in this work, similar observations in the context of monolingual SBERT have been thoroughly documented in [17].
3. SBERT models trained on synthetic corpus work well with real-world classification datasets We evaluate our sentence-BERT models on real-world news classification datasets to ensure that the models do not overfit the noise in the synthetic corpus. The results presented in Tables 2 and 3 indicate that SBERT models trained on translated corpora perform competitively compared to their original base models on classification datasets. The classification accuracy is neither improved nor deteriorated due to the two-step training.
Multilingual Indic-SBERT is competitive with monolingual SBERT models
Our experiments indicate that the multilingual IndicSBERT model demonstrates equivalent or better performance compared to monolingual SBERT models in terms of embedding similarity scores, as evidenced by Tables 1, 3. In 5, we observe that both the IndicSBERT-STS and two-step monolingual SBERT models perform comparably, with slight performance differences for certain languages. Except for Hindi, Marathi and Gujarati languages, the IndicSBERT-STS outperforms the SBERT models of the respective languages. This shows that the languages have assisting capabilities and the gains
5.
IndicSBERT works significantly better than existing multilingual models Figure 7, as well as Table 4, compare the zero-shot embedding similarities of mBERT, MuRIL, LASER, multilingual-mpnet-base, LaBSE, and IndicSBERT models on STSb for all 10 Indic languages, with the IndicSBERT based models clearly outperforming the others. Both IndicSBERT and IndicSBERT-STS produce richer embeddings than the publicly available LaBSE, which is shown in Table 4. Thus, the In-dicSBERT is the best-performing model among all the other publicly available multilingual models despite having the lowest number of trainable parameters.
6. IndicSBERT shows exceptional cross-lingual properties, outperforming the LaBSE The results presented in the Table 6 and Figure 6 demonstrate In-dicSBERT's robust cross-lingual performance across all language pairs, surpassing the performance of LaBSE by a significant margin. Overall, the multilingual IndicSBERT model demonstrates proficiency in processing both monolingual and multilingual datasets. This versatility enables the development of language-independent NLP applications that can seamlessly work across multiple Indian languages. In addition, IndicSBERT has the potential to enhance the precision and effectiveness of cross-lingual information retrieval systems and semantic search engines as it can handle queries and documents in multiple Indian languages. This characteristic holds particular importance for countries such as India, where multilingual communication is common, and organizations face the challenge of accommodating diverse language requirements.
7. Multilingual models are indeed cross-lingual learners, the enhancement of cross-lingual properties is generalizable to non-Indic languages
The performance of mBERT with mixed language NLI training on diverse languages like English, Hindi, German, and French is presented in Table 5. The results demonstrate a considerable improvement in the cross-lingual performance of the one-step trained model as compared to the vanilla mBERT. These findings support the effectiveness of the proposed mixed-language training technique in producing models with enhanced cross-lingual properties not only for Indic languages but also for other languages.
CONCLUSION
Our research addresses the crucial gap in the availability of highquality language models for low-resource Indian languages. We have presented a range of SBERT models for ten popular Indian languages, trained using synthetic corpus. They have been evaluated based on their embedding similarity with the translated standard STSb dataset and accuracies over text classification datasets. Our results demonstrate that the monolingual SBERT models outperform vanilla BERT models in terms of embedding similarity. Additionally, we have developed the multilingual IndicSBERT, which exhibits strong cross-lingual performance and outperforms existing multilingual models such as LaBSE and paraphrase-multilingual-mpnet-base-v2. This is a significant contribution to the field of IndicNLP, particularly in the context of the world becoming more globalized, and the need for accurate and efficient multilingual NLP models. While doing so we present a simple and clean approach to train cross-lingual sentence BERT models using only translated monolingual datasets and vanilla multilingual BERT. Indian languages pose a unique challenge, being diverse and having low-resource corpora. Our study highlights the effectiveness of the two-step training method in developing both monolingual SBERT models and the multilingual IndicSBERT. Its robust crosslingual capability makes IndicSBERT a superior choice for applications that require accurate and efficient multilingual NLP.
As part of this publication, we are releasing the monolingual SBERTs and the multilingual IndicSBERT, which will open up new possibilities for NLP research and applications in low-resource Indian languages. In summary, our research contributes to the development of high-quality language models for Indian languages and highlights the importance of combining the power of sentence-level embeddings with the ability to handle multiple languages to achieve optimal results in multilingual NLP applications.
Fig. 1 .
1An embarrassingly simple approach for learning cross-lingual sentence representations using synthetic monolingual corpus
Fig. 3 .
3Two-step (NLI + STS) training of the multilingual IndicSBERT models
Fig. 5 .
5Embedding similarity score comparison of MuRIL, IndicSBERT and monolingual SBERT models
• LASER[3] (Language-Agnostic SEntence Representations):This model from Facebook is trained on large parallel corpora for cross-lingual language understanding (XLU) task. It uses a multilingual encoder-decoder architecture, where the encoder is a five-layer bidirectional LSTM. It can generate superior-quality cross-lingual sentence embeddings for over 90 languages and outperforms other models on cross-lingualFig. 2. Two-step (NLI + STS) training of the monolingual SBERT modelsHindBERT
Hi
IndicXNLI
train set
HindSBERT
HindSBERT-STS
Hi
STSb
train set
Base model
One-Step (NLI)
trained model
Training
Fine-tuning
BengaliBERT
Bn
BengaliSBERT
BengaliSBERT-STS
Bn
Training
Fine-tuning
PunjabiBERT
Pa
PunjabiSBERT
PunjabiSBERT-STS
Pa
Training
Fine-tuning
Two-Step (NLI+STS)
trained model
Hi
IndicXNLI train sets
STSb train sets
Base model
One-Step (NLI)
trained model
Bn
Mr
Te
Pa
Combined & Shuffled
dataset
Ta
En
MuRIL
Training
Fine-tuning
IndicSBERT
Combined & Shuffled
dataset
Hi
Bn
Mr
Te
Pa
Ta
En
IndicSBERT-STS
Table 1 .
1Embedding similarity scores of monolingual BERT and SBERT modelsMultilingual base
Monolingual base
Base model:
mBERT
MuRIL
LaBSE
BERT
Training steps † †
0
1
2
0
1
2
0
1
2
0
1
2
Hindi (hi)
0.49
0.64
0.75
0.52
0.74
0.83
0.72
0.75
0.84
0.5
0.77
0.85
Bengali (bn)
0.5
0.65
0.75
0.55
0.74
0.82
0.71
0.75
0.81
0.5
0.72
0.81
Marathi (mr)
0.47
0.65
0.72
0.56
0.74
0.81
0.7
0.75
0.82
0.54
0.77
0.83
Telugu (te)
0.53
0.62
0.73
0.6
0.71
0.8
0.73
0.73
0.81
0.58
0.72
0.8
Tamil (ta)
0.49
0.65
0.75
0.6
0.72
0.8
0.72
0.74
0.82
0.59
0.72
0.8
Gujarati (gu)
0.47
0.65
0.74
0.58
0.72
0.8
0.73
0.73
0.82
0.55
0.74
0.82
Kannada (kn)
0.52
0.68
0.75
0.6
0.75
0.82
0.72
0.76
0.82
0.57
0.74
0.82
Odia (or) †
-
-
-
0.45
0.58
0.69
0.6
0.6
0.73
0.45
0.59
0.71
Malayalam (ml)
0.46
0.57
0.67
0.53
0.66
0.74
0.66
0.66
0.74
0.5
0.69
0.76
Punjabi (pa)
0.43
0.59
0.68
0.45
0.65
0.74
0.64
0.67
0.75
0.5
0.68
0.75
Table 2 .
2Classification accuracy of monolingual BERT and SBERT modelsMultilingual base
Monolingual base
Base model:
mBERT
MuRIL
LaBSE
BERT
Training steps † †
0
1
2
0
1
2
0
1
2
0
1
2
Hindi (hi)
0.62
0.6
0.62
0.67
0.7
0.69
0.68
0.64
0.65
0.7
0.69
0.68
Bengali (bn)
0.97
0.96
0.97
0.97
0.98
0.98
0.98
0.98
0.97
0.98
0.98
0.98
Marathi (mr)
0.98
0.97
0.97
0.97
0.98
0.98
0.98
0.99
0.99
0.98
0.98
0.99
Telugu (te)
0.98
0.97
0.97
0.99
0.99
0.99
0.99
0.99
0.99
0.99
0.95
0.99
Tamil (ta)
0.96
0.96
0.95
0.96
0.97
0.97
0.96
0.97
0.96
0.96
0.97
0.97
Gujarati (gu)
0.95
0.94
0.93
0.97
0.98
0.99
0.99
0.96
0.99
0.98
0.99
0.99
Kannada (kn)
0.96
0.95
0.94
0.97
0.97
0.97
0.96
0.96
0.97
0.97
0.97
0.97
Odia (or) †
-
-
-
0.97
0.97
0.97
0.98
0.97
0.97
0.98
0.97
0.97
Malayalam (ml)
0.85
0.86
0.84
0.9
0.92
0.91
0.92
0.9
0.9
0.92
0.92
0.92
Punjabi (pa)
0.94
0.96
0.92
0.96
0.96
0.96
0.97
0.96
0.96
0.96
0.95
0.96
20 https://huggingface.co/l3cube-pune/marathi-sentence-similarity-sbert
21 https://huggingface.co/l3cube-pune/hindi-sentence-similarity-sbert
22 https://huggingface.co/l3cube-pune/kannada-sentence-similarity-sbert
23 https://huggingface.co/l3cube-pune/telugu-sentence-similarity-sbert
24 https://huggingface.co/l3cube-pune/malayalam-sentence-similarity-sbert
25 https://huggingface.co/l3cube-pune/tamil-sentence-similarity-sbert
26 https://huggingface.co/l3cube-pune/gujarati-sentence-similarity-sbert
27 https://huggingface.co/l3cube-pune/odia-sentence-similarity-sbert
28 https://huggingface.co/l3cube-pune/bengali-sentence-similarity-sbert
29 https://huggingface.co/l3cube-pune/punjabi-sentence-similarity-sbert
† Odia language is not supported by mBERT
† † Training steps= 0 indicates the vanilla base model, 1 denotes single-step NLI training
over the base model, while 2 denotes the two-step trained model
Languages
Embedding similarity
0.35
0.45
0.55
0.65
0.75
0.85
H
in
d i ( h i)
B e n g a li
( b n )
M
a r a th
i ( m
r )
T e lu
g u
( te
) T a m
il
( ta
) G
u ja
r a ti
( g u )
K a n n a d a
( k n ) O
d ia
( o r )
M
a la
y a la
m
( m
l)
P u n ja
b i ( p a )
Vanilla model
One-step trained
Two-Step trained
Fig. 4. Embedding similarity score comparison of SBERT models having
monolingual BERT base
Table 3. IndicSBERT: Embedding similarity and classification accuracy re-
sults
Embedding Similarity
Classification Accuracy
IndicSBERT
IndicSBERT-STS
IndicSBERT
IndicSBERT-STS
Hindi (hi)
0.76
0.8
0.68
0.65
Bengali (bn)
0.76
0.81
0.98
0.97
Marathi (mr)
0.75
0.8
0.98
0.98
Telugu (te)
0.74
0.8
0.99
0.98
Tamil (ta)
0.74
0.8
0.96
0.95
Gujarati (gu)
0.76
0.81
0.99
0.99
Kannada (kn)
0.76
0.81
0.96
0.95
Odia (or)
0.66
0.73
0.97
0.95
Malayalam (ml)
0.7
0.76
0.91
0.89
Punjabi (pa)
0.7
0.76
0.95
0.96
Language
Embedding Similarity
0.35
0.45
0.55
0.65
0.75
0.85
H
in
d i
B e n g a li
M
a r a th
i
T e lu
g u
T a m
il
G
u ja
r a ti
K a n n a d a
O
d ia
M
a la
y a la
m
P u n ja
b i
MuRIL
IndicSBERT
IndicSBERT-STS
Two-Step mono SBERT
Table 4 .
4Zero-shot performance of multilingual modelsmBERT
MuRIL
LASER
mpnet-base
LaBSE
IndicSBERT
IndicSBERT-STS
Hindi
0.49
0.52
0.64
0.79
0.72
0.75
0.82
Bengali
0.5
0.55
0.68
0.66
0.71
0.76
0.82
Marathi
0.47
0.56
0.6
0.75
0.7
0.76
0.81
Telugu
0.53
0.6
0.59
0.64
0.73
0.74
0.81
Tamil
0.49
0.6
0.49
0.65
0.72
0.73
0.82
Gujarati
0.47
0.58
0.14
0.73
0.73
0.74
0.82
Kannada
0.52
0.6
0.17
0.65
0.72
0.76
0.83
Odia †
-
0.45
0.29
0.48
0.6
0.62
0.75
Malayalam
0.46
0.53
0.6
0.6
0.66
0.68
0.78
Punjabi
0.43
0.45
0.12
0.56
0.64
0.68
0.77
Table 5 .
5Cross-lingual performance of mBERT, single-step trained for 4 languages: Hindi, English, German and French. For every language-pair, the values reported from top to bottom correspond to One-step mBERT, and vanilla mBERT respectivelyHindi
English
German
French
Hindi
0.68
0.5
0.5
0.48
0.48
0.3
0.3
0.32
English
0.51
0.77
0.6
0.63
0.31
0.5
0.4
0.41
German
0.49
0.6
0.7
0.56
0.3
0.4
0.48
0.39
French
0.49
0.63
0.57
0.72
0.29
0.39
0.37
0.49
Table 6 .
6Cross-lingual performance of IndicSBERT-STS, IndicSBERT and LaBSE. For every language-pair, the values reported from top to bottom correspond to IndicSBERT-STS, IndicSBERT and LaBSE respectivelyEnglish
Hindi
Bengali
Marathi
Telugu
Tamil
Gujarati
Kannada
Oriya
Malayalam
Punjabi
English
0.85
0.8
0.8
0.8
0.79
0.8
0.79
0.8
0.72
0.77
0.76
0.8
0.72
0.73
0.7
0.7
0.71
0.7
0.72
0.64
0.67
0.68
0.72
0.68
0.68
0.69
0.7
0.69
0.7
0.68
0.63
0.63
0.65
Hindi
0.82
0.82
0.79
0.79
0.77
0.78
0.79
0.79
0.72
0.76
0.76
0.72
0.75
0.71
0.7
0.68
0.68
0.69
0.69
0.62
0.65
0.68
0.7
0.72
0.69
0.7
0.7
0.69
0.71
0.68
0.62
0.62
0.64
Bengali
0.82
0.79
0.82
0.79
0.77
0.77
0.79
0.79
0.73
0.76
0.76
0.73
0.7
0.76
0.7
0.68
0.68
0.7
0.7
0.63
0.65
0.67
0.69
0.69
0.71
0.69
0.7
0.69
0.71
0.69
0.64
0.64
0.66
Marathi
0.8
0.78
0.78
0.81
0.76
0.77
0.78
0.78
0.72
0.75
0.75
0.7
0.7
0.7
0.76
0.67
0.66
0.69
0.69
0.62
0.65
0.67
0.68
0.68
0.69
0.7
0.69
0.68
0.7
0.68
0.63
0.64
0.65
Telugu
0.79
0.77
0.77
0.76
0.81
0.77
0.76
0.78
0.71
0.74
0.73
0.72
0.68
0.68
0.68
0.74
0.68
0.67
0.69
0.6
0.64
0.65
0.7
0.7
0.7
0.7
0.73
0.7
0.71
0.69
0.63
0.64
0.66
Tamil
0.8
0.77
0.77
0.76
0.76
0.82
0.76
0.77
0.7
0.75
0.73
0.71
0.67
0.67
0.67
0.67
0.73
0.65
0.68
0.58
0.64
0.64
0.69
0.7
0.69
0.69
0.7
0.72
0.7
0.68
0.62
0.62
0.64
Gujarati
0.8
0.79
0.78
0.79
0.76
0.76
0.82
0.77
0.73
0.74
0.76
0.7
0.69
0.69
0.69
0.67
0.66
0.74
0.68
0.6
0.63
0.67
0.7
0.7
0.7
0.69
0.7
0.69
0.73
0.68
0.63
0.63
0.66
Kannada
0.8
0.77
0.77
0.77
0.77
0.77
0.76
0.83
0.7
0.75
0.73
0.71
0.68
0.69
0.68
0.68
0.67
0.66
0.76
0.59
0.65
0.64
0.68
0.67
0.68
0.68
0.69
0.67
0.69
0.72
0.62
0.62
0.64
Oriya
0.72
0.71
0.72
0.7
0.7
0.7
0.72
0.7
0.75
0.68
0.7
0.62
0.61
0.61
0.6
0.6
0.58
0.6
0.6
0.62
0.56
0.6
0.6
0.59
0.6
0.6
0.6
0.6
0.61
0.6
0.6
0.58
0.59
Malayalam
0.77
0.74
0.75
0.74
0.74
0.75
0.73
0.74
0.69
0.78
0.7
0.68
0.65
0.66
0.66
0.65
0.65
0.63
0.65
0.57
0.68
0.62
0.64
0.62
0.64
0.64
0.65
0.64
0.65
0.64
0.6
0.66
0.6
Punjabi
0.76
0.76
0.76
0.75
0.73
0.74
0.76
0.74
0.7
0.71
0.77
0.68
0.67
0.67
0.66
0.65
0.64
0.66
0.66
0.6
0.61
0.68
0.65
0.63
0.66
0.65
0.66
0.65
0.66
0.64
0.62
0.6
0.64
• S. Deode et al.
https://github.com/divyanshuaggarwal/IndicXNLI 4 https://huggingface.co/datasets/stsb_multi_mtWe also apply a series of preprocessing steps, such as removing punctuation, URLs, hashtags, Roman characters and blank spaces, to ensure that the data is suitable for our experiments.
https://huggingface.co/bert-base-multilingual-cased 6 https://huggingface.co/google/muril-base-cased 7 https://huggingface.co/setu4993/LaBSE 8 https://huggingface.co/sentence-transformers/paraphrase-multilingual-mpnetbase-v2
• S. Deode et al.
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arxiv |
Assessing Scientific Contributions in Data Sharing Spaces
2023. April 30-May 4, 2023
- Sci
T X Austin
Acm Reference Format: Kacy Adams
Fernando Spadea
Conor Flynn
Oshani Seneviratne
Assessing Scientific Contributions in Data Sharing Spaces
Companion Proceedings of the ACM Web Conference 2023 (WWW '23 Companion)
Austin, TX, USA2023. April 30-May 4, 202310.1145/3543873.3587608ACM ISBN 978-1-4503-9419-2/23/04. . . $15.00. ACM, New York, NY, USA, 8 pages. https: //doi.org/10.1145/3543873.3587608CCS CONCEPTS • Social and professional topics → Industry statistics• General and reference → Metrics• Information systems → Social rec- ommendationLearning to rank KEYWORDS Incentive Mechanisms, Author-level Metrics, Dataset Sharing, Peer- to-peer Sharing, Blockchain, Smart Contracts
In the present academic landscape, the process of collecting data is slow, and the lax infrastructures for data collaborations lead to significant delays in coming up with and disseminating conclusive findings. Therefore, there is an increasing need for a secure, scalable, and trustworthy data-sharing ecosystem that promotes and rewards collaborative data-sharing efforts among researchers, and a robust incentive mechanism is required to achieve this objective. Reputation-based incentives, such as the h-index, have historically played a pivotal role in the academic community. However, the h-index suffers from several limitations. This paper introduces the SCIENCE-index, a blockchain-based metric measuring a researcher's scientific contributions. Utilizing the Microsoft Academic Graph and machine learning techniques, the SCIENCEindex predicts the progress made by a researcher over their career and provides a soft incentive for sharing their datasets with peer researchers. To incentivize researchers to share their data, the SCIENCE-index is augmented to include a data-sharing parameter. DataCite, a database of openly available datasets, proxies this parameter, which is further enhanced by including a researcher's data-sharing activity. Our model is evaluated by comparing the distribution of its output for geographically diverse researchers to that of the h-index. We observe that it results in a much more even spread of evaluations. The SCIENCE-index is a crucial component in constructing a decentralized protocol that promotes trust-based data sharing, addressing the current inequity in dataset sharing. The work outlined in this paper provides the foundation for assessing scientific contributions in future data-sharing spaces powered by decentralized applications.
INTRODUCTION
There is a rising need for a secure, scalable, and trustless datasharing ecosystem that recommends, incentivizes, and rewards collaborative data-sharing efforts between researchers in many scientific disciplines. Such a protocol would require a robust incentive mechanism. In the academic space, reputation-based incentives rule, and since 2006, h-index [16] has reigned superior. It is a widely used bibliometric indicator that measures a scientist's publications' productivity and citation impact. The h-index, however, has several flaws. One of the main shortcomings of the h-index is that it does not account for the quality or impact of individual publications but considers all publications equally. Additionally, the h-index tends to favor established researchers with a long publication history, as it considers the total number of publications. The h-index may be subject to manipulation by self-citations or citation cartels, which can inflate an author's score. Therefore, it is important to use the h-index in conjunction with other metrics and to interpret it with caution.
If all research is to be fair and incentivized, researchers must mend these discrepancies as their reputations define their careers. We present a new reputation-based metric called the SCIENCEindex to incentivize and reward data aggregation and sharing.
We utilize the Microsoft Academic Graph (MAG) [30] to train an AI model to predict the researcher's progress over their career. We persist this model via smart contract on the Ethereum blockchain [7] and allow researchers to look up their and others' SCIENCE-indexes via web identifiers such as the Semantic Scholar ID. A blockchainbased mechanism provides several advantages, including increased security and transparency, as well as the ability to incentivize data sharing through the use of smart contracts. arXiv:2303.10476v1 [cs.CY] 18 Mar 2023
The Need for Data Sharing
Data is a significant part of modern evidence-based scientific research. Many studies rely heavily on collecting large amounts of data ranging from studies on human behaviors to machine learning. Collecting this data can be painstaking and time-consuming, taking up to seventeen years from bench to bedside in certain biomedical research studies [22]. Even in computer or information science research, a significant effort goes into collecting data. For example, several web science researchers 2012 conducted a study on buying unlicensed slimming drugs online, which required ethnographic data collection. They had to manually copy and paste parts of their data from the sites they scoured and held interviews with several stakeholders in UK regulatory agencies [31].
Despite data collection being very important, researchers are typically not incentivized to share their data. In fact, there are many reasons not to share data. Researchers face many challenges when it comes to intellectual property and confidentiality [1]. Others from less-endowed institutions may fear their work being scooped up by more prestigious institutions [3] or fear that others may use it to their advantage [12].
On top of this, existing academic incentives that reach further than a citation are scarce. One incentive, the use of open data badges 1 , has been tested in health and medical research [17], yet studies are unsure of their effectiveness [28]. Better incentives are necessary as data sharing is essential in modern academic research. In the case of protecting research participants, data sharing is an ethical necessity to protect human lives. In January 2016, one participant died, and four others were injured due to the first human testing of a fatty acid inhibitor [29]. With human lives at stake, data from studies like these must be made available.
In this work, we address the aforementioned inequity of dataset sharing. To incentivize researchers to share their data, we must build their reputation to include their data-sharing activity and the bibliometric reputation available through indexes such as the h-index. We augment our initial SCIENCE-index to include a datasharing parameter. We proxy this via DataCite [5], a database of openly available datasets, and widen our MAG dataset by including the number of times a researcher has shared their data. Such a metric would be pivotal in building a decentralized protocol that allows data sharing by adding trust between individuals who have not worked together prior to the collaboration event.
More Than The h-index
The h-index was created to measure the impact of a researcher's work. It represents the maximum number of "h" papers published by a researcher with at least "h" citations. This used to be a fairly accurate reflection of researchers' past impact, at least when tested against the reviews of the Bochringer Ingelheim Fonds organization [4]. However, the h-index's reliability has greatly diminished in recent years and no longer represents the scientific reputations of researchers [20].
Further, we express our concerns about the unfair playing field of scientific research in underdeveloped countries [2,25]. We believe that more robust reputation-based metrics would help to level the "playing field" among researchers with fewer resources compared to well-endowed researchers.
Data sharing is a massive service to the research community largely unconsidered by the h-index. A system is needed to measure an individual's contributions to science beyond that of publications. Then, researchers will be properly credited for their work and, thus, incentivized to continue to collect and share invaluable data.
A Data Sharing Space
The work outlined in this paper is part of a larger plan to create a data-sharing environment via blockchain. This environment would allow researchers to share data while being rewarded. We believe blockchain is an appropriate technology to use here because it is an effective ledger for keeping track of data sharing in a transparent and accountable manner. Its decentralized and public nature also makes it more transparent for the researchers who share and use data on it. To incentivize participation and the sharing of good quality data, we propose a new index, i.e., the SCIENCE-index, for assessing the impact of researchers' contributions with a model that will be hosted and persisted in the same decentralized manner.
We specifically target fields where research requires and produces large amounts of data. This is because the number of opportunities to share and use data varies greatly between fields, so there is no one-size-fits-all measure of one's contributions to data sharing. However, by specifically considering data-heavy fields, like, for example, biomedical research, we can create an accurate metric for those fields without downplaying the significance of work in other fields with fewer datasets.
We present the SCIENCE-index in Section 2, detailing a robust linear model for rating researchers in Section 2.2. In Section 2.3, we display a decentralized infrastructure for persisting this model in a public fashion. Next, we discuss our prior work in the decentralized data sharing space in Section 3 and augment our linear model to incentivize data sharing. We present results from the model and its augmentation in Section 4 and evaluate the efficacy of our model on geographically distributed researchers in Section 5. Finally, we discuss our work and future work in Section 6 and present related work in Section 7.
THE SCIENCE-INDEX
We present the SCIENCE-index, a self-sustaining metric for scientific reputation. The SCIENCE index encompasses an expressive, provenance-centric approach, and it is a recursive acronym for SCIENCE, Capability-based, Intention-centric, Experiment-oriented, Networked, Collaborative, Expression.
We bootstrap the SCIENCE-index via 21 million data points from the MAG that overlaps with entries from the data-sharing website DataCite. First, using a robust multiple linear regression across several academic career statistics, we predict a researcher's h-index and compare this to their actual h-index. This difference is then normalized to a scale from zero to ten. Five means expected, under five is below average, and over five is above average. We persist the model via smart contract on a public blockchain, allowing the model to exist publicly and continue to scale and update as researchers use it. We detail the model and its infrastructure below.
Data Aggregation
We utilize the MAG to aggregate a dataset of 21,660,755 authors, each with their corresponding publication count, citation count, h-index, and career length. We select and use publication count and citation count as they are straightforward indicators of productivity and are provided as part of the "authors" table of the MAG. With some simple calculations, we can build paper lists for each of our authors via the "papers" table of the MAG, and with this, we can assume career length as the years between their oldest paper and their newest paper and calculate their h-index. Although within the MAG and the academic research space, there are many more abstract parameters to use, these four were the most accessible via the web through tools such as the Semantic Scholar API 2 . This accessibility is an important piece of our goal to persist our metric in a decentralized manner.
The Model
The model of the SCIENCE-index takes in four different inputs: career length, paper count, citation count, and h-index.
1 = Career Length 2 = Paper Count 3 = Citation Count 4 = h-index
We calculate the predicted h-index value ( ) from these parameters and compare it to the actual h-index ( 4 ), extracting the SCIENCEindex from this difference. Since we have a narrow dataset, we use Multi-Linear Regression (MLR) to find our predicted h-index. After training the MLR on our 21 million data points, we derive the following equation
= 0 + 1 1 + 2 2 + 3 3 = 1.71933 + 0.06902 1 + 0.10867 2 + 0.00304 3
Finally, we scale for outliers that can occur anywhere > 60. This threshold, i.e., 60, is based on the notion that "an h index of 60 after 20 years, or 90 after 30 years, characterizes truly unique individuals" as stated by Hirsch [14]. Therefore, we give these researchers a bias to their SCIENCE-index calculation such that it will give them a higher score. If > 60, we apply the given function to scale it to an appropriate value.
If > 60: = 0.571 + (0.007 * ) Using these weights and our approximation for , we can then calculate the difference ( ) between the predicted and the actual h-index. After finding , we normalize it according to the entire dataset to find , our calculated performance factor comparable to any other data point's value. We then logarithmically regress the scaled delta to fit on a scale of one to ten for readability and easy comparison.
= 4 − = −
2 https://www.semanticscholar.org = 10 1 + − This calculated value of is the outputted SCIENCE-index. Any value of below 5 is deemed a below-average academic contribution by the researcher, and any value of above 5 signifies aboveaverage contributions.
Infrastructure
The SCIENCE-index lives as weights in a smart contract, making it publicly accessible and completely transparent. Upon call, a researcher provides their Semantic Scholar identifier, and the smart contract requests a Chainlink oracle [6], which requests the Semantic Scholar API to get the requesting researcher's statistics. Using these statistics, the smart contract adds the new data point to the model by updating the weights and then calculates and returns the researcher's SCIENCE-index. This sequence is shown in Figure 1. Chainlink is an industry-standard service for building blockchain oracles. Oracles allow smart contracts that live on the immutable blockchain to access the outside, mutable internet. We utilize a Chainlink external adapter to build a custom job that, when called, calculates and returns the career statistics used in our model.
SHARING SCIENCE
We narrow the scope of our incentive mechanism towards a hypothetical data-sharing ecosystem. The SCIENCE-index is part of the Sharing Science Ontology (SSO), a semantic model for a decentralized academic data-sharing application.
The Sharing Science Ontology (SSO)
We have discussed the need for a data-sharing ecosystem in a secure and scalable manner, and blockchain provides us with the peer-topeer platform to do so. The SSO describes a decentralized protocol to handle and incentivize peer-to-peer academic data sharing in a distributed environment.
The SSO handles peer-to-peer sharing events via what is called collaboration events. Collaboration events are between two parties and begin with a data request by the data seeker and end with a citation of the data sharer by the data seeker. Researchers are rewarded via incentive mechanisms for their honest and fair completion of collaboration events. The SSO is publicly available at a persistent URL at https://github.com/sharing-science/sharing-science-ontology.
The SCIENCE-index Augmented
In an attempt to incentivize the sharing of data, we introduce a new parameter to our SCIENCE-index model. We include the number of times a researcher has shared a dataset. In our discussed protocol, this would represent the number of collaboration events a researcher has participated in. We bootstrap the model again with approximately 3000 data points from the MAG. Using the DataCite API [5], we can count the times a researcher has published a publicly available dataset and establish this as a proxy for our collaboration events. Our parameters now include the following:
1 = Career Length 2 = Paper Count 3 = Citation Count 4 = Data Share Count 2 5 = h-index
We again regress on the h-index to predict a researcher's h-index and scale the difference. To further incentivize data sharing, we weigh the number of data shares by a power of two, and this gives enough weight to data sharing that researchers can effectively increase their SCIENCE-index through data-sharing activities.
RESULTS
With train and test sets from our initial data sets, we can visualize the results of our two proposed models.
The SCIENCE-index
Without the data sharing parameter, our initial SCIENCE-index presents a distribution as seen in Figure 2.
We can see that the model is conservative and leans forward after its density peaks just before 5. We further visualize the model in Figure 3.
We see that career length is correlated with the h-index, as the shade of blue gets lighter from left to right. However, career length is not correlated with the SCIENCE-index, which allows us to compare researchers of any age.
The SCIENCE-index with Data Sharing Data
Similar to before, once augmented by the data sharing data, we have a forward-leaning density of the metric shown in Figure 4 We can compare the original model to the now augmented model using our initial SCIENCE-index with our data-sharing proxy dataset. In Figure 5, we compare the density of the two models. The augmented SCIENCE-index has been shifted forward as each member has shared data.
Figure 5: Density Plots of Both Models Overlayed To Show the Effect of the Introduction of Data Sharing Data
The data sharing dataset has an average data share "count" of 6.6, giving us an average positive shift in each researcher's SCIENCEindex of 0.27.
EVALUATION
In the spirit of addressing inequality using the SCIENCE-index, we tailor our evaluation toward comparing geographically distributed researchers. We argue that researchers from less developed countries with fewer resources face larger challenges in building their academic reputations. We have compiled a brief dataset of researchers with half the dataset affiliated with universities located in the "global south, " i.e., resource-poor institutions, such as Rhodes University (https://www.ru.ac.za), University of Sao Paulo (https: //www.fearp.usp.br) and the other half located in the "global north, " i.e., resource-rich institutions, such as Stanford University, (https: //www.stanford.edu), Mcgill University (https://www.mcgill.ca), University of North Carolina (https://www.unc.edu), and Grenoble Alpes University (https://www.univ-grenoble-alpes.fr).
In Figure 6, we show the difference in the h-index between the two groups of researchers with a density plot. The mean of the northern-located researchers is an average of 2 points greater than the southern-located researchers.
We then run this group of researchers through our SCIENCEindex model trained on our original dataset. We show the results of this in Figure 7. The mean of the SCIENCE-index of each group converges to 5.1. This shows the ability of the SCIENCE-index to level the "playing field" and look objectively at a researcher's career progress.
DISCUSSION
The SCIENCE-index aims to be a valuable tool to credit researchers fairly for their contributions. It allows us to compare researchers at different points in their careers and levels of discrepancies in career statistics. Such a model has an incredibly diverse set of applications, as we can widen the training dataset as we desire. Different parameters can be used and weighted to augment the h-index in favor of specific activities, such as data sharing, as we've discussed, or other statistics, such as conference reviewing or institutional affiliations. This extensibility opens the door to incentive mechanisms in all aspects of academic research. More importantly, the SCIENCE-index predicts the future progress of a researcher based on their past contributions to science. This is important because it encourages researchers to continue contributing significantly to science and rewards those with a consistent track record.
Different Models
We consider future work examining different learning models to back the SCIENCE-index. This paper explores using multi-linear regression, but many different models could be trained on the data, including logistic regression, support vectors, or decision tree clustering. Yet, we initially aimed to avoid a black box model, as researchers are less likely to accept a metric they cannot interpret. We also note that there is no one size fits all solution for scientific contributions. For example, it is well known that the citation culture varies between areas, e.g., biology is different from computer science, so the model outlined in Section 2.2 may need tweaks to cater to the specific needs of the different scientific communities.
Bootstrapping the Models
A large amount of data is required to bootstrap any model. For the SCIENCE-index, we use a subset of the MAG. The MAG, however, has been deprecated, and for future training of models, we need a current, accurate date. Examining other parameters, such as data sharing, we must look to other sources. We also use DataCite [5], but future iterations require more robust and accurate aggregation of data-sharing statistics. Several other data-sharing resources covered in Section 7 could be used to aggregate prior data-sharing activity.
We acknowledge that adding a data component to the model in evaluating the SCIENCE-index can introduce some flaws as it may be easier to publish data than a paper, which could lead to artificially inflating one's SCIENCE-index for a useless dataset. However, the broader framework would allow the data reusers to review the dataset they have access to, and future iterations of the model would incorporate the citations to the datasets directly.
Another issue we foresee is the need for a canonical identifier. In our work, we utilized the Semantic Scholar ID. However, author disambiguation is a major issue in academic publishing, which will only worsen once datasets are factored in. We plan to utilize robust entity resolution mechanisms that will leverage a variety of identifiers, including Orcid [13] and decentralized identity mechanisms [27] championed by standards organizations such as the W3C.
Blockchain Usage
Our goal of persisting a reputation-based metric in a decentralized manner has come with challenges. Public blockchains present invaluable peer-to-peer networks that allow transparency and scalability, and they also present limitations on computability and cost. Our model must be computationally lightweight not to incur incredibly high transaction costs when computing on a public virtual machine but also be robust enough to rate researchers accurately. We must also consider the oracle problem [8], which presents difficulties accessing mutable data from an immutable state machine. Our use of Chainlink oracles described in Section 2.3 is our first take at tackling this problem. There are also questions about who takes on the burden of paying for the transaction fees. When a dataset is being requested for a collaboration event, the benefiting party is the dataset requester, and it seems only fair that they should pay the fees required. However, the subsequent citations benefit the original dataset sharer more than the requester unless there is a punitive mechanism for failing to provide proper data citations. Therefore, robust tokenomics should be defined in future work to benefit all parties involved.
RELATED WORK
We describe two main themes throughout our work. The former seeks to improve equity and fairness in author-level metrics, while the latter seeks to encourage and incentivize open academic data sharing. We explore related work among both of these.
Improving Reputation-based Metrics
The SCIENCE-index is not competing with other researcher metrics but rather looking to host a metric in a public data-sharing environment on a blockchain. Other research in this field is not a competitor as it is an opportunity to improve how the proposed framework can incentivize researchers to share their work.
The g-index takes the maximum number of "g" papers that collectively have 2 citations [11]. This makes the g-index more sensitive to impactful papers while avoiding letting insignificant papers have too much influence. The g-index would always be at least the value of the h-index, but the g-index adds the extra push from more important papers. Some argue that the two indexes do not replace each other but rather complement each other, where the h-index favors big paper producers and the g-index favors selective researchers [10]. Others, such as Google Scholar 3 , take a more straightforward approach with the i10-index [9]. This is simply the number of papers from a researcher with at least ten citations. Unlike the g-index, this is very simple to use and understand, but like the g-index, it has not risen to acceptance in the same way as the h-index.
We look at another author-level metric that tries to identify meaningful citations. The paper claims that the importance of citations can be measured with three objective measurements and that by identifying these, we can properly distinguish types of citations and credit researchers accordingly [32]. The significance of a researcher's citations is another important metric for us to consider when comparing researchers. Similarly, it is worthwhile to look into detecting the significance of cited data to the results of a paper. For example, data used to train a model successfully may be more significant than data used to test a model. Such a measurement could improve the SCIENCE-index greatly in the scope of data sharing.
Of course, there is significant work in creating better metrics for assessing research publications. Two major examples are the Field-Weighted Citation Impact (FWCI) and Relative Citation Ratio (RCR) [26]. The FWCI is an article-level metric that reflects the significance of a paper's citation count. It considers the publication type, year, and subject area to measure how a specific publication compares to others [26]. The RCR similarly divides the citation count of a publication by its expected value which is calculated with a quantile regression analysis of the citations of prior publications funded by the National Institutes of Health plotted against the field citation rate [26]. These article-level metrics can be collected to assess researchers, so they have significant promise and could be a useful source to improve the SCIENCE-index.
There is also other work in assessing researchers while considering their data contributions. One example is the data-index which takes into account data publications and citations [15]. Other examples include the s-index [19] and Data Citation Index (DCI) [23,24].
These are also promising projects in the same field as this paper, and each takes different approaches to reach a similar result. However, they are again not competitors but rather potential sources of improvement for the SCIENCE-index as an incentivizer. They could be useful in adjusting our model or even for bootstrapping purposes.
Distributed Data Sharing
Harvard's Dataverse project 4 [18] is a centralized approach to solving the problem of credited data sharing. It attempts to promote responsible data sharing by streamlining the process for researchers. It offers the option of archiving one's data in a "collection" and generating a unique academic citation with its own Document Object Identifier (DOI) that allows others to cite the data resource properly. The project aims to give researchers full control over their data. Data sharers can make data completely open or require that each person that wishes to look at it must ask them for permission to access the data. Researchers are also given the tools to add metadata to their data so that other researchers can find it in search engines. This allows researchers to maintain control over how their data is distributed while benefiting from institutional backing, such as in DataVerse. Our discussed data sharing protocol in Section 3 would share many traits of the Harvard Dataverse, such as rolebased access control over their data. Further, the ability to create DOIs specifically for datasets makes it much more feasible to credit researchers for their data contributions. The Dataverse shows us the importance of providing proper infrastructure and tools for researchers to handle, annotate, and share data easily.
The Ocean Protocol [21] is a decentralized data economy. Similar to the Dataverse, it promotes responsible data sharing by attempting to create a hub for researchers' data where they can be rewarded for their contributions. With the Ocean Protocol, one can publish their data as an NFT and then sell access tokens for their data. By tracking the usages of the data on a blockchain, researchers can be credited for their data contributions by citation while fiscally rewarding authors. Ocean Protocol's decentralized approach to data sharing promotes responsible data sharing while strongly incentivizing it, but it still fails to be the data ecosystem we seek to create. As researchers must pay for data, many will not have the resources to "buy" datasets.
CONCLUSION
Data sharing is a vital step towards more efficient and overall better research. SCIENCE-index addresses several flaws of the more major indexes, such as the inability to differentiate between highly cited but low-quality papers and low-cited but high-quality papers. The SCIENCE-index also considers the impact of data sharing, which is becoming increasingly important in scientific research. Our framework is a decentralized, self-governed, peer-to-peer datasharing protocol that would connect distributed researchers, decrease data reproduction, and increase research productivity. We build an ecosystem that fosters and rewards collaborations. However, this framework would only survive and scale if researchers were properly incentivized to participate. Our SCIENCE-index attempts to improve on current reputation-based metrics of measuring researchers, specifically the h-index, by augmenting it with data-sharing capabilities. We predict a given researcher's h-index based on 21 million other researchers and compare this to their actual h-index. This comparison gives us insight into their career progress compared to their peers. We also find that our model has a much more even spread of evaluations than the h-index when applied to geographically diverse researchers indicating that we have created a fair metric. We extend this to the scope of our data-sharing endeavors. By including data-sharing statistics as a parameter, we can reward researchers for their data sharing, thus incentivizing further data sharing. This incentive mechanism is necessary for distributed data sharing and encouraging more open science. This would increase the visibility of researchers who share their data and provide funding opportunities for those who share their data. While these incentives may not be perfect, they are a step in the right direction toward encouraging more data sharing in scientific research. Our initial SCIENCE-index levels the playing field among researchers with various amounts of resources at various points in their careers. Finally, we assert that the SCIENCE-index and its underlying infrastructure open the door for further discussion regarding how we rate and incentivize researchers.
Resource Contributions: We contribute the SCIENCE-index as an open-source repository, including our code for data gathering, model training, and visualization. We also include the smart contract code, which persists in our model, and the rest of our decentralized application. Our research artifacts are shared under the Apache 2.0 license. We maintain open-source Github repositories for all our artifacts at https://github.com/AI-and-Blockchain/F22_ SCIENCE_Index.
Figure 1 :
1High-level Sequence Diagram of Our Linear Model/Smart Contract/Oracle Design Pattern
Figure 2 :Figure 4 :
24Density Density Plot of the Augmented SCIENCE-Index Described In Section 3.2
Figure 6 :Figure 7 :
67Density Plot of the h-indexes of Our Hemisphere-Separated "Global South" and "Global North" Dataset Density Plot of the SCIENCE-Index of Our Hemisphere-Separated Dataset
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arxiv |
Temperature dependence of mycosubtilin homologues production in Bacillus subtilis ATCC6633
Patrick Fickers [email protected]érieleclère:[email protected]:[email protected]:[email protected]çoisecoucheney:[email protected]
Laboratoire de Procédés Biologiques
Génie Enzymatique et Microbien (ProBioGEM
UPRES EA
Polytech'Lille
Université des Sciences et Technologies de Lille
1026), F-59655Villeneuve d'Ascq CedexFrance
Centre d'Ingénierie des Protéines
Institut de Chimie
Laboratoire de Physiologie et Génétique Bactrienne
Université de Liège
Bat B6B-4000LiegeBelgique
Valérie Leclère
Laboratoire de Procédés Biologiques
Génie Enzymatique et Microbien (ProBioGEM
UPRES EA
Polytech'Lille
Université des Sciences et Technologies de Lille
1026), F-59655Villeneuve d'Ascq CedexFrance
Jean-Sébastien Guez
Laboratoire de Procédés Biologiques
Génie Enzymatique et Microbien (ProBioGEM
UPRES EA
Polytech'Lille
Université des Sciences et Technologies de Lille
1026), F-59655Villeneuve d'Ascq CedexFrance
Max Béchet
Laboratoire de Procédés Biologiques
Génie Enzymatique et Microbien (ProBioGEM
UPRES EA
Polytech'Lille
Université des Sciences et Technologies de Lille
1026), F-59655Villeneuve d'Ascq CedexFrance
Françoise Coucheney
Laboratoire de Procédés Biologiques
Génie Enzymatique et Microbien (ProBioGEM
UPRES EA
Polytech'Lille
Université des Sciences et Technologies de Lille
1026), F-59655Villeneuve d'Ascq CedexFrance
Bernard Joris [email protected]:[email protected]
Centre d'Ingénierie des Protéines
Institut de Chimie
Laboratoire de Physiologie et Génétique Bactrienne
Université de Liège
Bat B6B-4000LiegeBelgique
Philippe Jacques
Laboratoire de Procédés Biologiques
Génie Enzymatique et Microbien (ProBioGEM
UPRES EA
Polytech'Lille
Université des Sciences et Technologies de Lille
1026), F-59655Villeneuve d'Ascq CedexFrance
Patrick Fickers
Jean-Sébastien Guez
Max Béchet
Bernard Joris
Temperature dependence of mycosubtilin homologues production in Bacillus subtilis ATCC6633
1 *Correspondence and reprints 2Bacillus subtilis ATCC6633mycosubtilinodd-numbered fatty acidstemperature 3
Bacillus subtilis ATCC6633 produces mycosubtilin, a non-ribosomally synthesized lipopeptide of the iturin family which presents antagonistic activities against various phytopathogens. Different homologues with fatty acid moiety varying from C15 to C17 are usually co-produced with their biological activities increasing with the number of carbon in the fatty acid chain. In the present report, we highlight that growth temperature modulates either the level of the mycosubtilin production and the relative abundance of the different homologues. A 30-fold increase in mycosubtilin production was observed when the temperature was decreased from 37 °C to 25 °C for both strain ATCC6633 and its derivative BBG100, a constitutive mycosubtilin overproducer. However, no significant difference in both the expression of the mycosubtilin synthetase encoding genes and in the intracellular synthetase concentration could be found, suggesting that the observed phenotype originated from a higher mycosubtilin synthetase turnover at lower temperature. We also point out that a lower growth temperature leads to an increased proportion of odd-numbered fatty acid homologues as a consequence of the de novo synthesis of C17 anteiso fatty acid following the cell adaptation to low temperature.Keywords : Bacillus subtilis ATCC6633, mycosubtilin, odd-numbered fatty acids, temperature 1. Introduction Members of the Bacillus subtilis family produce a wide variety of antibacterial and antifungal antibiotics (for review see [32]). Some of them, such as bacilysin, chlorotetain, mycobacillin, difficidin and lipopeptides are formed by nonribosomal peptide synthetases and/or polyketide synthetases. The lipopeptides belonging to the surfactin, iturin and fengycin families [37] are amphiphilic cyclic peptides composed of seven α-amino acids (surfactins and iturins) or ten α-amino acids (fengycins) linked to a single β-amino fatty acid (iturins) or β-hydroxy fatty acids (surfactins and fengycins). The length of the fatty acid moiety may vary from C13 to C16 for surfactins, from C14 to C17 for iturins and from C14 to C18 in the case of fengycins. Different homologous compounds with a linear or branched fatty acid moiety are usually co-produced for each lipopeptide family [28]. B. subtilis ATCC6633 produces subtilin [31], subtilosin [30], rhizocticin [18], and two lipopeptides, surfactin and mycosubtilin, the latter being a member of the iturins family [4,5]. Mass spectrometry analyses of B. subtilis ATCC6633 supernatant cultured at 30 °C revealed that the two main mycosubtilins produced belong to C16 and C17 homologues [21]. The mycosubtilin gene cluster spans about 38 kb and consists of four ORFs designated fenF and mycA, mycB and mycC, all of them being under control of the myc promoter [4,5]. The subunits encoded by the three myc genes contain the seven modules necessary to synthesize the peptide moiety of mycosubtilin. They show strong similarity with members of the peptide synthetase family and display the ordered assembly of conserved condensation, adenylation, and thiolation domains. Iturins present a strong fungitoxic activity against different phytopathogens such as Botrytis cinerea, Fusarium oxysporum and Pythium aphanidernatum.
in the fatty acid chain. Indeed, C17 homologues are 20-fold more active against pathogens than the C14 isoform [14].
Of the biological control alternatives to chemical pesticides used for reducing plant diseases, the application of non-pathogenic soil bacteria living in association with plant roots is promising. These bacteria can antagonize fungal pathogens by producing low-molecularweight fungitoxic compounds, such as the above-mentioned lipopeptides [28]. We recently highlighted in biocontrol assays, conducted with tomato/Pythium pathosystem, that pretreatment of tomato seeds with the mycosubtilin-overproducing derivative BBG100 prior to planting led to enhanced seedling emergence. This demonstrated that overproduction of mycosubtilin gained protection against Pythium damping-off of tomato seedlings [21].
In the present study, we demonstrate that growth temperature modulates both the level of mycosubtilin production and the relative abundance of the different homologues produced without any influence on myc expression and synthetase intracellular concentration.
Construction of strain BBG117
To measure the expression of the myc operon, a lacZ-reporter gene was obtained using vector pDG1661 which contains the ribosomal binding site of the B. subtilis spoVG gene fused to a promoter-less lacZ cassette [11]. The myc promoter fragment to be tested was obtained by PCR using the primers pMYCfo and pMYCrev (Table 1), and then subcloned into pGEM-T Easy vector to generate pBG107. This latter was then sequenced using the universal T7 and SP6 primers to verify the absence of PCR mistake. pBG107 was EcoRI-BamHI digested and the resulting 942-bp fragment was subcloned into pDG1661 at the corresponding sites to yield pBG111. This reporter construct was then integrated into the B.
subtilis ATCC6633 amyE locus, giving rise to strain BBG117. Correct integration was verified by both the loss of amylase activity and by analytical PCR using primers AmyEfo and AmyERev (Table 1).
Construction of strain RFB136
The mycA cassette used to disrupt the myc operon was constructed as previously described [6]. First, the ~1,4 kb P and T fragment consisting to part of 3' and 5' mycA ORF were PCR amplified using primer pair RFO120/RFO121 and RFO122/RFO123 respectively, and B.
subtilis ATCC 6633 genomic DNA as a template. Primers RFO121 and RFO122 contain the rare meganuclease I-SceI recognition sequence. P-ISceI and ISceI-T fragments were then pooled and used as a template for amplification of the P-ISceI-T cassette with primers RFO120 and RFO123. The resulting fragment was then cloned into pGEM-T Easy vector to generate RFP119. The 1,6 kb fragment encoding a kanamycin resistance gene was rescued from RFP104 by ISceI digestion and subcloned into RFP119 at the corresponding restriction site to yield RFP120. The mycA cassette was used to transform B. subtilis ATCC6633 and transformants were selected on LB kanamycin plates. Integration by double crossing event was verified by analytical PCR using the primer pair RFO120/RFO123. The absence of mycosubtilin production in culture supernatant was verified by matrix-assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF) on a Bruker Ultreflex tof (Bruker Daltonics, Bremen, Gremany) as previously described [21]. The mycosubtilin non-producer derivative was named RFB136.
β-galactosidase assays
β-galactosidase activities were measured according to Fickers et al. [7] on cell extracts prepared by chloroform treatment and centrifugation. One unit of β-galactosidase activity is defined as the amount of enzyme that produces 1 nmole of o-nitrophenol min -1 at 37°C.
Reverse transcription-polymerase chain reaction (RT-PCR) and comparative PCR
Total RNAs were isolated from culture cells with RNeasy Protect Bacteria Mini Kit (Quiagen), with some modifications [12]. Total RNAs were quantified on a Nanodrop ND-100 spectrophotometer whereas their integrity was estimated by calculating the RNA 23S/16S ratio on an Agilent 2100 Bioanalyser equipped with a RNA 6000 nanoLabChip. Extracted RNAs were then subjected to reverse transcription with Superscript II reverse transcriptase as recommended by the manufacturer (Invitrogen, Carlsbad, CA, USA). fenF, rplL and cspB cDNAs were PCR-amplified using the primer pairs fenFfo/fenFrev, rplLFo/rplLrev and cspBFo/cspBrev, respectively. The PCR products were then analysed by gel electrophoresis on a 1.5% agarose gels and the median-base trimmed mean density (MTM) were determined by scanning densitometry using the Arrayvision software (GE Healthcare, Upsalla, Sweden).
MTM values of each DNA fragment corresponding to fenF and cspB were normalized for the level of the B. subtilis housekeeping gene rplL in the same sample.
Lipopeptide purification and identification
One ml culture samples were centrifuged at 13 000 x g before being loaded onto C18 homologues and the fatty acid isomers (branched or linear) were determined based on their retention time as described elsewhere [11]. For normalization, mycosubtilin productions were expressed as a percentage of the highest value obtained.
Preparation of the cell free extract and enzyme purification
Preparation of cell free extracts was performed as described elsewhere with some modifications [36]. Briefly, cells in late exponnential growth phase, cultured in 500 ml Landy medium, were resuspended in 10 ml Tris-HCl buffer 50 mM pH 7 containing 3 mM DTT, 3 mM EDTA, 2 mM benzamidine and 20 % sucrose. Protoplasts were generated by lysosyme treatment (5 mg ml -1 cell suspension) at 30 °C for 35 min. They were then freezed at -80 °C, thawed rapidly in a 30°C water bath and finaly sonicated 3 times for 30 sec on ice. Cell debris were removed by centrifugation at 25 000 x g for 30 min. Nucleic acids were precipitated by addition of streptomycin sulfate at a final concentration of 1 % (w/v) and stirring for 10 min.
Methods for protein analysis
Protein concentration was determined by the method of Bradford [1] using the Bio-Rad protein assay solution (Bio-Rad Laboratories, Hercules, CA, USA). SDS-PAGE was performed according to Laemmli [19] in a 5% (w/v) acrylamide gel at a constant current of 30 mA. The following marker proteins were used : β-galactosidase (116 kDa), apoferritin (443 kDa) and thyroglobulin (669 kDa). After electrophoresis, gels were colored using coomassie brilliant blue G-250 using standard procedures. For N-terminal sequence determination, proteins were electrobloted onto an Hybond polyvinyldiene difluoride membrane (GE healthcare) and submited to Edman degradation using a Procise proteins sequencer (Applied Biosystems, Weiterstadt, Germany). Scanning densitometry of SDS-PAGE were performed using the Quantity One software (Bio-Rad Laboratories). Concentration of protein bands corresponding to MycB were determined using standard solutions of thyroglobulin.
ATP/Ppi exchange
Substrate amino acid dependent ATP/PPi exchange reaction was measured as described elswhere [34] with some modifications. The reaction mixture consisted of 100 µl Tris-20 mM HCl pH 7.6, 1 mM MgCl2, 5mM ATP, 1 mM tetra-sodium pyrophosphate, 1 µCi tetra-sodium [ 32 P] pyrophosphate, 0,5 mM asparagine and 20 µl of purified fractions. The samples were incubated for 40 min at room temperature. The [ 32 P]-labeled ATP was quantified by liquid scintillation methods.
Results
Influence of the culture temperature on mycosubtilin production
Mycosubtilin was measured after 72 h of culture, which correspond to the maximal level of production, at different temperatures for the B. subtilis wild-type strain ATCC6633 and BBG100, its constitutive mycosubtilin overproducing derivative [21]. As shown in Fig. 1a, the mycosubtilin production decreased in a temperature-dependent manner for both strains. Indeed, the mycosubtilin concentration in the culture broth decreased from 28 mg (mg DW) -1 at 25 °C to 1 mg (mg DW) -1 at 37 °C for the wild-type strain ATCC6633, whereas it decreased from 123 mg (mg DW) -1 to 16 mg (mg DW) -1 for BBG100 under the same conditions. For both strains, a regression analysis between the mycosubtilin concentration and the temperature led to a coefficient of determination (R 2 ) of 0,99 and 0,98 for ATCC6633 and BBG100, respectively. This demonstrated a direct relation between the mycosubtilin production and the growth temperature. Additional analysis on normalized values highlighted a temperature dependency of the mycosubtilin production equivalent whatever its level of production (Fig 1b).
Influence of the culture temperature on myc expression
The multienzymatic complex involved in mycosubtilin biosynthesis is encoded by the myc operon. To further characterize the influence of growth temperature on mycosubtilin production, a myc-lacZ reporter gene fusion was constructed and the expression of myc was determined at different temperatures. As little information is available on the myc promoter sequence, the intergenic region between the upstream pbp gene and fenF, the first gene of the myc operon was cloned and fused to a spoVG-lacZ promoter-less cassette.
Strain BBG117, a B. subtilis ATCC6633 derivative containing the myc-lacZ reporter fusion, was grown at different temperatures and the β-galactosidase activity was measured. Samples were collected at different time periods corresponding : (i) to the exponential growth phase, (ii) to the transition from the exponential to the stationary phase and (iii) to the stationary phase. As shown in Fig. 2, the level of β-galactosidase activities was maximal in the samples collected during the transition phase, suggesting that myc expression is maximal at that culture period. The expression of myc then decreased during the stationary phase. However, for all the temperatures tested, the β-galactosidase activity profiles were not significantly different, indicating that myc expression seems to be temperature-independent.
To further characterize the influence of the temperature on myc expression, the mRNA of fenF, the first ORF of the myc operon, was quantified in cold stress condition and compared to that obtained at 37°C. Therefore, mRNAs of fenF and cspB, a gene encoding the major cold shock protein, were quantified by RT-PCR using rplL mRNA, encoding the ribosomal protein L12, for data normalization. As shown in Fig. 3, inducting a cold shock response by growing culture of B. subtilis ATCC6633 at 20 °C resulted in a clear increase in the mRNA level of cspB. However, at this low temperature, no significant difference in the fenF mRNA level could be found as compared to the one observed at 37°C. This demonstrated that myc expression is not affected at low temperature.
MycB purification and quantification
The mycosubtilin synthetase is a modular multienzymatic complex with a relative molecular mass of more than 1300 kDa. In an attempt to estimate its relative abundance in B. subtilis cells, MycB, its larger subunit, was isolated and purified. This was performed for BBG100 grown at either 25 °C or 37 °C and for the negative control strain RFB136. The latter is a mycosubtilin non-producer derivative of B. subtilis ATCC6633 obtained by disruption of the myc operon. Cell-free extracts were prepared from cultures in late logarithmic phase of growth by lysozyme treatment, sonication and nucleic acid precipitation. Proteins were precipitated with ammonium sulfate and fractionated by size exclusion chromatography.
Fractions from BBG100 cultures were analysed by SDS-PAGE and compared to those obtained from RFB106. Protein bands with an apparent molecular mass of approximately 610 kDa were detected in fractions of BBG100 samples whereas they were not observed in RFB106 fractions (Fig. 4a). This apparent molecular mass is in good agreement with the 612 kDa deduced from the published mycB sequence, suggesting that the purified protein corresponds to MycB [5]. This was confirmed by the determination of the three first Nterminal amino acids of the 610 kDA protein (data not shown). The protein concentration of the purified enzyme was then estimated by scanning densitometry using a standard solution of thyroglobulin. As shown in Fig. 4 b, with values equal to 65 and 48 µg ml -1 , respectively, MycB concentration was slightly higher when BBG100 was cultured at 25 °C compared to that obtained at 37 °C.
To further characterise the influence of growth temperature on the intracellular mycosubtilin synthetase concentration, its abundance was estimated using one of the enzymatic reaction catalysed by MycB. In NRPS mechanisms, amino acids are first activated as an adenylate derivative in the A-domain of the synthetase before being incorpored into the peptidic chain. Among the three amino acids incorpored by MycB, L-asparagine is the sole specific to the mycosubtilin peptidic chain (i.e. not present in others lipopeptides). Therefore, its adenylation was measured by an ATP/PPi exchange assay in samples purified from BBG100 cultured at 25 and 37 °C and from RFB136 cultures used as a negative control. As shown in Fig. 4b, a slight increased level of asparagine adenylation could be observed in samples obtained from culture performed at 25 °C compared to that obtained at 37 °C. Indeed, value of ATP-PPi exchange were equal to 675 10 3 cpm and 486 10 3 cpm for samples obtained from cells cultured at 25 and 37 °C, respectively. For RFB136 samples, adenylation of Lasparagine was never detected (data not shown).
All of these observation demonstated that mycosubtilin synthetase could be somewhat more abundant in the cells cultured at 25 °C. However, those discrepancies are not sufficient to explain the 30-fold increased mycosubtilin production at low temperature.
Mycosubtilin homologues production
Changes in membrane lipid composition, and thus in the intracellular pool of fatty acids is one of the known consequences of the cell adaptation to low temperature [10]. This suggests that the proportion of the different mycosubtilin homologues, containing various chain length fatty acid moieties, could be affected by the growth temperature. Therefore, the mycosubtilin homologue proportions were measured after 72 h of growth for B. subtilis strains ATCC6633 and BBG100 cultured at 25, 30 and 37°C. As shown in Fig. 5, a significant increase in the proportion of mycosubtilin with odd-numbered fatty acids (i.e., C15 and C17) could be observed at lower temperature for both strains. By contrast, the proportion of the C16 fatty acid homologue decreased under these conditions. Production of odd-numbered fatty acid homologues also seemed to be affected by the mycosubtilin production yield. Indeed, for the mycosubtilin overproducer BBG100, the C17 homologue represented 80% and 77% of the total mycosubtilin production at 25 and 30°C, respectively, whereas these proportions fall to 71 % and 63 % for the wild-type strain ATCC6633, under the same conditions (Table 2).
To confirm the increase in the production of mycosubtilin with long chain fatty acid at low temperature, we performed cold shock experiments. Both B. subtilis ATCC6633 and BBG100 strains were first cultured at 30°C for 8 h before being shifted to 25°C for 64 h.
Lipopeptide production and homologue composition were then analysed and compared to those obtained for the cultures performed at 25 and 30°C for 72 h. As it was previously observed, significant differences in the proportion of the mycosubtilin homologues produced occurred for the cultures performed for 72 h at 25 and 30°C ( Table 2). In addition, the differences in the C16 and C17 homologue proportions were more marked for the shifted culture than those obtained at 25°C (Table 2). Indeed, for BBG100, the C16/C17 homologue ratio were equal to 0.25 at 25°C and 0.12 for the shifted culture, whereas these ratio were 0.41 and 0.22 for the wild-type strain under the same conditions. This could result from a stronger response of the cell metabolism (i.e., a higher production of odd-numbered fatty acids) after cold stress. In all cases, the increase in odd-numbered homologues at 25 °C and after cold shock was mainly due to an increase in the synthesis of mycosubtilin with branched fatty acids moiety. Further cold shock experiments from 38 to 30°C performed with BBG100 yielded similar results (data not shown).
Discussion
In its natural environment, the surface layer of soil, B. subtilis is exposed to temperature fluctuations which induce modifications in its physiology and metabolism. At lower temperature, cells must face several problems, including low membrane fluidity, reduced enzyme activities, decreased initiation of translation due to stabilized secondary structures of mRNAs or slower protein folding [16]. Fatty acids are one of the most important building blocks of cellular materials. In B. subtilis, the membrane composition is characterized by a fatty acid profile dominated to a large extend by odd-numbered branchedchain fatty acids, with the major C15 and C17 species [16]. These latter were shown to play a major role in the correct physical state of the membrane lipids, which is required for optimal membrane structure and function. At lower temperature, the membrane fluidity must increase to avoid transition from a liquid crystalline into a gel-like phase state of the lipid bilayer.
Bacillus cells respond to a decrease in the growth temperature by desaturating the fatty acids of their membrane lipids though the activation of the Des pathway [3,8] and by increasing the proportion of ante-iso branched fatty acids which present a lower melting point [23,33].
However, it was shown that the deletion of the des genes does not lead to any detectable phenotype after cold shock, indicating that B. subtilis rather adapts to low temperature by modifying its iso-and anteiso-fatty acid membrane composition [17]. These modifications must involve de novo fatty acid synthesis. Anteiso-methyl branches cannot be added by methylation of existing fatty acids, but are introduced as part of the primer molecule during the initiation of fatty acid synthesis. Anteiso-branched C15 and C17 fatty acids are formed from -keto--methylvalerate and 2-methyl-butyryl-CoA, which both derive from isoleucine.
The modification of the membrane composition at low temperature were demonstrated in B.
subtilis JH642 [17]. These results suggested a de novo synthesis of anteiso C15:0 and C17:0 fatty acids from isoleucine or threonine present in the culture medium as a response to a cold stress.
In the first step of the mycosubtilin synthesis pathway, the acyl CoA-ligase domain of myc couples coenzyme A to a cytoplasmic long chain fatty acid. The activated fatty acid is then transferred to the 4-phosphopantetheine cofactor of the first acyl carrier domain of the mycosubtilin synthetase. In the subsequent reactions of condensation, adenylation and thiolation reactions, the mycosubtilin molecule is synthetized analogously to other nonribosomal peptides [5]. It was also recently demonstrated by pulse-chase experiments that some flexibility exists in the length of fatty acid group incorporated in mycosubtilin molecules [13]. Taken together, all of these findings suggest that the predominant fatty acid homologue present in the cytoplasmic pool compatible with the lipopeptide structure is preferentially incorporated. Therefore, the variation in the composition of this pool following a temperature modification should also influence the production of mycosubtilin in the same manner. Our results are clearly supportive with this hypothesis since both low growth temperature and cold shock, favouring de novo synthesis of odd-numbered fatty acids, led to an increased production of mycosubtilin with a C15 and C17 fatty acid moiety.
Low growth temperature was also shown to significantly increase the mycosubtilin production yield. Our results suggest that this increase was not due to an overexpression of myc at low temperature since (i) myc induction and the concentration of the resulting mRNA were not modified at low temperature; (ii) the increase of mycosubtilin production was also observed for the constitutive producer BBG100. Therefore, this increased level of production could rather result from a higher activity of the mycosubtilin synthetase at low temperature. This hypothesis is reinforced by the fact that the differences in the mycosubtilin synthetase concentration in the cells cultured at different temperature are not sufficient to explain the difference in mycosubtilin production yield. This overproduction at low temperature is consistent with previous findings observed for B. subtilis RB14 in solid-state fermentation for another member of the iturin family, iturin A [24,26].
Lipopeptides were reported to be a key parameter in biofilm formation and rhizosphere colonization [27,28]. Therefore, the production of the most active homologue in large amount facilitates these phenomena. When considering that the temperature in the rhizosphere is rather close to 20 than 37 °C [9], it is not surprising that the lipopeptide production mechanism seems to be adapted to this low temperature.
Maxi-Clean cartridges (1 ml bead volume, Alltech, Deerfield, IL, USA), washed successively with 8 ml of water and 8 ml of H2O/methanol mixture (1:1, v/v). Lipopeptides were then eluted with 5 ml pure methanol, dried under vacuum and resuspended in 100 µl of methanol. Mycosubtilin concentration was determined by RP-HPLC using a Vydac 218 TP C18 column (250 x 4.6 mm, 5 µM packing, Vydac, Hesperia, CA, USA). The mobile phase was an acetonitrile/H2O/trifluoroacetic acid mixture (40:60:0,5, v/v/v). Samples (20 µl) were eluted at a flow rate of 1 ml min -1 . Purified iturins standard were purchased from Sigma. Mycosubtilins were identified based on the second derivatives of their UV-visible spectra (Waters PDA 996 diode array; Millenium software, Milford, MA, USA). The different
They were then peletted by centrifugation at 25 000 x g for 45 min. Proteins in the supernatant were salted out with ammonium sulfate at 70 % saturation and dissolved in a minimum volume of 50 mM Tris-HCl buffer pH 7 containing 3 mM DTT, 3 mM EDTA and 10 % sucrose. Five hundred µl of the crude enzyme extract were loaded on an ultrogel AcA 34 column (Sigma, 15 x 1,5 cm) and eluted with the same buffer at a flow-rate of 0,3 ml min -1 . Fractions of 1 ml were collected.
Figures legends . Fig. 1 :
.1Mycosubtilin productivity by B. subtilis ATCC6633 (♦) and BBG100 (■) after 72 h of growth in Landy medium at 25, 30 and 37 °C. Raw values (A) and normalized values (B).Results are mean values of three independent experiments. Standard deviations were less than 10 % of average value.
Fig. 2 .
2β-galactosidase activity determined for B. subtilis BBG117 after 4, 6 and 8 h of growth in Landy medium at 25, 30 and 37 °C. Results are mean values of three independent experiments. Standard deviations were less than 10 % of average value.
Fig. 3 .
3Expression of rplL, fenF and cspB at 20 and 37 °C. Electrophoretic profile of purified mRNAs subjected to RT-PCR (A). MTM values of fenF and cspB gene expression obtained after scanning densitometry and correction for the level of rplL for a given sample (B). Displayed data are one representative result of two independent experiments.
Fig. 4 .
4Purification and caracterisation of MycB in cell-free extract of BBG100 and RFB136 strains cultured in Landy medium at 25 °C and 37 °C. (A) Coomassie-stained 5 % SDS polyacrylamide gel showing the protein composition of the fraction obtained after size exclusion chromatography and tested for adenylation reaction. Five µg of protein were loaded per lane. Fraction showing the most intense 610 kDa band are presented for BBG100 whereas the corresponding chromatographic fraction is presented for RFB136 (B) Determination of the MycB protein concentration in the corresponding fraction by scanning densitometry (grey) and ATP/PPi exchange assay (black). Displayed data are one representative result of two independent experiments.
Fig. 5 .
5Relative abundance of the C15 (black), C16 (white) and C17 (grey) mycosubtilin homologues determined for B. subtilis ATCC6633 (A) and the mycosubtilin overproducing strain BBG100 (B) after 72 h of growth in Landy medium. The data are mean values of three independent experiments. All the values of mycosubtilin homologue production were significantly different for a given strain according to a Student's t-test at P<0.05.
Figure 1
Figure 5
Table 1
1Plasmids and strains used in this studyRepartition of the C16 and C17 mycosubtilin homologues produced by B. subtilis ATCC6633 cultured for 72 h at 25, 30°C and during temperature shift experiments. B and L are for branched and linear isomers respectively whereas Tot are for the total amount detected. ND : nondetermined. All the values for a given strain were significantly different for each conditions tested according to a Student's t-test at P<0.05Plasmids
Description, structure or sequence (5'-3')
Source, restriction site
pGEM-T Easy
PCR fragments cloning vector
Promega
pDG1661
Integrative vector at amyE locus
[11]
pBG107
Pmyc promoter into pGEM-T Easy
This work
pBG111
Pmyc -lacZ into pDG1661
This work
RFP104
IsceI-Kan R -IsceI fragment into pGEM-T Easy
Fickers et al
unpublished results
RFP119
mycA PT cassette into pGEM-T Easy
This work
RFP120
Kan R fragment into RFP119 at ISceI site
This work
Primers
pMYCfo
CGTCAAGAATTCTTTATCATTCCATATATACG
EcoRI
pMYCrev
ATTCATTGGATCCCTCCAATCTTTTCGAACGG
BamHI
AmyEfo
GGAAGCGGAAGAATGAAGTAAGAGGG
AmyERev
GCCAGGCTGATTCTGACCGGGCAC
FenFfo
CAAAATGCAGATCCTGAGCA
FenFrev
GGCATAGTCATGTGCGTTTG
rplL fo
GCTTCCGTTAAAGAAGCAACTG
RplLrev
AGAAGCGCCAACTTCTTCAA
cspB Fo
AAAAGGTTTCGGATTCATCG
cspBrev
Acknowledgements
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The lipopeptides mycosubtilin and surfactin enhance spreading of Bacillus subtilis strains by their surface-active properties. V Leclère, R Marti, M Béchet, P Fickers, P Jacques, Arch. Microbiol. 186Leclère, V., Marti, R., Béchet, M., Fickers, P., Jacques, P. (2006) The lipopeptides mycosubtilin and surfactin enhance spreading of Bacillus subtilis strains by their surface-active properties. Arch. Microbiol. 186, 475-485.
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Medium optimization of antifungal lipopeptide, iturin A, production by Bacillus subtilis in solid-state fermentation by response surface methodology. S Mizumoto, M Shoda, Appl. Microbiol. Biotechnol. 76Mizumoto, S., Shoda, M. (2007) Medium optimization of antifungal lipopeptide, iturin A, production by Bacillus subtilis in solid-state fermentation by response surface methodology. Appl. Microbiol. Biotechnol. 76, 101-108.
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Effect of temperature on production of lipopeptide antibiotics, iturin A and surfactin by a dual producer, Bacillus subtilis RD14, in solidstate fermentation. A Ohno, T Ano, M Shoda, J. Ferment. Bioeng. 80Ohno, A., Ano, T., Shoda, M. (1995) Effect of temperature on production of lipopeptide antibiotics, iturin A and surfactin by a dual producer, Bacillus subtilis RD14, in solid- state fermentation. J. Ferment. Bioeng. 80, 517-519.
Involvement of fengycin-type lipopeptides in the multifaceted biocontrol potential of Bacillus subtilis. M Ongena, P Jacques, Y Touré, J Destain, A Jabrane, P Thonart, Appl. Microbiol. Biotechnol. 69Ongena, M., Jacques, P., Touré, Y., Destain, J., Jabrane, A., Thonart, P. (2005) Involvement of fengycin-type lipopeptides in the multifaceted biocontrol potential of Bacillus subtilis. Appl. Microbiol. Biotechnol. 69, 29-38.
Bacillus subtilis lipopeptides: versatile weapons for plant disease biocontrol. M Ongena, P Jacques, Trends in microbiology. 16Ongena, M., Jacques P. (2008) Bacillus subtilis lipopeptides: versatile weapons for plant disease biocontrol. Trends in microbiology, 16, 115-125.
Molecular cloning :a laboratory manual. J Sambrook, E Frittsch, T Maniatis, NY: Cold Spring HarborSambrook, J., Frittsch, E., Maniatis, T. (1989) Molecular cloning :a laboratory manual, NY: Cold Spring Harbor.
Subtilosin production by two Bacillus subtilis subspecies and variance of the sbo-alb cluster. T Stein, S Dusterhus, A Stroh, K D Entian, Appl. Environ. Microbiol. 70Stein, T., Dusterhus, S., Stroh, A., Entian, K. D. (2004) Subtilosin production by two Bacillus subtilis subspecies and variance of the sbo-alb cluster. Appl. Environ. Microbiol. 70, 2349-2353.
Dual control of subtilin biosynthesis and immunity in Bacillus subtilis. T Stein, S Borchert, P Kiesau, S Heinzmann, S Klöss, C Klein, M Helfrich, K D Entian, Mol. Microbiol. 44Stein, T., Borchert, S., Kiesau, P., Heinzmann, S., Klöss, S., Klein, C., Helfrich, M., Entian, K.D. (2002) Dual control of subtilin biosynthesis and immunity in Bacillus subtilis. Mol. Microbiol. 44, 403-416
Bacillus subtilis antibiotics: structures, syntheses and specific functions. T Stein, Mol. Microbiol. 56Stein, T. (2005) Bacillus subtilis antibiotics: structures, syntheses and specific functions. Mol. Microbiol. 56, 845-857.
Unsaturated and branched chain-fatty acids in temperature adaptation of Bacillus subtilis and Bacillus megaterium. M Suutari, S Laakso, Biochim. Biophys. Acta. 1126Suutari, M., Laakso, S. (1992) Unsaturated and branched chain-fatty acids in temperature adaptation of Bacillus subtilis and Bacillus megaterium, Biochim. Biophys. Acta. 1126, 119-124.
Cloning and characterization of a Streptomyces single module type non-ribosomal peptide synthetase catalyzing a blue pigment synthesis. H Takahashi, T Kumagai, K Kitani, M Mori, Y ( Matoba, J. Biol. Chem. 282Takahashi, H., Kumagai, T., Kitani, K, Mori, M., Matoba, Y (2007) Cloning and characterization of a Streptomyces single module type non-ribosomal peptide synthetase catalyzing a blue pigment synthesis. J. Biol. Chem. 282, 9073-9081
Role of lipopeptides produced by Bacillus subtilis GA1 in the reduction of grey mould disease caused by Botrytis cinerea on apple. Y Toure, M Ongena, P Jacques, A Guiro, P Thonart, J. Appl. Microbiol. 96Toure, Y., Ongena, M., Jacques, P., Guiro, A., Thonart, P. (2004) Role of lipopeptides produced by Bacillus subtilis GA1 in the reduction of grey mould disease caused by Botrytis cinerea on apple. J. Appl. Microbiol. 96, 1151-1160.
Cell-free biosynthesis of surfactin, a cyclic lipopeptide produced by Bacillus subtilis. Ch Ullrich, B Kluge, Z Palacz, J Vater, Biochemistry. 30Ullrich, Ch., Kluge, B., Palacz, Z., Vater, J. (1991) Cell-free biosynthesis of surfactin, a cyclic lipopeptide produced by Bacillus subtilis. Biochemistry, 30, 6503-6508.
Bacillus subtilis and other gram-positive bacteria. P Zuber, M Nakano, M Marahiel, A. Sonenshein, J. Hoch, R. LosikAmerican Society for MicrobiologyWashington D.C.Zuber, P., Nakano, M., Marahiel, M. (1993) In : A. Sonenshein, J. Hoch, R. Losik (Eds), Bacillus subtilis and other gram-positive bacteria. American Society for Microbiology, Washington D.C., pp. 897-916.
| {'fraction_non_alphanumeric': 0.045400930273963455, 'fraction_numerical': 0.03251507800502971, 'mean_word_length': 4.827259758786149, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 6, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Bacillus subtilis ATCC6633 produces mycosubtilin, a non-ribosomally synthesized lipopeptide of the iturin family which presents antagonistic activities against various phytopathogens. Different homologues with fatty acid moiety varying from C15 to C17 are usually co-produced with their biological activities increasing with the number of carbon in the fatty acid chain. In the present report, we highlight that growth temperature modulates either the level of the mycosubtilin production and the relative abundance of the different homologues. A 30-fold increase in mycosubtilin production was observed when the temperature was decreased from 37 °C to 25 °C for both strain ATCC6633 and its derivative BBG100, a constitutive mycosubtilin overproducer. However, no significant difference in both the expression of the mycosubtilin synthetase encoding genes and in the intracellular synthetase concentration could be found, suggesting that the observed phenotype originated from a higher mycosubtilin synthetase turnover at lower temperature. We also point out that a lower growth temperature leads to an increased proportion of odd-numbered fatty acid homologues as a consequence of the de novo synthesis of C17 anteiso fatty acid following the cell adaptation to low temperature.Keywords : Bacillus subtilis ATCC6633, mycosubtilin, odd-numbered fatty acids, temperature 1. Introduction Members of the Bacillus subtilis family produce a wide variety of antibacterial and antifungal antibiotics (for review see [32]). Some of them, such as bacilysin, chlorotetain, mycobacillin, difficidin and lipopeptides are formed by nonribosomal peptide synthetases and/or polyketide synthetases. The lipopeptides belonging to the surfactin, iturin and fengycin families [37] are amphiphilic cyclic peptides composed of seven α-amino acids (surfactins and iturins) or ten α-amino acids (fengycins) linked to a single β-amino fatty acid (iturins) or β-hydroxy fatty acids (surfactins and fengycins). The length of the fatty acid moiety may vary from C13 to C16 for surfactins, from C14 to C17 for iturins and from C14 to C18 in the case of fengycins. Different homologous compounds with a linear or branched fatty acid moiety are usually co-produced for each lipopeptide family [28]. B. subtilis ATCC6633 produces subtilin [31], subtilosin [30], rhizocticin [18], and two lipopeptides, surfactin and mycosubtilin, the latter being a member of the iturins family [4,5]. Mass spectrometry analyses of B. subtilis ATCC6633 supernatant cultured at 30 °C revealed that the two main mycosubtilins produced belong to C16 and C17 homologues [21]. The mycosubtilin gene cluster spans about 38 kb and consists of four ORFs designated fenF and mycA, mycB and mycC, all of them being under control of the myc promoter [4,5]. The subunits encoded by the three myc genes contain the seven modules necessary to synthesize the peptide moiety of mycosubtilin. They show strong similarity with members of the peptide synthetase family and display the ordered assembly of conserved condensation, adenylation, and thiolation domains. Iturins present a strong fungitoxic activity against different phytopathogens such as Botrytis cinerea, Fusarium oxysporum and Pythium aphanidernatum.', 'arxivid': '2009.10375', 'author': ["Patrick Fickers [email protected]érieleclère:[email protected]:[email protected]:[email protected]çoisecoucheney:[email protected] \nLaboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance\n\nCentre d'Ingénierie des Protéines\nInstitut de Chimie\nLaboratoire de Physiologie et Génétique Bactrienne\nUniversité de Liège\nBat B6B-4000LiegeBelgique\n", "Valérie Leclère \nLaboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance\n", "Jean-Sébastien Guez \nLaboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance\n", "Max Béchet \nLaboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance\n", "Françoise Coucheney \nLaboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance\n", "Bernard Joris [email protected]:[email protected] \nCentre d'Ingénierie des Protéines\nInstitut de Chimie\nLaboratoire de Physiologie et Génétique Bactrienne\nUniversité de Liège\nBat B6B-4000LiegeBelgique\n", "Philippe Jacques \nLaboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance\n", 'Patrick Fickers ', 'Jean-Sébastien Guez ', 'Max Béchet ', 'Bernard Joris '], 'authoraffiliation': ["Laboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance", "Centre d'Ingénierie des Protéines\nInstitut de Chimie\nLaboratoire de Physiologie et Génétique Bactrienne\nUniversité de Liège\nBat B6B-4000LiegeBelgique", "Laboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance", "Laboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance", "Laboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance", "Laboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance", "Centre d'Ingénierie des Protéines\nInstitut de Chimie\nLaboratoire de Physiologie et Génétique Bactrienne\nUniversité de Liège\nBat B6B-4000LiegeBelgique", "Laboratoire de Procédés Biologiques\nGénie Enzymatique et Microbien (ProBioGEM\nUPRES EA\nPolytech'Lille\nUniversité des Sciences et Technologies de Lille\n1026), F-59655Villeneuve d'Ascq CedexFrance"], 'corpusid': 28679077, 'doi': '10.1016/j.resmic.2008.05.004', 'github_urls': [], 'n_tokens_mistral': 14515, 'n_tokens_neox': 11867, 'n_words': 6546, 'pdfsha': '18a20d604fa5ea1e8cd19b66a52f67e79d043fb1', 'pdfurls': ['https://arxiv.org/pdf/2009.10375v1.pdf'], 'title': ['Temperature dependence of mycosubtilin homologues production in Bacillus subtilis ATCC6633', 'Temperature dependence of mycosubtilin homologues production in Bacillus subtilis ATCC6633'], 'venue': []} |
arxiv |
More Infinite Products: Thue-Morse and the Gamma function
Sep 2017
J.-P Allouche [email protected]
School of Computer Science
CNRS
IMJ-PRG Université P. et M. Curie Case
247, 4 Place JussieuF-75252Paris Cedex 05France
S Riasat [email protected]
University of Waterloo
N2L 3G1WaterlooONCanada
J Shallit [email protected]
University of Waterloo
N2L 3G1WaterlooONCanada
More Infinite Products: Thue-Morse and the Gamma function
Sep 2017
Letting (t n ) denote the Thue-Morse sequence with values 0, 1, we note that the Woods-Robbins product n≥0 2n + 1 2n + 2 (−1) tn = 2 −1/2 involves a rational function in n and the ±1 Thue-Morse sequence ((−1) tn ) n≥0 . The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the ±1 Thue-Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions R for which the infinite product R(n) tn also has an expression in terms of known constants.
Introduction
Several infinite products involving the sum of binary digits of the integers were inspired by the discovery of the Woods and Robbins infinite product (see [16,12]). More precisely, letting t n denote the sum, modulo 2, of the binary digits of the integer n, the sequence (t n ) n≥0 = 0 1 1 0 1 0 0 1 1 0 0 1 . . . is called the Thue-Morse sequence with values 0 and 1 (see, e.g., [5] and the references therein). The Woods-Robbins product identity is n≥0 2n + 1 2n + 2
(−1) tn = 1 √ 2 ·(1)
Several infinite products inspired by (1) were discovered later (see, e.g., [3,4,1,6,10]). They all involve, as exponents, sequences of the form (−1) u w,b (n) where u w,b (n) is the number, reduced modulo 2, of occurrences of the word (the block) w in the b-ary expansion of the integer n. But none of these products are in terms of 0-1-sequences (u w,b (n)) n≥0 alone. In particular, none of them are in terms of the binary sequence (t n ) n≥0 = (u 1,2 (n)) n≥0 given above. Furthermore, there has been no attempt up to now to find explicitly-given large classes of rational functions R for which the infinite product R(n) (−1) tn has an expression in terms of known constants.
The purpose of this paper is thus twofold. First, to find other infinite products of the form R(n) (−1) tn admitting an expression in terms of known constants. Second, to find infinite products of the form R(n) tn also having an expression in terms of known constants. Two examples that we find are n≥0 4n + 1 4n + 3
(−1) tn = 1 2 · n≥0
(4n + 1)(4n + 4) (4n + 2)(4n + 3)
tn = π 3/4 √ 2 Γ(1/4) · 2 Products of the form R(n) (−1) t n
We start with a lemma about the convergence of infinite products involving the sequence ((−1) tn )).
Lemma 2.1. Let t n be the sum, reduced modulo 2, of the binary digits of the integer n. Let R ∈ C(X) be a rational function such that the values R(n) are defined for n ≥ 1. Then the infinite product n R(n) (−1) tn converges if and only if the numerator and the denominator of R have same degree and same leading coefficient.
Proof. If the infinite product converges, then R(n) must tend to 1 when n tends to infinity. Thus the numerator and the denominator of R have the same degree and the same leading coefficient. Now suppose that the numerator and the denominator of R have the same leading coefficient and the same degree. Decomposing them into factors of degree 1, it suffices, for proving that the infinite product converges, to show that infinite products of the form n≥1 n+b n+c (−1) tn converge for complex numbers b and c such that n + b and n + c do not vanish for any n ≥ 1. Since the general factor of such a product tends to 1, this is equivalent, grouping the factors pairwise, to proving that the product
n≥1 2n + b 2n + c (−1) t 2n 2n + 1 + b 2n + 1 + c (−1) t 2n+1
converges. Since (−1) t 2n = (−1) tn and (−1) t 2n+1 = −(−1) tn we only need to prove that the infinite product
n≥1 (2n + b)(2n + 1 + c) (2n + c)(2n + 1 + b) (−1) tn
converges. Taking the (principal determination of the) logarithm, we see that
log (2n + b)(2n + 1 + c) (2n + c)(2n + 1 + b) = O(1/n 2 ),
which gives the convergence result.
In order to study the infinite product n≥1 R(n) (−1) tn , it suffices, using Lemma 2.1 above, to study products of the form n n+a n+b
(−1) tn
where a and b belong to C \ {−1, −2, −3, . . .}.
Theorem 2.2. Define f (a, b) := n≥1 n + a n + b (−1) tn and g(x) := f ( x 2 , x+1 2 ) x + 1
for a, b, x complex numbers that are not negative integers. Then
f (a, b) = g(a) g(b) ·
Furthermore, g satisfies the functional equation
(1 + x)g(x) = g( x 2 ) g( x+1 2 ) ∀x ∈ C \ {−1, −2, −3, . . .}.
In particular we have g(1/2) = 1 and g(1) = √ 2/2.
Proof. Recall that (−1) t 2n and (−1) t 2n+1 = −(−1) tn . Hence
f (a, b) = n≥1 n + a n + b (−1) tn = n≥1 2n + a 2n + b (−1) t 2n n≥0 2n + 1 + a 2n + 1 + b (−1) t 2n+1 = n≥1 (2n + a)(2n + 1 + b) (2n + b)(2n + 1 + a) (−1) tn 1 + b 1 + a = n≥1 (n + a 2 )(n + 1+b 2 ) (n + a+1 2 )(n + b 2 ) (−1) tn 1 + b 1 + a = f ( a 2 , a+1 2 ) f ( b 2 , b+1 2 ) 1 + b 1 + a = g(a) g(b) ·
Now taking a = x 2 and b = x+1 2 in the equality g(a)
g(b) = f (a, b) yields g( x 2 ) g( x+1 2 ) = f ( x 2 , x+1 2 ) = (x + 1)g(x),
which is the announced functional equation.
Finally putting x = 0 in this functional equation, and noting that g(0) = 0, yields g( 1 2 ) = 1, while putting x = 1 gives g(1) 2 = 1 2 , hence g(1) = 1
i) n≥1 (n + a)(2n + a + 1)(2n + b) (2n + a)(n + b)(2n + b + 1) (−1) tn = b + 1 a + 1 (ii) n≥1 (n + a)(2n + a + 1) 2 (2n + a)(2n + a + 2)(n + a + 1) (−1) tn = a + 2 a + 1 (iii) n≥1 (2n + 2a)(2n + a + 1) (2n + a)(2n + 1) (−1) tn = 1 a + 1 · and, for a ∈ C \ {0, −1, −2, −3, . . .} ∪ {−1/2, −3/2, −5/2, . . .}), (iv) n≥1 (2n + a + 1)(2n + 2a − 1) (2n + a)(2n + 4a − 2) (−1) tn = 2a a + 1 ·
Proof. (i) is proved by writing its left side, say A, in terms of values of f and applying Theorem 2.2:
A = f (a, a 2 )f ( a+1 2 , b)f ( b 2 , b+1 2 ) = g(a) g( a 2 ) g( a+1 2 ) g(b) g( b 2 ) g( b+1 2 ) = b + 1 a + 1 · (ii) is obtained from (i) by taking b = a + 1. (iii) is obtained from (i) by taking b = 0. (iv) is obtained from (i) by taking b = 2a − 1.
We give examples with particular values of the parameters in the next corollary.
Corollary 2.4. We have the following equalities.
(a) n≥0 2n + 1 2n + 2
(−1) tn = √ 2 2 (W.-R.) (b) n≥0 4n + 1 4n + 3 (−1) tn = 1 2 (c) n≥1 (2n − 1)(4n + 1) (2n + 1)(4n − 1) (−1) tn = 2 (d) n≥0 (n + 1)(2n + 1) (n + 2)(2n + 3) (−1) tn = 1 2 (e) n≥0 (2n + 2)(4n + 3) (2n + 3)(4n + 5) (−1) tn = √ 2 2 (f ) n≥0 (n + 1)(4n + 5) (n + 2)(4n + 3) (−1) tn = 1 (g) n≥0 (n + 1)(2n + 2) (n + 2)(2n + 3) (−1) tn = √ 2 2 (h) n≥0 (n + 1)(4n + 5) (n + 2)(4n + 1) (−1) tn = 2 (i) n≥0 (2n + 2)(4n + 1) (2n + 3)(4n + 5) (−1) tn = √ 2 4 (j) n≥0 (2n + 1)(4n + 1) (2n + 3)(4n + 5) (−1) tn = 1 4 (k) n≥0 (4n + 1)(8n + 7) (4n + 2)(8n + 3) (−1) tn = 1 (l) n≥0 (8n + 1)(8n + 7) (8n + 3)(8n + 5) (−1) tn = 1 2 · Proof. Corollary 2.3 (ii) with a = 0 yields n≥1 (2n + 1) 2 (2n + 2) 2 (−1) tn = 2.
Taking the square root, and multiplying by the value of 2n+1 2n+2 for n = 0, we obtain the Woods-Robbins identity (a).
Corollary 2.3 (iii) with a = 1 2 gives n≥1 2n + 3 2 2n + 1 2 (−1) tn = 2 3 ·
This implies (b) (note the different range of multiplication again).
Corollary 2.3 (iii) with a = 1 2 gives n≥1 (2n − 1)(2n + 1 2 ) (2n − 1 2 )(2n + 1) (−1) tn = 2,
which implies (c). (n + 1)(2n + 2) 2 (2n + 1)(n + 2)(2n + 3)
(−1) tn = 3 2 ·
We obtain (d) after multiplying by the factor corresponding to n = 0, then by the square of n≥0 2n+1 2n+2 (−1) tn (this square is equal to 1 2 from the identity of Woods and Robbins).
Corollary 2.3 (i) with a = 1 and b = 3 2 yields n≥1 (n + 1)(2n + 2)(2n + 3 2 ) (2n + 1)(n + 3 2 )(2n + 5 2 ) (−1) tn = 5 4 ·
Equality (e) is then obtained by multiplying by the factor corresponding to n = 0 and then multiplying by n≥0 2n+1 2n+2
(−1) tn , (which again is equal to √ 2 2 ). Corollary 2.3 (i) with a = 2 and b = 3 2 gives n≥1 (n + 2)(2n + 3)(2n + 3 2 ) (2n + 2)(n + 3 2 )(2n + 5 2 ) (−1) tn = 5 6 ·
We simplify by (2n + 3), multiply by the factor corresponding to n = 0, and we obtain (the inverse of) Equality (f). Remark 2.5. The proofs that we give, e.g., in Corollary 2.4, provide infinite products whose values are rational: in the case of the Woods-Robbins infinite product P = n≥0 2n+1 2n+2 (−1) tn we actually obtain the value of P 2 (= 1/2). We finally get √ 2 2 only because the product we first obtain involves the square of a rational function.
3 More remarks on the function g As we have seen above, the function g defined by g(
x) := f ( x 2 , x+1 2 ) x+1
has the property that
f (a, b) = g(a)
g (b) . It satisfies the functional equation
g( x 2 ) g( x+1 2 ) = (1 + x)g(x) for
x not equal to a negative integer. This functional equation has some resemblance with the celebrated duplication formula for the Γ function:
Γ( z 2 )Γ( z+1 2 ) = 2 1−z √ π Γ(z).
We also point out the cancellation of g(0) when we computed g(1/2). In particular, we have not been able to give the value of g(0) in terms of known constants. The quantity g(0) = and it easily follows that
ϕ = 2 −1/2 e γ g(0) ·
Finally, we will prove that the function g is decreasing. Actually we have the stronger result given in Theorem 3.2 below. We first state and slighty extend a lemma (Lemmas 3.
Let A = {G : R + → R, G is C ∞ , ∀x ≥ 0, (−1) r G (r) (x) > 0}. Then
• for all k ≥ 0,one has T k A ⊂ A. Furthermore, if G belongs to A;
• if the series n≥0 T G(n) converges, then all the series n≥0 T k G(n) converge and R(n, T k G) := j≥n (−1) t j T k G(j) has the sign of (−1) tn .
Proof. See [2, Lemmas 3.2 and 3.2] where everything is proved, except that the last assertion about the sign of R(n, T k G) is stated only for k = 0, but clearly holds for all k ≥ 0.
Theorem 3.2. The function x → f ( x 2 , x+1
2 ) is decreasing on the nonnegative real numbers. Proof. G(x) = log x+a x+b for x ≥ 0. Then, G (r) (x) = (−1) r−1 (r − 1)!((x + a) −r − (x + b) −r ) for r ≥ 1 so that G belongs to A. Now applying Lemma 3.1 to T G and n = 1 yields n≥1 (−1) tn (log 2n+a 2n+b − log 2n+1+a 2n+1+b ) < 0, which is the same as saying that
f ( a 2 , a+1 2 ) f ( b 2 , b+1
Products of the form R(n) t n
We let again (t n ) n≥0 = 0 1 1 0 1 0 0 1 1 0 0 1 . . . denote the 0-1-Thue-Morse sequence. We have seen that several infinite products of the form R(n) (−1) tn admit a closed-form expression, but it might seem more natural (or at least desirable) to have results for infinite products of the form R(n) tn . Our first result deals with the convergence of such products.
Lemma 4.1. Let t n be the sum, reduced modulo 2, of the binary digits of the integer n. Let R ∈ C(X) be a rational function such that the values R(n) are defined for n ≥ 1. Then the infinite product n R(n) tn converges if and only if the numerator and the denominator of R have the same degree, the same leading coefficient, and the same sum of roots (in C).
Proof. If the infinite product R(n) tn converges, then R(n) must tend to 1 when n tends to infinity (on the subsequence for which t n = 1). Hence the numerator and denominator of R have the same degree and the same leading coefficient. But then, as we have seen, the product R(n) (−1) tn converges, and so does the product R(n) 2tn+(−1) tn . But this product is equal to R(n), which is known to converge if and only if the sum of the roots of the numerator is equal to the sum of the roots of the denominator. Now if the numerator and denominator of R have the same degree, the same leading coefficient, and the same sum of roots, then both infinite products R(n) (−1) tn and R(n) converge, which implies the convergence of the infinite product R(n) 1−2tn = R(n) (−1) tn . Now we give three equalities for products of the form R(n) tn .
Γ(1/4) = π 3/2 √ 2 Γ(1/4) 2 ,
where the last equality uses the reflection formula Γ(x)Γ(1 − x) = π/ sin(πx) for x / ∈ Z.
To compute the denominator, we start from the Woods-Robbins product and split the set of indices into even and odd indices, so that
√ 2 2 = n≥0 2n + 1 2n + 2 (−1) tn = n≥0 4n + 1 4n + 2 (−1) t 2n 4n + 3 4n + 4 (−1) t 2n+1 .
Using that t 2n = t n and t 2n+1 = 1 − t n , we thus have
√ 2 2 = n≥0 (4n + 1)(4n + 4) (4n + 2)(4n + 3) (−1) tn .
Gathering the results for the numerator and for the denominator we deduce n≥0 (4n + 1)(4n + 4) (4n + 2)(4n + 3)
tn 2 = 2π 3/2 Γ(1/4) 2 ,
hence the first assertion in our theorem.
The proof of the second assertion goes along the same lines. We start from Equality (h) in Corollary 2.4 n≥0 (n + 1)(4n + 5) (n + 2)(4n + 1) Note that this equality can also be obtained by telescopic cancellation in the finite product 0≤n≤N (n+1)(4n+5) (n+2)(4n+1) . Thus n≥0 (n + 1)(4n + 5) (n + 2)(4n + 1) tn 2 = n≥0 (n + 1)(4n + 5) (n + 2)(4n + 1) 1−(−1) tn = 2; hence n≥0 (n + 1)(4n + 5) (n + 2)(4n + 1)
(−1) tn = 2.tn = √ 2.
The proof of the third assertion is similar. We start from Equality (l) in Corollary 2.4: n≥0 (8n + 1)(8n + 7) (8n + 3)(8n + 5)
(−1) tn = 1 2 ·
Now, as previously, From this we obtain n≥0 (8n + 1)(8n + 7) (8n + 3)(8n + 5)
1−(−1) tn = 2 √ 2 − 2.
Thus, as claimed in the second assertion of the theorem, n≥0 (8n + 1)(8n + 7) (8n + 3)(8n + 5)
tn = 2 √ 2 − 2.
Remark 4.3. Several other closed-form expressions for infinite products R(n) tn can be obtained. For example one can use closed-form expressions for infinite products R(n) (−1) tn where R(n) satisfies the hypotheses of Lemma 4.1 and the classical result about R(n). Another possibility is to start from an already known product A = n≥0 S(n) (−1) tn where S satisfies the hypotheses of Lemma 2.1 and note that (splitting the indexes into even and odd)
A = n≥0 (S(2n) (−1) t 2n S(2n + 1) (−1) t 2n+1 ) = n≥0 S(2n) S(2n + 1) (−1) tn .
As easily checked the rational function satisfies the hypotheses of Lemma 4.1. The reader can observe that this generalizes the method used to prove the first assertion (i.e., Equality (1)) of Theorem 4.2.
Generalization to another block counting sequence
In this section we give an example of two products of the kind of those in Corollary 2.4 and in Theorem 4.2 that involve the Golay-Shapiro sequence (also called the Rudin-Shapiro sequence). Let us recall that this sequence (v n ) n≥0 (in its binary version) can be defined as follows: v 0 = 0, and for all
n ≥ 0, v 2n = v n , v 4n+1 = v n , v 4n+3 = 1 − v 2n+1 .
Another definition is that v n is the number, reduced modulo 2, of (possibly overlapping) 11's in the binary expansion of the integer n. The ± version ((−1) vn ) n≥0 of this sequence was introduced independently the same year (1951) by Shapiro [14] and by Golay [9], and rediscovered in 1959 by Rudin [13] who acknowledged Shapiro's priority.
An infinite product involving the Golay-Shapiro sequence was given in [3] (also see [4]): n≥1 (2n + 1) 2 (n + 1)(4n + 1)
(−1) vn = √ 2 2 ·
Here we prove the following theorem. Proof. Let R be a rational function in C(X) such that its numerator and denominator have same degree and same leading coefficient. Suppose furthermore that R(n) is defined for any integer n ≥ 1. Then it is not difficult to see that the infinite product n≥1 R(n) (−1) vn converges (summation by part, given the well-known property that the partial sum 1≤k≤n (−1) v k is O( √ n)). Now, using the recursive definition of (v n ) n≥0 one has
n≥1 R(n) (−1) vn = n≥1 R(2n) (−1) v 2n n≥0 R(2n + 1) (−1) v 2n+1 = n≥1 R(2n) (−1) vn n≥0 R(4n + 1) (−1) v 4n+1 n≥0 R(4n + 3) (−1) v 4n+3 = n≥1 R(2n) (−1) vn n≥0 R(4n + 1) (−1) vn n≥0 R(4n + 3) −(−1) v 2n+1 .
Thus n≥1 R(n) R(2n)R(4n + 1) 2 ), some are transcendental (e.g., n≥0
(−1) vn n≥0 R(4n + 3) (−1) v 2n+1 = R(1) (−1) v 1 = R(1). But n≥0 R(4n + 3) (−1) v 2n+1 = n≥0 R(2n + 1) (−1) vn n≥0 R(4n + 1) (−1) v 2n = n≥1 R(2n + 1) (−1)
(4n+1)(4n+4) (4n+2)(4n+3) tn = π 3/4 √ 2 Γ(1/4) whose transcendency is a consequence of the algebraic independence of π and Γ(1/4) proved byČudnovs'kiȋ; see [7]). As in [1] Another question is to generalize the results for the Thue-Morse sequence to other sequences counting certain patterns in the base-b expansion of integers: the example given in Section 5 is a first step in this direction.
implies many identities, including the original one of Woods-Robbins (W.-R.).
Corollary 2 . 3 .
23Let a and b belong to C \ {−1, −2, −3, . . .}. Then the following equalities hold.
(
0
0Corollary 2.3(ii) with a = 1 gives (g) with the usual manipulations (multiplying by the factor for n =
Corollary 2.3 (i) with a = 3 4 and b = 1 4
4gives (l) (multiply by the factor corresponding to n = 0 and use (b)).
Lemma 3 . 1 .
31For every function G, define the operator T by T G(x) := G(2x) − G(2x + 1).
properties values of the infinite products that we have obtained can be quite different. Some are rational (e.g.,
some of these values could be proved transcendental if one admits the Rohrlich conjecture. A still totally open question is the arithmetical nature of the Flajolet-Martin constant(s) (see the beginning of Section 3), namely ϕ and R, where ϕ :
). Alternatively (g) can be obtained by multiplying (e) and (f).Equality (h) is obtained by dividing (f) by (b). The inverse of Equality (i) is obtained by dividing (h) by (g). Equality (j) is obtained by multiplying (i) by n≥02n+1
2n+2
(−1) tn
, (which
is equal to
√
2
2 ).
Corollary 2.3 (iv) with a = 3
4 yields
n≥1
(2n + 7
4 )(2n + 1
2 )
(2n + 3
4 )(2n + 1)
(−1) tn
=
6
7
,
which implies (k).
Theorem 4.2. The following three equalities hold:n≥0
(4n + 1)(4n + 4)
(4n + 2)(4n + 3)
tn
=
π 3/4 √
2
Γ(1/4)
(2)
n≥0
(n + 1)(4n + 5)
(n + 2)(4n + 1)
tn
=
√
2
(3)
n≥0
(8n + 1)(8n + 7)
(8n + 3)(8n + 5)
tn
= 2
√
2 − 2.
(4)
Proof. As above we have 2t n = 1 − (−1) tn . Then we write
n≥0
(4n + 1)(4n + 4)
(4n + 2)(4n + 3)
tn 2
=
n≥0
(4n + 1)(4n + 4)
(4n + 2)(4n + 3)
n≥0
(4n + 1)(4n + 4)
(4n + 2)(4n + 3)
(−1) tn ·
The computation of the numerator is classical (see, e.g., [17, Section 12-13]):
n≥0
(4n + 1)(4n + 4)
(4n + 2)(4n + 3)
=
n≥0
(n + 1/4)(n + 1)
(n + 1/2)(n + 3/4)
=
Γ(1/2)Γ(3/4)
Γ(1/4)Γ(1)
=
√ πΓ(3/4)
Theorem 5.1. The following two equalities hold.n≥0
4(n + 2)(2n + 1) 3 (2n + 3) 3
(n + 3)(n + 1) 2 (4n + 3) 4
(−1) vn
= 1
(5)
n≥0
4(n + 2)(2n + 1) 3 (2n + 3) 3
(n + 3)(n + 1) 2 (4n + 3) 4
vn
=
16Γ(3/4) 4
π 6
·
(6)
) < 1.
AcknowledgmentThis paper is an extended version of[11]. While we were preparing this extended version, we found the paper[15]which has interesting results on finite (and infinite) sums involving the sum of digits of integers in integer bases.
Paperfolding infinite products and the gamma function. J.-P Allouche, J. Number Theory. 148J.-P. Allouche, Paperfolding infinite products and the gamma function, J. Number The- ory 148 (2015), 95-111.
Dirichlet series and curious infinite products. J.-P Allouche, H Cohen, Bull. London Math. Soc. 17J.-P. Allouche, H. Cohen, Dirichlet series and curious infinite products, Bull. London Math. Soc. 17 (1985), 531-538.
De nouveaux curieux produits infinis. J.-P Allouche, H Cohen, M France, J O Shallit, Acta Arith. 49J.-P. Allouche, H. Cohen, M. Mendès France, J. O. Shallit, De nouveaux curieux pro- duits infinis, Acta Arith. 49 (1987), 141-153.
Infinite products associated with counting blocks in binary strings. J.-P Allouche, J Shallit, J. London Math. Soc. 39J.-P. Allouche, J. Shallit, Infinite products associated with counting blocks in binary strings, J. London Math. Soc. 39 (1989), 193-204.
The ubiquitous Prouhet-Thue-Morse sequence. J.-P Allouche, J Shallit, Sequences and their Applications. Singapore; LondonSpringerJ.-P. Allouche, J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in Sequences and their Applications (Singapore, 1998), 1-16, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 1999.
Infinite products with strongly B-multiplicative exponents. J.-P Allouche, J Sondow, Errata: Ann. Univ. Sci. Budapest. Sect. Comput. 28Ann. Univ. Sci. Budapest. Sect. Comput.J.-P. Allouche, J. Sondow, Infinite products with strongly B-multiplicative exponents, Ann. Univ. Sci. Budapest. Sect. Comput. 28 (2008), 35-53. [Errata: Ann. Univ. Sci. Budapest. Sect. Comput. 32 (2010), 253.]
Algebraic independence of constants connected with the exponential and the elliptic functions. G V Čudnovs'kiȋ, Dokl. Akad. Nauk Ukrain. SSR Ser. A. 8767in RussianG. V.Čudnovs'kiȋ, Algebraic independence of constants connected with the exponential and the elliptic functions, (in Russian), Dokl. Akad. Nauk Ukrain. SSR Ser. A 8 (1976), 698-701, 767.
Probabilistic counting algorithms for data base applications. P Flajolet, G N Martin, J. Comput. Sys. Sci. 31P. Flajolet, G. N. Martin, Probabilistic counting algorithms for data base applications, J. Comput. Sys. Sci. 31 (1985), 182-209.
Statistic multislit spectrometry and its application to the panoramic display of infrared spectra. M J E Golay, J. Optical Soc. America. 41M. J. E. Golay, Statistic multislit spectrometry and its application to the panoramic display of infrared spectra, J. Optical Soc. America 41 (1951), 468-472.
Patterns in numbers and infinite sums and products. Y Hu, J. Number Theory. 162Y. Hu, Patterns in numbers and infinite sums and products, J. Number Theory 162 (2016), 589-600.
Infinite products involving binary digit sums, to appear. S Riasat, Proc. AMMCS-2017 International Conference. AMMCS-2017 International ConferenceWaterloo, OntarioWilfrid Laurier UniversityS. Riasat, Infinite products involving binary digit sums, to appear, Proc. AMMCS-2017 International Conference, Wilfrid Laurier University, Waterloo, Ontario, 2017.
Solution to problem E 2692. D Robbins, Amer. Math. Monthly. 86D. Robbins, Solution to problem E 2692, Amer. Math. Monthly 86 (1979), 394-395.
Some theorems on Fourier coefficients. W Rudin, Proc. Amer. Math. Soc. 10W. Rudin, Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), 855-859.
Extremal Problems for Polynomials and Power Series. H S Shapiro, Massachusetts Institute of Technology, Department of Mathematics. H. S. Shapiro, Extremal Problems for Polynomials and Power Series, Thesis (M. S.), Massachusetts Institute of Technology, Department of Mathematics, 1951, available at http://dspace.mit.edu/handle/1721.1/12198
Finite generating functions for the sum of digits sequence. C Vignat, T Wakhare, ArXiv. C. Vignat, T. Wakhare, Finite generating functions for the sum of digits sequence, ArXiv, 2017, available at https://arxiv.org/abs/1708.06479.
Elementary problem proposal E 2692. D R Woods, Amer. Math. Monthly. 8548D. R. Woods, Elementary problem proposal E 2692, Amer. Math. Monthly 85 (1978), 48.
E T Whittaker, G N Watson, A Course of Modern Analysis, Fourth Edition, reprinted. CambridgeCambridge University PressE. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Fourth Edition, reprinted, Cambridge University Press, Cambridge, 1996.
| {'fraction_non_alphanumeric': 0.12268659310105134, 'fraction_numerical': 0.06421061777739161, 'mean_word_length': 3.2392265193370164, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Letting (t n ) denote the Thue-Morse sequence with values 0, 1, we note that the Woods-Robbins product n≥0 2n + 1 2n + 2 (−1) tn = 2 −1/2 involves a rational function in n and the ±1 Thue-Morse sequence ((−1) tn ) n≥0 . The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the ±1 Thue-Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions R for which the infinite product R(n) tn also has an expression in terms of known constants.', 'arxivid': '1709.03398', 'author': ['J.-P Allouche [email protected] \nSchool of Computer Science\nCNRS\nIMJ-PRG Université P. et M. Curie Case\n247, 4 Place JussieuF-75252Paris Cedex 05France\n', 'S Riasat [email protected] \nUniversity of Waterloo\nN2L 3G1WaterlooONCanada\n', 'J Shallit [email protected] \nUniversity of Waterloo\nN2L 3G1WaterlooONCanada\n'], 'authoraffiliation': ['School of Computer Science\nCNRS\nIMJ-PRG Université P. et M. Curie Case\n247, 4 Place JussieuF-75252Paris Cedex 05France', 'University of Waterloo\nN2L 3G1WaterlooONCanada', 'University of Waterloo\nN2L 3G1WaterlooONCanada'], 'corpusid': 119169700, 'doi': '10.1007/s11139-017-9981-7', 'github_urls': [], 'n_tokens_mistral': 9601, 'n_tokens_neox': 8339, 'n_words': 4160, 'pdfsha': 'f31561c85e4272ad1fd109a134c033e5ea5b7919', 'pdfurls': ['https://arxiv.org/pdf/1709.03398v2.pdf'], 'title': ['More Infinite Products: Thue-Morse and the Gamma function', 'More Infinite Products: Thue-Morse and the Gamma function'], 'venue': []} |
arxiv |
Forward-backward b-quark asymmetry at the Z pole: QCD uncertainties redux
David D'enterria
CERN, EP Department
CH-1211Geneva 23Switzerland
Cynthia Yan
CERN, EP Department
CH-1211Geneva 23Switzerland
Department of Physics
Harvey Mudd College
CA91711ClaremontUSA
Forward-backward b-quark asymmetry at the Z pole: QCD uncertainties redux
10.1142/9789813224568
The forward-backward asymmetry of b-quarks measured at LEP in e + e − collisions at the Z pole, A 0,b fb | exp = 0.0992 ± 0.0016, remains today the electroweak precision observable with the largest disagreement (2.8σ) with the Standard Model theoretical prediction, A 0,b fb | th = 0.1037 ± 0.0008. The dominant systematic uncertainties are due to QCD effects -b, c-quark showering and fragmentation, and B, D meson decay models -which have not been revisited in the last 20 years. We reassess the QCD uncertainties of the eight original LEP measurements of A 0,b fb , using modern parton shower simulations based on pythia 8 and pythia 8 + vincia with different tunes of soft and collinear radiation as well as of hadronization. Our analysis indicates QCD uncertainties, of order ±0.4% and ±1% for the jet-charge and lepton-charge based analyses, that are overall slightly smaller but still consistent with the original ones. Using the updated QCD systematic uncertainties, we obtain A 0,b fb = 0.0996 ± 0.0016.
Introduction
In the Standard Model (SM), the Z boson mediates weak neutral currents between fermions of the same generation. The Z couples to both left-and right-handed chiral states with different strengths depending on weak-isospin and electromagnetic charges. The vector and axial-vector Z couplings for a fermion of type f are g f V = (g f L +g f R ) = I f 3 −2Q f sin 2 θ W and g f A = (g f L −g f R ) = I f 3 respectively, where I 3 is the third component of the weak isospin of the fermion, Q f its charge (related to the former via the hypercharge Y f : Q f = I f 3 + Y f /2), and sin 2 θ W ≈ 0.23 is the weak mixing angle that controls the γ-Z mixing and provides a relationship between the coupling constants of the electroweak theory: g sin θ W = g cos θ W = e. From the expressions above, the varying strengths of the Z-fermion couplings for the (ν e , ν µ , ν τ ), (e, µ, τ ), (u, c, t), and (d, s, b) lepton/quark groups are explained. The mixed Z vector and axial-vector couplings not only affect the total e + e − → f f cross section but induce asymmetries in the angular distributions of the final-state fermions produced in the process. Angular asymmetries in the e + e − → f f final-state are ultimately driven by the fermions' charge Q and the weak mixing angle:
A f = (g f L ) 2 − (g f R ) 2 (g f L ) 2 + (g f R ) 2 = 2 g f V /g f A 1 + (g f V /g f A ) 2 , with g f V g f A = 1 − 4|Q f | sin 2 θ f eff .(1)
Experimentally, forward-backward asymmetries in e + e − → f f are determined from the ratio of the number of forward-(backward-)going (anti)fermions measured in the hemisphere defined by the direction of the e + (e − ) beams:
A f FB = N F − N B N F + N B , where F = 1 0 dσ dΩ dΩ, B = 0 −1 dσ dΩ dΩ,(2)
The forward-backward asymmetry of b quarks (A 0,b fb ) in the process e + e − → Z → bb at √ s = m Z is the one most accurately measured among all quarks at LEP, given that b-quarks are the easiest jets to identify. The value A 0,b fb | exp = 0.0992 ± 0.0016, obtained from the combination arXiv:1806.00141v2 [hep-ex] 6 Jun 2018
of eight measurements at √ s = 91.21-91.26 GeV using two different (lepton-and jet-charge based) methods, shows today the largest discrepancy (2.8σ) with respect to the theoretical SM prediction, A 0,b fb | th = 0.1037 ± 0.0008 (and so does the value of sin 2 θ W derived from them) 1 . We reanalyze here the original studies to see if such a discrepancy could be explained by a potential underestimation of the associated systematic uncertainties.
2 LEP b-quark forward-backward asymmetry data Table 1 lists the eight A 0,b fb measurements with the breakdown of their uncertainties. In four measurements, the original b,b quarks are identified from the charge of the leading lepton inside each b-jet (through the fragmentation b → B, b → c → D and subsequent B, D → decay), whereas in the other four, the b charge is reconstructed from the jet constituent particles. The statistical uncertainties of A 0,b fb dominate, being twice bigger than the systematic ones, while the QCD uncertainties account for about half of the latter (and are assumed to be fully-correlated among measurements). The QCD-related biases on A 0,b fb depend strongly on the experimental selection procedure and are related to: (i) hard gluon radiation, and (ii) smearing of the b-jet (thrust) axis due to b and (b →)c soft radiation and hadronization, and subsequent B and D hadron decay models. Whereas the first bias is theoretically well controlled through next-tonext-to-leading-order perturbative QCD (plus massive b-quark) corrections 2 , the uncertainties of the latter were estimated using Monte Carlo (MC) parton shower simulations 3 that have not been revisited in 20 years. At future high-luminosity e + e − machines, such as the FCC-ee with 10 5 times more data collected at the Z pole than at LEP 12 , statistical uncertainties will be totally negligible, and the latter QCD effects will dominate the systematics of the A 0,b fb measurement.
Simulation of the LEP b-quark forward-backward asymmetry measurements
The eight original LEP measurements of A 0,b fb have been implemented in a MC event simulation based on pythia 8.226 13 with seven different parton-shower and hadronization tunes, as well as based on two alternative (dipole antenna) shower approaches from pythia 8.210 combined with vincia 1.1 and 2.2 (with uncertainties given by 12 variations of the vincia parameter set) 14 . Ten million e + e − → Z(bb) events are thereby generated at √ s = 92.4 GeV with QED radiation on, and analysed as done in the original experiments. The whole MC setup effectively corresponds to nine different modelings of the underlying QCD effects (bottom-and charmquark gluon radiation and fragmentation functions, and B, D semileptonic decays). Tune-7 and vincia 2.2 include proton-proton data whereas all other models are based on LEP data alone. For all analyses, the b-jets are first reconstructed with the JADE algorithm from the list of final-state particles and the thrust axis of the event is computed as a proxy of the bb direction. Each original y cut and M jet jet selection criteria, and (transverse) momenta (p T ) p cuts on the final electron and muons, are applied. On the one hand, the lepton-based analyses determine the b-quark charge from that of the hardest charged lepton in the event, and then extract A obs,b fb by fitting the corresponding distribution of polar angles θ between the e − and the thrust axis, dN/d cos θ = 3/8 [1 + cos 2
θ + 8/3 A obs,b fb (1 − 2χ B ) cos θ], where χ B ≈ 0.12
is the B 0 B 0 effective mixing parameter. On the other, in the jet-charge-based analyses, b,bquarks are identified via their measured jet charge Q jet = p κ L Q/ p κ L (where p L is the longitudinal momentum of the final-state particles, with charge Q, with respect to the thrust axis, and the power κ varies between 0.4 and 0.6), and A obs,b fb is derived by fitting the distribution
Q F − Q B / Q b − Qb = 8/3 A obs,b fb (1 + C) cos θ/(1 + cos 2 θ), where Q F (Q B )
are the jet charges in the forward (backward) hemisphere, and the C factor is a ∼3.5% correction for missing higherorder QCD terms and for the difference between the thrust axis and the b-quark direction 1,3 .
Results and conclusions
Through the procedure describe above, we extract 9 different MC values of A obs,b fb for each one of the eight experimental setups, which we compare among themselves and against the experimental data in Fig. 1 and 2 for lepton-and jet-charge analyses. The central A obs,b fb values plotted differ slightly from the A 0,b fb values quoted in Table 1, since the latter are obtained correcting for radiative effects, γ exchange, Z-γ interference, and shifted to the pole m Z = 91.187 GeV mass. The first (leftmost) MC point corresponds to the pythia 8 tune-1 result obtained with the 1990 jetset parameter set, very similar to the one used to obtain the original LEP QCD uncertainties 3 . The red band around the MC points is the standard deviation of the predictions, which we take as indicative of the associated QCD systematic uncertainty for each measurement. It amounts to about 1% (0.4%) for the lepton (jet) charge-based measurements, and is found to be overall slightly smaller but still fully consistent with the original QCD uncertainties (last column of Table 1). Using the updated QCD systematics, we obtain 15 a new weighted-average b-quark forward-backward asymmetry, A 0,b fb = 0.0996 ± 0.0016, very similar to the current one. Figure 2 -b-quark forward-backward asymmetry extracted from jet-charge analyses of e + e − → bb simulations based on seven pythia 8 and two pythia 8+vincia tunes (squares with red band), compared to the corresponding experimental results (rightmost data point, with QCD, in red, and uncorrelated, in blue, systematic uncertainty bands) measured by ALEPH (top left) 8 , DELPHI (top right) 9 , L3 (bottom left) 10 , and OPAL (bottom right) 11 .
Figure 1 -
1b-quark forward-backward asymmetry extracted from lepton-charge analyses of e + e − → bb simulations based on seven pythia 8 and two pythia 8+vincia tunes (squares with red band), compared to the corresponding experimental results (rightmost data point, with QCD, in red, and uncorrelated, in blue, systematic uncertainty bands) measured by ALEPH (top left) 4 , DELPHI (top right) 5 , L3 (bottom left)6 , and OPAL (bottom right) 7 .
Table 1 :
1LEP measurements of A 0,b fb and associated statistical, total systematic, and QCD-systematic uncertainties (with the newly-computed QCD systematics quoted in parentheses).Measurement
A 0,b
fb
uncertainties
stat.
total syst. QCD syst. (new)
ALEPH lepton (2002) 4
0.1003 ± 0.0038 ± 0.0017 4.1%
1.7%
0.6% (0.8%)
DELPHI lepton (2004-5) 5 0.1025 ± 0.0051 ± 0.0024 6.4%
2.4%
1.5% (1.3%)
L3 lepton (1999) 6
0.1001 ± 0.0060 ± 0.0035 6.9%
3.4%
1.8% (0.8%)
OPAL lepton (2003) 7
0.0977 ± 0.0038 ± 0.0018 4.3%
1.5%
1.1% (1.4%)
ALEPH jet-charge (2001) 8 0.1010 ± 0.0025 ± 0.0012 2.7%
1.1%
0.5% (0.5%)
DELPHI jet-charge (2005) 9 0.0978 ± 0.0030 ± 0.0015 3.3%
1.5%
0.5% (0.4%)
L3 jet-charge (1998) 10
0.0948 ± 0.0101 ± 0.0056 10.8%
5.9%
4.1% (0.4%)
OPAL jet-charge (2002) 11 0.0994 ± 0.0034 ± 0.0018 3.7%
1.8%
1.5% (0.3%)
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. D Enterria, C Yan, in preparationD. d'Enterria and C. Yan, in preparation
| {'fraction_non_alphanumeric': 0.07365917286887133, 'fraction_numerical': 0.06291721015882759, 'mean_word_length': 3.737913486005089, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The forward-backward asymmetry of b-quarks measured at LEP in e + e − collisions at the Z pole, A 0,b fb | exp = 0.0992 ± 0.0016, remains today the electroweak precision observable with the largest disagreement (2.8σ) with the Standard Model theoretical prediction, A 0,b fb | th = 0.1037 ± 0.0008. The dominant systematic uncertainties are due to QCD effects -b, c-quark showering and fragmentation, and B, D meson decay models -which have not been revisited in the last 20 years. We reassess the QCD uncertainties of the eight original LEP measurements of A 0,b fb , using modern parton shower simulations based on pythia 8 and pythia 8 + vincia with different tunes of soft and collinear radiation as well as of hadronization. Our analysis indicates QCD uncertainties, of order ±0.4% and ±1% for the jet-charge and lepton-charge based analyses, that are overall slightly smaller but still consistent with the original ones. Using the updated QCD systematic uncertainties, we obtain A 0,b fb = 0.0996 ± 0.0016.', 'arxivid': '1806.00141', 'author': ['David D'enterria \nCERN, EP Department\nCH-1211Geneva 23Switzerland\n', 'Cynthia Yan \nCERN, EP Department\nCH-1211Geneva 23Switzerland\n\nDepartment of Physics\nHarvey Mudd College\nCA91711ClaremontUSA\n'], 'authoraffiliation': ['CERN, EP Department\nCH-1211Geneva 23Switzerland', 'CERN, EP Department\nCH-1211Geneva 23Switzerland', 'Department of Physics\nHarvey Mudd College\nCA91711ClaremontUSA'], 'corpusid': 119359320, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4798, 'n_tokens_neox': 3845, 'n_words': 2184, 'pdfsha': '5c4be63e20630b2afe52d3da1f36bef65c66eb31', 'pdfurls': ['https://arxiv.org/pdf/1806.00141v2.pdf'], 'title': ['Forward-backward b-quark asymmetry at the Z pole: QCD uncertainties redux', 'Forward-backward b-quark asymmetry at the Z pole: QCD uncertainties redux'], 'venue': []} |
arxiv |
Practical Guidance for Bayesian Inference in Astronomy
10 February 2023
Gwendolyn M Eadie
David A. Dunlap Department of Astronomy & Astrophysics
University of Toronto
M5S 3H4TorontoCanada
Department of Statistical Sciences
University of Toronto
M5S 3G3TorontoCanada
2★Joshua S Speagle
David A. Dunlap Department of Astronomy & Astrophysics
University of Toronto
M5S 3H4TorontoCanada
Department of Statistical Sciences
University of Toronto
M5S 3G3TorontoCanada
University of Toronto
Dunlap Institute for Astronomy & Astrophysics
M5S 3H4TorontoCanada
Jessi Cisewski-Kehe
Department of Statistics
University of Wisconsin-Madison
53706MadisonWIUSA
Daniel Foreman-Mackey
Center for Computational Astrophysics
Flatiron Institute
160 5th Ave10010New YorkNYUSA
Daniela Huppenkothen
SRON Netherlands Institute for Space Research
Niels Bohrlaan 42333 CALeidenNetherlands
David E Jones
Department of Statistics
Texas A&M University
77843College StationTXUSA
Aaron Springford
Cytel, TorontoOntarioCanada
Hyungsuk Tak
Department of Statistics
Pennsylvania State University
16802University ParkPAUSA
Department of Astronomy & Astrophysics
Pennsylvania State University
16802University ParkPAUSA
Institute for Computational and Data Sciences
Pennsylvania State University
16802University ParkPAUSA
Practical Guidance for Bayesian Inference in Astronomy
RASTI 000
000010 February 2023Preprint 10 February 2023 Compiled using rasti L A T E X style file v3.0astrostatistics -computational methods -parallax
In the last two decades, Bayesian inference has become commonplace in astronomy. At the same time, the choice of algorithms, terminology, notation, and interpretation of Bayesian inference varies from one sub-field of astronomy to the next, which can lead to confusion to both those learning and those familiar with Bayesian statistics. Moreover, the choice varies between the astronomy and statistics literature, too. In this paper, our goal is two-fold: (1) provide a reference that consolidates and clarifies terminology and notation across disciplines, and (2) outline practical guidance for Bayesian inference in astronomy. Highlighting both the astronomy and statistics literature, we cover topics such as notation, specification of the likelihood and prior distributions, inference using the posterior distribution, and posterior predictive checking. It is not our intention to introduce the entire field of Bayesian data analysis -rather, we present a series of useful practices for astronomers who already have an understanding of the Bayesian "nuts and bolts" and wish to increase their expertise and extend their knowledge. Moreover, as the field of astrostatistics and astroinformatics continues to grow, we hope this paper will serve as both a helpful reference and as a jumping off point for deeper dives into the statistics and astrostatistics literature.
INTRODUCTION
Over the past two decades, Bayesian inference has become increasingly popular in astronomy. On NASA's Astrophysics Data System (ADS), a search using "keyword:statistical" and "abs:bayesian" yields 2377 refereed papers, and shows exponential growth since the year 2000, with over 237 papers in 2021.
Bayesian analyses have become popular in astronomy due to several key advantages over traditional methods. First, an estimate of the posterior distribution of model parameters provides a more complete picture of parameter uncertainty, joint parameter uncertainty, and parameter relationships given the model, data, and prior assumptions than traditional methods. Second, the interpretation of Bayesian probability intervals is often closer to what scientists desire, and is an appealing alternative to point estimates with confidence intervals which often rely on the sampling distribution of the estimator. Third, Bayesian analysis easily allows for marginalization over nuisance parameters, incorporation of measurement uncertainties through measurement error models, and inclusion of incomplete ★ E-mail: [email protected] data such as missing and censored data. Fourth, astronomers often have prior knowledge about allowable and realistic ranges of parameter values (e.g., through physical theories and previous observations/experiments) which can naturally be included in prior distributions and thereby improve the final inference.
Importantly, in addition to the aforementioned advantages of the Bayesian approach, efficient and increased computing power, along with easy-to-use or out-of-the-box algorithms, have brought Bayesian methodology to astronomers in convenient practical forms (e.g., emcee (Foreman-Mackey et al. 2013), Rstan (Stan Development Team 2020), PyStan (Riddell et al. 2017), PyMC3 (Salvatier et al. 2016), BUGS (Lunn et al. 2000), NIMBLE (de Valpine et al. 2017), and JAGS (Plummer et al. 2003)).
Interestingly, the surge in popularity of Bayesian statistics comes in spite of the fact that Bayesian methods are rarely taught in undergraduate astronomy and physics programs, and has only recently been introduced at a basic level in astronomy graduate courses (Eadie et al. 2019b,a). Some challenges faced by both new and seasoned users of Bayesian inference are the varied notation, terminology, interpretation, and choice of algorithms available in the astronomy and statistics literature.
Being well-versed in best practices and common pitfalls associated with the Bayesian framework is important if these methods are to be used to advance the field of astronomy. Here, users of Bayesian inference in astronomy face challenges too, since undergraduate and graduate program training is still catching up to the state-of-the-art Bayesian inference methods.
The goal of this paper is two-fold. Our first goal is to provide a "translation" between terminology and notation used for Bayesian inference across the fields of astronomy and statistics. Our second goal is to illustrate useful practices for the Bayesian inference process which we hope will be a valuable contribution to astronomers who are familiar with and/or use Bayesian statistics in research. To achieve these goals, we deal with the following topics in the main body of the paper: notation (Section 2.1), interpreting and determining the likelihood (Section 2.2), choosing and assessing prior distributions (Section 2.3), evaluating and making inference from the posterior distribution (Section 2.4), and performing posterior predictive checks (Section 2.5).
This work is not meant to be a comprehensive introduction to Bayesian inference, but rather an unveiling of Bayesian statistics as both an extensive topic and an active research area. We focus our efforts on identifying common mistakes and misunderstandings related to Bayesian inference, and use these as jumping off points for highlighting important topics for further study. Indeed, many valuable topics and subtopics arise which we do not cover, but we make a point of providing key references. For example, throughout the paper we touch on areas such as Bayesian design, posterior predictive checking, hierarchical modeling, and Bayesian computations (Craiu & Rosenthal 2014;Robert 2014), and also recommend books on Bayesian data analysis from statistics and astrostatistics (Gelman et al. 2013;Carlin & Louis 2008;Hilbe et al. 2017).
To help the narrative, we use a running example at the end of each section -inferring the distance to a star through its parallax measurement. The specific problem of distance estimation from parallax in the Bayesian context is explored closely in other studies (Bailer-Jones 2015; Astraatmadja & Bailer-Jones 2016a,b) and applied to the Gaia second data release (Gaia DR2) (Lindegren, L. et al. 2018;Bailer-Jones et al. 2018;Schönrich et al. 2019), and we refer the reader to these papers for a deep exploration on this topic. Here, we employ this example because of its generality, because it provides some interesting challenges and potential pitfalls, and because it provides a nice framework to illustrate sound practices in Bayesian analysis. Along the way, we also identify how our advice applies to other areas in astronomy.
SPECIFYING A BAYESIAN MODEL
Notation & Bayes Theorem
A number of different notation practices for Bayesian inference are used in both the astronomy and statistics literature. This section is meant to clarify some of these differences, while also providing a "translation" so that astronomers can more easily follow statistics papers (e.g., recognize notation for random variables, probability distribution functions, etc.).
We use (vectorized form) to represent the observed data of a random variable , and to represent the parameter(s) of interest. The posterior distribution is defined by Bayes' theorem as
( | ) = ( | ) ( ) ( )(1)
where ( | ) is the sampling distribution for given (Section 2.2), ( ) is the prior density (Section 2.3), and ( ) is the prior predictive density (Schervish 1995). With the data in hand, ( | ) is often viewed as a function of called the likelihood function (which is not a probability density), and ( ) is a normalizing constant that does not depend on (which is often referred to in astronomy as the model evidence). In the probability and Bayesian computation literature, the posterior probability distribution of interest is usually denoted and referred to as the target distribution with the notation . There are also differences in notation for the likelihood across disciplines, which we discuss in Section 2.2.
For smooth translation between sub-disciplines of astronomy and statistics, it is beneficial to use explicit statements about model choices and parameter definitions. For example, a list of all model parameters, notation, and their associated prior probability distributions in the form of a table is very useful to the reader. Moreover, we stress the importance of fully specifying any Bayesian model in papers to increase reproducibility (e.g., via a detailed appendix, open code). In this spirit, we provide the full Bayesian model for our running example in Table 1, explicitly define notation next, and provide our open source code 1 .
Parallax Example: Defining Notation
For the running example in this paper, we infer the distance to a star from a parallax measurement. The true but unknown distance in kiloparsecs (kpc) is related to the true but unknown parallax in milliarcseconds (mas) through
[kpc] = 1 [mas] .(2)
Our data is a measurement of the parallax, and has some fixed uncertainty that we treat as known. Thus, in the Bayesian framework, we wish to infer the parameter given the data , and we seek to find the posterior distribution,
( | ) ∝ ( | ) ( ).(3)
In Section 2.4.2, we extend this example to infer the distance to a star cluster from the parallax measurements of many stars within the cluster. The true but unknown distance to the cluster, cluster , is related to the true but unknown parallax of the cluster through Equation 2, too. In this case, we express the corresponding posterior density as follows:
( cluster | ) ∝ ( | cluster ) ( cluster ),(4)
where represents a vector of parallax measurements of its stars. Table 1 summarizes this model specification.
Likelihood function
Differences in notation and language between statistics and astronomy can lead to confusions regarding the likelihood. In Bayesian statistics, the capital letters and Θ often denote random variables. For example, both Gelman et al. (2013) in their applied statistics text and Schervish (1995) in his statistics theory text first write down a joint probability density (Θ, ), and then specify the likelihood function as ( = | ) (sometimes written L ( ) elsewhere), where is the fixed, observed value of (i.e., the data), and is the argument of the likelihood function. That is, the likelihood is a function of the parameters , given the data (Gelman et al. 2013;Schervish 1995 ; Table 1. Bayesian models for inferring (1) a star's distance parameter from its parallax measurement , assuming a Gaussian distribution for its true parallax (top panel), and (2) a star cluster's distance parameter cluster from the parallax measurement of stars (bottom panel).
Inferring a single star's distance parameter
Sampling density / likelihood (parallax)
( | ) = ( | 1/ , 2 ), where = 1/ Prior (distance) ( ) ∝ 2 − / if min < < max with a constant 0 otherwise Posterior on distance ( | , ) ∝ 2 exp − − ( −1/ ) 2 2 2
Inferring a star cluster's distance parameter cluster
Sampling density / likelihood ( parallax measurements)
( | cluster ) = =1 ( | 1/ cluster , 2 )
Prior (on parallax of cluster)
( cluster ) = ( cluster | , )
Posterior on distance to cluster ( cluster | ) ∝ ( cluster ) exp − ( −1/ cluster ) 2 2 2 2 In both the upper and lower box, and : = 1, 2, . . . , are assumed to be known. Throughout ( | , ) denotes the Gaussian density function of with mean and variance . Carlin & Louis 2008;Berger & Wolpert 1988;Casella & Berger 2002). However, in astronomy, it is not unusual to see phrases such as "the likelihood function of the data given the model parameters", which might be misconstrued as treating the likelihood function as a function of data. Finally, we note that the likelihood function of is not a probability density function of .
In Table 2, we summarize common notation for the likelihood found in both the statistics and astronomy literature, which ranges from being very explicit (e.g., ( | )) to quite simplified (e.g., ( | )). We note that a subtlety sometimes missed in astronomy is the difference between and . In statistics, and are often used for probability density functions (pdf) of continuous random variables, and or are used to denote probabilities of discrete events (probability mass functions (pmf) are an exception, and are often denoted using , , or ). A capital is usually reserved for the cumulative distribution function (cdf).
Determining what the likelihood function should be in a given astronomy problem can be challenging, and care must be taken to choose an appropriate sampling distribution. The likelihood is often taken to be a product of independent and identically distributed (i.i.d.) Gaussian random variables with known variance. While this choice is sometimes plausible, there are also many cases in which it is inappropriate, and has a material effect on inference. For instance, when describing the brightness of a high energy source, a discrete distribution such as the Poisson distribution is usually more appropriate than a Gaussian. In other cases, uncertainty in the variance of the data might lead us to use a likelihood function based on the -distribution or another non-Gaussian parametric family. A further consideration is whether the data being modeled are collected as a function of space or time, in which case the assumption of exchangeabilitythat data can be reordered without affecting the likelihood -is generally unwarranted. In these cases, an expanded model that includes correlation among observations should be considered.
Best practice includes all non-negligible contributors to the measurement process in the likelihood function. For example, it is important to account for substantial truncation and censoring issues when present, because these can strongly influence parameter inference in some cases (Rubin 1976;Eadie et al. 2021). Other common issues to check for and address are measurement uncertainty, correlated errors, measurement bias, sampling bias, and missing data. There are a number of valuable references in the statistics literature on these topics (Rubin 1976;Little & Rubin 2019). We recommend writing down enough mathematical details to uniquely determine the likelihood by defining (algebraically) not only the physical process of interest but also the sampling/measurement process that generated the data.
Parallax Example: the likelihood function
The Gaia spacecraft has measured parallaxes for over a billion stars (Gaia et al. 2018;Collaboration et al. 2018). These parallaxes have been shown empirically, through simulations (Holl et al. 2012;Lindegren et al. 2012), to follow a normal distribution with mean equal to the true underlying parallax so that
( | ) = 1 √ 2 2 exp − ( − ) 2 2 2 or equivalently,(5)| , ∼ ( , 2 )(6)
where is the measured parallax and is the associated (assumed known) measurement uncertainty (Hogg 2018). The parameter of interest is the distance , so we rewrite Equation 6 as
( | ) = 1 √ 2 2 exp − ( − 1/ ) 2 2 2 or equivalently,(7)| , ∼ (1/ , 2 ).(8)
We note that a similar Gaussian model assumption is widely applicable to various sub-fields in observational astronomy such as detecting exoplanets by RV (Danby 1988;Mayor et al. 2011;Pepe et al. 2011;Fischer et al. 2013;Butler et al. 2017) or by transit (Konacki et al. 2003;Alonso et al. 2004;Dragomir et al. 2019), inferring the true brightness of a source (Tak et al. 2017), or estimating the Hubble constant (Hubble 1929). This is because the statistical details are analogous; the observation is measured with Gaussian measurement error, estimated measurement error uncertainty is treated as a known constant, and the mean model can be written as a deterministic function of other parameters, e.g., = 1/ in Equation 6. On the other hand, as mentioned already, in each new setting it is important to carefully consider which model is most appropriate;
[t!]
Notation Description Context
( | ) distribution function of the random variable , statistics but viewed as a function of , with fixed ( | ) format used in this paper (common) statistics and astronomy L ( ; ) or L ( ) explicit notation for the likelihood with specific statistics topics, e.g., argument maximum likelihood estimation ( | ) represents the data astronomy ( | ) represents the model assumption, astronomy, model selection implicitly suggests parameters
( | , )
represents a particular proposed model astronomy, model selection
Gaussian-based models are (i) sometimes misused, (ii) overused, or (iii) sometimes inappropriate.
Prior Distributions
The prior probability distribution, or the prior, captures our initial knowledge about the model parameters before we have seen the data. Priors may assign higher probability (or density) to some values of the model parameters over others. Priors are often categorized into two classes: informative and non-informative. The former type summarizes knowledge gained from previous studies, theoretical predictions, and/or scientific intuition. The latter type attempts to include as little information as possible about the model parameters. Informative priors can be conjugate (Diaconis & Ylvisaker 1979), mixtures of conjugate priors (Dalal & Hall 1983), scientifically motivated (Tak et al. 2018; Lemoine 2019), based on previous data, or in the case of empirical Bayes, based on the data at hand (often called data-driven priors) (Carlin & Louis 2000;Maritz 2018). Non-informative priors can be improper or "flat", weakly-informative, Jeffrey's priors (Tuyl et al. 2008), or other reference distributions. Conjugate priors are sometimes defined to be non-informative. One popular choice for a non-informative prior is an improper prior -a prior that is not a probability distribution and in particular does not integrate to one. Good introductions to improper priors are available in the statistics literature (Gelman et al. 2013. An example of an improper prior is a flat prior on an unbounded range, e.g., Unif(0, ∞) or Unif(−∞, ∞). When an improper prior has been adopted, it is imperative to check whether the resulting posterior is a proper probability distribution before making any inference. Without posterior propriety the analysis has no probability interpretation. Empirical checks may not be sufficient; posterior samples may not reveal any evidence of posterior impropriety, forming a seemingly reasonable distribution even when the posterior is actually improper (Hobert & Casella 1996;Tak et al. 2018).
Research on quantifying prior impact is active (e.g., effective prior sample size Clarke 1996; Reimherr et al. 2014;Jones et al. 2020) as is the discussion on choosing a prior in the context of the likelihood (Reimherr et al. 2014;Gelman et al. 2017;Jones et al. 2020).
In astronomy, there is a tendency for scientists to adopt noninformative prior distributions, perhaps because informative priors are perceived as too subjective or because there is a lack of easily quantifiable information about the parameters in question. However, all priors provide some information about the likely values of the model parameter(s), even a "flat" prior. Notably, a flat prior is non-flat after a transformation. For instance, in our example (Section 2.3.1) a "non-informative" uniform prior distribution on the parallax of a star is actually quite informative in terms of distance (third panel, Figure 1). Thus, we recommend carefully considering what direct or indirect information is available about the value of a parameter before resorting to default or non-informative priors; in astronomy, we usually have at least a little information about the range of allowed or physically reasonable values. Even when a non-informative prior does seem appropriate, checking that the chosen distribution is consistent with known physical constraints is essential.
Complete descriptions and mathematical forms of prior distributions, including the values of hyperparameters defining these distributions, help promote reproducibility and open science. Unfortunately, a recent meta-analysis of the astronomical literature showed that prior definitions are often incomplete or unstated (Tak et al. 2018), making it difficult for others to interpret results.
To summarize, good practices in the context of priors are: (1) choosing informative priors when existing knowledge is available, (2) choosing priors with caution if there is no prior knowledge, (3) testing the influence of alternative priors (see the discussion of sensitivity analyses in Section 2.4), and (4) explicitly specifying the chosen prior distributions for clarity and reproducibility.
Parallax Example: choosing a prior
A naive choice of prior on the true parallax is ( ) ∝ constant, an improper prior that assigns equal density to all values of from (0, +∞). A straightforward way to be more informative and proper is to instead define a truncated uniform prior, where is uniformly distributed between = ( min , max ) so that
( ) ∝ constant min < < max 0 otherwise ,(9)
or equivalently,
∼ Unif( min , max )(10)
Here, min and max are hyperparameters set by the scientist (e.g., using some physically-motivated cutoff for min and the minimum realistic distance to the star for max ). Thus the prior in Equation 9 can be regarded as weakly-informative because some physical knowledge is reflected in the bounds. Similarly, we could instead define a uniform prior on distance:
( ) ∝ constant min < < max 0 otherwise ,(11)
or equivalently,
∼ Unif( min , max )(12)
where min = 1/ max and max = 1/ min . Like Equation 9, this prior can also be regarded as weakly-informative. However, both display drastically different behavior as a function of (see Figure 1), which highlights how the interpretation of non-informative (or weakly-informative) priors may change depending on the choice of parameterization. While the prior in Equation 11 may appear non-informative (in a sense that it is uniform), it actually encodes a strong assumption about the number density of stars as a function of distance. The prior implies that we are just as likely to observe stars at large distances as we are at smaller distances. However, as we look out into space, the area of the solid angle defined by the distance increases, and this in turn implies that the stellar number density is decreasing with distance. Thus, Equation 11, which says that all distances are equally likely, implies that there are fewer stars per volume at large distances than stars per volume at small distances.
Bailer-Jones et al. (2018) introduced a better prior for the parallax inference problem, which we outline briefly and reproduce here. The physical volume d probed by an infinitesimal solid angle on the sky dΩ at a given distance scales as the size of a shell so that dΩ ∝ 2 . This means that, assuming a constant stellar number density everywhere, a prior behaving as ( ) ∝ 2 is more appropriate. However, we can go one step further -we know that our Sun sits in the disk of the Galaxy, and that the actual stellar density as we go radially outward in the disk should decrease as a function of distance. Assuming we are looking outward, and that the stellar density decreases exponentially with a length scale (so that for a given distance we have ( | ) ∝ − / ) the prior on distance is
( ) ∝ 2 − / min < < max 0 otherwise ,(13)
which is the density function of a truncated Gamma(3, ) distribution. The scientist using Equation 13 would need to choose and define the three hyperparameters min , max , and . Equation 13 is the exponentially decreasing space density prior of previously presented in Bailer- Jones et al. (2018). Figure 1 illustrates all three priors discussed here.
Posterior distributions
The posterior distribution of Equation 1 is the focus of Bayesian inference. Once the prior distribution(s) and the likelihood function are specified, the posterior distribution is uniquely determined. Often, the denominator quantity ( ) is not available analytically. In this case, ( ) can be estimated by numerical integration. Samples drawn from ( | ) can be used to estimate properties of the posterior. A popular approach for obtaining samples is to construct a Markov Chain whose stationary distribution is designed to match the target distribution ( | ), which is known as Markov chain Monte Carlo (MCMC). The canonical example is the Metropolis-Hasting algorithm (Metropolis & Ulam 1949;Metropolis et al. 1953;Hastings 1970;Gelman et al. 2013), but there are many variations, some of which are designed to address specific challenges such as sampling high-dimensional or multi-modal target distributions; see Brooks et al. (2011) for details. The posterior distribution enables inference of model parameters or of quantities that can be derived from model parameters. For example, the posterior mean ( | ) is a point summary for . The posterior distribution can also be used to define credible intervals for parameters that provide a range of probable values. We stress that credible intervals are not confidence intervals; a 95% credible interval suggests that there is a 95% probability that the parameter lies within the specified range given our prior beliefs, the model, and the data, whereas a 95% confidence interval suggests that if similar intervals are properly constructed for multiple datasets then 95% of them are expected to contain the true (fixed) parameter value.
Depending on the characteristics of the posterior distribution, we emphasize that point summaries and intervals may not provide a complete description of uncertainty (e.g., for multi-modal posteriors). Here, visualizations of the posterior can provide a more comprehensive picture (see Figure 2). Recommendations and open source software packages containing visualization tools for Bayesian analysis can be found in the statistics literature (Gabry et al. 2019;Gabry & Mahr 2019;Kumar et al. 2019;Vehtari et al. 2020). Projections of the joint posterior distribution into two parameter dimensionsalso colloquially referred to as a corner plot in astronomy literature -is the most common visualization tool. Drawing credible regions or contours on these types of visualizations are also helpful, although defaulting to a "1-sigma" credible region is not always appropriate (i.e., when the distributions are non-Gaussian).
The posterior distribution also provides a useful way to obtain estimates and credible intervals for other quantities of physical interest. For example, if a model has parameters = ( , , ) and there is some physical quantity described by e.g., = 2 / , then for every sample of , a sample of can be calculated. Thus, a distribution of the physically interesting quantity is obtained, which can also be used to obtain point estimates and credible intervals. In other words, in the Bayesian paradigm, uncertainties in each model parameter are naturally propagated to uncertainties in derived physical quantities in a coherent way.
Posterior distributions can be complicated in shape (e.g., asymmetric, with multiple modes). This can create computational challenges in cases where the posterior cannot be derived in closed form. Fortunately, many algorithms have been developed for approximating the posterior distribution. Different algorithms perform well for different characteristics of the posterior, and therefore prior knowledge of what we might expect the posterior to look like, as well preliminary explorations, are often valuable in practice. In addition to the MCMC sampling algorithms already mentioned, a number of other techniques have been developed. For example, integrated nested Laplace approximations (Rue et al. 2009) approximates posterior distributions, variational Bayes methods (Jordan et al. 1999;Blei et al. 2003;Hoffman et al. 2013), and approximate Bayesian computation (Beaumont et al. 2009;Marin et al. 2012;Weyant et al. 2013;Akeret et al. 2015;Ishida et al. 2015;Beaumont 2019) are possible alternatives when the likelihood functions are too complicated or expensive to be evaluated.
Parallax Example: inferring the distance to a star
In our running example, we are interested in inferring the parameter for the distance = 1/ given the measured parallax and its associated measurement uncertainty (which we treat as known).
Parallax [mas]
Distance From Bayes' theorem, the posterior is
( | ) ∝ ( | ) ( ).(14)
For the three priors discussed previously, this corresponds to the following posteriors:
Equation 9 ⇒ ( | ) ∝ 1 2 exp − ( − 1/ ) 2 2 2(15)Equation 11 ⇒ ( | ) ∝ exp − ( − 1/ ) 2 2 2(16)Equation 13 ⇒ ( | ) ∝ 2 exp − − ( − 1/ ) 2 2 2 ,(17)
for min < < max (and 0 otherwise). While none of these have analytic solutions for point estimates or credible intervals, they can be computed using computational techniques. Approximations to these three posterior distributions are show in Figures 1 and 2. In this illustration, though each resulting posterior distribution is right-skewed, the shape is notably different for each considered prior distribution.
Extended Example: inferring the distance to a cluster of stars
We now extend our example to infer the distance to a cluster of stars, based on the collection of parallax measurements of each individual star. Assuming that there are stars located at approximately the same distance cluster and that the measured parallaxes = { 1 , 2 , . . . , } to each star are independent given cluster , our combined likelihood is the product of the individual likelihoods
( 1 , 2 , . . . , |1/ cluster ) = =1 ( |1/ cluster ),(18)
where the individual likelihoods are defined following Equation 6. We assume that the measurement uncertainties are known constants. Our posterior is
( cluster | ) ∝ ( cluster ) =1 ( |1/ cluster ).(19)
The product of independent Gaussian densities with known variances is a Gaussian density with precision parameter −2 = =1 −2 and mean parameter 1/ cluster . The observed parallaxes can be combined to obtain an effective parallax eff = 2 =1 / 2 , and thus,
( cluster | ) ∝ ( cluster ) exp − ( eff − 1/ cluster ) 2 2 2 .(20)
The estimated posterior distribution over cluster = 1/ cluster for a nearby cluster of stars (M67) using data from Gaia DR2, and using a conjugate Gaussian prior for cluster , is shown in Figure 3. The top panel of Figure 3 shows the individual parallax measurements of stars in M67, sorted by their signal-to-noise values. The bottom panel shows the assumed prior distribution (narrow left panel) and the (estimated) posterior distribution for the cluster's parallax, as more and better data are added to the analysis.
Note: A prior over ( cluster ), which governs the distribution of clusters of stars, is not the same as a prior over ( ), which governs the distribution of individual stars. While it might be reasonable to assume these are similar, they are not interchangeable quantities and may indeed follow different distributions. Realizing the differences in priors between various scenarios such as these is key to building good models and subsequently making good inferences.
Posterior Predictive Checking
After obtaining the posterior distribution, it is recommended to assess the adequacy of the model using posterior predictive checks (Gelman et al. 1996), which compare the empirical distribution of the data to the distribution described by the Bayesian model. The posterior predictive distribution is the posterior distribution of hypothetical future data (˜) under the chosen model and given the previously collected data:
(˜| ) = ∫ (˜, | ) .(21)
We find that posterior predictive checking is underused in astronomy but can be very useful. Posterior predictive checks not only assess the the adequacy of the model but also simultaneously check any approximations to the posterior. They are a valuable tool for diagnosing issues with computational sampling methods. In most cases, the posterior predictive distribution is not available in closed form. However, it is possible to generate simulated observations from the posterior predictive distribution and compare these to the original data. For example, for each of the posterior samples of , draw a random sample of˜and compare these samples to the real data. Significant or systematic differences between the distributions of the real and simulated data may suggest a problem with the model.
It is good practice to perform quantitative and/or graphical comparison between the simulated data and the real data. For graphical comparison, an overlaid density plot could be used (top panel of Figure 4), but in general this is a poor choice because it is difficult to judge differences between the overlaid densities visually. For graphical comparison, we recommend instead using a quantile-quantile (Q-Q) plot to characterize any differences (bottom panel of Figure 4).
To construct a Q-Q plot, it suffices to compute the quantiles from the original data and from the posterior predictive distribution (or from data simulated from the posterior predictive distribution), and to plot the pairs one against the other (bottom panel, Figure 4). If the empirical distribution and the posterior predictive distribution match, then their quantiles should lie along a 1:1 line. Functions to display Q-Q plots are common in statistical computing software languages.
In our parallax example, parallax values in the tails of the simulated and real data distributions show some disagreement (Figure 4). The Q-Q plot shows this more explicitly that the density plot, as the quantiles do not follow the 1:1 line in the tails of the distribution (below ∼ 10th percentile and above ∼ 70th percentile). Differences in either end of a Q-Q plot can be due to chance, but strong deviations from the 1:1 line are usually worth investigating.
There are also other valuable approaches for checking a Bayesian model and the quality of approximations to the posterior distribution. For example, one may set a portion of the data aside, or obtain additional data, and then compare the resulting inference to that from the original data. One may also use multiple methods for approximating the posterior distribution, and compare results. This can help diagnose situations where one or more sampling algorithms did not explore the full parameter space, and consequently fail to include high-probability regions in the posterior samples.
In addition to model and posterior checking, it is important to consider the influence of the prior distribution(s). The rightmost panel of Figure 1 shows three posteriors: each one used one of the three priors discussed in Section 2.3.1. While the posteriors are vaguely similar in shape (e.g., right-skewed), the inferred summary statistics can be quite different (Figure 2).
More generally, investigating how the analysis compares for several different prior distributions is an important technique, often referred to as a sensitivity analysis. A sensitivity analysis directly assesses the impact of the prior distribution on the posterior, and for this reason we recommend them -particularly when the information available to construct a prior is limited. On the other hand, sensitivity analyses can be somewhat ad hoc (e.g., which priors are tried, how they are compared) making it difficult to summarize and compare the prior impact across multiple analyses, instruments, and models. More principled approaches may therefore be preferred or complementary in some scenarios. One such method is to quantify the effective prior sample size (EPSS), i.e., the number of data points that the information provided by the prior distribution corresponds to.
The EPSS is simple to compute for conjugate models. For example, if we have the data ∼ ( , 2 ), for = 1, . . . , , and the conjugate prior distribution ∼ ( 0 , 2 / ), with known 2 , then the posterior distribution of has variance 2 /( + ). Thus, the effect of the prior is equivalent to that of samples, and we say that the EPSS is . The statistics literature includes proposals of several methods for extending this idea beyond conjugate models (Clarke 1996;Morita et al. 2008;Reimherr et al. 2014;Jones et al. 2020) and how to additionally account for location discrepancies, e.g., the value of |¯− 0 | in the preceding example. Clarke (1996) and Morita et al. (2008) use EPSS to quantify the information in the prior in isolation from the data, while Reimherr et al. (2014) Figure 1 illustrating how to infer the distance to an open cluster (M67) based on parallax measurements of many stars. Top: Parallax measurements (gray) for likely cluster members (based on proper motions), sorted by their observed signal-to-nose ratio obs / . Bottom: The joint likelihood (gray) and posterior (blue) for the cluster parallax cluster = 1/ cluster as more and more stars are added to our analysis. The (Gaussian) prior distribution on the cluster's parallax is illustrated in the narrow left panel. When there is only a small number of stars, the location of the prior has a substantial impact on the posterior. However, as more stars are added, the information from the data dominates.
performed. The latter is typically more relevant in science and more closely coincides with sensitivity analyses.
Good practices outlined in this section can be summarized as (1) using multiple ways to summarize the posterior inference, (2) quantitatively and graphically checking the posterior distribution (e.g., using posterior predictive checks, Q-Q plots), and (3) providing evidence that diagnostic checks were completed.
Extended Example: posterior predictive checks
We investigate the validity of our model for the distance to a cluster of stars by computing the posterior predictive distribution for the observed stellar parallaxes. While in this case the posterior predictive can be written in closed form (since it is a Gaussian distribution), we also approximate it by simulating values of cluster from the posterior and then subsequently simulating values for the predicted parallax measurements pred, given cluster .
In Figure 4, we compare both the distribution and quantiles estimated for the simulated dataset and the observed dataset via a density and Q-Q plot respectively. While there are differences, especially in the tails of the distribution, overall the cluster model reproduces most of the observed properties of the data. It would be worth investigating whether these differences persist under different models -for example, a model in which the distance to each star is not assumed to be identical, or a model in which measurement uncertainty is not assumed to be known exactly.
Conclusion
We hope that this article has identified, clarified, and illuminated fundamental Bayesian inference notation and techniques from the statistics literature, and in particular, has made a case for fully specifying the model, posterior predictive checking, and the use of underused aids such as the Q-Q plot. In summary, we highlight sound practices for conducting Bayesian inference in astronomy as follows:
• Be explicit about notation, and use appropriate terminology for the interpretation of concepts such as the likelihood and credible intervals, which will help interdisciplinary collaboration and reproducibility.
• Describe the likelihood as a function of the parameters, given the data.
• Use informative priors whenever possible and justified. Carefully consider what direct or indirect information is available about the parameters.
• Use non-informative priors carefully, and assess their properties under parameter transformations.
• Test the sensitivity of the posterior distribution to different prior distributions.
• Fully specify the Bayesian model in terms of the likelihood, prior, and posterior, and provide open-source code whenever possible.
• Perform posterior predictive checks of the model, using visualizations such as Q-Q plots where appropriate.
• Strive to include all non-negligible contributors to the measurement process.
We hope that there is a continued growth of interdisciplinary collaborations between astronomers and statisticians in the future. Data Figure 3.
Top: The distribution of parallax measurements from the data (gray) and simulated values from the posterior predictive (light blue). The posterior mean is indicated using the dashed dark blue line. The distributions appear relatively consistent with each other by eye, but a quantile-quantile (Q-Q) plot is more informative and suggests otherwise. Bottom: The Q-Q plot of the quantiles from the posterior predictive simulated parallax data ( -axis) and of the observed parallaxes ( -axis). If the real and simulated data followed the same distribution, then the quantiles would lie on the one-to-one line. However, strong discrepancies are apparent below ∼ 10th percentile and above ∼ 70th percentile.
from cutting-edge telescopes such as the Vera Rubin Observatory, the James Webb Space Telescope, and many others, have the potential to drive the field of astronomy, but this new information is best understood in the context of existing knowledge and careful statistical inference. Bayesian inference provides a framework in which this type of analysis and discovery can occur. Areas of astronomy where prior information and non-Gaussian based likelihoods are common can especially benefit from Bayesian methods, for example X-ray and gamma-ray astronomy. Bayesian inference is a broad topic, and many subtopics were not covered in this article. Ultimately, we hope that this article not only serves as a useful resource, but will also be the inception for a series of more specific papers on Bayesian methods and techniques in astronomy and physics.
Figure 1 .Figure 2 .
12Bayesian inference of the distance = 1/ to a star based on the measured parallax . Far left: The likelihood for parallax is normal with variance assumed known. Center left: A transformation of parameters from to gives a non-normal PDF. Note that a non-negativity constraint was applied to the distribution of . Center right: We highlight three possible priors ( ) over the distance: uniform in parallax = 1/ (blue), uniform in distance (red), and a physically-motivated prior (Bailer-Jones et al. 2018) (orange). Far right: The posteriors that correspond to each of the three priors. The three posterior distributions corresponding to each prior distribution shown inFigure 1: uniform in parallax prior (blue curve), uniform in distance prior (red curve), and physically-motivated prior (orange curve). Also shown are one summary statistic for each posterior: the mode (blue dotted line), the median (red dotted-dashed line), and the 90% credible interval (orange dashed lines and shaded region).
Figure 3 .
3and Jones et al. (2020) concentrate on the impact of the prior on the specific analysis Extended Example for Open Cluster M67. An extension of the example shown in
Figure 4 .
4Quantile-Quantile (Q-Q) Plot. This figure demonstrates a way to perform posterior predictive checking for the model shown in
Table 2 .
2Likelihood notation found in different contexts.
[kpc] Likelihood
Observed
units
( )
Physical
units
(d = 1/ )
Prior
Uniform (parallax)
Uniform (distance)
Physically-motivated
Posterior
Distance [kpc]
Distance [kpc]
G.M.Eadie et al.
https://github.com/joshspeagle/nrp_astrobayes
RASTI 000, 1-10 (0000)
ACKNOWLEDGEMENTS GME acknowledges the support of a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC, RGPIN-2020-04554). JCK gratefully acknowledges support from NSF under Grant Numbers AST 2009528 and DMS 2038556. DH is supported by the Women In Science Excel (WISE) programme of the Netherlands Organisation for Scientific Research (NWO).DATA AVAILABILITYData used in the running example is provided with permission and courtesy of Phill Cargile (Center for Astrophysics | Harvard & Smithsonian).
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. T E X/L A T E X File, 000This paper has been typeset from a T E X/L A T E X file prepared by the author. RASTI 000, 1-10 (0000)
| {'fraction_non_alphanumeric': 0.053712892835379125, 'fraction_numerical': 0.03550100113171411, 'mean_word_length': 4.546156817303978, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 0, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In the last two decades, Bayesian inference has become commonplace in astronomy. At the same time, the choice of algorithms, terminology, notation, and interpretation of Bayesian inference varies from one sub-field of astronomy to the next, which can lead to confusion to both those learning and those familiar with Bayesian statistics. Moreover, the choice varies between the astronomy and statistics literature, too. In this paper, our goal is two-fold: (1) provide a reference that consolidates and clarifies terminology and notation across disciplines, and (2) outline practical guidance for Bayesian inference in astronomy. Highlighting both the astronomy and statistics literature, we cover topics such as notation, specification of the likelihood and prior distributions, inference using the posterior distribution, and posterior predictive checking. It is not our intention to introduce the entire field of Bayesian data analysis -rather, we present a series of useful practices for astronomers who already have an understanding of the Bayesian "nuts and bolts" and wish to increase their expertise and extend their knowledge. Moreover, as the field of astrostatistics and astroinformatics continues to grow, we hope this paper will serve as both a helpful reference and as a jumping off point for deeper dives into the statistics and astrostatistics literature.', 'arxivid': '2302.04703', 'author': ['Gwendolyn M Eadie \nDavid A. Dunlap Department of Astronomy & Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoCanada\n\nDepartment of Statistical Sciences\nUniversity of Toronto\nM5S 3G3TorontoCanada\n', '2★Joshua S Speagle \nDavid A. Dunlap Department of Astronomy & Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoCanada\n\nDepartment of Statistical Sciences\nUniversity of Toronto\nM5S 3G3TorontoCanada\n\nUniversity of Toronto\nDunlap Institute for Astronomy & Astrophysics\nM5S 3H4TorontoCanada\n', 'Jessi Cisewski-Kehe \nDepartment of Statistics\nUniversity of Wisconsin-Madison\n53706MadisonWIUSA\n', 'Daniel Foreman-Mackey \nCenter for Computational Astrophysics\nFlatiron Institute\n160 5th Ave10010New YorkNYUSA\n', 'Daniela Huppenkothen \nSRON Netherlands Institute for Space Research\nNiels Bohrlaan 42333 CALeidenNetherlands\n', 'David E Jones \nDepartment of Statistics\nTexas A&M University\n77843College StationTXUSA\n', 'Aaron Springford \nCytel, TorontoOntarioCanada\n', 'Hyungsuk Tak \nDepartment of Statistics\nPennsylvania State University\n16802University ParkPAUSA\n\nDepartment of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA\n\nInstitute for Computational and Data Sciences\nPennsylvania State University\n16802University ParkPAUSA\n'], 'authoraffiliation': ['David A. Dunlap Department of Astronomy & Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoCanada', 'Department of Statistical Sciences\nUniversity of Toronto\nM5S 3G3TorontoCanada', 'David A. Dunlap Department of Astronomy & Astrophysics\nUniversity of Toronto\nM5S 3H4TorontoCanada', 'Department of Statistical Sciences\nUniversity of Toronto\nM5S 3G3TorontoCanada', 'University of Toronto\nDunlap Institute for Astronomy & Astrophysics\nM5S 3H4TorontoCanada', 'Department of Statistics\nUniversity of Wisconsin-Madison\n53706MadisonWIUSA', 'Center for Computational Astrophysics\nFlatiron Institute\n160 5th Ave10010New YorkNYUSA', 'SRON Netherlands Institute for Space Research\nNiels Bohrlaan 42333 CALeidenNetherlands', 'Department of Statistics\nTexas A&M University\n77843College StationTXUSA', 'Cytel, TorontoOntarioCanada', 'Department of Statistics\nPennsylvania State University\n16802University ParkPAUSA', 'Department of Astronomy & Astrophysics\nPennsylvania State University\n16802University ParkPAUSA', 'Institute for Computational and Data Sciences\nPennsylvania State University\n16802University ParkPAUSA'], 'corpusid': 256697625, 'doi': None, 'github_urls': ['https://github.com/joshspeagle/nrp_astrobayes'], 'n_tokens_mistral': 16420, 'n_tokens_neox': 13734, 'n_words': 8568, 'pdfsha': 'b9eb345d1920491274bb1b2e617f41cb7197e4cd', 'pdfurls': ['https://export.arxiv.org/pdf/2302.04703v1.pdf'], 'title': ['Practical Guidance for Bayesian Inference in Astronomy', 'Practical Guidance for Bayesian Inference in Astronomy'], 'venue': ['RASTI 000']} |
arxiv |
Thermalization and dephasing in collisional reservoirs
Jorge Tabanera-Bravo
Departamento de Estructura de la Materia
Física Térmica y Electrónica and GISC
Universidad Complutense de Madrid
Pl. de las Ciencias 128040MadridSpain
Juan M R Parrondo
Departamento de Estructura de la Materia
Física Térmica y Electrónica and GISC
Universidad Complutense de Madrid
Pl. de las Ciencias 128040MadridSpain
Massimiliano Esposito
Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit
University of Luxembourg
L-1511LuxembourgG.D. Luxembourg
Felipe Barra
Departamento de Física
Facultad de Ciencias Físicas y Matemáticas
Universidad de Chile
837.0415SantiagoChile
Thermalization and dephasing in collisional reservoirs
(Dated: April 12, 2023)
We introduce a wide class of quantum maps that arise in collisional reservoirs and are able to thermalize a system if they operate in conjunction with an additional dephasing mechanism. These maps describe the effect of collisions and induce transitions between populations that obey detailed balance, but also create coherences that prevent the system from thermalizing. We combine these maps with a unitary evolution acting during random Poissonian times between collisions and causing dephasing. We find that, at a low collision rate, the nontrivial combination of these two effects causes thermalization in the system. This scenario is suitable for modeling collisional reservoirs at equilibrium. We justify this claim by identifying the conditions for such maps to arise within a scattering theory approach and provide a thorough characterization of the resulting thermalization process.
Collisional reservoirs are becoming an essential tool for the study of open quantum systems and quantum thermodynamics [1][2][3][4]. The term applies to situations in which a system interacts sequentially with the internal degrees of freedom of particles extracted from a reservoir. The effect of each interaction is described by a quantum map that can be obtained under some simplifying assumptions, like supposing that the particle and the system interact for a given time. However, this approach turns out to be thermodynamically inconsistent in certain situations of interest because switching on and off the interaction involves the performance of a work that, for example, prevents the system from thermalizing when the reservoir is a thermal bath at equilibrium [1,[4][5][6]. This drawback imposes some limitations when applying collisional models to fundamental problems in quantum thermodynamics, such as the thermalization of spatially extended systems [7] and of systems with noncommuting conserved quantities [8][9][10].
One way to restore thermodynamic consistency is to consider the particle's spatial degrees of freedom and analyze the collision as an autonomous event. In this case, there is no longer a need for an external agent to switch on and off the interaction [6]. In this context, using a scattering theory approach, two necessary conditions for thermalization have been found: i) the velocity of the particles must be distributed according to the effusion distribution at a given temperature and ii) the interaction must be time reversible [6,11,12]. Moreover, in the quantum case, the dispersion of the momentum of the incident particles must be small enough to cancel out the coherences among the eigenstates of the system's Hamiltonian [6]. For incident wave packets with a nonnegligible momentum dispersion, the collision can induce coherences that prevent the system from thermalizing.
An open question is whether the combination of these collisions with some dephasing mechanism can yield a repeated-interaction scheme that is thermodynamically consistent. This is the question that we address in this Letter. We first analyze the problem of thermalization in a generic repeated-interaction scheme given by a quantum map whose transition probabilities obey detailed balance but, at the same time, generate coherences that drive the system out of equilibrium. Then we apply this generic analysis to quantum maps derived within scattering theory. Consider a quantum system with Hamiltonian H S and eigenstates H S |j = e j |j . When it interacts with an auxiliary system for a given time or is bombarded by particles coming from a reservoir, the density matrix of the system changes as ρ → ρ = Sρ. The super-operator S can be written in tensorial form S jk j k in the eigenbasis of H S :
ρ j k = jk S jk j k ρ jk(1)
with ρ jk ≡ j|ρ|k . The term S jj j j is the transition probability from eigenstate |j to |j . In this Letter, we analyze quantum maps with transition probabilities that obey detailed balance:
e −βej S jj j j = e −βe j S j j jj(2)
with respect to an inverse temperature β. Such a map S can induce thermalization by itself. A relevant example is when the only nonzero entries of the tensor are the transition probabilities S jj j j and S jk jk , with |S jk jk | < 1 for j = k. Then coherences (the off-diagonal entries of the density matrix ρ in the eigenbasis of H S ) decay and populations (the diagonal entries of ρ in the eigenbasis of H S ) evolve as a Markov chain and thermalize. This is the case of systems bombarded by wave packets with a small momentum dispersion [6,12].
However, in some relevant situations, such as systems bombarded by broad packets in momentum representation coming from thermal baths at equilibrium, populations can couple to coherences through nonzero terms S jj j k with j = k . In these cases, even though the transition probabilities obey detailed balance, coherences prevent the system from thermalizing. Concatenation of the map S is not enough to thermalize the system and must be complemented by a dephasing mechanism. The most trivial one consists of intercalating a full dephasing superoperator D that kills all off-diagonal terms of the density matrix. In the eigenbasis of the Hamiltonian:
D jk j k = δ jj δ kk δ jk .(3)
It is straightforward to prove that the detailed balance condition (2) and the complete dephasing D are sufficient to thermalize the system for any initial condition ρ 0 :
lim n→∞ [DS] n ρ 0 = e −βH S Z ≡ ρ Therm ,(4)
Z = Tr[e −βH S ] being the partition function. However, to devise realistic scenarios, one has to consider more specific dephasing mechanisms. A candidate is the random phase added to the off-diagonal terms of the density matrix if the system is bombarded at random times and evolves under the Hamiltonian H S between collisions [13]. The density matrix after n collisions is
ρ n ≡ SU τn SU τn−1 . . . SU τ2 SU τ1 ρ 0(5)
where U τ ρ = e −iH S τ / ρe iH S τ / is the super-operator corresponding to the Hamiltonian unitary evolution and τ 1 , . . . , τ n are random variables. The density matrix ρ(t) at time t is given by the average
ρ(t) = ρ n(6)
taken over all possible values of n and τ k (k = 1, 2, . . . , n) such that t = k τ k . If the collisions are Poissonian events occurring at a rate Γ, then the probability of a collision in an interval [t, t + ∆t] is Γ∆t, independently of past events. Hence,
ρ(t + ∆t) [1 − Γ∆t] U ∆t ρ(t) + Γ∆t Sρ(t)(7)
yielding the master equation [1,13]
dρ(t) dt = − i [H S , ρ(t)] + Γ(S − I)ρ(t).(8)
The corresponding steady state verifies
Detailed balance for the transition probabilities (2) is not sufficient for thermalization, i.e., for having ρ ss = ρ Therm , even with Poissonian collisions. The reason is that, if S generates coherences that subsequently affect populations, then Sρ Therm = ρ Therm . However, the generation of coherences can be reduced if the collision rate Γ is very small. To see this, let us solve Eq. (9) perturbatively by inserting
ρ ss = ρ (0) + Γρ (1) + Γ 2 ρ (2) + . . .(10)
The first-order terms yield
− i [H S , ρ (0) ] = 0 − i [H S , ρ (1) ] + (S − I)ρ (0) = 0 − i [H S , ρ (2) ] + (S − I)ρ (1) = 0.(11)
Multiplying these equations by j| on the left and |k on the right, we get :
− i ∆ jk ρ (0) jk = 0 − i ∆ jk ρ (1) jk + j k S j k jk ρ (0) j k − ρ (0) jk = 0 − i ∆ jk ρ (2) jk + j k S j k jk ρ (1) j k − ρ (1) jk = 0(12)
where ∆ jk ≡ e j − e k . For simplicity, we assume that the eigenstates of H S are non-degenerate:
∆ jk = 0 ⇔ j = k.
In this case, the first equation in (12) implies that ρ (0) is diagonal in the eigenbasis of H S and the second one, particularized for k = j, determines the diagonal terms or populations. They fulfill the following equation:
j S j j jj ρ (0) j j = ρ (0) jj(13)
whose solution is ρ Therm if the transition probabilities verify the detailed balance condition (2). The offdiagonal terms are of order Γ with
ρ (1) jk = − i ∆ jk j S j j jk ρ (0) j j(14)
for j = k. The third equation in (12) for k = j determines the first-order correction to the diagonal terms:
j k S j k jj ρ (1) j k = ρ (1) jj(15)
which are no longer thermal. We see that the corrections to the thermal state are of order Γ /|∆ jk |. That is, the system thermalizes if the average time between collisions 1/Γ is much longer than the evolution time of the phases of the off-diagonal terms of the density matrix, which are /|∆ jk |. Now we investigate the conditions under which this type of map results from the interaction between a system and a particle or unit U extracted from a reservoir.
As in [6], we study the case of a system colliding with a one-dimensional quantum particle of mass m. The Hamiltonian of the global setup reads (16) wherep andx are the momentum and position operators of the particle, respectively, V is an operator acting on the system, and f (x) is a function with finite support, which is the scattering region where the system and the particle interact. Here, we assume for simplicity that the scatterer is symmetric under spatial inversion, f (x) = f (−x), and that the particle has no internal degrees of freedom. They can be incorporated in a straightforward manner, following [12]. We bombard the system with units prepared in a generic mix state ρ U . In this case, the resulting scattering map is [6,14]
H =p 2 2m + H S + V f (x)S jk j k = α=± ∞ p inf dp ρ U (p, π(p)) p π(p) s (α) j j (E p + e j ) × s (α) k k (E p − ∆ j j + e k ) *(17)
where ρ U (p, p ) = p|ρ U |p is the density matrix of the particle in the momentum representation,
E p = p 2 /(2m), π(p) = p 2 − 2m(∆ j j − ∆ k k ), and the lower limit of the integral obeys p 2 inf /(2m) = max{0, ∆ j j , ∆ j j −∆ k k }. The quantities s (α)
j j (E) are the complex entries of the scattering matrix, which are related to the amplitudes of the reflecting (α = −) and transmitted (α = +) plane waves of scattering states [6] with total energy E. They are obtained by solving the time-independent Schrödinger equation for scattering states, which behave as plane waves asymptotically [6,14] (see also [12] for exact and approximate expressions of the scattering matrix in terms of transfer matrices).
Following similar steps as in [6,12], we can obtain sufficient conditions for the scattering map (17) to obey the detailed balance condition (2). The first condition requires that the diagonal of the state of the particles in momentum representation coincides with the effusion distribution:
ρ U (p, p) = µ eff (p) = βp m e −βp 2 /(2m) .(18)
The second one is micro-reversibility:
s (α) j j (E) = s (α) jj (E).(19)
The proof is straightforward. Inserting (18) in (17) for k = j and k = j , we obtain
S jj j j = α=± ∞ p inf dp βp m e −βp 2 /(2m) s (α) j j p 2 2m + e j 2(20)
with p inf = 2m∆ j j if ∆ j j > 0 and zero otherwise.
With the change of variable ε = p 2 /(2m) + e j , we get
S jj j j = βe βej α=± ∞ max{ej ,e j } dε e −βε s (α) j j (ε) 2(21)
which, together with Eq. (19), immediately yields the detailed balance condition (2). Micro-reversibility is fulfilled by the exact scattering matrix in any collision described by a Hamiltonian of the form (16). In [12], we have developed several approximations of the scattering matrix that still satisfy this condition and can be used to design simple quantum repeated-interaction thermostats. We use one of these approximations to analyze an explicit example below [15].
However, if ρ U (p, p ) = 0 for p = p , then the scattering map can create coherences [6]. For example, if ρ U (p, p 2 − 2m∆ j k ) = 0 for j = k and the amplitudes of the transitions j → j and j → k are nonzero, then the term S jj j k is nonzero and couples the off-diagonal term ρ j k to the population ρ jj .
In particular, for particles in a pure state, the density matrix is ρ U (p, p ) = φ(p)φ * (p ), where φ(p) is the wave function of the pure state in the momentum representation. Then, a diagonal state ρ U in momentum representation can be obtained only by using plane waves with |φ(p)| 2 ∝ δ(p − p 0 ). These plane waves are completely delocalized in space and do not induce individual collision events, but rather a continuous-time evolution of the state of the system [16]. In [6], we have shown that this condition can be relaxed to narrow wave packets whose momentum dispersion σ p is small enough to avoid overlapping between outgoing packets with different energies, i.e., ρ U (p, p 2 − 2m∆ j k ) 0 for all j = k . On the other hand, for very broad packets, one can even obtain a unitary scattering map that preserves the entropy of the system. Consequently, in this case, the collision is a work source from a thermodynamic point of view [17].
We now analyze whether the dephasing induced by Poissonian collisions can restore thermalization even in the case of broad packets. As we have shown above, this is the case if the rate of collisions is low enough. Now, we check this statement using a specific example. We consider a qubit bombarded by particles that are localized in space around an initial position x 0 with dispersion ∆x, and whose momentum is distributed according to the effusion distribution (18). These two requirements can be implemented using the following Wigner function [17]:
W (p, x) = µ eff (p) 1 √ 2π∆x 2 e −(x−x0) 2 /(2∆x 2 )(22)
which is valid if 4π∆x m/β ≥ [18]. From the Wigner function, we can obtain the density matrix in momentum representation [17]:
ρ U (p, p ) = dx W p + p 2 , x e −i(p−p )x/ = µ eff p + p 2 exp − ∆x 2 (p − p ) 2 2 2 − i (p − p )x 0 .(23)
Notice that the density matrix is diagonal in the momentum representation only for ∆x → ∞ (the Wigner function (22) in this case is not valid because the resulting state ρ U is a mixture of plane waves, which are not proper states). This is equivalent to the narrow packets considered in [6,12]. If one imposes the localization of the particle, which is necessary to have well-defined isolated collisions, then, the density matrix ρ U is no longer diagonal in the momentum representation. This is equivalent to the broad packets that exhibit non-negligible momentum dispersion when one localizes the packet in space, due to the Heisenberg uncertainty principle. The total Hamiltonian of our system is (16), with H S = (∆/2)σ z and V = λ (σ x + σ y ), σ i being the three Pauli matrices of the qubit. The difference between the energy levels is ∆ and λ is the intensity of the interaction. The scattering region is the interval [0, L] and f (x) is the indicator function of this interval: f (x) = 1 if x ∈ [0, L] and zero otherwise.
First, we show in Fig. 1 the time evolution of the population of the ground state of the qubit when we when the evolution is given by random collisions, as in Eq. (5) (blue curves), for different values of the bombarding rate Γ. To obtain the time evolution, we calculate the amplitudes s (α) j j (E) using the approximation introduced in [12,15], which neglects the reflecting waves but fulfills micro-reversibility. From the scattering amplitudes and the incident density matrix (23), we get the scattering map S, using Eq. (17). Fig. 1 shows the population of the ground state ρ 00 for a given realization of collision times τ 1 , τ 2 , . . . . The populations are plotted as a function of Γt, which is the average number of collisions up to time t. We see that the system does not thermalize for values of Γ well above the Bohr frequency ∆ = 0.6 ( = 1). For comparison, we also include the population when the dephasing operator (3) is applied in each collision for Γ = 10. In this case, the evolution is almost deterministic and drives the qubit to the thermal state. The steady state of the evolution can be calculated analytically by solving Eq. (9). We plot the solution as a function of Γ (for both populations and coherences) in Fig. 2. The two plots confirm our results and show that thermalization is achieved when Γ is of the same order as the qubit frequency ∆/ = 0.6 or lower.
There are three key parameters that determine whether the qubit thermalizes or not: The Bohr energy ∆, the bombardment rate Γ, and the dispersion of the particle's position ∆x that determine the magnitude of the off-diagonal terms of the density matrix ρ U in momentum representation, as shown by Eq. (23). The interplay among the three parameters is not trivial, as shown in Fig. 3, but is qualitatively captured by the following expression [15]:
|ρ 01 | ∼ Γ ω e −βm∆x 2 ω 2 /2(24)
where ω ≡ ∆/ is the Bohr frequency of the qubit. For a 1 GHz qubit, for example, one could observe coherence if effusion occurs at a rate of 10 9 particles per second or greater, which is achievable [19] and compatible with the condition ΓL kT /m, warranting that there is a single particle in the scattering region at any time (the presence of two or more particles could induce nonlinear effects that have been explored in the context of cavity QED [20]). The exponential factor in Eq. (24) imposes a more involved condition on the spatial dispersion of the bombarding particles, but it does not seem very restrictive either: for the 1 GHz qubit, molecules with a mass of 1000 protons could generate coherences at room temperature if they are localized in an interval of the order of 50 nanometers or smaller.
To conclude, we have established the necessary conditions for thermalization in a wide class of quantum maps combined with a unitary evolution that lasts a random Poissonian time. We have determined when one could expect a deviation from thermalization due to quantum effects. Our results are useful to design repeatedinteraction reservoirs that are thermodynamically consistent and could help to clarify current open problems. One example is whether the presence of non-commuting conserved observables hinders or boosts thermalization [8][9][10], a problem that has been partially addressed using generic collisional reservoirs [21]. In addition, they show that the system exhibits a high sensitivity to the kinetic characteristics of the bombarding particles in situations that can be reproduced in experimental setups, such as those used in cavity QED [22], and could shed light on the problem of spatial decoherence of macromolecules [23][24][25].
I. TRANSITION AMPLITUDES AND SIMULATIONS
In this section, we summarize the methods used to solve the model, perform the simulations shown in Fig. 1, and obtain the steady state depicted in Figs. 2 and 3 of the main text.
We start by evaluating the scattering map S jk j k in Eq. (17) of the main text. It contains the energy-dependent transition amplitudes s (α) j j (E). In [? ] we provide an approximate expression for these quantities:
s (+) j j (E) e −iL(k j +k j )/2 j |e iLK(E) |j E ≥ e max , δ j j E < e max .
(S1)
In j j (E) 0. With this aspproximation, we evaluate the scattering map S jk j k in Eq.(17) for the given ρ U (p, p ) in the main text Eq. (23). The stochastic evolution in Fig. 1 of the main text is obtained by concatenating the scattering map with unitary evolution U τ ρ = e −iH S τ / ρe iH S τ / with Poissonian distributed times τ , and bombarding rate Γ. We initialize the system in a diagonal state with ρ 00 = 0 and ρ 11 = 1.
We obtain the results in Figs. 2 and 3 numerically solving Eq. (9). In order to do this, we write Eq. (9) in the eigenbasis of H S ,
− i ∆ jk ρ jk + Γ j k S jk j k − δ j j δ k k ρ j k = 0. (S2)
For a single qubit density matrix, the last equation becomes a system of 2×2 linear equations that can be treated with standard techniques.
II. STATIONARY DENSITY MATRIX
Here, we obtain a rough approximation of the off-diagonal term ρ 01 of the stationary density matrix in the example discussed in the main text: a single qubit with energies * [email protected] † [email protected] ‡ [email protected] § [email protected] ±∆/2. According to Eqs. (10) and (14) In Figs. S1 and S2, we plot the value of |ρ 01 | in different situations obtained from the exact solution of Eq. (9) in the main text, which gives the steady state for Poissonian bombardment at a rate Γ, and from the estimate given by (S10) with s = 0.01. Although the estimation is very rough, it captures the qualitative behaviour of the off-diagonal term and provides a rule of thumb to determine whether the system thermalizes.
− i [H S , ρ ss ] + Γ(S − I)ρ ss = 0.
FIG. 1 .
1Time evolution of the diagonal element ρ00 with Poissonian bombarding with rates Γ = 10, 5, 1 and when we intercalate the dephasing super-operator D (red). The black dashed line represents the population in the thermal equilibrium state. ∆ = 0.6, β = 0.1, m = 0.1, λ = L = = 1, ∆x = 1 and x0 = −10.
FIG. 2 .
2Numerical solution of(9) in the case of a single qubit as a function of Γ: ρ00 (blue), ρ11 (red) and |ρ10| (green). The black dashed lines represent the population in the thermal equilibrium state. ∆ = 0.6, β = 0.1, m = 0.1, λ = L = = 1, ∆x = 1 and x0 = −10.
FIG. 3 .
3Modulus of the off-diagonal element of the steady state, |ρ01|, from the numerical solution of (9) in the case of a single qubit: Left: as a function of Γ and ∆ for ∆x = 1 . Middle: as a function of ∆x and ∆ for Γ = 5. Right: as a function of ∆x and Γ for ∆/2 = 0.6. β = 0.1, m = 0.1, λ = L = = 1 and x0 = −10.
JT-B and JMRP acknowledge financial support from the Spanish Government (Grant FLUID, PID2020-113455GB-I00) and from the Foundational Questions Institute Fund, a donor advised fund of Silicon Valley Community Foundation (Grant number FQXi-IAF19-01). ME is funded by the Foundational Questions Institute Fund (Grant number FQXi-IAF19-05). F. B. thanks Fondecyt project 1191441 and ANID -Millennium Science Initiative Program-NCN19-170.Supplemental MaterialJorge Tabanera-Bravo, 1, * Juan M. R. Parrondo, 1, † Massimiliano Esposito, 2, ‡ and FelipeBarra 3,
this expression, the operator K(E) = 2m(E − H tot ) depends on the complete Hamiltonian H tot = H S + V . The wave vectors are k j = 2m(E − e j ), and e max is the maximum eigenvalue of H S and H tot . This approximation is valid for high incident kinetic energies and neglects the reflection amplitudes s (−)
FIG
. S1. Off-diagonal element |ρ 01 | from the numerical solution of Eq.(9) in the main text in the case of a single qubit (left) as a function of ∆ and Γ with ∆x = 1 (middle) as a function of ∆ and ∆x with Γ = 5 and (right) as a function of Γ and ∆x with ∆/2 = 0.6. λ = L = = 1, β = 0.1, m = 0.1 and x 0 = −10.
FIG
. S2. Off-diagonal element |ρ 01 | from the estimation (S10) and with s = 0.01 and same parameters as in figure S1.
, its modulus up to first order in Γ reads|ρ 01 |
Γ
∆
|S 00
01 ρ
(0)
00 + S 11
01 ρ
(0)
11 |.
(S3)
Populations ρ (0) jj are of order 1 and the two entries of the collisional map S 00 01 and S 11 01 are of the same order. Hence, the order of magnitude of |ρ 01 | isThe term of the collisional map is given by Eq.(17), which readswith π(p) = p 2 + 2m∆. The density matrix of the particle in the momentum representation is given by Eq.(23):The effusion distribution µ eff (p) is peaked around its maximum at p max = m/β. Then, the integral in (S5) is dominated by values of the momentum close to p max . Furthermore, if p 2 max /m = 1/β is much larger than ∆, thenAssuming that the scattering amplitudes are not very sensitive to the momentum, we can replace the sum over α by a single dimensionless constant s 2 . Then we getand |ρ 01 | ∼ s 2 Γ ∆ e −βm∆x 2 ∆ 2 /(2 2 ) = s 2 Γ ω e −βm∆x 2 ω 2 /2 (S10)where ω ≡ ∆/ is the Bohr frequency of the qubit.
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Non-abelian symmetry can increase entanglement entropy. S Majidy, A Lasek, D A Huse, N. Yunger Halpern, Phys. Rev. B. 10745102S. Majidy, A. Lasek, D. A. Huse, and N. Yunger Halpern, Non-abelian symmetry can increase entanglement en- tropy, Phys. Rev. B 107, 045102 (2023).
Parrondo, Micro-reversibility and thermalization with collisional baths. J Ehrich, M Esposito, F Barra, J M R , Physica A: Statistical Mechanics and its Applications. 552122108J. Ehrich, M. Esposito, F. Barra, and J. M. R. Par- rondo, Micro-reversibility and thermalization with col- lisional baths, Physica A: Statistical Mechanics and its Applications 552, 122108 (2020).
Quantum collisional thermostats. J Tabanera, I Luque, S L Jacob, M Esposito, F Barra, J M R Parrondo, New Journal of Physics. 2423018J. Tabanera, I. Luque, S. L. Jacob, M. Esposito, F. Barra, and J. M. R. Parrondo, Quantum collisional thermostats, New Journal of Physics 24, 023018 (2022).
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Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model. S N Filippov, G N Semin, A N Pechen, Phys. Rev. A. 10112114S. N. Filippov, G. N. Semin, and A. N. Pechen, Quantum master equations for a system interacting with a quan- tum gas in the low-density limit and for the semiclassical collision model, Phys. Rev. A 101, 012114 (2020).
Quantum scattering as a work source. S L Jacob, M Esposito, J M R Parrondo, F Barra, Quantum. 6750S. L. Jacob, M. Esposito, J. M. R. Parrondo, and F. Barra, Quantum scattering as a work source, Quan- tum 6, 750 (2022).
M Hillery, R O'connell, M Scully, E Wigner, Distribution functions in physics: Fundamentals. 106121M. Hillery, R. O'Connell, M. Scully, and E. Wigner, Dis- tribution functions in physics: Fundamentals, Physics Reports 106, 121 (1984).
A cold cesium atomic beam produced out of a pyramidal funnel. A Camposeo, A Piombini, F Cervelli, F Tantussi, F Fuso, E Arimondo, Optics Communications. 200231A. Camposeo, A. Piombini, F. Cervelli, F. Tantussi, F. Fuso, and E. Arimondo, A cold cesium atomic beam produced out of a pyramidal funnel, Optics Communica- tions 200, 231 (2001).
Multiatom effects in cavity qed with atomic beams. H J Carmichael, B C Sanders, Phys. Rev. A. 602497H. J. Carmichael and B. C. Sanders, Multiatom effects in cavity qed with atomic beams, Phys. Rev. A 60, 2497 (1999).
Nonabelian quantum transport and thermosqueezing effects. G Manzano, J M Parrondo, G T Landi, PRX Quantum. 310304G. Manzano, J. M. Parrondo, and G. T. Landi, Non- abelian quantum transport and thermosqueezing effects, PRX Quantum 3, 010304 (2022).
S Haroche, J.-M Raimond, Exploring the Quantum: Atoms, Cavities, and Photons. OxfordOxford University PressS. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University Press, Oxford, 2006).
Collisional decoherence observed in matter wave interferometry. K Hornberger, S Uttenthaler, B Brezger, L Hackermüller, M Arndt, A Zeilinger, 10.1103/physrevlett.90.160401Physical Review Letters. 90K. Hornberger, S. Uttenthaler, B. Brezger, L. Hack- ermüller, M. Arndt, and A. Zeilinger, Collisional deco- herence observed in matter wave interferometry, Phys- ical Review Letters 90, 10.1103/physrevlett.90.160401 (2003).
Decoherence of matter waves by thermal emission of radiation. L Hackermüller, K Hornberger, B Brezger, A Zeilinger, M Arndt, Nature. 427711L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger, and M. Arndt, Decoherence of matter waves by thermal emission of radiation, Nature 427, 711 (2004).
Collisional decoherence reexamined. K Hornberger, J E Sipe, 10.1103/phys-reva.68.012105Physical Review A. 68K. Hornberger and J. E. Sipe, Collisional decoher- ence reexamined, Physical Review A 68, 10.1103/phys- reva.68.012105 (2003).
| {'fraction_non_alphanumeric': 0.06314584591904868, 'fraction_numerical': 0.031032399605735908, 'mean_word_length': 3.9860494610019024, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We introduce a wide class of quantum maps that arise in collisional reservoirs and are able to thermalize a system if they operate in conjunction with an additional dephasing mechanism. These maps describe the effect of collisions and induce transitions between populations that obey detailed balance, but also create coherences that prevent the system from thermalizing. We combine these maps with a unitary evolution acting during random Poissonian times between collisions and causing dephasing. We find that, at a low collision rate, the nontrivial combination of these two effects causes thermalization in the system. This scenario is suitable for modeling collisional reservoirs at equilibrium. We justify this claim by identifying the conditions for such maps to arise within a scattering theory approach and provide a thorough characterization of the resulting thermalization process.', 'arxivid': '2302.06429', 'author': ['Jorge Tabanera-Bravo \nDepartamento de Estructura de la Materia\nFísica Térmica y Electrónica and GISC\nUniversidad Complutense de Madrid\nPl. de las Ciencias 128040MadridSpain\n', 'Juan M R Parrondo \nDepartamento de Estructura de la Materia\nFísica Térmica y Electrónica and GISC\nUniversidad Complutense de Madrid\nPl. de las Ciencias 128040MadridSpain\n', 'Massimiliano Esposito \nComplex Systems and Statistical Mechanics, Physics and Materials Science Research Unit\nUniversity of Luxembourg\nL-1511LuxembourgG.D. Luxembourg\n', 'Felipe Barra \nDepartamento de Física\nFacultad de Ciencias Físicas y Matemáticas\nUniversidad de Chile\n837.0415SantiagoChile\n'], 'authoraffiliation': ['Departamento de Estructura de la Materia\nFísica Térmica y Electrónica and GISC\nUniversidad Complutense de Madrid\nPl. de las Ciencias 128040MadridSpain', 'Departamento de Estructura de la Materia\nFísica Térmica y Electrónica and GISC\nUniversidad Complutense de Madrid\nPl. de las Ciencias 128040MadridSpain', 'Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit\nUniversity of Luxembourg\nL-1511LuxembourgG.D. Luxembourg', 'Departamento de Física\nFacultad de Ciencias Físicas y Matemáticas\nUniversidad de Chile\n837.0415SantiagoChile'], 'corpusid': 256827208, 'doi': '10.1103/physrevlett.130.200402', 'github_urls': [], 'n_tokens_mistral': 9966, 'n_tokens_neox': 8831, 'n_words': 5372, 'pdfsha': '8e79d4cdcd653ba6dc38a261e59da0c725cd9863', 'pdfurls': ['https://export.arxiv.org/pdf/2302.06429v3.pdf'], 'title': ['Thermalization and dephasing in collisional reservoirs', 'Thermalization and dephasing in collisional reservoirs'], 'venue': []} |
arxiv |
THE DE SITTER/ANTI-DE SITTER BLACK HOLES PHASE TRANSITION?
arXiv:gr-qc/0112066v1 26 Dec 2001
Shin ' Nojiri [email protected]
Department of Applied Physics National Defence Academy
Tomsk State Pedagogical University Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato
Hashirimizu Yokosuka, Lomas del Bosque 103, Apdo. Postal E-143239-8686, 37150LeonJAPAN, MEXICO
Sergei D Odintsov [email protected]
Department of Applied Physics National Defence Academy
Tomsk State Pedagogical University Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato
Hashirimizu Yokosuka, Lomas del Bosque 103, Apdo. Postal E-143239-8686, 37150LeonJAPAN, MEXICO
THE DE SITTER/ANTI-DE SITTER BLACK HOLES PHASE TRANSITION?
arXiv:gr-qc/0112066v1 26 Dec 2001Black hole thermodynamics(anti-)de Sitter spacehigher derivative gravity
We investigate the Schwarzschild-Anti-deSitter (SAdS) and SdS BH thermodynamics in 5d higher derivative gravity. The interesting feature of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. The appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur or where definition of entropy should be modified.
BH thermodynamics is quite attractive, as it provides the understanding of gravitational physics at extremal conditions. Moreover, it has been realized recently that AdS BH may be relevant in the study of AdS/CFT correspondence. Hence, there appears nice way to decribe strong coupling gauge theories via their gravitational duals. In the present work we discuss the thermodynamics of dS and AdS BHs in higher derivative gravity. The fundamental issue of entropy for such objects leads to some interesting conclusions.
We start from the following action of d dimensional R 2 -gravity with cosmological constant. The action is given by:
S = d d+1 x √ −g aR 2 + bR µν R µν + cR µνξσ R µνξσ + 1 κ 2 R − Λ . (1)
We discuss the relation between SdS and SAdS BHs based on entropy considerations. For simplicity, we consider c = 0 case in (1) for most results. When c = 0, Schwarzschild-anti de Sitter space is an exact solution:
ds 2 = −e 2ν(r) dt 2 + e −2ν(r) dr 2 + r 2 d−1 i,j=1g ij dx i dx j , e 2ν = e 2ν 0 ≡ 1 r d−2 −µ + kr d−2 d − 2 + r d l 2 .(2)
Hereg ij is the metric of the Einstein manifold, which is defined bỹ R ij = kg ij .R ij is the Ricci curvature given byg ij and k is a constant. For example, one has k = d − 2 for d − 1-dimensional unit sphere, k = −(d − 2) for d − 1-dimensional unit hyperboloid, and k = 0 for flat surface. The curvatures have the following form:R = − d(d+1) l 2 and R µν = − d l 2Ĝµν . In (1), µ is the parameter corresponding to mass and the scale parameter l is given by solving the following equation:
0 = d 2 (d + 1)(d − 3)a l 4 + d 2 (d − 3)b l 4 − d(d − 1) κ 2 l 2 − Λ .(3)
When the solution for l 2 (3) is positive (negative), the spacetime is asymptotically anti-de Sitter (de Sitter) space and especially if µ = 0, the solution expresses the pure anti-de Sitter or de Sitter space. If Λ = 0, there is asymptotically flat (Minkowski) solution, where 1 l 2 = 0. In the following we concentrate on the case of d = 4.
The calculation of thermodynamical quantites like free energy F , the entropy S and the energy E may be done with the help of the following method: After Wick-rotating the time variable by t → iτ , the free energy F can be obtained from the action S (1) where the classical solution is substituted : F = −T S. Substituting eq.(3) into (1) in the case of d = 4 with c = 0, one gets
S = − d 5 x √ −G 8 l 2 κ 2 − 320a l 4 − 64b l 4 = − V 3 T ∞ r H drr 3 8 l 2 κ 2 − 320a l 4 − 64b l 4(4)
Here V 3 is the volume of 3d sphere and we assume τ has a period of 1 T . The expression of S contains the divergence coming from large r. In order to subtract the divergence, we regularize S (3) by cutting off the integral at a large radius r m and subtracting the solution with µ = 0:
S reg = − V 3 T rm r H drr 3 − e ν(r=rm) e ν(r=rm;µ=0) rm 0 drr 3 8 κ 2 l 2 − 320a l 4 − 64b l 4 (5)
The factor e ν(r=rm) /e ν(r=rm;µ=0) is chosen so that the proper length of the circle which corresponds to the period 1 T in the Euclidean time at r = r m concides with each other in the two solutions. Taking r m → ∞, one finds, as found in [1],
F = −V 3 l 2 µ 8 − r 4 H 4 8 l 2 κ 2 − 320a l 4 − 64b l 4 (6)
The horizon radius r h is given by solving the equation e 2ν 0 (r H ) = 0 in (1):
r 2 H = − kl 2 4 + 1 2 k 2 4 l 4 + 4µl 2 .(7)
The Hawking temperature T H is
T H = k 4πr H + r H πl 2 (8)
where ′ denotes the derivative with respect to r. From the above equation (8), r H can be rewritten in terms of T H as
r H = 1 2 πl 2 T H ± (πl 2 T H ) 2 − kl 2(9)
In (9), the plus sign corresponds to k = −2 or k = 0 case and the minus sign to k = 2 case. 1 One can also rewrite µ by using r H as µ =
r 4 H l 2 + kr 2 H 2 . Then we can rewrite F using r H as F = − V 3 8 r 2 H r 2 H l 2 − k 2 8 κ 2 − 320a l 2 − 64b l 2 .(10)
Then the entropy S = − dF dT H and the thermodynamical energy E = F + T S can be obtained as follows [1]:
S = 4V 3 πr 3 H 1 κ 2 − 40a l 2 − 8b l 2 , E = 3V 3 µ 1 κ 2 − 40a l 2 − 8b l 2 ,(11)
This seems to indicate that the contribution from the R 2 -terms can be absorbed into the redefinition:
1 κ 2 = 1 κ 2 − 40a l 2 − 8b l 2 ,(12)
although this is not true for c = 0 case. On the other hand, by using the surface counter term method [2] , one gets the following expression for the conserved mass M :
M = 3l 2 16 V 3 1 κ 2 − 40a l 2 − 8b l 2 − 4c l 2 k 2 + 16µ l 2 .(13)
One can also start from the expression for M with c = 0 as the thermodynamical energy E:
E = 3V 3 1 κ 2 − 40a l 2 − 8b l 2 k 2 l 2 16 + µ l 2(14)
The expression of energy E (14) is different from that in (11) by a first µ-independent term, which comes from the AdS background. Using the thermodynamical relation dS = dE T , we find
S = dE T H = dr H dE dµ dµ dr H 1 T H = V 3 πr 3 H 2 8 κ 2 − 320a l 2 − 64b l 2 + S 0 .(15)
Here S 0 is a constant of the integration. Up to the constant S 0 , the expression (15) is identical with (11). We should note that the entropy S (11) becomes negative, when
8 κ 2 − 320a l 2 − 64b l 2 < 0. .(16)
This is true even for the expression (15) for the black hole with large radius r H since S 0 can be neglected for the large r H . We now investigate in more detail what happens when Eq.(16) is satisfied. First we should note l 2 is determined by (3), which has, in case of d = 4, the following form:
0 = 80a + 16b l 4 − 12 κ 2 l 2 − Λ ,(17)
There are two real solutions for l 2 when 6 κ 2 + (80a + 16b) Λ ≥ 0 and the solutions are given by
1 l 2 = 6 κ 2 ± 6 κ 2 + (80a + 16b) Λ 80a + 16b .(18)Suppose κ 2 > 0. Then if (80a + 16b) Λ > 0 ,(19)
one solution is positive but another is negative. Therefore there are both of the asymptotically AdS solution and asymptotically dS one. Let us denote the positive solution for l 2 by l 2 AdS and the negative one by −l 2 dS :
l 2 = l 2 AdS , −l 2 dS , l 2 AdS , l 2 dS > 0 .(20)
Then when the asymptotically AdS solution is chosen, the entropy (15) has the following form:
S AdS = V 3 πr 3 H 2 8 κ 2 − 320a + 64b l 2 AdS .(21)
Here we have chosen S 0 = 0. On the other hand, when the solution is asymptotically dS, the entropy (15) has the following form:
S dS = V 3 πr 3 H 2 8 κ 2 + 320a + 64b l 2 dS .(22)
When 8
κ 2 − 320a + 64b l 2 AdS < 0 ,(23)
the entropy S AdS (21) is negative! There are different points of view to this situation. Naively, one can assume that above condition is just the equation to remove the nonphysical domain of theory parameters. However, it is difficult to justify such proposal. Why for classical action on some specific background there are parameters values which are not permitted? Moreover, the string/M-theory and its compactification would tell us what are the values of the theory parameters.
From another side, one can conjecture that classical thermodynamics is not applied here and negative entropy simply indicates to new type of instability in asymptotically AdS black hole physics. Indeed, when Eq.(23) is satisfied, since 80a + 16b > 0 (same range of parameters!), the entropy S dS (22) for asymptotically dS solution is positive. In other words, may be the asymptotically dS solution would be preferrable?
On the other hand, when
8 κ 2 + 320a + 64b l 2 dS < 0 ,(24)
the entropy S dS in (22) is negative and the asymptotically dS solution is instable (or does not exist). In this case, since 80a + 16b < 0, the entropy S AdS in (21) for asymptotically AdS solution is positive and the asymptotically AdS solution would be preferrable. Expression for the AdS black hole mass in (14) tells that when 8 κ 2 − 320a+64b l 2 AdS = 0, the AdS black hole becomes massless then there would occur the condensation of the black holes, which would make the transition to the dS black hole. On the other hand, when 8 κ 2 + 320a+64b l 2 dS = 0, the dS black hole becomes massless then there would occur the condensation of the black holes and the AdS black hole would be produced. Note that above state with zero entropy (and also zero free energy and zero conserved BH mass) is very interesting. Perhaps, this is some new state of BHs. As we saw that is this state which defines the border between physical SAdS (SdS) BH with positive entropy and SdS (SAdS) BH with negative entropy.
Hence, there appeared some indication that some new type of phase transition (or phase transmutation) between SdS and SAdS BHs in higher derivative gravity occurs. Unfortunately, we cannot suggest any dynamical formulation to describe explicitly such phase transition (it is definitely phase transition not in standard thermodynamic sense).
Let us consider now the entropy for Gauss-Bonnet case, where a = c and b = −4c in (1). For this purpose, we use the thermodynamical relation dS = dE T . For the Gauss-Bonnet case, the energy (13) has the following form [2]:
E = M = 3l 2 16 V 3 1 κ 2 − 12c l 2 k 2 + 16µ l 2(25)
Here S 0 is a constant of the integration, which could be chosen to be zero if we assume S = 0 when r H = 0. When ǫ = 0 (c = 0), the expression reproduces the standard result
S → 4πV 3 r 3 H κ 2 .(30)
The entropy (28) becomes negative (at least for the large black hole even if S 0 = 0) when 1 12 < ǫ < 1 4 .
Therefore the unitarity might be broken in this region but it might be recovered when ǫ > 1 4 . Even in case ǫ < 0 (k = 2), the entropy becomes negative when
r 2 H < −12ǫ ,(32)
if S 0 = 0. Then the small black hole might be unphysical. The fact discovered here-that entropy for S(A)dS BHS in gravity with higher derivatives terms may be easily done to be negative by the corresponding choice of parameters is quite remarkable. It is likely that thermodynamics for black holes with negative entropies should be reconsidered. In this respect one possibility would be to redefine the gravitational entropy for higher derivative gravity.
AcknowledgmentsThe authors are grateful to M. Cvetič for collaboration. SDO would like to thank the organizers of First Mexican Meeting on Math. and Exp. Physics for hospitality. The work by SN is supported in part by the Ministry of Education, Science, Sports and Culture of Japan under the grant n. 13135208. Notes 1. When k = 2, as we can see from(7)and(8), r H , and also T H , are finite in the limit of l → ∞, which corresponds to the flat background. Therefore we need to choose the minus sign in (9) for k = 2 case.We also found[2]Here ǫ ≡ cκ 2 l 2 . Then using (25), (26), and the expression of the Hawking temperature,the entropy can be obtained as
. S Nojiri, S D Odintsov, S Ogushi, hep-th/0108172Int.J.Mod.Phys. A. Phys.Rev. DS. Nojiri, S.D. Odintsov and S. Ogushi, hep-th/0105117, to appear in Int.J.Mod.Phys. A; hep-th/0108172, to appear in Phys.Rev. D.
. M Cvetič, S Nojiri, S D Odintsov, hep-th/0112045M. Cvetič, S. Nojiri and S.D. Odintsov, hep-th/0112045.
| {'fraction_non_alphanumeric': 0.05940922668436774, 'fraction_numerical': 0.05152671755725191, 'mean_word_length': 3.4723562152133582, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We investigate the Schwarzschild-Anti-deSitter (SAdS) and SdS BH thermodynamics in 5d higher derivative gravity. The interesting feature of higher derivative gravity is the possibility for negative (or zero) SdS (or SAdS) BH entropy which depends on the parameters of higher derivative terms. The appearence of negative entropy may indicate a new type instability where a transition between SdS (SAdS) BH with negative entropy to SAdS (SdS) BH with positive entropy would occur or where definition of entropy should be modified.', 'arxivid': 'gr-qc/0112066', 'author': ['Shin ' Nojiri [email protected] \nDepartment of Applied Physics National Defence Academy\nTomsk State Pedagogical University Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato\nHashirimizu Yokosuka, Lomas del Bosque 103, Apdo. Postal E-143239-8686, 37150LeonJAPAN, MEXICO\n', 'Sergei D Odintsov [email protected] \nDepartment of Applied Physics National Defence Academy\nTomsk State Pedagogical University Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato\nHashirimizu Yokosuka, Lomas del Bosque 103, Apdo. Postal E-143239-8686, 37150LeonJAPAN, MEXICO\n'], 'authoraffiliation': ['Department of Applied Physics National Defence Academy\nTomsk State Pedagogical University Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato\nHashirimizu Yokosuka, Lomas del Bosque 103, Apdo. Postal E-143239-8686, 37150LeonJAPAN, MEXICO', 'Department of Applied Physics National Defence Academy\nTomsk State Pedagogical University Tomsk, RUSSIA and Instituto de Fisica de la Universidad de Guanajuato\nHashirimizu Yokosuka, Lomas del Bosque 103, Apdo. Postal E-143239-8686, 37150LeonJAPAN, MEXICO'], 'corpusid': 15620030, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4330, 'n_tokens_neox': 3618, 'n_words': 2372, 'pdfsha': 'd2324383771d9ee998980fe8451ba28c8e8683a9', 'pdfurls': ['https://arxiv.org/pdf/gr-qc/0112066v1.pdf'], 'title': ['THE DE SITTER/ANTI-DE SITTER BLACK HOLES PHASE TRANSITION?', 'THE DE SITTER/ANTI-DE SITTER BLACK HOLES PHASE TRANSITION?'], 'venue': []} |
arxiv |
Construction and Iteration-Complexity of Primal Sequences in Alternating Minimization Algorithms
10 Nov 2015
Quoc Tran-Dinh
Construction and Iteration-Complexity of Primal Sequences in Alternating Minimization Algorithms
10 Nov 2015Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)
We introduce a new weighted averaging scheme using "Fenchel-type" operators to recover primal solutions in the alternating minimization-type algorithm (AMA) for prototype constrained convex optimization. Our approach combines the classical AMA idea in[18]and Nesterov's prox-function smoothing technique without requiring the strong convexity of the objective function. We develop a new non-accelerated primal-dual AMA method and estimate its primal convergence rate both on the objective residual and on the feasibility gap. Then, we incorporate Nesterov's accelerated step into this algorithm and obtain a new accelerated primal-dual AMA variant endowed with a rigorous convergence rate guarantee. We show that the worst-case iteration-complexity in this algorithm is optimal (in the sense of first-oder black-box models), without imposing the full strong convexity assumption on the objective.Keywords Alternating minimization algorithm · smoothing technique · primal solution recovery · accelerated first-oder method · constrained convex optimization 1 Introduction This paper studies a new weighted-averaging strategy in alternating minimizationtype algorithms (AMA) to recover a primal solution of the following constrained convex optimization problem:where g : R p 1 → R ∪ {+∞} and h : R p 2 → R ∪ {+∞} are both proper, closed and convex (not necessarily strongly convex), (p 1 + p 2 = p, A ∈ R n×p 1 , B ∈ R n×p 2 , c ∈ R n , and U ⊂ R p 1 and V ⊂ R p 2 are two nonempty, closed and convex sets. Problem (1) surprisingly covers a broad class of constrained convex programs, including composite convex minimization, general linear constrained convex optimization problems, and conic programs.
Primal-dual methods handle problem (1) together with its dual formulation, and generate a primal-dual sequence so that it converges to a primal and dual solution of (1). Research on primal-dual methods has been extensively studied in the literature for many decades, see, e.g., [4,17,19] and the references quoted therein. However, such methods have attracted a great attention in the past decade due to new applications in signal and image processing, economics, machine learning, and statistics. Various primal-dual methods have been rediscovered and extended, not only from algorithmic perspectives, but also from theoretical convergence guarantees. Despite of this great attempt in the algorithmic development, the corresponding supporting theory has not been well-developed, especially, the algorithms with rigorous convergence guarantees and low complexity-per-iteration.
Perhaps, applying first order methods to the dual is the most nature approach to solve constrained problems of the form (1). By means of the Lagrange duality theory, we can formulate the dual problem of (1) as a convex problem, where existing convex optimization techniques can be applied to solve it. Depending on the structure assumptions imposing on (1), the dual problem possesses useful properties that can be exploited to develop algorithms for the dual. For instance, we can use subgradient, gradient, proximal-gradient, as well as other proximal and splitting techniques to solve this problem. Then, the primal solutions of (1) can be recovered from the dual solutions [10,20]. Among many other primal-dual splitting methods, alternating minimization algorithm (AMA) proposed by Tseng [18] becomes one of the most popular and powerful methods to solve (1) when g and h are nonsmooth and convex, and either g or h is strongly convex. Unfortunately, to the best of our knowledge, there has existed no optimization scheme to recover primal solutions of (1) in AMAs with convergence rate guarantees on both the primal objective residual and the feasibility gap.
If g and h are nonsmooth, then numerical methods for solving (1) often rely on the proximal operators of g and h. Mathematically, a proximal operator of a proper, closed, and convex function ϕ : R p → R ∪ {+∞} is defined as: prox ϕ (x) := argmin z ϕ(z) + (1/2) z − x 2 .
(
If prox ϕ can be computed efficiently, i.e., by a closed form or by a polynomial time algorithm, then we say that ϕ has a "tractable proximity" operator. There exist many smooth and nonsmooth convex functions with tractable proximity operators as indicated in, e.g., [6,14]. The proximal operator is in fact a special case of the resolvent in monotone inclusions [16]. Principally, the optimality condition for (1) can be cast into a monotone inclusion [1,8]. By mean of proximity operators and gradients, splitting approaches in monotone inclusions can be applied to solve such a problem [7,5,8]. However, due to this generalization, the convergence guarantees and the convergence rates of these algorithms often achieve via a primal-dual gap or residual metric joined both the primal and dual variables. Such convergence guarantees do not reveal the complexity bounds of the primal sequence for (1) at intermediate iterations when we terminate the algorithm at a desired accuracy. Our approach in this paper is briefly described as follows. First, since we work with non-strongly convex objectives g and h, we employ Nesterov's smoothing technique via prox-functions [13] to partially smooth the dual function. Then, we apply the forward-backward splitting method to solve the smoothed dual problem, which is exactly the AMA method in [18]. Next, we introduce a new weighted averaging scheme using the Fenchel-type operators (c.f. (7)) to generate the primal sequence simultaneously with the dual one. We then prove convergence rate guarantees for (1) in the primal variable as opposed to the dual one as in [9]. Finally, by incorporating Nesterov's acceleration step into the forward-backward splitting method, we obtain an accelerated primal-dual variant for solving (1) with a primal convergence rate guarantee. Interestingly, we can show that the primal sequence converges to an optimal solution of (1) with the O(1/k 2 )-optimal rate provided that only g or h is strongly convex, but not the whole function f as in accelerated dual gradient methods [10], where k is the iteration counter.
Our contributions: Our specific contributions can be summarized as follows: a) We propose to combine Nesterov's smoothing technique, the alternating minimization idea, and the weighted-averaging strategy to develop a new primaldual AMA algorithm for solving (1) without strong convexity assumption on g or h. We characterize the convergence rate on the absolute primal objective residual |f (x k ) − f ⋆ | and feasibility gap Aū k + Bv k − c for the averaging primal sequence x k . By an appropriate choice of the smoothness parameter, we provide the worst-case iteration-complexity of this algorithm to obtain an ǫ-primal solution. b) By incorperatiing Nesterov's accelerated step, we develop a new accelerated primal-dual AMA variant for solving (1), and characterize its worst-case iterationcomplexity which is optimal in the sense of first-oder black-box models [12]. c) When either g or h is strongly convex, we recover the standard AMA algorithm as in [9], but with our averaging strategy, we obtain the O(1/k 2 )-convergence rate on |f (x k ) − f ⋆ | and Aū k + Bv k − c separably for the primal problem (1), not for its dual. Let us emphasize the following points of our contributions. First, we can view the algorithms presented in this paper as the ISTA and FISTA schemes [2] applied to the smoothed dual problem of (1) instead the original dual of (1) as in [9]. The convergence rate on the dual objective residual is well-known and standard, while the convergence rates on the primal sequence are new. Second, we adapt the weights in our averaging primal sequence (c.f. (9)) to the local Lipschitz constant via a back-tracking line-search, which potentially increases the empirical performance of the algorithms. Third, the averaging primal sequence is computed via an additional sharp-operator of h V (c.f. (7)) instead of the current primal iterate. This computation can be done efficiently (e.g., in a closed form) when h V has a decomposable structure. Paper organization: The rest of this paper is organized as follows. Section 2 briefly describes standard Lagrange duality framework for (1), and shows how to apply Nesterov's smoothing idea to the dual problem. The main results are presented in Sections 3 and 4, where the two new algorithms and their convergence are provided. Section 5 is devoted to investigating the strongly convex case. Concluding remarks are given in Section 6, while technical proof is moved to the appendix.
2 Primal-dual framework and smoothing technique First, we briefly present the Lagrange duality framework for (1). Then we show how to apply Nesterov's smoothing technique to smooth the dual function of (1).
The Lagrange primal-dual framework
Let x := (u, v) denote the primal variables, and D := {x ∈ U × V : Au + Bv = c} denote the feasible set of (1). We define the Lagrange function of (1) corresponding to the linear constraint Au + Bv = c as L(x, λ) := g(u) + h(v) + λ, c− Au − Bv , where λ is the vector of Lagrange multipliers. Then, we can define the dual function d of (1) as
d(λ) := min u∈U ,v∈V {g(u) + h(v) + λ, c − Au − Bv } .(3)
Clearly, d can be split into three terms
d(λ) = d 1 (λ) + d 2 (λ) + c, λ , where d 1 (λ) := min u∈U g(u) − A T λ, u , d 2 (λ) := min v∈V h(v) − B T λ, v .(4)
Using d, we can define the dual problem of (1) as
d ⋆ := max λ∈R n d(λ).(5)
We say that problem (1) satisfies the Slater condition if
ri(X ) ∩ {Au + Bv = c} = ∅,(6)
where X := U × V and ri(X ) is a the relative interior of X [17].
In this paper, we require the following blanket assumptions, which are standard in convex optimization. Assumption A.1 The functions g and h are both proper, closed, and convex (not necessarily strongly convex). The solution set X ⋆ of (1) is nonempty. The Slater condition (6) holds for (1).
It is well-known that, under Assumption A.1, strong duality in (1) and (5) holds, i.e., we have zero duality gap which is expressed as f ⋆ − d ⋆ = 0. Moreover, for any feasible point (x, λ) ∈ dom(f ) × R n and any primal-dual solution (
x ⋆ , λ ⋆ ) with x ⋆ := (u ⋆ , v ⋆ ) ∈ X ⋆ we have: L(x ⋆ , λ) ≤ L(x ⋆ , λ ⋆ ) = f ⋆ = d ⋆ ≤ L(x, λ ⋆ )
for all x ∈ X and λ ∈ R n . Now, let us consider the components d 1 and d 2 of (4). Indeed, we can write these components as
d 1 (λ) = − max u∈U A T λ, u − g(u) = −g * U (A T λ), d 2 (λ) = − max v∈V B T λ, v − h(v) = −h * V (B T λ),
where g * U and h * V are the Fenchel conjugate of g U := g + δ U and h V := h + δ V , respectively [17]. If we define two multivalued maps
u # (s) := argmax u∈U { s, u − g(u)} , and v # (s) := argmax v∈V { s, v − h(v)} , (7) then the solution u * (λ) of d 1 in (4) is given by u * (λ) ∈ u # (A T λ) ≡ ∂g * U (A T λ). Similarly, the solution v * (λ) of d 2 in (4) is given by v * (λ) ∈ v # (B T λ) ≡ ∂h * V (B T λ)
. We call u # and v # the sharp-operator of g and h, respectively [20]. Each oracle call to d queries one element of the sharp-operators u # and v # at a given λ ∈ R n . By using the saddle point relation, we can show that
f * ≤ L(x, λ ⋆ ) = f (x) − Au + Bv − c, λ ⋆ ≤ f (x) + Au + Bv − c λ ⋆ for any x ∈ X . Hence, we have − λ ⋆ Au + Bv − c ≤ f (x) − f ⋆ ≤ f (x) − d(λ).(8)
In this paper, we only assume that the second dual component d 2 defined by (4) satisfies the following assumption.
Assumption A.2 The dual component d 2 defined by (4) is finite.
This assumption holds in particular when V is bounded. Moreover, v * (λ) is welldefined for any λ ∈ R n . Throughout this paper, we assume that Assumptions A.1 and A.2 holds without referring to them again.
The primal weighted averaging sequence
Given a sequence of the primal approximation x k k≥0 , wherex k := (ũ k ,ṽ k ) ∈ X . We define the following weighted averaging sequence x k withx k := (ū k ,v k ) as
u k := S −1 k k i=1 w iũ i ,v k := S −1 k k i=0 w iṽ i , and S k := k i=0 w i ,(9)
where {w i } i≥0 ⊂ R ++ is the corresponding weights.
To avoid storing the whole sequence ũ k ,ṽ k ) in our algorithms, we can compute x k recursively as follows:
u k := (1 − τ k )ū k−1 + τ kũ k , andv k := (1 − τ k )v k−1 + τ kṽ k , ∀k ≥ 1, (10) where τ k := w k S k ∈ [0, 1],ū 0 :=ũ 0 , andv 0 :=ṽ 0 . Clearly, for any convex function f , we have f (x k ) ≤ S −1 k k i=0 w i f (x i )
by the well-known Jensen inequality.
Approximate solutions: Our goal is to approximate a solution x ⋆ of (1) by x ⋆ ǫ in the following sense:
Definition 1 Given an accuracy level ǫ > 0, a point x ⋆ ǫ := (u ⋆ ǫ , v ⋆ ǫ ) ∈ X is said to be an ǫ-solution of (1) if |f (x ⋆ ǫ ) − f ⋆ | ≤ ǫ and Au ⋆ ǫ + Bv ⋆ ǫ − c ≤ ǫ.(11)
Here, we call |f (x ⋆ ǫ )−f ⋆ | the [absolute] primal objective residual and Au ⋆ ǫ +Bv ⋆ ǫ − c the primal feasibility gap. The condition x ⋆ ǫ ∈ X is in general not restrictive since, in many cases, X is a simple set (e.g., a box, a simplex, or a conic cone) so that the projection onto X can exactly be guaranteed.
Smoothing the dual component
As mentioned earlier, we first focus on the non-strongly convex functions g and h. In this case, we can not directly apply the standard AMA [18] to solve (1). We smooth g by using a prox-function as follows.
A continuous and strongly convex function p U with the strong convexity parameter µ p > 0 is called a prox-function for U if U ⊆ dom(p U ) [13]. We consider the following smoothed function d 1 γ for d 1 :
d 1 γ (λ) := min u∈U {g(u) − λ, Au + γp U (u)} ,(12)
where γ > 0 is a smoothness parameter. It is well-known that d 1 γ is concave and smooth. Moreover, as shown in [13], its gradient is given by ∇d 1
γ (λ) = −Au * γ (λ), which is Lipschitz continuous with the Lipschitz constant L γ d 1 := A 2 γµ p , where u * γ (λ)
is the unique solution of the minimization problem in (12). In addition, we have the following estimate
d 1 γ (λ) − γD U ≤ d 1 (λ) ≤ d 1 γ (λ), ∀ λ ∈ R n ,(13)
where D U is the prox-diameter of U , i.e.,
D U := sup u∈U p U (u).(14)
In order to develop algorithms, we require the following additional assumption.
Assumption A.3 The quantity D U defined by (14) is finite, i.e., 0 ≤ D U < +∞.
Clearly, if U is bounded, then Assumption A.3 is automatically satisfied. Under Assumption A.3, we consider the following convex problem:
d ⋆ γ := max λ∈R n d γ (λ) := d 1 γ (λ) + d 2 (λ) + c, λ .(15)
Using (13), we can see that d ⋆ γ converges to d ⋆ as γ ↓ 0 + . Hence, (15) can be considered as an approximation to the dual problem (5). We call (15) the smoothed dual problem of (1).
The non-accelerated primal-dual alternating minimization algorithm
Since d 1 γ is Lipschitz gradient, we can apply the proximal-gradient method (ISTA [2]) to solve (15). This leads to the AMA scheme presented in [9,18].
The main iteration of the alternating minimization algorithm (AMA) [18] applying to the corresponding primal problem of (15) can be written as
û k+1 := argmin u∈U g(u) − A Tλ k , u + γp U (u) = ∇g * γ (A Tλ k ), v k+1 := argmin v∈V h(v) − B Tλ k , v + η k 2 c − Aû k+1 − Bv 2 , λ k+1 :=λ k + η k (c − Aû k+1 − Bv k+1 ),(16)
whereλ k ∈ R n is given, η k > 0 is the penalty parameter, and g γ (·) := g(·)+γp U (·).
We define the quadratic surrogate of d 1 as follows:
Q γ L k (λ;λ k ) := d 1 γ (λ k ) + ∇d 1 γ (λ k ), λ −λ k − L k 2 λ −λ k 2 .(17)
Then the following lemma provides a key estimate to prove the convergence of the algorithms in the sequel, whose proof can be found in Appendix A. (12) is concave and smooth. It satisfies the following estimate
Lemma 1 The smoothed dual component d 1 γ defined byd 1 γ (λ) + ∇d 1 γ (λ),λ − λ − L d 1 2 λ − λ 2 ≤ d 1 (λ), ∀λ,λ ∈ R n ,(18)
where L γ d 1 := A 2 γµ p . Let λ k+1 be the point generated by (16) fromλ k and η k . Then, (16) is equivalent to the forward-backward splitting scheme applying to the smoothed dual problem (15), i.e.,
λ k+1 := prox (−η k d 2 ) λ k + η k ∇d 1 γ (λ k ) .(19)
In addition, with Q γ L k defined by (17), if the following condition holds
d 1 γ (λ k+1 ) ≥ Q γ L k (λ k+1 ;λ k ),(20)
then, for any λ ∈ R n , the following estimates hold
d γ (λ k+1 ) ≥ ℓ γ k (λ) + 1 η k λ k+1 −λ k , λ −λ k + 1 η k − L k 2 λ k − λ k+1 2 ≥ d γ (λ) + 1 η k λ k+1 −λ k , λ −λ k + 1 η k − L k 2 λ k − λ k+1 2 , (21) where ℓ γ k (λ) := d 1 γ (λ k ) + ∇d 1 γ (λ k ), λ −λ k + d 2 (λ k+1 ) + ∇d 2 (λ k+1 ), λ − λ k+1 + c, λ , and ∇d 2 (λ k+1 ) ∈ ∂d 2 (λ k+1 ) is a subgradient of d 2 at λ k+1 .
Our next step is to recover an approximate primal solutionx k := (ū k ,v k ) of (1) using the weighted averaging scheme (9). Combing this strategy and (16) we can present the new primal-dual AMA algorithm is as in Algorithm 1 below.
Algorithm 1 (Primal-dual alternating minimization algorithm)
Initialization: (7). 8. Update S k := S k−1 + w k , with w k := η k , and τ k := w k S k . 9. Updateū k := (1 − τ k )ū k−1 + τ kũ k andv k := (1 − τ k )v k−1 + τ kṽ k . end for Output: The sequence x k withx k := (ū k ,v k ).
1. Choose γ := ǫ 2D U , and L such that 0 < L ≤ L γ d 1 := A 2 γµ p . 2. Choose an initial point λ 0 ∈ R n . 3. Set S −1 := 0,ū −1 := 0 andv −1 := 0. for k := 0 to k max do 4. Computeũ k =û k+1 = u * γ (λ k ) defined in (12). 5. Choose η k ∈ 0, 1 L γ d 1 and computê v k+1 := arg min v∈V h(v) − B T λ k , v + η k 2 c − Aũ k − Bv 2 . 6. Update λ k+1 := λ k + η k c − Aû k+1 − Bv k+1 . 7. Computeṽ k := v * (λ k+1 ) ∈ v ♯ B T λ k+1 defined in
In fact, we can use the Lipschitz constant L γ d 1 = A 1 γµ p to compute the constant step η k as η k := 1
L γ d 1 at
Step 5. However, we can adaptively choose η k = L −1 k via a back-tracking line-search procedure in Algorithm 1 to guarantee the condition (20), and this usually performs better in practice than the constant step-size.
Algorithm 1 requires one more sharp operator query of v at Step 7. As mentioned earlier, when h V has decomposable structures, computing this sharp operator can be done efficiently (e.g., closed form or parallel/distributed manner).
The following theorem shows the bounds on the objective residual f (x k ) − f ⋆ and the feasibility gap Aū k + Bv k − c of (1) atx k .
Theorem 1 Let x k withx k := (ū k ,v k ) be the sequence generated by Algorithm 1 and L d 1 := A 2 µ p . Then, the following estimates hold:
|f (x k ) − f ⋆ | ≤ max L d 1 λ 0 2 γ(k+1) + γD U , 2L d 1 λ ⋆ λ 0 −λ ⋆ γ(k+1) + λ ⋆ L d 1 D U k+1 , Aū k + Bv k − c ≤ 2L d 1 λ 0 −λ ⋆ γ(k+1) + L d 1 D U k+1 .(22)
Consequently, if we choose γ := ǫ 2D U , which is optimal, then the worst-case iterationcomplexity of Algorithm 1 to achieve the ǫ-solutionx k of (1) in the sense of Def-
inition 1 is O L d 1 D U ǫ 2 R 2 0 , where R 0 := max 2, 3 λ ⋆ , 2 λ 0 , 2 λ 0 − λ ⋆ . Proof Since 0 < η i ≤ 1 L γ d 1
by
Step 5 of Algorithm 1, for any λ ∈ R n , it follows from (21) that
d γ (λ i+1 ) ≥ ℓ γ i (λ) + 1 η i λ i+1 − λ i , λ − λ i + 1 2η i λ i+1 − λ i 2 = ℓ γ i (λ) + 1 2η i λ i+1 − λ 2 − λ i − λ 2 ,(23)
where
ℓ γ i (λ) := d 1 γ (λ i ) + ∇d 1 γ (λ i ), λ −λ i + d 2 (λ i+1 ) + ∇d 2 (λ i+1 ), λ − λ i+1 + c, λ and ∇d 2 (λ i+1 ) ∈ ∂d 2 (λ i+1 ) is a subgradient of d 2 at λ i+1 .
Next, we consider ℓ γ i (λ). We first note that, for any i = 0, · · · , k, we have
d 1 γ (λ i )+ ∇d 1 γ (λ i ), λ−λ i = g(û i+1 )+γp U (û i+1 ) − Aû i+1 , λ i − Aû i+1 , λ − λ i = g(û i+1 ) − Aû i+1 , λ + γp U (û i+1 ).(24)
Second, by Step 6 of Algorithm 1, we haveṽ i ∈ v ♯ (B T λ i+1 ), which implies
d 2 (λ i+1 ) + ∇d 2 (λ i+1 ), λ − λ i+1 = h(ṽ i ) − Bṽ i , λ i+1 − Bṽ i , λ − λ i+1 = h(ṽ i ) − Bṽ i , λ .(25)
Summing up (24) and (25) and using the definition of ℓ γ i , we obtain
ℓ γ i (λ) = g(ũ i )+h(ṽ i )− Aũ i + Bṽ i −c,λ i + c−Aũ i − Bṽ i , λ −λ i +γp U (ũ i ) = f (x i ) − Aũ i + Bṽ i − c, λ + γp U (ũ i ).(26)
By (13), we have d γ (λ) ≤ d(λ) + γD U ≤ d ⋆ + γD U :=d ⋆ γ for any λ ∈ R n . Substituting (26) into (23), subtracting tod ⋆ γ , and summing up the result from i = 0 to k, we obtain
k i=0 η i d ⋆ γ − d γ (λ i+1 ) ≤ k i=0 η i d ⋆ γ − f (x i ) + Aũ i + Bṽ i − c, λ − γp U (ũ i ) + 1 2 λ 0 − λ 2 − λ k+1 − λ 2 .(27)
On the one hand, we note that d(λ) ≤ d ⋆ = f ⋆ ≤ L(x, λ ⋆ ) = f (x) − Au + Bv − c, λ ⋆ for any λ ∈ R n and x ∈ X due to strong duality. Hence,
Aū k + Bv k − c, λ ⋆ ≤ f (x k ) − d ⋆ . Moreover,d ⋆ γ − d γ (λ i+1 ) ≥ 0.
On the other hand, using the
convexity of f we have S k f (x k ) ≤ k i=0 w i f (x i ) and S k Aū k + Bv k − c, λ = k i=0 w i Aũ i + Bṽ i − c, λ for w i := η i .
Combining these expressions into (27), and noting that 0 ≤ p U (ũ i ) ≤ D U , we can derive
0 ≤ k i=0 w i d ⋆ γ − f (x i ) + Aũ i + Bṽ i − c, λ − γp U (ũ i ) + 1 2 λ 0 − λ 2 ≤ S k d ⋆ − f (x k ) + Aū k + Bv k − c, λ + γD U + 1 2 λ 0 − λ 2 , which implies Aū k +Bv k −c, λ ⋆ ≤ f (x k )−d ⋆ ≤ Aū k +Bv k −c, λ + 1 2S k λ 0 −λ 2 +γD U . (28)
Hence, we obtain
Aū k + Bv k − c, λ ⋆ − λ − 1 2S k λ 0 − λ 2 − γD U ≤ 0,(29)
for all λ ∈ R n . Since (29) holds for all λ ∈ R n , we can show that
max λ∈R n Aū k + Bv k − c, λ ⋆ − λ − 1 2S k λ 0 − λ 2 − γD U ≤ 0,(30)
By optimizing the left-hand side over λ ∈ R n and using λ 0 =λ 0 , we obtain
S k Aū k + Bv k − c 2 + 2 Aū k + Bv k − c + r, λ 0 − λ ⋆ − γD U ≤ 0.
Using the Cauchy-Schwarz inequality, we have Aū k +Bv k −c, λ 0 −λ ⋆ ≤ Aū k + Bv k − c λ 0 − λ ⋆ . Hence, the last inequality leads to
Aū k + Bv k − c ≤ λ 0 −λ ⋆ + λ 0 −λ ⋆ 2 +γS k D U S k ≤ 2 λ 0 −λ ⋆ S k + γD U S k .(31)
Now, since w i = η i ≥ γ L d 1 for i = 0 to k, where L d 1 := A 2 µ p . Hence, S k ≥ γ(k+1) L d 1 . Substituting this bound into (31), we obtain the second inequality of (22).
To prove the first inequality of (22), we note from (28) and f ⋆ = d ⋆ that
f (x k ) − f ⋆ ≤ Aū k + Bv k − c, λ + 1 2S k λ 0 − λ 2 + γD U .
Taking λ = 0 n into this inequality, we get
f (x k ) − f ⋆ ≤ 1 2S k λ 0 2 + γD U ≤ L d 1 γ(k + 1) λ 0 2 + γD U .
Combining this inequality, (8), and the second estimate of (22), we obtain the first estimate of (22).
Let us choose γ such that
2L d 1 r 0 γ(k+1) = L d 1 D U k+1 , where r 0 := max λ 0 − λ ⋆ , λ 0 . Then, γ = 2r 0 √ L d 1 √ D U (k+1)
. Substituting this expression into (22), we obtain
|f (x k ) − f ⋆ | ≤ max 2r 0 √ L d 1 D U √ k+1 , 3 λ ⋆ √ L d 1 D U √ k+1 ≤ ǫ Aū k + Bv k − c ≤ 3 √ L d 1 D U √ k+1 ≤ ǫ.
Consequently, we obtain the worst-case complexity of Algorithm 1 from the last estimates, which is O
L d 1 D U ǫ 2 R 2 0 , where R 0 := max 2, 3 λ ⋆ , 2 λ 0 , 2 λ 0 − λ ⋆ .
In this case, we can also show that γ = ǫ 2D U . Remark 1 If we apply a back-tracking line-search with a bi-section strategy on η k , then we have 0 < η k ≤ 2
L γ d 1 at
Step 5 of Algorithm 1. In this case, the bounds in
Theorem 1 still hold with L d 1 = 2 A 2 µ p instead of L d 1 = A 2 µ p . 4
The accelerated primal-dual alternating minimization algorithm In this section, we incorperate Nesterov's accelerated step into Algorithm 1 as done in [9], but applying to (15) to obtain a new accelerated primal-dual AMA variant. Clearly, this algorithm can be viewed as the FISTA scheme [2] applying to the smoothed dual problem (15). Let t 0 := 1 andλ 0 := λ 0 ∈ R n . The main step at the iteration k of the accelerated AMA method is presented as follows:
û k+1 := argmin u∈U g(u) − A Tλ k , u + γp U (u) = ∇g * γ (A Tλ k ), v k+1 := argmin v∈V h(v) − B Tλ k , v + η k 2 c − Aû k+1 − Bv 2 , λ k+1 :=λ k + η k c − Aû k+1 − Bv k+1 , t k+1 := 1 2 1 + 1 + 4t 2 k , λ k+1 := λ k+1 + t k −1 t k+1 λ k+1 −λ k ,(32)
where, again, g γ (·) := g(·) + γp U (·). We now combine the accelerated AMA step (32) and the weighted averaging scheme (9) to construct a new accelerated primaldual AMA method as presented in Algorithm 2 below. Similar to Algorithm 1, if we know the Lipschitz constant L γ d 1 a priori, we can use η k := 1 L γ d 1 . However, we can also use a backtracking line-search to adaptively choose η k := L −1 k such that the condition (20) holds. We note that the complexityper-iteration of Algorithm 2 essentially remains the same as in Algorithm 1.
The following theorem provides the bound on the absolute objective residual and the primal feasibility gap at the iterationx k for Algorithm 2.
Theorem 2 Let {x k } be the sequence generated by Algorithm 2 and L d 1 := A 2 µ p . Then, the following estimates hold:
|f (x k ) − f ⋆ | ≤ max 2L d 1 λ 0 2 γ(k+1)(k+2) + γD U , 8L d 1 λ ⋆ λ 0 −λ ⋆ γ(k+1)(k+2) + λ ⋆ 4L d 1 D U (k+1)(k+2) , Aū k + Bv k − c ≤ 8L d 1 λ 0 −λ ⋆ γ(k+1)(k+2) + 4L d 1 D U (k+1)(k+2) .
(33) Consequently, if we choose γ := ǫ D U , which is optimal, then the worst-case iterationcomplexity of Algorithm 2 to achieve an ǫ-solutionx k of (1) in the sense of Defi-
nition 1 is O √ L d 1 D U ǫ R 0 , where R 0 := max 4, 9 2 λ 0 , 9 2 λ 0 − λ ⋆ , 4 λ ⋆ .
Algorithm 2 (Accelerated primal-dual alternating minimization algorithm) Initialization: 1. Choose γ := ǫ D U , and L such that 0 < L ≤ L γ d 1 := A 2 γµ p . 2. Choose an initial point λ 0 ∈ R n . 3. Set t 0 := 1 andλ 0 := λ 0 . Set S −1 := 0,ū −1 := 0 andv −1 := 0.
for k := 0 to k max do 4. Computeũ k =û k+1 = u * γ (λ k ) defined in (15).
5. Choose η k ∈ 0, 1 L γ d 1 and computê v k+1 := arg min v∈V h(v) − B Tλ k , v + η k 2 Aũ k + Bv − c 2 . 6. Update λ k+1 :=λ k + η k (c − Aû k+1 − Bv k+1 ).
7. Update t k+1 := 0. (7). 9. Update S k := S k−1 + w k , with w k := η k t k , and τ k :
5 1 + (1 + 4t 2 k ) 1/2 andλ k+1 := λ k+1 + t k −1 t k+1 (λ k+1 −λ k ). 8. Computeṽ k := v * (λ k+1 ) ∈ v ♯ (B T λ k+1 ) defined in= w k S k . 10. Updateū k := (1 − τ k )ū k−1 + τ kũ k andv k := (1 − τ k )v k−1 + τ kṽ k . end for Output: The primal sequence x k withx k := (ū k ,v k ).
Proof If we define τ k := 1 t k , then τ 0 = 1, and by Step 7 of Algorithm 2, one has
τ 2 k+1 = (1 − τ k+1 )τ 2 k . Moreover, if we defineλ k := 1 τ k λ k − (1 − τ k )λ k , thenλ 0 =λ 0 = λ 0 . Using Step 7 of Algorithm 2, we can also deriveλ k+1 = 1 τ k+1 λ k+1 − (1 − τ k+1 )λ k+1 ) =λ k − 1 τ k λ k+1 −λ k . By (13), we have d γ (λ) ≤ d(λ)+γD U ≤ d ⋆ +γD U :=d ⋆ γ . Hence,d ⋆ γ −d γ (λ) ≥ 0 for any λ ∈ R n . For i = 0, · · · , k, let ℓ γ i (λ) := d 1 γ (λ i ) + ∇d 1 γ (λ i ), λ −λ i + d 2 (λ i+1 )+ ∇d 2 (λ i+1 ), λ−λ i+1 + c, λ . Then, from (21) with 0 < η i ≤ γL −1 d 1 , and ℓ γ i (λ i ) = d 1 γ (λ i )+ ∇d 1 γ (λ i ), λ i −λ i +d 2 (λ i+1 )+ ∇d 2 (λ i+1 ), λ i −λ i+1 + c, λ ≥ d 1 γ (λ i ) + d 2 (λ i ) + c, λ = d γ (λ i ), we havē d ⋆ γ −d γ (λ i+1 ) ≤d ⋆ γ −ℓ γ i (λ) − η −1 i λ i+1 −λ i , λ−λ i − 1 2η i λ i+1 −λ i 2 , d ⋆ γ − d γ (λ i+1 ) ≤d ⋆ γ −d γ (λ i )−η −1 i λ i+1 −λ i , λ i −λ i − 1 2η i λ i+1 −λ i 2 .(34)
Multiplying the first inequality of (34) by τ i and the second one by (1 − τ i ) for τ i ∈ (0, 1) and summing the results up, we obtain
d ⋆ γ − d γ (λ i+1 ) ≤ (1 − τ i )[d ⋆ γ − d γ (λ i )] + τ i [d ⋆ γ − ℓ γ i (λ)] + 1 η i λ i+1 −λ i ,λ i − (1 − τ i )λ i − τ i λ − 1 2η i λ i+1 −λ i 2 2 = (1 − τ i ) d ⋆ γ − d γ (λ i ) + τ i d ⋆ γ − ℓ γ i (λ) + τ i 2η i λ i − λ 2 − λ i − 1 τ i (λ i+1 −λ i ) − λ 2 ,(35)whereλ i := 1 τ i λ i − (1 − τ i )λ i . Now, letλ i+1 =λ i − 1 τ i (λ i+1 −λ i ) as stated
above. Then, (35) leads tō
d ⋆ γ −d γ (λ i+1 ) ≤ (1−τ i ) d ⋆ γ −d γ (λ i ) +τ i d ⋆ γ −ℓ γ i (λ) + τ 2 i 2η i λ i −λ 2 − λ i+1 −λ 2 . Now, since τ 2 i = (1 − τ i )τ 2 i−1 and η i ≤ η i−1 , we have η i (1−τ i ) τ 2 i ≤ η i−1 τ 2 i−1 . Then, sincē d ⋆ γ − d γ (λ i ) ≥ 0, the last inequality implies η i τ 2 i d ⋆ γ − d γ (λ i+1 ) ≤ η i−1 τ 2 i−1 d ⋆ γ − d γ (λ i ) + η i τ i d ⋆ γ − ℓ γ i (λ) + 1 2 λ i − λ 2 − λ i+1 − λ 2 .
Summing up this inequality from i = 0 to k, and using the fact that τ 0 = 1, we obtain
η k τ k d ⋆ γ − d γ (λ k+1 ) ≤ η 0 (1 − τ 0 ) τ 2 0 d ⋆ γ − d γ (λ k ) + k i=0 η i τ i d ⋆ γ − ℓ γ i (λ) + 1 2 λ 0 − λ 2 − λ k+1 − λ 2 ≤ k i=0 η i τ i d ⋆ γ − ℓ γ i (λ) + 1 2 λ 0 − λ 2 .(36)
Similar to the proof of (26), we have
ℓ γ i (λ) = g(ũ i ) + h(ṽ i ) − Aũ i + Bṽ i − c, λ + γp U (ũ i ).
Next, using the convexity of g and h, and p U (ũ i ) ≥ 0, the last inequality implies
k i=0 η i τ i d ⋆ γ − ℓ γ i (λ) = k i=0 η i τ i d ⋆ γ − g(ũ i ) − h(ṽ i ) + Aũ i + Bṽ i − c, λ − γp U (ũ i ) ≤ S k d ⋆ γ − g(ū k ) − h(v k ) + Aū k + Bv k − c, λ .(37)
Substituting (37) into (36) and noting thatd ⋆
γ ≥ d γ (λ k+1 ), f (x k ) = g(ū k ) + h(v k ) and f ⋆ = d ⋆ =d ⋆ γ − γD U , we have f (x k ) − f ⋆ ≤ Aū k + Bv k − c, λ + 1 2S k λ 0 − λ 2 + γD U .(38)
Moreover, we have f ⋆ ≤ L(x, λ ⋆ ) = f (x) − Au + Bv − c, λ ⋆ for x ∈ X . Substituting x :=x k , u :=ū k and v :=v k into this inequality we get
f ⋆ ≤ f (x k ) − Aū k + Bv k − c, λ ⋆ .(39)
Combining (38) and (39), we obtain
Aū k + Bv k − c, λ ⋆ − λ − 1 2S k λ 0 − λ 2 − γD U ≤ 0, ∀λ ∈ R n .(40)
Hence, by maximizing the left-hand side over λ ∈ R n , we finally get
max λ∈R n Aū k + Bv k − c, λ ⋆ − λ − 1 2S k λ 0 − λ 2 − γD U ≤ 0,
Solving the maximization problem in this inequality, we can show that
Aū k + Bv k − c ≤ 2 λ 0 − λ ⋆ S k + γD U S k .(41)
We note that t k updated by Step 6 satisfies: k+1
2 ≤ t k ≤ k+1, and 0 < η k ≤ γL −1 d 1 . Hence, S k = k i=0 w i = k i=0 t i η i ≥ γ k i=0 i+1 2L d 1 = γ(k+1)(k+2) 4L d 1 .
Using this estimate into (41), we get the second estimate of (33).
To prove the first estimate of (33), we note from (38) with λ := 0 n that
f (x k ) − f ⋆ ≤ 1 2S k λ 0 2 + γD U ≤ 2L d 1 γ(k + 1)(k + 2) λ 0 2 + γD U .
Combining this estimate, the second estimate of (33), and (8), we obtain the first estimate of (33).
Let us choose γ > 0 such that
8L d 1 r 0 γ(k+1)(k+2) = 4L d 1 D U (k+1)(k+2) , where r 0 := max λ 0 , λ 0 − λ ⋆ . Then, γ = 4r 0 √ L d 1 √ D U (k+1)(k+2)
. Substituting this γ into (33), we obtain
|f (x k ) − f ⋆ | ≤ max 9r 0 √ L d 1 D U 2 √ (k+1)(k+2) , 4 λ ⋆ √ L d 1 D U √ (k+1)(k+2) ≤ ǫ Aū k + Bv k − c ≤ 4 √ L d 1 D U √ (k+1)(k+2) ≤ ǫ.
Hence, the worst-case complexity of Algorithm 2 to achieve the ǫ-solutionx k is
O √ L d 1 D U ǫ R 0 , where R 0 := max 4, 9 2 λ 0 , 9 2 λ 0 − λ ⋆ , 4 λ ⋆ .
In this case, we also have γ = ǫ D U . Remark 2 We note that the bounds in Theorems 1 and 2 only essentially depend on the prox-diameter D U of U , but not of V. Since we can exchange g and h in the alternating step, we can choose U or V that has smaller prox-diameter in our algorithms to smooth its corresponding objective.
Application to strongly convex objectives
We assume that either g or h is strongly convex. Without loss of generality, we can assume that g is strongly convex with the convexity parameter µ g > 0 but h remains non-strongly convex, then the dual component d 1 is concave and smooth. Its gradient ∇d 1 (λ) = −Au * (λ) is Lipschitz continuous with the Lipschitz constant L d 1 := A 2 µ g . In this case, we can modified Algorithms 1 and 2 at the following steps to capture this assumption. (7).
-Step 1: Choose L such that 0 < L ≤ L d 1 := A 2 µ g . -Step 4: Computeũ k =û k+1 = u * (λ k ) = u ♯ (A Tλ k ) defined by-Step 5: Choose η k ∈ (0, L −1 d 1 ].
We call this modification the strongly convex variant of Algorithms 1 and 2, respectively. In this case, we obtain the following convergence result, which is a direct consequence of Theorems 1 and 2.
Corollary 1 Let g be strongly convex with the convexity parameter µ g > 0. Assume that x k is the sequence generated by the strongly convex variant of Algorithm 1. Then
|f (x k ) − f ⋆ | ≤ A 2 µ g (k+1) max λ 0 2 , 2 λ ⋆ λ 0 − λ ⋆ , Aū k + Bv k − c ≤ 2 A 2 λ 0 −λ ⋆ µ g (k+1)
.
(42)
Consequently, the worst-case iteration-complexity of this variant to achieve an ǫ-
solutionx k of (1) is O A 2 R 0 µ g ǫ , where R 0 := max λ 0 2 , 2 λ ⋆ λ 0 − λ ⋆ .
Alternatively, assume that x k is the sequence generated by the strongly convex variant of Algorithm 2. Then
|f (x k ) − f ⋆ | ≤ 2 A 2 µ g (k+1)(k+2) max λ 0 2 , 4 λ ⋆ λ 0 − λ ⋆ , Aū k + Bv k − c ≤ 8 A 2 λ 0 −λ ⋆ µ g (k+1)(k+2) .(43)
Consequently, the worst-case iteration-complexity of this variant to achieve an ǫ-
solutionx k of (1) is O A R 0 µ g ǫ , where R 0 := max 2 λ 0 2 , 8 λ ⋆ λ 0 − λ ⋆ .
Remark 3 It is important to note that, even h is not strongly convex, our accelerated primal-dual AMA algorithm still achieves the O(1/ √ ǫ)-worst case iterationcomplexity, which is different from existing dual accelerated schemes [3,11,10,15]. In addition, if h is also strongly convex, then the sharp-operator v ♯ (·) of h V is well-defined and single-valued without requiring Assumption A.2.
We note that our results present in Corollary 1 can be considered as the primaldual variants of the AMA methods in [9], while the result presented in Theorems 1 and 2 is an extension to the non-strongly convex case.
Concluding remarks
We have introduce a new weighted averaging scheme, and combine the AMA idea and Nesterov's smoothing technique to develop new primal-dual AMA methods, Algorithm 1 and Algorithm 2, for solving prototype constrained convex optimization problems of the form (1) without strong convexity assumption. Then, we have incorporated Nesterov's accelerated step into Algorithm 1 to improve the worstcase iteration-complexity of the primal sequence from O 1/ǫ 2 (resp., O (1/ǫ) to O (1/ǫ) (resp., O (1/ √ ǫ). Our complexity bounds are directly given for the primal objective residual and the primal feasibility gap of (1), which are new. Interestingly, the O (1/ √ ǫ)-complexity bound is archived with only the strong convexity of g or h, but not both of them. We will extend this idea to other splitting schemes such as alternating direction methods of multipliers and other sets of assumptions such as the Höder continuity of the dual gradient in the forthcoming work.
A Appendix: The proof of Lemma 1 The concavity and smoothness of d γ 1 is trivial [13]. In addition, the equivalence between the AMA scheme (16) and the forward-backward splitting method was proved in, e.g., [18,9].
Let g U ,γ := g γ + δ U and h V := h + δ V . We first write the optimality condition for the two convex subproblems in (16) as ∇g U ,γ (û k+1 ) − A Tλ k = 0, and ∇h V (v k+1 ) − B Tλ k − η k B T (c − Aû k+1 − Bv k+1 ).
Using the third line of (16) we obtain from the last expressions that ∇g U ,γ (û k+1 ) = A Tλ k , and ∇h V (v k+1 ) = B T λ k+1 , which are equivalent tô u k+1 = ∇g U ,γ * (A Tλ k ), andv k+1 = ∇h V * (B T λ k+1 ).
Multiplying these expressions by A and B, respectively, and adding them together, and then subtracting to c, we finally obtain
η −1 k (λ k+1 −λ k ) = c − Aû k+1 − Bv k+1 = c − A∇g U ,γ * (A Tλ k ) − B∇h V * (B T λ k+1 ).(44)
Now, from the definition (4) of d 1 γ and d 2 , we have A∇g U ,γ * (A Tλ k ) = ∇d 1 (λ k )
and B∇h V * (B T λ k+1 ) = ∇d 2 (λ k+1 ). Substituting these relations into (44), we get
η −1 k (λ k+1 −λ k ) = c − ∇d 1 γ (λ k ) − ∇d 2 (λ k+1 ).(45)
Next, under the condition (20), we can derive
d 1 γ (λ k ) + ∇d 1 γ (λ k ), λ −λ k = d 1 γ (λ k )+ ∇d 1 γ (λ k ), λ k+1 −λ k + ∇d 1 γ (λ k ), λ−λ k+1 = Q γ L k (λ k+1 ;λ k ) + ∇d 1 γ (λ k ), λ − λ k+1 + L k 2 λ k+1 −λ k 2 (20) ≤ d 1 γ (λ k+1 ) + ∇d 1 γ (λ k ), λ − λ k+1 + L k 2 λ k+1 −λ k 2 .(46)
Let ℓ γ k (λ) := d 1 γ (λ k ) + ∇d 1 γ (λ k ), λ −λ k + d 2 (λ k+1 ) + ∇d 2 (λ k+1 ), λ − λ k+1 + c, λ . Using this experesion in (46), and then combining the result with (45) and d γ (·) = d 1 γ (·) + d 2 (·) + c, · , we finally get
ℓ γ k (λ) ≤ d γ (λ k+1 ) + ∇d 1 γ (λ k ) + ∇d 2 (λ k+1 ) − c, λ − λ k+1 + L k 2 λ k+1 −λ k 2 = d γ (λ k+1 ) − η −1 k (λ k+1 −λ k ), λ − λ k+1 + L k 2 λ k+1 −λ k 2 = d γ (λ k+1 ) − η −1 k (λ k+1 −λ k ), λ −λ k − 1 η k − L k 2 λ k+1 −λ k 2 ,
which is the first inequality of (21). The second inequality of (21) follows from the first one, d 1 γ (λ k ) + ∇d 1 γ (λ k ), λ −λ k ≥ d 1 γ (λ) and d 2 (λ k+1 ) + ∇d 2 (λ k+1 ), λ − λ k+1 ≥ d 2 (λ) due to the concavity of d 1 γ and d 2 , respectively.
Primal Solution Recovery in Alternating Minimization Algorithms
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| {'fraction_non_alphanumeric': 0.10227741262642187, 'fraction_numerical': 0.036913972496422594, 'mean_word_length': 3.043145714846048, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 92, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We introduce a new weighted averaging scheme using "Fenchel-type" operators to recover primal solutions in the alternating minimization-type algorithm (AMA) for prototype constrained convex optimization. Our approach combines the classical AMA idea in[18]and Nesterov\'s prox-function smoothing technique without requiring the strong convexity of the objective function. We develop a new non-accelerated primal-dual AMA method and estimate its primal convergence rate both on the objective residual and on the feasibility gap. Then, we incorporate Nesterov\'s accelerated step into this algorithm and obtain a new accelerated primal-dual AMA variant endowed with a rigorous convergence rate guarantee. We show that the worst-case iteration-complexity in this algorithm is optimal (in the sense of first-oder black-box models), without imposing the full strong convexity assumption on the objective.Keywords Alternating minimization algorithm · smoothing technique · primal solution recovery · accelerated first-oder method · constrained convex optimization 1 Introduction This paper studies a new weighted-averaging strategy in alternating minimizationtype algorithms (AMA) to recover a primal solution of the following constrained convex optimization problem:where g : R p 1 → R ∪ {+∞} and h : R p 2 → R ∪ {+∞} are both proper, closed and convex (not necessarily strongly convex), (p 1 + p 2 = p, A ∈ R n×p 1 , B ∈ R n×p 2 , c ∈ R n , and U ⊂ R p 1 and V ⊂ R p 2 are two nonempty, closed and convex sets. Problem (1) surprisingly covers a broad class of constrained convex programs, including composite convex minimization, general linear constrained convex optimization problems, and conic programs.', 'arxivid': '1511.03305', 'author': ['Quoc Tran-Dinh '], 'authoraffiliation': [], 'corpusid': 119576375, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16590, 'n_tokens_neox': 14409, 'n_words': 8706, 'pdfsha': 'effe16d97571053c4abe2cac70f1f0cc2558b6f2', 'pdfurls': ['https://arxiv.org/pdf/1511.03305v1.pdf'], 'title': ['Construction and Iteration-Complexity of Primal Sequences in Alternating Minimization Algorithms', 'Construction and Iteration-Complexity of Primal Sequences in Alternating Minimization Algorithms'], 'venue': []} |
arxiv |
Dynamic prediction and analysis based on restricted mean survival time in survival analysis with nonproportional hazards §
2021
Zijing Yang
Department of Biostatistics
Southern Medical University
GuangzhouChina
Hongji Wu
Department of Biostatistics
Southern Medical University
GuangzhouChina
Yawen Hou
Department of Statistics
Jinan University
GuangzhouChina
Hao Yuan
Department of Biostatistics
Southern Medical University
GuangzhouChina
Zheng Chen
Department of Biostatistics
Southern Medical University
GuangzhouChina
Dynamic prediction and analysis based on restricted mean survival time in survival analysis with nonproportional hazards §
Computer Methods and Programs in Biomedicine
207106155202110.1016/j.cmpb.2021.1061551 * Corresponding author: Zheng Chen § This is the initial submission version of this manuscript. The Version of Record of the manuscript has been published and is available in Computer Methods and Programs in Biomedicine. 2021, 207: 106155, https:// 3survival analysistime-dependent covariatesconditional restricted mean survival timedynamic predictionnonproportional hazards
In the process of clinical diagnosis and treatment, the restricted mean survival time (RMST), which reflects the life expectancy of patients up to a specified time, can be used as an appropriate outcome measure. However, the RMST only calculates the mean survival time of patients within a period of time after the start of follow-up and may not accurately portray the change in a patient's life expectancy over time. The life expectancy can be adjusted for the time the patient has already survived and defined as the conditional restricted mean survival time (cRMST). A dynamic RMST model based on the cRMST can be established by incorporating time-dependent covariates and covariates with time-varying effects. We analysed data from a study of primary biliary cirrhosis (PBC) to illustrate the use of the dynamic RMST model. The predictive performance was evaluated using the C-index and the prediction error. The proposed dynamic RMST model, which can explore the dynamic effects of prognostic factors on survival time, has better predictive performance than the RMST model. Three PBC patient examples were used to illustrate how the predicted cRMST changed at different prediction times during follow-up. The use of the dynamic RMST model based on the cRMST allows for optimization of evidence-based decision-making by updating personalized dynamic life expectancy for patients.
Introduction
Time-to-event outcomes, such as overall survival or progression-free survival, are often used as the primary endpoint for clinical trials in many diseases. In this context, survival curves are estimated by the Kaplan-Meier method, and comparisons are performed by the log-rank test. The hazard ratio (HR) obtained from the Cox proportional hazards (PHs) regression model is used to quantify treatment effects. However, the Cox model must satisfy the PHs assumption that the HR is constant over time, which often fails during long-term follow-up 1,2 . Furthermore, as the ratio of hazard rates (or hazard functions) in the two groups, the HR is difficult to interpret and hard to translate into clinical benefits in terms of a prolonged survival time [3][4][5] . As an alternative, the restricted mean survival time (RMST) is a good summary of the survival distribution, and the treatment effect can be quantified by the difference in the RMST between two treatment groups [6][7][8] .
Generally, after being diagnosed (such as at the time of diagnosis or after a period of treatment), one of the key questions that is often asked by patients is "How long will I live?". This question can be answered by estimating the mean survival time. For example, It is worth noting that patients may want to know their prognosis at any time during follow-up, which requires the continuous prediction of life expectancy at a different
2.Methods
Data sources
This example comes from the PBC data collected by the Mayo Clinic from January 1974 to May 1984. Follow-up was extended to April 30, 1988. A total of 312 patients participated in the study, of whom 158 (50.6%) were randomly assigned to receive D-penicillamine and 154 to receive a placebo. Patients had on average 6.23 visits, resulting in a total of 1945 observations. The outcome of this analysis was overall survival, which was calculated in years from the time of referral to death.
There were nine baseline and time-dependent covariates that were included in the dynamic RMST model. Predictors measured at baseline were the drug (D-penicillamine, placebo), sex (female, male) and age (years). The time-dependent covariates were the serum bilirubin value (mg/dl), edema (yes, no), serum albumin value (g/dl), prothrombin time (seconds), histologic stage of disease (I/II, III, IV) and serum glutamic oxaloacetic transaminase (SGOT) level (U/ml).
Statistical analysis
To obtain the dynamic prediction of
Results
The number of patients used for this analysis was 312, with a median follow-up of 6
Effects of prognostic factors
1 1 0 1 1 ( ) ( / 5) 0.004 0.272 ( / 5), [0,5] j s s s s β β β = = + × =− − × ∈ ,
that is, the md(s,5) between the D-penicillamine group ( 1 ( ) 1 Z s = ) and the placebo group
( 1 ( ) 0 Z s = ).
The change in md(s,5) over time based on the drug is depicted in Fig. 3A. It can be seen that there was no significant difference (95% CI of md(0,5) contains 0) in 5-year life expectancy between patients treated with different drugs when s=0, but the adverse effects of D-penicillamine increased with increasing prediction time s (the upper limit of the 95% CI was less than 0). This may be due to the serious side effects of D-penicillamine, resulting in an increased incidence of adverse events and an increased risk of death 23 .
In addition, serum bilirubin was an important prognostic factor in PBC patients, and high serum bilirubin levels negatively affected the life expectancy of patients (Fig. 3B).
Female patients had a longer life expectancy than male patients (Fig. 3C), which may be due to the relatively larger proportion of older patients (more than 60 years) among males (42.9%) than females (15.2%). The occurrence of edema decreased the 5-year life expectancy, but the effect decreased with increasing prediction time s (Fig. 3D). High albumin levels appeared to have a protective effect with regard to the 5-year life expectancy, with the md(s,5) increasing from the start of follow-up (Fig. 3E)
Individual dynamic prediction
In addition to exploring the dynamic effects of covariates on the 5-year cRMST, another important role of the dynamic RMST model is to provide individual dynamic predictions for patients. Three patients were selected from the dataset analyzed herein (see Table 3 for details). Fig. 4 (the solid lines) shows the 5-year cRMSTs of these patients at different prediction times, as derived from the dynamic RMST model. Patient A visited the clinic at the time of referral (s=0), that is, no time-dependent covariates were generated.
Model assessment
The Table 3 by the dynam ion error in ndicates
Fig. 1
1shows the survival curve of patients with primary biliary cirrhosis (PBC) from a clinical trial 9 , and the area under the entire curve is their mean survival time. However, the mean survival time cannot be estimated unless follow-up is continued until each subject has experienced the event of interest (or in the presence of censoring, until the survival curve has reached zero) 3 . In Fig. 1, the follow-up time of this trial was actually 14.31 years, and it was impossible to observe survival after the end of follow-up. At this time, the area under the survival curve up to 14.31 years can be calculated, that is, the 14.31-year RMST. It is readily interpretable as the mean survival time or "life expectancy" between the start of follow-up and a specific time point ( 14.31)
the 5-year (w=5) cRMST, a set of landmark time points ( l s ) were chosen from the prediction times: in the current model, ( 0,1,...,25) l s l = were selected every 0.2 years from the start of follow-up. For each landmark time point l s , the corresponding landmark dataset l R was constructed by selecting all patients still alive and undergoing follow-up at l s . Then, ˆ( , ) i l m s w , the estimator of the cRMST corresponding to each individual i (i=1,2,…, l n ) in l R , could be calculated (see Supplementary File S1) and used as a dependent variable for a generalized are the values of the covariates at l s . All these models were then combined into a dynamic RMST model: ˆ( , for time-varying covariate effects, interactions between covariates and s were then included in the dynamic RMST model: a one-unit increase in the jth ( 1, 2,...,9) j = covariates at s (i.e., md(s,w)). Initially, all interactions were included in the model, after which the quadratic time Computer Methods and Programs in Biomedicine. 2021, 207: 106155 interactions were tested and removed if they had no significant effect. The covariates with nonsignificant quadratic time interactions were then tested for linear time interactions. Similarly, only the significant interactions were retained. For numeric stability, established for comparison with the dynamic RMST model in application. The predictive performances of different models were evaluated by Harrell's C-index 20 and the prediction error 21 . The C-index measures the probability of concordance between the predicted order and the observed order, while the prediction error is the difference between the predicted value and the observed value. A Monte-Carlo cross-validation was used to avoid overoptimism 22 . The data were divided into a training set (a 70% random sample) and a test set (the remaining 30%). Then, the dynamic RMST model was fitted to the training set and used to predict ( ,5) l m s for these patients who were still at risk at l s in the test set. Performance measures (Harrell's C-index and prediction error) were calculated separately for each l s . The above steps were repeated 200 times to obtain average C-index and prediction error values. All statistical tests were performed at a two-sided significance level of 0.05, and all analyses were performed using R software (version 3.6.1). The data underlying this article are open source and available in the R package 'JM'. Supplementary File S2 details the R code used to perform the process.
. The prothrombin time also demonstrated a significant time-varying effect on the 5-year cRMST, with the md(s,5) decreasing from the start of follow-up but increasing 3 years after the time of referral (Fig. 3F). As expected, advanced histologic stage (III and IV) were associated with a reduced 5-year life expectancy compared with early stage (I/II). However, the md(s,5) between these groups decreased with increasing prediction time (Fig. 3G1-2). In contrast, the RMST model cannot reflect the time-varying effects of these covariates. Furthermore, age and SGOT level demonstrated time-constant effects on the 5-year cRMST in the dynamic RMST model, although the SGOT level was not statistically significant in the RMST model (Z=-1.826, P=0.068).
The 5 -
5year life expectancy of this patient remained basically unchanged (m(0,5)=2.66, m(5,5)=2.93), indicating that her condition was stable(Fig. 4A). Patient B visited the clinic two times (s=0 and s=0.665). In the time between 0 and 0.665 years, the observed values of some variables changed (i.e., time-dependent covariates were generated), which reduced the 5-year life expectancy of this patient(Fig. 4B). Patient C made annual visits to the Mayo Clinic after her initial referral until her death. She had a total of 6 visits, and the observed values of the time-dependent covariates were different at each visit, which had an impact on the patient's survival(Fig. 4C).The (s+5)-year RMSTs calculated by the RMST model are also shown inFig. 4(dashed lines) (e.g., the horizontal axis s=0 corresponds to the 5-year RMST from the start of follow-up, and s=5 corresponds to the 10-year RMST from the start of follow-up). Since only the information at the start of follow-up (s=0) was considered, the trend in the changes in the RMST remains the same under different situations and does not reflect the change in life expectancy over prediction time.
model assessment measures (Harrell's C-index and prediction error) were obtained by the 5-year (w=5) cRMST in the dynamic RMST model from each landmark time point l s (solid lines in Fig. 5). Meanwhile, the predictive performances of these corresponding RMST models ( l s w τ = + ) were also evaluated (dashed lines in Fig. 5). Compared with the RMST models, the advantages of the dynamic RMST model (a higher C-index and a lower prediction error) are more obvious with increasing prediction time s.4. DiscussionSurvival prediction is an indispensable integral part of current clinical practice; it can help determine optimal treatment strategies for individual patients and avoid overtreatment and the associated waste of medical resources. Compared with the survival rate, hazard rate and so on, the RMST is directly based on the concept of time, reflecting the life expectancy of patients up to the specified time, and therefore is a more appropriate evaluation measure24 . In addition, the difference in the RMSTs measures the impact of different treatments on survival and can be a practical and useful alternative to the HR 7,25 . However, the RMST only calculates the mean survival time of patients within a period of time after the start of follow-up (s = 0) and may not accurately portray the change in a patient's life expectancy over time. Taking the perspective of a patient who has already survived a number of years, the cRMST, which is the measure proposed in this article that is based on the RMST, provides more relevant information by adjusting the life expectancy for the time the patient has already survived. In a sense, cRMST can also be understood as the restricted mean residual life 26 . Generally, after considering the concept of condition, the estimated value of measures (such as conditional survival and cRMST) will increase as the number of years survived increases. This relationship is usually even more obvious in patients with advanced-stage disease 27,28 . For example, in this dataset, the 5-year cRMST of patients with histologic stage IV disease was 3.44 years at the time of referral (i.e. m(0,5)=3.44). If the patient was still alive at 3 or even 5 years after referral, the 5-year cRMST would change to 3.93 (m(3,5)=3.93) years and 4.07 (m(5,5)=4.07) years. This means an approximately 0.63-year increase in the 5-year life expectancy of patients who have been followed up for 5 years compared with those who have just been referred. This relationship actually reflects a natural selection effect 29 : due to the existence of individual differences in prognosis, patients with a high risk of death are very likely to experience their endpoint events in the initial years after the start of follow-up. Over time, as these patients expire, the surviving population becomes "healthier" and has a longer life expectancy. The concept of the cRMST is a way to quantify this phenomenon and make it easier for clinicians and patients to comprehend. Therefore, for patients who have been alive for a period of time, the cRMST provides valuable and relevant information on how their life expectancy develops over time. This knowledge can help motivate a patient to continue treatment, improve compliance, and ultimately improve survival. In this paper, based on the cRMST, a dynamic RMST model was established by incorporating time-dependent covariates and allowing for time-varying effects, enabling the updating of the 5-year cRMST for PBC patients at any of the cRMST during follow-up allows for the optimization of evidence-based decision-making and may improve the personalization of the treatment options for patients with progressive disease. In addition, compared with the RMST models that only use the patients' baseline (s=0) risk factors, the dynamic RMST model has better predictive performance, as assessed by the C-index and prediction error. However, we must pay attention to several points when applying the dynamic RMST model. First, the time window w used depends on the severity of the disease. For severe diseases, w=1 or w=2 years is relevant, while for milder diseases with longer follow-up times, such as cirrhosis, w=5 or even w=10 years is reasonable. Second, the selection of landmark time points l s , which implicitly defines the weighting of the prediction time, is independent of the actual event time. The simplest approach is taking these points equidistantly in the selected interval be sufficient 17 . Finally, the functional form, such as the quadratic functions used in this study, of ( ) s α (how to interpret changes over s) and ( ) s β (time-varying covariate effect) should be prespecified in practice.In summary, predicting patient survival is a complex decision-making process involving the patient's own factors, the disease itself, treatment programs, living environment and other factors. Although prediction models can help clinicians improve the accuracy of prediction, the prediction results cannot be blindly accepted. As Lau 30 said, "every patient is unique, one can only observe and not determine the final journey".
Fig
Figure 3
3Figure 3. D
Abbr A hig better
Computer Methods and Programs inBiomedicine. 2021, 207: 106155 prediction times, represented by s. As shown inFig. 2, a PBC patient started follow-up at if the early postoperative period can be successfully survived. Third, some clinical, biochemical and histological indicators (e.g., coagulation indicators) are often measured in subjects at each follow-up visit; these response data give rise to time-dependent covariates (or longitudinal data). Changes in these indicators will also have an impact on life expectancy.In view of this, the continually updating life expectancy or mean survival time depending on the prediction time s is defined as the conditional restricted mean survival0
s s
= and underwent liver transplantation at
1
s s
= . The question "How long will I live?"
is equally pressing at
1
s s
= as it was at the start of follow-up (
0
s s
= ). However, the
patient's life expectancy may vary at different prediction times. First, in the time between
0
s and 1
s , important events have taken place, such as surgical treatment, that may alter a
patient's life expectancy. Second, some variables that have an impact on the outcome may
exhibit time-varying effects, resulting in a change in life expectancy as time progresses 12-14 .
For instance, due to the possibility of postoperative infection and/or transplant rejection, the
life expectancy of this patient will be reduced at
1
s s
= but then greatly improved at
2
s s
=
time (cRMST), represented by m(s,w), that is,
( )
( , )
( )
s w
s
S t dt
m s w
S s
+
=
,
where ( )
S t denotes the survival function, s is the prediction time (more precisely, the time
of the prediction) and w is the time window. For example, m(0,5) represents the life
expectancy of the patient in the next 5 years from the start of follow-up, which is equivalent
to the 5-year RMST, while m(3,5) means the life expectancy in the next 5 years of a patient
who had already survived for 3 years from the start of follow-up. The difference in
cRMSTs between groups is represented by md(s,w). This concept of obtaining/updating the
life expectancy at different prediction times by considering time-dependent covariates and
covariates with time-varying effects is called "dynamic prediction" 15,16 .
To illustrate the clinical applicability of dynamic prediction based on the cRMST, we
utilized a dataset from a well-known clinical study conducted at Mayo Clinic on the
treatment of liver disease 9 . A dynamic prediction model (i.e., dynamic RMST model) was
developed by landmarking 15,17,18 to explore the dynamic effects of prognostic factors on
survival time. Specific patient examples were used to illustrate how the predicted cRMST
changed at different prediction times during follow-up.
.30 years (range: 0.11~14.31 years). During the follow-up period, 140 individuals (55.1%) died.The overall 5-year survival rate was 71.2% (95% CI: 66.3%-76.5%) and 10-year survival
Computer Methods and Programs in Biomedicine. 2021, 207: 106155
rate was 47.9% (95% CI: 41.3%-55.4%).
Table 1
1shows the regression coefficients together with the standard error of the covariates included in the dynamic RMST model, andFig. 3shows the dynamic coefficients (i.e., difference in 5-year cRMST md(s,5) curves (w=5) with 95% confidence intervals). For reference,Table 2describes the results of the RMST model.Drug was not statistically significant in the RMST model (Z=-0.554, P=0.579) in that only the patients' referral or baseline (s=0) values for risk factors were used (Table 2). In contrast, in the dynamic RSMT model, patients treated with D-penicillamine had a lower 5-year life expectancy than those taking the placebo. The dynamic coefficient of this covariate can be calculated by the following formula(Table 1):
Table 1 .
1The results of the dynamic RMST model (w=5 years) Abbreviations: RMST: restricted mean survival time; CI: confidence interval; SGOT: serum glutamic oxaloacetic transaminase.Variable
No. ( Deaths ) Time function a Coefficient
SE
P
(Intercept)
1
7.772
0.541
<0.001
s/5
-13.624
2.047
<0.001
( s/5) 2
11.783
2.094
<0.001
Drug (ref: placebo)
154(69)
D-penicillamine
158(71)
1
-0.004
0.047
0.925
s/5
-0.272
0.094
0.004
Sex (ref: male)
36(26)
Female
276(114)
1
0.221
0.124
0.075
s/5
1.738
0.689
0.012
( s/5) 2
-2.353
0.726
0.001
SerBilir (per 1 mg/dl)
312(140)
1
-0.118
0.010
<0.001
s/5
-0.147
0.051
0.004
(s/5) 2
0.164
0.055
0.003
Edema (ref: no)
247(96)
Yes
65(44)
1
-0.566
0.079
<0.001
s/5
0.311
0.150
0.038
Albumin (per 1 gm/dl)
312(140)
1
0.278
0.074
<0.001
s/5
0.482
0.152
0.001
Prothrombin(per 1 second)
312(140)
1
-0.261
0.043
<0.001
s/5
0.997
0.182
<0.001
( s/5) 2
-0.945
0.188
<0.001
Histologic (ref: Ⅰ/Ⅱ)
83(22)
Ⅲ
120(48)
1
-0.145
0.046
0.001
s/5
0.280
0.104
0.007
Ⅳ
109(70)
1
-0.567
0.060
<0.001
s/5
0.516
0.118
<0.001
Age (per 1 year)
312(140)
1
-0.021
0.002
<0.001
SGOT (per 10 U/ml)
312(140)
1
-0.011
0.002
<0.001
Abbreviations: RMST: restricted mean survival time; No: number; SE: standard error;
SGOT: serum glutamic oxaloacetic transaminase.
a
: The effects for covariates are calculated by the following formula:
2
0
1
2
( )
( / 5)
( / 5)
j
j
j
j
s
s
s
β
β
β
β
=
+
+
,
and the intercept for this model is calculated as
2
0
1
2
( )
( / 5)
( / 5)
s
s
s
α
α α
α
= +
+
.
Table 3 .
3The definition of example patients Abbreviations: SGOT: serum glutamic oxaloacetic transaminase. a : The observation time.Patient
Computer Methods and Programs inBiomedicine. 2021, 207: 106155
Conflict of Interest Statement:The authors declare no competing or financial interest in this work.
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No new clinical data was gathered or used. There is thus no need for an ethical approval. Funding: This research was supported by the National Natural Science Foundation of China. The data in this study is available in R package JM. grant numbers 81673268, 81903411Ethical approval: The data in this study is available in R package JM. No new clinical data was gathered or used. There is thus no need for an ethical approval. Funding: This research was supported by the National Natural Science Foundation of China [grant numbers 81673268, 81903411];
Natural Science Foundation of Guangdong Province [grant number 2018A030313849] and the Guangdong Basic and Applied Basic Research Foundation. grant number 2019A1515011506Natural Science Foundation of Guangdong Province [grant number 2018A030313849] and the Guangdong Basic and Applied Basic Research Foundation [grant number 2019A1515011506].
| {'fraction_non_alphanumeric': 0.04911985127519898, 'fraction_numerical': 0.04217742403996979, 'mean_word_length': 4.311169392162912, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "In the process of clinical diagnosis and treatment, the restricted mean survival time (RMST), which reflects the life expectancy of patients up to a specified time, can be used as an appropriate outcome measure. However, the RMST only calculates the mean survival time of patients within a period of time after the start of follow-up and may not accurately portray the change in a patient's life expectancy over time. The life expectancy can be adjusted for the time the patient has already survived and defined as the conditional restricted mean survival time (cRMST). A dynamic RMST model based on the cRMST can be established by incorporating time-dependent covariates and covariates with time-varying effects. We analysed data from a study of primary biliary cirrhosis (PBC) to illustrate the use of the dynamic RMST model. The predictive performance was evaluated using the C-index and the prediction error. The proposed dynamic RMST model, which can explore the dynamic effects of prognostic factors on survival time, has better predictive performance than the RMST model. Three PBC patient examples were used to illustrate how the predicted cRMST changed at different prediction times during follow-up. The use of the dynamic RMST model based on the cRMST allows for optimization of evidence-based decision-making by updating personalized dynamic life expectancy for patients.", 'arxivid': '2106.10625', 'author': ['Zijing Yang \nDepartment of Biostatistics\nSouthern Medical University\nGuangzhouChina\n', 'Hongji Wu \nDepartment of Biostatistics\nSouthern Medical University\nGuangzhouChina\n', 'Yawen Hou \nDepartment of Statistics\nJinan University\nGuangzhouChina\n', 'Hao Yuan \nDepartment of Biostatistics\nSouthern Medical University\nGuangzhouChina\n', 'Zheng Chen \nDepartment of Biostatistics\nSouthern Medical University\nGuangzhouChina\n'], 'authoraffiliation': ['Department of Biostatistics\nSouthern Medical University\nGuangzhouChina', 'Department of Biostatistics\nSouthern Medical University\nGuangzhouChina', 'Department of Statistics\nJinan University\nGuangzhouChina', 'Department of Biostatistics\nSouthern Medical University\nGuangzhouChina', 'Department of Biostatistics\nSouthern Medical University\nGuangzhouChina'], 'corpusid': 235216805, 'doi': '10.1016/j.cmpb.2021.106155', 'github_urls': [], 'n_tokens_mistral': 10716, 'n_tokens_neox': 8893, 'n_words': 5275, 'pdfsha': '2f466a4a916171b2792c8082c43dc352f9ecf9b7', 'pdfurls': ['https://arxiv.org/pdf/2106.10625v1.pdf'], 'title': ['Dynamic prediction and analysis based on restricted mean survival time in survival analysis with nonproportional hazards §', 'Dynamic prediction and analysis based on restricted mean survival time in survival analysis with nonproportional hazards §'], 'venue': ['Computer Methods and Programs in Biomedicine']} |
arxiv |
More on cosmological constraints on spontaneous R-symmetry breaking models
14 Jan 2014
Yuta Hamada
Department of Physics
Kyoto University
606-8502KyotoJapan
Kohei Kamada
Deutsches Elektronen-Synchrotron DESY
Notkestrasse 85D-22607HamburgGermany
Institut de Théorie des Phénomènes Physiques
École Polytechnique Fédérale de Lausanne
CH-1015LausanneSwitzerland
Tatsuo Kobayashi
Department of Physics
Kyoto University
606-8502KyotoJapan
Yutaka Ookouchi
Faculty of Arts and Science
Kyushu University
819-0395FukuokaJapan
More on cosmological constraints on spontaneous R-symmetry breaking models
14 Jan 2014
We study the spontaneous R-symmetry breaking model and investigate the cosmological constraints on this model due to the pseudo Nambu-Goldstone boson, R-axion. We consider the R-axion which has relatively heavy mass in order to complement our previous work. In this regime, model parameters, R-axions mass and R-symmetry breaking scale, are constrained by Big Bang Nucleosynthesis and overproduction of the gravitino produced from R-axion decay and thermal plasma. We find that the allowed parameter space is very small for high reheating temperature. For low reheating temperature, the U(1) R breaking scale f a is constrained as f a < 10 12−14 GeV regardless of the value of R-axion mass.Recent discovery of a Higgs boson at the LHC [1] may suggest that the stop mass is around O(10) TeV without introducing an artificial new mechanism [2], in particular, in models of gauge mediation, which is the main topic in this paper. Although it is not at the right scale for solving the naturalness problem, 1 other good points of SUSY encourage us to study it further. Therefore, we focus on relatively high-scale SUSY-breaking scenarios in this paper.If high-scale SUSY is realized in nature, it would be interesting to seek for a connection between the SUSY breaking scale and cosmological observations, which are quite useful tools to probe the high-scale physics beyond the TeV scale. Among many other scenarios of high-scale SUSY breaking, gauge mediation models with spontaneously broken R-symmetry, which is a specific symmetry in supersymmetric theories, is one of the models that has recently experienced striking progress on model building[4], (see for reviews[5,6,7]). As is emphasized in Refs.[8,9,10,11], R-symmetry opens up interesting windows into the connection between SUSY breaking and cosmological aspects. In particular, cosmology with the Nambu-Goldstone boson, called R-axion, which is generated and acquires a mass term in coupling to the gravity theory because the constant term in superpotential breaks R-symmetry explicitly is an interesting working place; R-axions are produced at some time in the cosmic history and their decays may affect the standard cosmological scenario, which, in turn, constrains the model parameters[8].Depending on its mass scale, various decay modes are allowed. In our previous study [8], we focused on relatively light and long-lived R-axions since such parameter regions are favored in the context of "low-scale gauge mediation"[12], where various cosmological constraints including the Big Bang Nucleosynthesis (BBN), the Cosmic Microwave Background (CMB), cosmic γ-ray and the re-ionization can be imposed in the late epoch of the expanding universe. In addition to these constraints, we here point out that the abundance of heavier R-axions with shorter lifetime, which can explain the 125 GeV Higgs more easily, can be constrained in a wide range of parameter space by two cosmological constraints: One is coming from hadronic decays of R-axions. When the mass scale of the R-axion is much larger than O(1) 1 In a certain scenario, a heavy stop mass such as several TeV is still natural[3].
Introduction
Supersymmetry (SUSY) is one of the most promising candidates of the physics beyond the standard model (SM) because it can relax the naturalness problem and suggests the gauge coupling unification. It is also widely believed to be one of the key ingredients for constructing a consistent string theory encompassing the SM. Since SUSY has not been observed in experiments yet, it has to be broken somewhere between the weak scale and the Planck scale.
GeV, the R-axion can efficiently decay into various hadrons. For sufficiently heavy R-axions, they immediately turn to hadronic jets, which affects successful BBN. This phenomenon can constrain the parameter space of R-axions. The other constraint comes from gravitino production. In this short note, we mainly focus on the regime in which gauge mediation is the dominant contribution to the mediation of SUSY breaking. In this case, the gravitino is a stable particle and can be over-produced via R-axion decay. In fact, as we will show below, overproduction of such gravitinos yields a condition which is complementary to the one for thermal production of gravitino.
In this paper, we investigate cosmological effects of heavy R-axion whose mass scale is not covered in the previous work [8]. Especially, we focus the mass scale heavier than 3 GeV to avoid subtlety of non-relativistic decay into pions which requires careful treatment because it does not necessarily destroy the light elements constructed by BBN. Also, we assume that all superparticles (except gravitino) are heavier than the R-axion because R-axion decay into superparticles requires highly model dependent argument. If the decay channel into superparticles of R-axions opens, the constraint would become more severe. The organization of this paper is as follows: In section 2, we briefly review the R-symmetry breaking model. Then, we examine the lifetime of the R-axion and the branching ratio of hadronic decays. In section 3, we firstly review the mechanisms of the R-axion production and its abundance. Then we impose constraints on R-axion abundance from the BBN and gravitino overproduction. As we will show that the R-axion mass and R-symmetry breaking scale are severely constrained. Section 4 is devoted to summary and discussion.
Hadronic decay of R-axion
In this section, we briefly review the simple model with spontaneously broken R-symmetry studied in Ref. [8]. Let us focus on the R-charged light SUSY-breaking field, X, and consider a low-energy scale where all other fields including messenger fields are integrated out. The effective superpotential is, then, assumed to be
W eff = Λ 2 eff X + W 0 . (2.1)
Here the constant term W 0 is required to realize vanishing cosmological constant. Note that this class of models is common in various F -term supersymmetry breaking models [4]. We assume non-canonical Kähler potential yielding the following potential,
V (X) = λ 4 |X| 2 − f 2 a 2 − 2 W 0 Λ 2 eff M 2 pl X + h.c. + · · · ,(2.2)
which realizes a potential minimum with spontaneously broken R-symmetry. Here M pl is the reduced Planck mass. The first term comes from our assumption about Kähler potential. f a is turned out to be the "axion decay constant" and we assume f a ≪ M pl . The second term appears from the Planck-suppressed interaction in supergravity theory. It breaks R-symmetry explicitly and generates the mass term for the R-axion, the phase component of the X field.
In coupling to the gravity, vanishing cosmological constant requires
Λ 4 eff = 3W 2 0 /M 2 pl . (2.3)
Though we have introduced an ad-hoc Kähler potential for the SUSY-braking vacuum with spontaneously broken R-symmetry, we believe that it is a simple toy model which reveals various aspects in spontaneous R-symmetry breaking models 2 [4]. Dividing X into VEV and fluctuation,
X = s + √ 2f a √ 2 exp(ia/ √ 2f a ),(2.4)
we have the potential for s and a as
V (s, a) = λ 2 f 2 a s 2 − 4 W 0 Λ 2 eff M 2 pl f a cos a √ 2f a + 1 2 √ 2 m 2 a f a s cos a √ 2f a . (2.5)
Hereafter we call s R-saxion and a R-axion. From the potential, we can read off the R-axion mass m a and R-saxion mass m s as
m 2 a = 2W 0 Λ 2 eff f a M 2 pl , m 2 s = λf 2 a . (2.6)
We expect that R-symmetry breaking potential Eq.(2.2) would be related to SUSY breaking and
V (0) ≃ λf 4 a ≃ F X 2 = Λ 4 eff . (2.7)
With this assumption, we find the relation of m a and m s as
m 2 s ≃ M pl f a m 2 a . (2.8)
Meanwhile the gravitino mass m 3/2 is written by
m 2 3/2 ≃ f a 2 √ 3M pl m 2 a . (2.9)
Since the gravitino is lighter than R-axion while R-saxion is heavier than R-axion, R-axion can decay into gravitinos but not R-saxions. Now we evaluate the interactions of R-axions, which is necessary to investigate the production and decay rate of R-axions. First, let us consider the interaction with the gauge fields.
R-axions couple with the SM gauge fields through anomaly couplings,
C i g 2 i 32π 2 f a aF G i µνF G i µν , with C i = Tr U(1) R G 2 i ,(2.10)
where G i for i = 3, 2 and 1 represent the SM gauge groups SU(3) C , SU(2) L and U(1) Y , respectively and g i s are the corresponding gauge couplings. From these couplings, we obtain the decay rates of R-axion to each pair of gauge bosons as 11) where θ w is the Weinberg angle and m Z and m W are the Z-boson and W-boson masses, respectively. Since the anomaly coefficients are model-dependent parameters of the order of the unity, we take C i = 1 for all G i in the following. Taking other values of the order of the unity does not change our results significantly.
Γ(a → 2g) = C 2 3 2π g 3 4π 4 m a f a 2 m a , Γ(a → 2γ) = (C 2 sin 2 θ w g 2 2 + C Y cos 2 θ w g 2 Y ) 2 16π(4π) 4 m a f a 2 m a , Γ(a → 2Z) = 1 16π(4π) 4 (C 2 cos 2 θ w g 2 2 + C Y sin 2 θ w g 2 Y ) 2 m a f a 2 m a 1 − 4m 2 Z m 2 a 3/2 , Γ(a → 2W ) = C 2 2 8π g 2 4π 4 m a f a 2 m a 1 − 4m 2 W m 2 a 3/2 , Γ(a → γZ) = cos 2 θ w sin 2 θ w 8π(4π) 4 (C 2 g 2 2 − C Y g 2 Y ) 2 m a f a 2 m a 1 − m 2 Z m 2 a 3 ,(2.
Secondly, the R-axion can also couple with the SM fermions through the mixing between the R-axion and the Higgs bosons [12]. Couplings with up type quarks, down type quarks, charged leptons and neutrinos are expressed as the effective interactions, λ f af γ 5 f, with the coupling constants
λ u = i m u f a κ cos 2 β, λ d = i m d f a κ sin 2 β, λ ℓ = i m ℓ f a κ sin 2 β, λ ν = i m ν f a κ cos 2 β, (2.12) where κ = v/( √ 2f a ) with v = 246
GeV and m f denotes the mass of each fermion f . tan β is the ratio of the vacuum expectation values of the up-type Higgs boson H u and the down-type Higgs boson H d . From these couplings, R-axions can decay into a pair of fermions with the decay rates,
Γ(a → ff ) = |λ f | 2 8π m a 1 − 4m 2 f /m 2 a 1/2 × 3 for f = u, d 1 for f = l, ν . (2.13)
Note that tan β as well as other parameters such as the stop mass determines the Higgs mass.
For the 125 GeV Higgs, tan β 10 is favored for the stop mass with a few TeV [2]. Finally, R-axions can decay to a pair of gravitinos through supergravity effect,
W * M 2 pl ψ µ σ µν ψ ν + h.c. ∋ −i Λ 2 eff a √ 2M 2 pl ψ µ σ µν ψ ν + h.c. (2.14)
Decay rate is given by
Γ(a → 2ψ) ≃ 1 2 √ 3π m 3 a M pl f a ,(2.15)
for m a ≫ m 3/2 . Here we have used the relation Λ 2 eff = 3 1/4 f a M pl /2 m a and taken into account that for the light gravitino, the decay into the spin 1/2 goldstino component ψ with ψ µ ∼ i 2/3 ∂ µ ψ/m 3/2 dominates over the decay into the spin 3/2 component [13].
We plot the lifetime τ a of the R-axion for tan β = 30 in Fig.1. For 3GeV m a 8GeV, decay into tau pairs dominates; for 8GeV m a 100GeV, decay into bottom pairs dominates; and for m a 100GeV, decay into gluon or gravitino pairs dominates. Note that for large f a , the decaying into the gravitino pairs dominates other channels for the heavy R-axion mass since the suppression factor f a /M pl is not so small that overwhelms the loop factors. From this figure, We can see that the lifetime is shorter than 10 13 s for m a 3 GeV.
In the case that the energy injected quarks are high enough, right after the axion decay, quarks and gluons immediately turn into hadronic jets 3 . Hence, the process does not depend on the first particles created by axion decay. For the BBN constraint, then, only the branching ratio decaying into hadronic particles determines the constraint. We present the hadronic branching ratio of the R-axion B h for tan β = 30 in Fig.2, where B h is the sum of the branching ratio of the R-axion to colored particles. B h becomes small if we take larger f a since the branching ratio of the R-axion decay into gravitinos becomes sizable. It is
found that B h becomes constant for large m a 10 3 GeV, Γ(a → 2g)/(Γ(a → 2g) + Γ(a → 2ψ)) ∼ min.{1, (g 3 /4π) 4 M pl /f a }. Numerically B h is of O(10 −1 ) for f a ≃ 10 16 GeV. For
10GeV m a 10 2 GeV, decay channel into bottom pairs dominates the total decaying ratio, we have B h ∼ 1 regardless of f a . For 3GeV m a 10 GeV, B h becomes small once more because the decay channel to taus dominates the total decay ratio. In this range B h is of O(10 −2 ). Summarizing the above, B h is bigger than 10 −2 for m a 3 GeV, f a 10 16 GeV.
Thus, the constraint with B h = 1 gives the stringent constraint whereas that with B h = 10 −3 gives the conservative one, as we will see in the next section. 3 Cosmological constraints on R-axion abundance
R-axion production
Let us consider the R-axion cosmology. First we study the R-axion production. The R-axion production depends on cosmological scenarios, and we here simply suppose that the U(1) R breaking occurs after inflation. In this case, cosmic R-string forms at the time of the phase transition [9]. The cosmic R-strings enter the scaling regime quickly, and the cosmic string loops that are produced continuously emit R-axions, which is the first source of R-axions [14].
Gradually the explicit U(1) R breaking effect can no longer be neglected, and string network If we take f a < 10 13 GeV, we obtain the value of B h which is the same as one for f a = 10 13 GeV.
turns into string-wall system, which is unstable and decay to R-axions immediately [15]. This is the second source of R-axions. At the same time, coherent oscillation of R-axions also starts, which is the third source of R-axions [16]. In addition to these R-axion production from the dynamics, there are thermal production [17] and production from R-saxion decay of R-axions. Note that it can be shown that production from R-saxion decay is negligible (see Appendix of Ref. [8]). Here we summarize the R-axion production from the dynamics (the coherent oscillation, the decay of cosmic string, decay of the string-wall system) and thermal bath referring the result of Ref. [8]. The R-axion abundance produced from dynamics is given by
ρ a,dyn s ≃ 9.4 × 10 −7 GeV m a 10GeV 1/2 f a 10 10 GeV 2 for H osc < H R 1.7 × 10 −10 GeV f a 10 10 GeV 2 T R 10 6 GeV for H osc > H R ,(3.1)
where T R is reheating temperature. H R and H osc are Hubble parameters evaluated at the time of reheating and of beginning of the R-axion oscillation, respectively. Note that the result changes only by numerical factors, but parameter dependence does not change if we consider the scenario without cosmic string formation but only with the coherent oscillation of R-axion [8]. The abundance of R-axions produced thermally is given by where T D = 10 6 GeVg −6 3 C −2 3 (f a /10 10 GeV) 2 is the decoupling temperature. This value is fixed when R-axions become non relativistic. Note that for T R > T D , R-axions are once thermalized and the R-axion abundance becomes independent of the reheating temperature.
Constraints for parameter space
The standard BBN scenario can explain the light elements in the present Universe elegantly.
However, if massive exotic particle decays occur during or after BBN epoch, light elements would be broken by the decay products, which would abandon the successful BBN [18,19]. In particular, if the decay includes hadronic decay with hadronic jets, a lot of 4 He's are destroyed and 3 He, D, and T are produced from 4 He dissociation, which gives much stringent constraint. Hence the amount of hadronic decay product must be small enough, and in turn, R-axion abundance is constrained. The abundance of R-axions is constrained with respect to their decay rate and hadronic branching ratio. For B h = 10 −3 , this constraint [19] is given by
ρ a s
10 −17/2 (τ a /1s) −5/2 GeV for 10 −1 s < τ a < 1s 10 −17/2 GeV for 1s < τ a < 10 2 s 10 −6 (τ a /1s) −5/4 GeV for 10 2 s < τ a < 10 4 s 10 −11 GeV for 10 4 s < τ a < 10 6 s 10 −2 (τ a /1s) −3/2 GeV for 10 6 s < τ a < 10 8 s 10 −14 GeV for 10 8 s < τ a < 10 10 s .
(3.3)
The constraint for B h = 1 [19] is also given by
ρ a s
10 −16 (τ a /1s) −5 GeV for 10 −2 s < τ a < 10 −1 s 10 −23/2 (τ a /1s) −1/2 GeV for 10 −1 s < τ a < 10s 10 −12 GeV for 10s < τ a < 10 2 s 10 −10 (τ a /1s) −1 GeV for 10 2 s < τ a < 10 4 s 10 −14 GeV for 10 4 s < τ a < 10 5 s 10 −33/2 (τ a /1s) 1/2 GeV for 10 5 s < τ a < 10 7 s 10 −6 (τ a /1s) −1 GeV for 10 7 s < τ a < 10 8 s 10 −14 GeV for 10 8 s < τ a < 10 10 s .
(3.4)
Since we have seen B h 10 −2 from Fig. 2, we obtain the conservative bound if we use (3.3). The larger B h is, the more severe constraint becomes. Then, (3.4) for B h = 1 gives more severe bound than (3.3). Note that B h = 1 is the good approximation for m a 10 GeV and f a 10 13 GeV. Later, we will show the results of both cases.
There is another cosmological constraint on the R-axion abundance. Since R-axions can decay into the stable gravitinos, their abundance from R-axion decay may overwhelm the present abundance of dark matter (DM). Since the gravitino abundance should not exceed that of DM, we have the following constraint as 3.5) where Br ψ is branching ratio of gravitino. Note that the present gravitino abundance has a suppression factor Br ψ (2m 3/2 /m a ), since a R-axion decays into two relativistic gravitinos with the total energy m a and gradually becomes nonrelativistic. Since this constraint comes from the present Universe, it exists regardless of the lifetime of R-axion if τ a < τ 0 . We should also note the constraint from the gravitino produced thermally. The thermally produced gravitino abundance is given by [21]
Br ψ 2m 3/2 m a ρ a s < 4.7 × 10 −10 GeV Ω m h 2 0.13 ,(ρ (th) 3/2 s ≃ 6.3 × 10 −10 GeV mg 10 TeV 2 m 3/2 10 GeV −1 T R 10 6 GeV ,(3.6)
which must be smaller than, again, 4.7 × 10 −10 GeV(Ω m h 2 /0.13), where mg is the gluino mass. Therefore, we need relatively large gravitino mass to avoid the overclosure problem, depending on the gravitino mass. In other words, relatively large f a and m a are required, see
Eq. (2.9). In Fig. 3 and Fig. 4, we present the cosmological constraints on model parameters discussed above, in terms of m a and f a with various cases of the reheating temperature. Here we have set mg = 10TeV. If the reheating temperature is high, T R ≃ 10 6 GeV, thermal production of R-axion is so large that the gravitino abundance from the R-axion decay exceeds that of DM for m a 10 3 GeV and f a 10 9 GeV. Since the gravitinos produced thermally overclose the Universe for light gravitinos, which means small m a and f a , the allowed parameter space is very small. Allowed parameter region enlarges for lower reheating temperature. In particular, the gravitino overclosure problem is almost absent for T R < 1 GeV. In this case, model parameters are constrained only by the BBN. We can see that R-symmetry breaking scale f a is constrained from above regardless of the reheating temperature, f a 10 12−14 GeV. Note that we find that the constraint would not change so much if we use the precise value of B h (between 10 −3 to 1) by comparing the constraints of B h = 1 and 10 −3 .
We comment on the generality of the constraints shown in Fig. 3 and Fig. 4. R-axions couple with SM gauge fields through anomaly couplings and couple with the SM fermions through the mixing between the R-axion and the Higgs bosons which comes from B-term in Higgs potential. These couplings are general in spontaneous R-symmetry breaking models up to numerical factor. On the other hand, it would be possible to change the gravitino mass relation Eq. (2.9) and the R-axion gravitino coupling Eq. (2.15) if we consider more complicated superpotential. In order to avoid the constraint from gravitinos, we need a model to change Eq. (2.9) or Eq. (2.15).
It would be interesting to further explore heavier R-axion such as m a 10TeV. In this case, various decay modes into superparticles are open. Thus, arguments become highly model dependent. We will study some examples elsewhere.
Summary and discussions
In this paper, we considered the spontaneous R-symmetry breaking model and investigated the cosmological constraints of heavy R-axion which can decay to hadrons. This work complements our previous one [8]. We estimated the abundance of the R-axion produced by decay of R-string and domain wall, vacuum misalignment and thermal plasma. Such abundance is constrained by BBN and the gravitino overproduction. We showed cosmological constraints on model parameters, R-axion mass and R-symmetry breaking scale. As a result, we found that U(1) R breaking scale is constrained as f a < 10 12−14 GeV for low reheating temperature regardless of the value of R-axion mass. For high reheating temperature, BBN, gravitinos from R-axion decay, and gravitinos from thermal plasma constrain different parameter regions and the allowed parameter space is very small. In conclusion, even in the heavy R-axion regime, it has poor compatibility with relatively high reheating temperature T R 10 6 GeV. The constraints we showed in this paper can apply to many spontaneous R-symmetry breaking models and are important for phenomenological model building.
Finially it would be useful to re-interpret out results shown above as a constraint for messenger scale by using a simple gauge mediation model, taking the following simple messenger sector,
W mess = λ ′ XΦΦ + M ΦΦ Φ,(4.1)
where Φ andΦ represent messenger fields. In this set up, the stop mass m 0 is given by
m 0 = α s 4π λ ′ √ λ f 2 a M Φ = α s 4π λ ′ m a M pl f a M Φ . (4.2)
In the second line, we used Eq.(2.6) and Eq. (2.8). From this expression we obtain the following formula, (4.3)
In order to realize the appropriate Higgs mass, we take m 0 ≃ 10 TeV [20]. Therefore, from Eq.(4.3) M Φ 10 10−11 GeV is required for m a ≃ 10 GeV, λ ′ ≃ 1 to obtain f a 10 12−14 GeV. Colored region is excluded by BBN or gravitino overproduction. The abundance of the gravitino produced from thermal plasma(R-axion decay) exceeds that of DM in the blue(red) region. Here we have set mg = 10TeV.
Figure 1 :
1R-axion lifetime τ a with various values of f a . Yellow, blue, purple and red lines correspond to f a = 10 6 GeV, 10 10 GeV, 10 13 GeV, and 10 16 GeV, respectively. Here we use tan β = 30.
Figure 2 :
2Hadronic branching ratio of R-axion. Purple and red lines correspond to f a = 10 13 GeV, and 10 16 GeV. Here we use tan β = 30.
2
26 × 10 −2 GeV m a 10GeV for T R > T D 2.0 × 10 −GeVg GeV for T R < T D ,(3.2)
Figure 3 :
3Cosmological constraints on the model parameters, m a and f a with T R = 10 −2 GeV, 1 GeV, 10 3 GeV and 10 6 GeV. Here we use the BBN constraint (3.3) for B h = 10 −3 . Colored region is excluded by BBN or gravitino overproduction. The abundance of the gravitino produced from thermal plasma(R-axion decay) exceeds that of DM in the blue(red) region. Here we have set mg = 10TeV.
Figure 4 :
4Cosmological constraints on the model parameters, m a and f a with T R = 10 −2 GeV, 1 GeV, 10 3 GeV and 10 6 GeV. Here we use the BBN constraint(3.4) for B h = 1.
As we will see later, in order to put cosmological constraints, we need to know the interactions and the abundance of R-axions. In spontaneous R-symmetry breaking models, these are characterized by R-axion mass m a and decay constant f a for any Kähler potential. Therefore, taking m a and f a as free parameters, our discussion of ad-hoc Kähler potential is applicable to many spontaneous R-symmetry breaking models where the R-symmetry breaking is realized by general non-canonical Kähler potential.
If the axion mass is around GeV, axion dominantly decay into non-or semi-relativistic pions. In this case the detail of the decay process matters and the analysis is subtle and complicated, and hence we do not consider such cases.
AcknowledgmentThe authors would like to thank M. Ibe for useful comments and discussions. Especially, we would like to thank K. Kohri for his lectures on the Big Bang Nucleosynthesis and for crucial comments on hadronic decay of R-axion. The work of Y.
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| {'fraction_non_alphanumeric': 0.07757785047861383, 'fraction_numerical': 0.07339755240518599, 'mean_word_length': 3.555402235407755, 'pattern_counts': {'":': 0, '<': 36, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 31, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We study the spontaneous R-symmetry breaking model and investigate the cosmological constraints on this model due to the pseudo Nambu-Goldstone boson, R-axion. We consider the R-axion which has relatively heavy mass in order to complement our previous work. In this regime, model parameters, R-axions mass and R-symmetry breaking scale, are constrained by Big Bang Nucleosynthesis and overproduction of the gravitino produced from R-axion decay and thermal plasma. We find that the allowed parameter space is very small for high reheating temperature. For low reheating temperature, the U(1) R breaking scale f a is constrained as f a < 10 12−14 GeV regardless of the value of R-axion mass.Recent discovery of a Higgs boson at the LHC [1] may suggest that the stop mass is around O(10) TeV without introducing an artificial new mechanism [2], in particular, in models of gauge mediation, which is the main topic in this paper. Although it is not at the right scale for solving the naturalness problem, 1 other good points of SUSY encourage us to study it further. Therefore, we focus on relatively high-scale SUSY-breaking scenarios in this paper.If high-scale SUSY is realized in nature, it would be interesting to seek for a connection between the SUSY breaking scale and cosmological observations, which are quite useful tools to probe the high-scale physics beyond the TeV scale. Among many other scenarios of high-scale SUSY breaking, gauge mediation models with spontaneously broken R-symmetry, which is a specific symmetry in supersymmetric theories, is one of the models that has recently experienced striking progress on model building[4], (see for reviews[5,6,7]). As is emphasized in Refs.[8,9,10,11], R-symmetry opens up interesting windows into the connection between SUSY breaking and cosmological aspects. In particular, cosmology with the Nambu-Goldstone boson, called R-axion, which is generated and acquires a mass term in coupling to the gravity theory because the constant term in superpotential breaks R-symmetry explicitly is an interesting working place; R-axions are produced at some time in the cosmic history and their decays may affect the standard cosmological scenario, which, in turn, constrains the model parameters[8].Depending on its mass scale, various decay modes are allowed. In our previous study [8], we focused on relatively light and long-lived R-axions since such parameter regions are favored in the context of "low-scale gauge mediation"[12], where various cosmological constraints including the Big Bang Nucleosynthesis (BBN), the Cosmic Microwave Background (CMB), cosmic γ-ray and the re-ionization can be imposed in the late epoch of the expanding universe. In addition to these constraints, we here point out that the abundance of heavier R-axions with shorter lifetime, which can explain the 125 GeV Higgs more easily, can be constrained in a wide range of parameter space by two cosmological constraints: One is coming from hadronic decays of R-axions. When the mass scale of the R-axion is much larger than O(1) 1 In a certain scenario, a heavy stop mass such as several TeV is still natural[3].', 'arxivid': '1310.0118', 'author': ['Yuta Hamada \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n', 'Kohei Kamada \nDeutsches Elektronen-Synchrotron DESY\nNotkestrasse 85D-22607HamburgGermany\n\nInstitut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland\n', 'Tatsuo Kobayashi \nDepartment of Physics\nKyoto University\n606-8502KyotoJapan\n', 'Yutaka Ookouchi \nFaculty of Arts and Science\nKyushu University\n819-0395FukuokaJapan\n'], 'authoraffiliation': ['Department of Physics\nKyoto University\n606-8502KyotoJapan', 'Deutsches Elektronen-Synchrotron DESY\nNotkestrasse 85D-22607HamburgGermany', 'Institut de Théorie des Phénomènes Physiques\nÉcole Polytechnique Fédérale de Lausanne\nCH-1015LausanneSwitzerland', 'Department of Physics\nKyoto University\n606-8502KyotoJapan', 'Faculty of Arts and Science\nKyushu University\n819-0395FukuokaJapan'], 'corpusid': 119180057, 'doi': '10.1088/1475-7516/2014/01/024', 'github_urls': [], 'n_tokens_mistral': 13200, 'n_tokens_neox': 10578, 'n_words': 5796, 'pdfsha': '652c2e323a5a3fc14417e40816161525f60f16b1', 'pdfurls': ['https://arxiv.org/pdf/1310.0118v2.pdf'], 'title': ['More on cosmological constraints on spontaneous R-symmetry breaking models', 'More on cosmological constraints on spontaneous R-symmetry breaking models'], 'venue': []} |
arxiv |
WEB TABLE EXTRACTION, RETRIEVAL AND AUGMENTATION: A SURVEY A PREPRINT
February 6, 2020
Shuo Zhang [email protected]
University of Stavanger
University of Stavanger
Krisztian Balog [email protected]
University of Stavanger
University of Stavanger
WEB TABLE EXTRACTION, RETRIEVAL AND AUGMENTATION: A SURVEY A PREPRINT
February 6, 2020Table extraction · table search · table retrieval · table mining · table augmentation · table interpretation
Tables are a powerful and popular tool for organizing and manipulating data. A vast number of tables can be found on the Web, which represent a valuable knowledge resource. The objective of this survey is to synthesize and present two decades of research on web tables. In particular, we organize existing literature into six main categories of information access tasks: table extraction, table interpretation, table search, question answering, knowledge base augmentation, and table augmentation. For each of these tasks, we identify and describe seminal approaches, present relevant resources, and point out interdependencies among the different tasks.
Introduction
Tables are a practical and useful tool in many application scenarios. Tables can be effectively utilized for collecting and organizing information from multiple sources. With the help of additional operations, such as sorting, filtering, and joins, this information can be turned into knowledge and, ultimately, can be used to support decision-making. Thanks to their convenience and utility, a large number of tables are being produced and are made available on the Web. These tables represent a valuable resource and have been a focus of research for over two decades now. In this survey paper, we provide a systematic overview of this body of research.
Tables on the Web, referred to as web tables further on in this paper, differ from traditional tables (that is, tables in relational databases and tables created in spreadsheet programs) in a number of ways. First, web tables are embedded in webpages. There is a lot of contextual information, such as the embedding page's title and link structure, the surrounding text, etc. that can be utilized. Second, web tables are rather heterogeneous regarding their quality, organization, and content. For example, tables on the Web are often used for layout and navigation purposes. Among the different table types, relational tables (also referred to as genuine tables) are of special interest. These describe a set of entities (such as people, organizations, locations, etc.) along with their attributes (Wang and Hu, 2002a;Cafarella et al., 2008b;Crestan and Pantel, 2011;Zhang, 2018). Relational tables are considered to be of high quality, because of the relational knowledge contained in them. However, unlike from tables in relational databases, these relationships are not made explicit in web tables; uncovering them is one of the main research challenges.
We organize relevant literature based on the task that is being addressed into six main categories. These are: table extraction (Sect. 3), table interpretation (Sect. 4), table search (Sect. 5), question answering on tables (Sect. 6), knowledge base augmentation (Sect. 7) and table augmentation (Sect. 8). The relationship between the different tasks is shown in Fig. 1.
• ; Chen and Cafarella (2013); Cafarella et al. (2008b); Balakrishnan et al. (2015); Cafarella et al. (2009); ; Bhagavatula et al. (2015) Wang and Hu (2002b,a); ; Chen and Cafarella (2013); Cafarella et al. (2008b); Crestan and Pantel (2011) Table augmentation Table Table Das (2013); ; Wang et al. (2015a); Zhang and Balog (2019a) identifying what table columns are about, recognizing and disambiguating entity mentions in table cells, and uncovering the relationships between table columns.
• Table search (or table retrieval) is the task of answering a search query with a ranked list of tables. The search query may be a sequence of keywords or it may be a table itself.
• Question Answering utilizes structured data in tables for answering natural language questions.
• Knowledge base augmentation leverages tabular data for exploring, constructing, and augmenting knowledge bases.
• For each of these tasks, summarized in Table 1, we identify seminal work, describe the key ideas behind the proposed approaches, discuss relevant resources, and point out interdependencies among the different tasks.
Web Table Search Table Extraction Table Interpretation Table Augmentation Question Answering
Knowledge Base Augmentation High level applications
Low-level tasks [:,j] ), and table entities (T E ).
The remainder of this paper is organized as follows. Next, in Sect. 2, we introduce the different table types and table corpora. Sections 3-8 are dedicated to the six main table tasks we have identified above. Finally, we conclude with a discussion of past progress and future research directions in Sect. 9.
Table Types and Corpora
In this section, we formally introduce tables (Sect. 2.1), present various types of tables (Sect. 2.2), and provide an overview of publicly available datasets (Sect. 2.3).
The Anatomy of a Table
A table T is grid of cells arranged in rows and columns. Tables are used as visual communication patterns, and as data arrangement and organization tools. In this paper, our primary focus is on web tables, that is, tables embedded in webpages. Below, we define elements of a web table. We refer to Fig. 2 for an illustration. [i,j] is specified with the row index i and column index j. Table cells hold (possibly empty) values and are considered as atomic units in a table.
Types of Tables
A number of table classification schemes have been proposed in the literature. We start by reviewing those, then propose a normalized categorization based on the main aspects these share.
In early work, Wang and Hu (2002a) make a distinction between genuine and non-genuine tables:
• Genuine tables are leaf tables, i.e., do not contain other tables, lists, forms, images or other non-text formatting tags in a cell. Furthermore, they contain multiple rows and columns. • Non-genuines tables refer to all those that are not leaf tables. Cafarella et al. (2008b) classify web tables into five main categories:
• Extremely small tables are those having fewer than two rows or columns.
• HTML forms are used for aligning form fields for user input.
• Calendars are a specific table type, for rendering calendars.
• Non-relational tables are characterized by low quality data, e.g., used only for layout purposes (many blank cells, simple lists, etc.). • Relational tables contain high-quality relational data. Crestan and Pantel (2011) develop a fine-grained classification taxonomy, organized into a multi-layer hierarchy.
• Relational knowledge tables contain relational data.
-Listings refer to tables consisting a series of entities with a single attribute. In terms of layout direction, these are further classified as vertical listings or horizontal listings. -Attribute/value tables describe a certain entity along with its attributes.
-Matrix tables have the same value type for each cell at the junction of a row and a column. Calendars, for example, can be regarded as matrix tables. -Enumeration tables list a series of objects that have the same ontological relation (e.g., hyponomys or siblings). -Form tables are composed of input fields for the user to input or select values.
• Layout tables do not contain any knowledge and are used merely for layout purposes.
-Navigational tables are meant for navigating within or outside a website.
-Formatting tables are used for visually organizing content. Lautert et al. (2013) refine the classification scheme of Crestan and Pantel (2011).
• Relational knowledge tables -Horizontal tables place attribute names on top (column header). Each column corresponds to an attribute.
-Vertical tables place attribute names on the left (row header). Each row represents an attribute.
-Matrix tables are three dimensional data sets, where headers are both on the top and on the left.
• Layout tables, as before, are subdivided into navigational tables and formatting tables.
Relational knowledge tables are further classified according to a secondary type taxonomy. Tables for navigational purposes Formatting Tables for visual organization of elements • Concise tables contain merged cells (i.e., cells with the same value conflated together) to avoid value repetition.
• Nested tables contain a table in a cell.
• Multivalued tables refer to tables containing multiple values in a single cell. If all values in one cell come from one domain, they are named as simple multivalued web tables, if not, they are called composed multivalued value tables.
• Splitted tables present sequentially ordered repetitions in row/column headers (i.e., each label is repeated in every x cell).
With a particular focus on web spreadsheets, Chen and Cafarella (2013) define the following type taxonomy:
• Data frame spreadsheets contain data frames, each consisting of two regions: data (numeric values) and headings (attribute names). These are further classified based on how they are arranged:
-Hierarchical left spreadsheets place attributes on the left of the data region.
-Hierarchical top spreadsheets put attributes on top of the data region.
• Non-data frame (flat) spreadsheets do not contain a data frame.
-Relation spreadsheets can be converted into the relational model (Codd, 1970).
-Form spreadsheets are designed for human-computer interaction.
-Diagram spreadsheets are for visualization purposes.
-List spreadsheets consist of non-numeric tuples.
-Other spreadsheets include schedules, syllabi, scorecards, and other files without a clear purpose. distinguish tables along two dimensions: content and layout. In terms of content, they adopt the classification scheme by Wang and Hu (2002a). Considering layout purposes, they sort tables according to their logical structure into the following categories:
• Horizontal listings align cells horizontally.
• Vertical listings align cells vertically.
• Matrix tables refer to numerical tables. distinguish between three main types of tables:
• Relational tables contain a set of entities, which could exist in rows (horizontal) or columns (vertical); the remainder of the cells contain their descriptive attributes.
• Entity tables describe a certain entity.
• Matrix tables refer to tables with numerical values only.
The above categorization systems are quite diverse, which is not surprising considering that each was designed with a different use-case in mind. Nevertheless, we can observe two main dimensions along which tables are distinguished: content and layout. We propose a normalized classification scheme, which is presented to
Table Corpora
A number of table corpora have been developed in prior work, which are summarized in Table 3.
WDC Web Table Corpus
There are two versions of WDC Web
Scientific Tables
Scientific tables are a particular type of table, which contain valuable knowledge and are available in large quantities. The TableArXiv corpus 6 consists of 341,573 tables, extracted from physics e-prints on arxiv.org. Along with the corpus, 105 information needs and corresponding relevance judgements are also provided for the task of scientific table search.
Table Extraction
A vast number of tables can be found on the Web, produced for various purposes and storing an abundance of information. These tables are available in heterogenous format, from HTML tables embedded in webpages to files created by spreadsheet programs (e.g., Microsoft Excel). To conveniently utilize these resources, tabular data should be extracted, classified, and stored in a consistent format, resulting ultimately in a table corpus. This process is referred to as table extraction. In this section, we present approaches for the table extraction task, organized around three main types of tables: web tables, Wikipedia tables, and spreadsheets.
Web Table Extraction
Table extraction is concerned with the problem of identifying and classifying tables in webpages, which encompasses a range of more specific tasks, such as relational
Relational table classification
The identification of tables on the Web is usually straightforward based on HTML markup. Tables, however, are also used extensively for formatting and layout purposes. Therefore, web table extraction involves a data cleaning subtask, i.e., identifying and filtering out "bad" tables (where "bad" usually denotes non-relational tables). Relational table classification (also known as identifying high-quality or genuine tables) refers to the task of predicting whether a web table contains relational data.
One of the pioneering works utilizing tables on the Web is the WebTables project (Cafarella et al., 2008a,b Tables 4 and 5. As for word group features, Wang and Hu (2002b) treat each table as a document and compute word frequency statistics. In follow-up work, the authors also experiment with other machine learning methods (Naive Bayes and weighted kNN), using the same set of features (Wang and Hu, 2002a). Building on (Wang and Hu, 2002b), carry out relational table classification as well as classification according to layout type (vertical listings, horizontal listing, and matrix tables). Their first method performs classification along both dimensions simultaneously, using a single layer. Their second approach separates the two tasks into two layers, where the first layer executes table detection and, subsequently, the second layer determines the layout type. Various machine learning methods are employed, including decision trees, Random Forests, and SVMs, using a combination of global and local features; a selection of features are listed in Table 5. As a result, classify millions of tables and generate the Dresden Web
Header detection
To extract data in a structured format, the semantics of tables need to be uncovered to some extent. One question of particular importance is whether the table contains a header row or column. This is known as the task of header detection. Headers may be seen as a particular kind of table metadata. Header detection is commonly addressed along with the other two tasks and uses similar features (cf. Tables 4 and 5).
Table type classification
Another type of metadata that can help to uncover table semantics is table type. Tables 4 and 5 are used in for both relational table classification and table type classification. Instead of directly classifying tables as relational or not, this can also be done indirectly by saying that a table is relational if relational information can successfully be extracted from it (Chen and Cafarella, 2013). Table extraction is also involved in a number of other studies, but these datasets are not publicly available. For example, with the purpose of data integration, Wang et al. (2012) use a rule-based filtering method to construct a corpus of 1.95 billion tables. For a type-classification study, Crestan and Pantel (2011) extract a corpus of 8.2 billion tables. Using a more fine-grained type taxonomy (see Sect. 2.2), table type classification is approached as a multi-class classification problem. Crestan and Pantel (2011) propose a rich set of features, including global layout features, layout features, and content features. Global layout features include the maximum number of rows, cols, and maximum cell length. Layout features include average length of cells, length variance, and the ratio of row/column span. Content features include HTML features (based on HTML tags) and lexical features (based on cell content). As a follow-up work, Lautert et al. (2013) additionally consider the category obtained in (Crestan and Pantel, 2011) as one features to further classify tables into a multi-layer taxonomy. The first layer of classification is similar to the one in Crestan and Pantel (2011
Wikipedia Table Extraction
Wikipedia tables may be regarded as a special case of web tables. They are much more homogeneous than regular web tables and are generally of high quality. Therefore, no additional data cleaning is required. Bhagavatula et al.
Spreadsheet Extraction
The Web contains a great variety and number of Microsoft Excel spreadsheets. Spreadsheets are often roughly relational. Chen and Cafarella (2013) design an automatic system to extract relational data, in order to support data integration operations, such as joins. A data frame is defined as a block of numerical data. Chen and Cafarella (2013) extract 410,554 Microsoft Excel files from the ClueWeb09 Web crawl by targeting Excel-style file endings that contain a data frame. Within a data frame, the attributes might lie on the left or top. Chen and Cafarella (2013) find that 50.5% of the spreadsheets contain a data frame and 32.5% of them have hierarchical top or left attributes (the rest are called flat spreadsheets). Among the 49.5% non-data frame spreadsheets, 22% are relational, 10.5% are forms, 3.5% are diagrams, 3% are lists, and 10.5% are other spreadsheets. For each spreadsheet, the extraction system firstly finds the data frame, then extracts the attribute hierarchy (top or left), and finally builds relational tuples (see Sect. 4.3 for more details). (1) column type identification, that is, associating a table column with the type of entities or relations it contains, (2) entity linking, which is the task of identifying mentions of entities in cells and linking them to entries in a reference knowledge base, and (3) relation extraction, which is about associating a pair of columns in a table with the relation that holds between their contents. See Figure 3 as the task illustration. Table 6 provides an overview of studies addressing either or all of these tasks.
Column Type Identification
In relational tables, the core column (also referred to as subject column, name column, or entity column ) is a special column that contains entities. Commonly, this is the leftmost column in a table (and other table columns correspond to attributes or relationships of these entities). The identification of the core column is a central pre-processing step for entity linking, table augmentation, and relation extraction. Most of the existing work assumes the presence of a single core column. Such tables are also known as single-concept relational tables. However, in some cases, a relational table might have multiple core columns that may be located at any position in the table (Braunschweig et al., 2015a), called a multi-concept relational table. Braunschweig et al. (2015a) extend a single-concept method, which utilizes table headings as well as intrinsic data correlations, with more features, like the correlation with the left neighbor, to determine all the core columns. We focus on single-concept relational tables in the remainder of this section.
Generally, column type identification is concerned with determining the types of columns, including locating the core column. This knowledge can then be used to help interpret a table. Table 7 displays a summary of the methods, (2017) Freebase Unsupervised featured-based method which we shall discuss below. Venetis et al. (2011) argue that the meaning of web tables is "only described in the text surrounding them. Header rows exist in few cases, and even when they do, the attribute names are typically useless" (Venetis et al., 2011). Therefore, they add annotations to tables to describe the sets of entities in the table (i.e., column type identification). This is accomplished by leveraging an IS-A database of entity-class pairs. This IS-A database is created by aggregating all the entity-class e, C pairs that are mined from the Web (100 million English documents using 50 million anonymized queries) using the pattern "C [such as|including] e [and|,|.]." A class label is assigned to a column if a certain fraction of entities in that column is identified with that label in the IS-A database. Venetis et al. (2011) conclude that using a knowledge base (YAGO) results in higher precision, while annotating against the IS-A database has better coverage, i.e., higher recall. Mulwad et al. (2010) map each cell's value in a column to a ranked list of classes, and then selects a single class which best describes the whole column. To get the ranked list of classes, a complex query, based on cell values, is submitted to the Wikitology knowledge base (Syed, 2010). Possible class labels are obtained by utilizing the relevant entities in the knowledge base. Then, a PageRank-based method is used to compute a score for the entities' classes, from which the one with the highest score is regarded as the class label.
Mapping each column to one of the four types ("Person", "Place", "Organization," and "Other"), Mulwad et al. (2010) achieve great success on "Person" and "Places," and moderate success on "Organization" and "Other" types, due to their sparseness in the reference knowledge base.
Because of the inherent semantic heterogeneity in web tables, not all tables can be matched to a knowledge base using pure machine learning methods. Fan et al. (2014) propose a "two-pronged" approach for matching web tables' columns to a knowledge base. First, a concept-based method is used to map each column to the best knowledge base concept. Specifically, they employ Freebase as the concept catalog. Second, a hybrid human-machine framework discerns the concepts for some exceptional columns manually. The matches between table columns and their candidate concepts are represented as a bipartite graph, where relationships correspond to edges. Fan et al. (2014) employ crowdsourcing for this task, and find that a higher payment leads to better accuracy.
A table corpus is constructed in (Wang et al., 2012) and it is classified according to a probabilistic taxonomy called Probase, which is able to understand entities, attributes, and cells in tables. To get the The candidate concepts assist to detect entities in a given column by computing the maximum number of common concepts. In turn, the entity column type is obtained based on the confidence of the concepts. Wang et al. (2012) demonstrate that table headers can help to understand the columns as well as to identify the core column. propose a categorization scheme for web table columns that distinguishes the different types of relations that appear in tables on the Web. First, a binary relation is a relation that holds between the core column and the values in another column, e.g., populations of cities. Second, an N-ary relation is a relation that holds between the core column and additional entities and values in other columns. Third, an independent column is one that has no direct relation with the core column. propose a feature-based classifier that distinguishes between these three types of relations for better table interpretation.
Zhang (2017) presents TableMiner+ for semantic table interpretation, where core column detection and type identification linking are executed at the same stage. Zhang (2017) first simply uses regular expressions and classifies cells as "empty," "entities," "numbers," "data," "text," or "other." Then, evidence is gathered from the Web for each column to predict the likelihood of it being the subject (core) column. Specifically, a keyword query is composed from all text content in each row, and the subject entity in this row is detected by recognizing the top-ranked page. Finally, an unsupervised feature-based method is employed to find the core column and type by aggregating evidence across all rows. Features include the fraction of empty cells, the fraction of cells with unique content, context match score (heading frequency within surrounding text), and web search score. The main differences between TableMiner+ and other methods are twofold: (1) TableMiner+ uses context outside the tables while others not, and (2) it adopts an iterative process to optimize and enforce the interdependence between different annotation tasks (entity linking and relation extraction).
The above methods work well for string values and static attributes but perform poorly for numeric and time-varying attributes. Zhang and Chakrabarti (2013) build a semantic graph over web tables suited for numeric and time-varying attributes by annotating columns with semantic labels, like timestamp, and converting columns by comparing with columns from other tables. While this method is designed for entity augmentation, it can also be utilized for column type identification.
Entity Linking
Recognizing and disambiguating specific entities (such as persons, organizations, locations, etc.), a task commonly referred to as entity linking, is a key step to uncovering semantics (Bhagavatula et al., 2015). Since many web tables are relational, describing entities, entity linking is a key step to understanding what the table is about. A number of table-related tasks, such as row population (Zhang and Balog, 2017b;Wang et al., 2015a), column population (Zhang and Balog, 2017b), and table search (Zhang and Balog, 2018a), rely on entity linking in tables. Table 8 compares the tasks we will discuss below. Limaye et al. (2010) pioneered research on table entity linking. They introduce and combine five features, namely, the TF-IDF scores between cell text and entity label, the TF-IDF scores between the column header and the type label, the compatibility between column type and cell entity, compatibility between relation and pair of column types, and the compatibility between relation and entity pairs. Their idea of a factor graph based entity linking approach influenced later research. For example, Bhagavatula et al. (2015) design a system called TabEL for table entity linking. TabEL employs a graphical model that "assigns higher likelihood to sets of entities that tend to co-occur in Wikipedia documents and tables" (Bhagavatula et al., 2015). Specifically, it uses a supervised learning approach and annotated mentions in tables for training. TabEL focuses on Wikipedia table and executes mention identification for each table cell, then obtains a set of candidate entities for disambiguation. The disambiguation technique is based on the assumption that entities in a given row and column tend to be related. They use a collective classification technique to optimize a global coherence score for a set of entities in a given table. By comparing against traditional entity linking methods for unstructured text, Bhagavatula et al. (2015) demonstrate the need for entity linking methods designed specifically for tables.
Unlike most methods, which consider a single knowledge base, propose an entity linking method for web tables that considers multiple knowledge bases to ensure good coverage. From each knowledge base, entities whose names share at least one word with the content of a given table cell are taken as candidates. Then, an entity disambiguation graph is constructed, consisting of mention nodes, entity nodes, mention-entity edges, and entity-entity edges. The method utilizes entity linking "impact factors," which are two probabilities, for ranking candidates and for disambiguating entities, based on mention nodes and edges. To incorporate multiple knowledge bases, "same-As" relations between entities from different knowledge bases are leveraged to reduce errors and to improve coverage. This system shares many similarities with TabEL. TabEL, however, does not consider synonyms and deals with a single KB. Efthymiou et al. (2017) propose three unsupervised annotation methods for matching web tables with entities. The first is a lookup-based method, which relies on the minimal entity context from the tables to discover correspondences to the knowledge base. A second method exploits a vectorial representation of the rich entity context in a knowledge base to identity the most relevant subset of entities in web tables. The third method is based on ontology matching, and exploits schematic and instance information of entities available both in a knowledge base and in a web table. Efthymiou et al. (2017) find that hybrid methods that combine the second and third methods (in any order) tend to perform best. The column type identification component of TableMiner+ (Zhang, 2017) has already been discussed earlier, in Sect. 4.1. Building on this, TableMiner+ uses the partial annotations from column type identification for all columns to guide entity linking in the rest of the table. It re-ranks table rows under the assumption that some cells are easy to disambiguate, i.e., they have more candidates or the text is less ambiguous (candidate sampling). In each iteration of this so-called learning phase, it searches new candidates and compares the feature representation of each candidate entity (web search results) against all the feature representations of that cell (using the same features as for column type identification). The associated concepts with the highest scoring entity are gathered as candidate concepts for the column. These are further compared against those from the previous iteration in the learning phase (optimization). The process is repeated until convergence is reached. Mulwad et al. (2010) exploit the predicted class labels for columns (see Sect. 4.1) as additional evidence, to link entities in table cells. A knowledge base is queried to construct a feature vector, which comprises the entity's retrieval score, Wikipedia page length, PageRank, etc., which are used for computing the similarity score against the table cell's value. The feature vectors are input to an SVMRank classifier, which outputs a ranked list of entities. The top-ranked entity is selected and is used to introduce two more features for a final classification (the SVM rank score for the top-ranked entity and the score difference between the top two entities). The final classification yields a binary outcome whether the entity should be linked or not. Similar to the column type identification task, this method performs very well on the "Person" and "Place" entity types, achieves moderate accuracy on "Organization," and low accuracy on "Other" (for the same reason of sparseness, as before). A similar approach is taken by , but they perform entity linking in table cells first, using the Google Knowledge Graph, and then use this information for getting class labels for columns.
Another study on knowledge base matching in Ibrahim et al. (2016) aims to overcome the problem of table matching and aggregation by making sense of entities and quantities in web tables. Ibrahim et al. (2016) map the table elements of table headers, entity tables cells, and numeric table cells to different knowledge bases. Specifically, (1) tables headers denote classes or concepts and are linked to a taxonomic catalog or to Wikipedia pages, (2) named entities are mapped to a knowledge base (YAGO), and (3) numeric cells, which denote quantities, are mapped to normalized representations. An interesting observation made about quantity linking is that many of the linking errors are (1) due to the absence of specific measures or units and (2) because of ambiguous headings, like "Nat."
As mentioned in Sect. 3.1, a relational table refers to an entity-attribute table, where a set of entities and their attributes are listed. propose an instance-based schema mapping method to map entity-attribute tables to a knowledge base. In , an entity-attribute table is supposed to have a key column, which contains a set of entities. Each tuple is an entity with its attributes. Then, memory-based indexes are used to judge whether a tuple contains candidate entities, resulting in an evidence mapping vector. This vector is then used for finding a table-to-KB schema mapping, which essentially serves as a bridge between web tables and knowledge bases.
The choice of the knowledge base for uncovering table semantics is important. Hassanzadeh et al. (2015) give a detailed study on the utility of different knowledge bases, including DBpedia, Schema.org, YAGO, Wikidata, and Freebase. The method of concept linking in Hassanzadeh et al. (2015) is tagging columns with entity types (classes) in the knowledge base. Specifically, they firstly get the basic statistical distribution of tables sizes and values. Then, with the help of the selected knowledge base, the distribution of overlap scores in the ontology is obtained. Finally, these scores can give an indication of how well the table's content is covered by the given knowledge base. Ritze and Bizer (2017) study the utility of different features for entity linking in tables. These features are extracted from the table itself (such as entity label, table, URL, page title, and surrounding text) or from the knowledge base (such as instance label and classes). They introduce a specific similarity linker for each feature, resulting in similarity matrices, representing feature-specific results. These matrix predictors can be used to decide which features to use for which web table. implement the T2K Match framework to map the WDC Web corpus to DBpedia, for knowledge base extension (entity linking happens the same time with, and rely on, schema matching and table type identification). Taking table content as evidence, the incomplete and unclear values of DBpedia can be filled and corrected. They find that "only 1.3% of all tables that were extracted from the Web crawl contained relational data. Out of these relational tables, about 3% could be linked to DBpedia" . However, the
Relation Extraction
Relation extraction refers to the task of associating a pair of columns in a table with the relation that holds between their contents and/or extracting relationship information from tabular data and representing them in a new format (e.g., RDF). Table 9 summarizes the methods we will discuss below. Venetis et al. (2011) add annotations to tables to describe the binary relationships represented by columns. This is accomplished by leveraging a relations database of (argument1, predicate, argument2) triples. For binary relationships, the relationship between columns A and B is labeled with R if a substantial number of pairs of values from A and B occur in the relations database. Venetis et al. (2011) are only able to annotate a small portion of a whole table corpus (i.e., low recall). They discover that the vast majority of these tables are either not useful for answering entity-attribute queries, or can be labeled using a handful of domain-specific methods. Mulwad et al. (2010) propose a preliminary method for relation extraction, which utilizes the results of entity linking and column type prediction. Specifically, the method generates a set of candidate relations by querying DBpedia using SPARQL. Each pair of strings in two columns vote for the candidate relation. The normalized scores are used for ranking candidate relations and the highest one is taken as the column relation. In follow-up work, Mulwad et al. (2013) implement an improved semantic message passing method to extract RDF triples from tables. The semantic message passing first pre-processes the input table, separated by table elements such as column headers, cell values, columns, etc. Then, the processed table is passed to a query and rank module, which turns to knowledge bases from Linked Open Data to find candidates for each table element. Finally, a joint inference step uses a probabilistic graph model to rank candidate relations that were identified for the table elements. Mulwad et al. (2013) point out that current methods rely on semantically poor and noisy knowledge bases and can only interpret part of a table (low recall). Moreover, systems for numeric values remain challenging, which is consistent with (Ibrahim et al., 2016).
TableMiner+ (Zhang, 2017) interprets relations between the core column and other columns on each row independently. It computes an individual confidence score for each candidate relation from each row. The candidate set of relations for two columns is derived by collecting the winning relations on all rows. A final confidence score of a candidate relation adds up its instance and context score computed based on context overlap. It is used to find the relation with the highest confidence. A key finding in (Zhang, 2017) is that a system that is based on partial tabular data can be as good as systems that use the entire (2013) realize it in an automatic manner. Generally, the system detects attributes and values, identifies the hierarchical structure of attributes, and generates relational tuples from spreadsheet data. Specifically, the so-called frame finder module of their system aims to identify the data frame regions within a spreadsheet. These data frames consist of attribute and value regions. First, it labels each row with one of the categories: title, header, data, or footnote. Then, the data frame regions are created, which are passed to the hierarchy extractor for recovering the attribute hierarchies by finding all parent-child pairs in an attribute region. See Fig. 4 for an illustration, where Airplane pilots and Airline transport would be annotated as a parent-child attribute pair. Finally, a series of parent-child candidates are generated and the true parent-child pairs are identified through classification. Alternatively, a so-called enforced-tree classification is proposed, which constructs a strict hierarchical tree for attributes. In the end, relational tuples are generated from the value region, whose value is annotated with one attribute from the attribute hierarchy.
Other Tasks
Data translation is concerned with the problem of mapping raw data, collected from heterogenous sources, to a transformed version for the end user He et al. (2018). Tables encode a large number of mapping relationships as column pairs, e.g., person and birthday, which can be useful data assets for data translation. Wang and He (2017) propose to automatically synthesize mapping relationships using table corpora by leveraging the compatibility of tables based on co-occurrence statistics. Braunschweig et al. (2015a) propose a method to normalize web tables in cases where multiple core columns and mixed concepts are detected in one Govindaraju et al. (2013), where NLP tools, like part-of-speech tagging, dependency paths, and named-entity recognition, are explored to mine surrounding texts for understanding table semantics. Braunschweig et al. (2015b) propose a heuristic approach that extracts text snippets from the context of a web table, i.e., caption, headline, surrounding text, and full text, which describe individual columns in the table and link these new labels to columns. As a follow-up, Braunschweig et al. (2016) propose a contextualization method of splitting table context into paragraphs with consistent topics, providing a similarity measure that is able to match each paragraph to the table in question. Paragraphs are then ranked based on their relevance.
Table Search
Keyword-based Search
Given a keyword query, the process of returning a ranked list of tables is referred to as keyword-based search (or keyword query search). One of the first published methods is by Cafarella et al. (2008a), who implement keyword table (2017) search on top of an existing web search engine. Specifically, they extract the top-k tables from the returned webpages. In follow-up work, a similar system called OCTOPUS (Cafarella et al., 2009) extends the same method (referred to as SimpleRank) with a reranking mechanism (SCPRank) that considers attribute co-occurrences.
Later works search directly within a designated table corpus. Methods may be divided into document-based and feature-based approaches. According to the first group of approaches, a document-based representation is created for each table. This might contain all text included in the table or only certain elements of the table (e.g., caption or header labels). Then, these document-based representations may be ranked using traditional retrieval models, such as TF-IDF (Pimplikar and Sarawagi, 2012 Instead of relying on a single keyword query as input, Pimplikar and Sarawagi (2012) take q columns, each described by a set of keywords Q 1 , . . . , Q q , as input (e.g., Q 1 ="chemical element," Q 2 ="atomic number," and Q 3 ="atomic weight"), and return a table with q columns as the answer. First, they rank tables using the union of words in Q 1 , . . . , Q q . Then, each table column is labeled with the query column it maps to. Finally, relevant columns and rows are merged into a single table, by considering the table-level relevance scores and the column-level mapping confidence scores.
Reference
Application
T E T H T [:,j] T p T [i,j]
Ahmadov et al. (2015) Data completion Das Sarma et al. (2012) Schema complement Entity complement Relation join Limaye et al. (2010) Table cell retrieval Nargesian et al. (2018) Table union search Nguyen et al. (2015) Diverse table search Yakout et al. (2012) Table augmentation Zhang and Balog (2019b) Table recommendation To decide if two rows are duplicates of each other, they employ the method in (Gupta and Sarawagi, 2009). Zhang and Balog (2018a) perform semantic matching between queries and tables for keyword table search. Specifically, they (1) represent queries and tables in multiple semantic spaces (both discrete sparse and continuous dense vector representations) and (2) introduce various similarity measures for matching those semantic representations. For the former, both queries and tables are represented as bag-of-entities, bag-of-categories, word embeddings (trained on Google news) and graph embeddings respectively. As for the latter, matching methods, they employ the early and late fusion patterns (Zhang and Balog, 2017a). They consider all possible combinations of semantic representations and similarity measures and use these as features in a supervised learning model. They demonstrate significant and substantial improvements over a state-of-the-art feature-based baseline. Most recently, Deng et al. (2019) train word embeddings utilizing the Wikipedia table corpus and achieve comparable results. Table search is not limited to keyword queries. The input may be also be a table, in which case the task of returning related tables is referred to as table-based search (or query by table). At its core, this task boils down to computing a similarity score between the input and candidate tables, which we shall refer to as table matching. Search by table may be performed for different goals: (1) to be presented to the user to answer her information need (Das Sarma et al., 2012;Limaye et al., 2010;Nguyen et al., 2015) and (2) to serve as an intermediate step that feeds into other tasks, like table augmentation Yakout et al., 2012;Nargesian et al., 2018).
Table-based Search
One group of approaches addresses the table matching task by using certain table elements as a keyword query, and scoring tables using keyword-based methods. For example, use table entities and table headings as queries to retrieve a ranked list of tables for data completion (to be detailed in Sect. 8.3). The two ranked lists are then intersected afterwards in order to arrive at a more complete result set.
More commonly, table matching is tackled by dividing tables into various elements (such as table caption, table entities, column headings, cell values), then computing element-level similarities. Table 11 provides an overview of the table elements that have been utilized in past work. It is worth pointing out that in most of these cases, table search is not the ultimate goal, it is only used as a component in a larger application. The Mannheim Search Join Engine seeks to extend the input table with additional attributes. It utilizes table headings by comparing the column headings between the input table and candidate tables. Specifically, they first filter tables that share at least one column heading with the input table, using exact term matching. Then, the table matching score is computed by (1) building an edit distance similarity matrix between the input and candidate tables' column headings, and (2) calculating the Jaccard similarity of the two tables using the matrix's maximum weighted bipartite matching score. Similar to the Mannheim Search Join Engine that is based on table headings, Nargesian et al. (2018) search tables that are likely unifiable with the seed table, which is called attribute union ability. Nargesian et al. (2018) formalize three statistical models to estimate the likelihood that two attributes contain values that are in the same domain. The simplest case, named set domains, uses the size of the intersection of values between two columns. The second case, called semantic domains, measures the semantic similarity between the values by mapping the columns to classes, e.g., entities. For values that are expressed in natural language, the third case of natural language domains measures semantics based on natural langue rather than on ontologies. They use word embeddings trained based on Wikipedia documents to define natural language domains and statistical tests between the vectors are used to evaluate the likelihood that two attributes are from the same domain. Das Sarma et al. (2012) aim to find related tables for augmenting the input table with additional rows or columns, referred to as entity complement and schema complement, respectively. Entity complement considers the relatedness between entity sets of the input and candidate tables. Relatedness between two entities is estimated by representing entities as weighed label sets (from a knowledge base or from a table corpus) and taking their dot product. Das Sarma et al. (2012) propose multiple methods to aggregate pairwise entity relatedness scores for computing relatedness between two sets of entities. Schema complement combines two element-wise similarities: table entities and column headings. The former considers the overlap between table entities. The latter estimates the benefits of adding a column from the candidate table to the input table by determining the consistency between the new column and the existing columns of the input table. Yakout et al. (2012) propose InfoGather, a holistic method for matching tables in order to support three core operations: augmentation by column headings, augmentation by example, and column heading discovery. They consider element-wise similarities, including table context, URL, tuples, column headings, column values, and table data, as well as cross-element similarity between table and context. Similarity is measured using the vector product of TF/IDF-weighted term vectors. Then, element-level similarity scores are combined as features in a machine learned model. In follow-up work, InfoGather is extended as InfoGather+ Zhang and Chakrabarti (2013) to incorporate tables with numeric and time-varying attributes. Zhang and Balog (2019b) perform table matching by representing table elements in multiple semantic spaces, and then combining element-level similarities using a discriminative learning model. Nguyen et al. (2015) consider the diversity of the returned tables. They focus on two table elements: column headings and table data. The former is similar in spirit to the Mannheim Search Join Engine . The latter works by measuring the similarity between table columns, which are represented as term frequency vectors.
Unlike the above methods, which consider tables as the unit of retrieval, Limaye et al. (2010) return a ranked list of cells as result. They train a machine learning method for annotating (1) entities in tables cells, (2) columns with types, and (3) relations between columns. Then, search is performed by issuing an automatically generated structured query.
Question Answering on Tables
Tables are a rich source of knowledge that can be utilized for answering natural language questions. This problem has been investigated in two main flavors: (1) where the table, which contains the answer to the input question, is given beforehand (Pasupat and Liang, 2015), and (2) where a collection of tables are to be considered Sun et al. (2016). The latter variant shares many similarities with traditional question answering (QA), while the former requires different techniques. One of the main challenges of QA on tables, shared by both scenarios, is how to match the unstructured query with the (semi-)structured information in tables. Question answering on tables is also closely related to work on natural language interfaces to databases, where the idea is that users can issue natural language queries, instead of using formal structured query languages (like SQL), for accessing databases Androutsopoulos et al. (1995); Li and Jagadish (2014); Li et al. (2005); Popescu et al. (2003). Semantic parsing is the task of parsing natural language queries into a formal representation. Semantic parsing is often used in question answering, by generating logical expressions that are executable on knowledge bases (Berant et al., 2013;Fader et al., 2014).
Using a Single Table
We first discuss approaches that take a single table as input (sometimes referred to as knowledge base table (Yin et al., 2016)), and seek the answer to the input question in that table. The basic idea is to regard the input table as a knowledge base, which poses a number of challenges. First, knowledge bases contain a canonicalized set of relations, while tabular data is much more noisy. Second, traditional semantic parsing sequentially parses natural language queries into logical forms and executes them against a knowledge base. To make them executable on tables, special logical forms are required. Lastly, semantic parsing and query execution become complicated for complex questions as carefully designed rules are needed to be able to parse them into logical forms. Pasupat and Liang (2015) propose to answer complex questions, involving operation such as comparison, superlatives, aggregation, and arithmetics in order to address above problems. They convert the input table into a knowledge graph by taking table rows as row nodes, strings as entity nodes, and columns as directed edges. The column headings are used as predicates. Numbers and strings are normalized following a set of manual rules. Being one of the earliest works addressing this task, Pasupat and Liang (2015) follow a traditional parser design strategy. A semantic parser is trained based on a set of question-answer pairs. Given a table and a question, a set of candidate logical forms is generated by parsing the question. Then, logical forms are ranked using a feature-based representation, and the highest ranked one is applied on the knowledge graph to obtain the answer. Pasupat and Liang (2015) develop a dataset, called WikiTableQuestion, which consists of a random sample of 2,100 tables from Wikipedia and 22,000 question-answer pairs. The proposed approach is found to suffer from a coverage issue, i.e., it is able to answer only 20% of the queries that have answers in Freebase.
Different from semantic parsing methods that require predefined logical operations, Yin et al. (2016) propose a neural network architecture, named Neural Enquirer, for semantic parsing with a specific table. Neural Enquirer is a fully neural system that generates distributed representations of queries and tables (called query encoder and table encoder, respectively). Then, the question is executed through a series of query operations, called executors, with intermediate execution results computed in the form of table annotations at different levels. Training can be performed in an end-to-end fashion or carried out using step-by-step supervision (for complex queries). They use query answers as indirect supervision, but jointly perform semantic parsing and query execution in distributional spaces. The distributed representations of logical forms are learned in end-to-end tasks, which is based on the idea of adopting the results of the query execution as indirect supervision to train the parser. It is worth pointing out that this model makes a number of strong assumptions. For example, they consider four specific types of queries, provide the logical form template for each, and carefully and manually select a table that is supplied as part of the input. Similar to (Yin et al., 2016), Neelakantan et al. (2015) attempt to solve the task of question answering on tables using neural networks, a system called Neural Programmer. Neural Programmer runs for T steps and the final result is formed step by step. The model adopts a Recurrent Neural Network (RNN) architecture to process the input question, a selector to assign probabilities to a set of possible operations and data segments at each step, and a history RNN to remember the previous operation and data selections. Providing a small set of basic operations, they also take the result of correct executions as indirect supervision. During the training, adding random noise to the gradient greatly improves performance as the operations and data selections are quite heterogenous.
Using a Collection of Tables
Another line of work focuses on answering questions using a collection of tables. These approaches are more similar to traditional question answering on text, comprising of candidate identification, query type prediction, and ranking components. The main differences are twofold. One is schema matching, which is the same as before (in Sect. 6.1), but there is an additional normalization issue across tables here. The other is the need for extracting quantity values from tables. Sarawagi and Chakrabarti (2014) show that over 40% of table columns contain numeric quantities, and propose a collective extraction framework to extract quantities from raw web tables based on a consensus model. Their system, called QEWT, extends the work of Banerjee et al. (2009) to tables. They employ keyword-based table ranking in order to fetch table candidates. This corresponds to the candidate snippet/answer identification step in traditional QA. QEWT can answer quantity-target queries with a ranked list of quantity distributions, which are taken from the tables. It uses a table column unit annotator based on probabilistic context free grammars for easily extracting quantities from table columns to deal with ambiguity (both for headings and for values). From an information retrieval perspective, quantity queries on web tables is the task of returning a ranked list of quantities for a query. QEWT employs a quantity response model for this task.
Inspired by classical textual QA, Sun et al. (2016) decompose table cells into relational chains, where each relational chain is a two-node graph connecting two entities. Specifically, each row of a table represents relations among the cells. They construct a pseudo-node to connect all the cells and take the headings to label relationships. Any pair of cells in the same row form a directional relational chain. The input query is also represented as a two-node graph question chain, by identifying the entities using an entity linking method. The task then boils down to finding the relational chains that best match the question chain. This matching is performed using deep neural networks, in particular the Convolutional Deep Structured Semantic Model (C-DSSM) (Shen et al., 2014), to overcome the vocabulary gap limitation of bag-of-words models. They find that combining the deep features with some shallow features, like term-level similarity between query and table chains, achieve the best performance. Sun et al. (2016) conclude that their method can complement KB-based QA methods by improving their coverage.
Knowledge Base Augmentation
The knowledge extracted from tabular data can be used for enriching knowledge bases. First, we present an approach that is devised for exploring the knowledge contained in web tables. Then, we discuss methods for knowledge base augmentation using tabular data.
Tables for Knowledge Exploration
The knowledge contained in web tables can be harnessed for knowledge exploration. Knowledge Carousels Chirigati et al. (2016) is the first system addressing this, by providing support for exploring "is-A" and "has-A" relationships. These correspond to two kinds of entity-seeking queries (queries searching for attributes and relationships of entities), called "sideways" and "downwards," respectively. Given an input entity, Chirigati et al. (2016) utilize web tables to select the carousel type, create a set entities of this carousel, generate human-readable titles, and rank carousels based on popularity and relatedness extracted from tables. See Fig. 5 for an illustration.
Knowledge Base Augmentation and Construction
Tabular data on the Web can be used to construct new or augment existing knowledge bases.
Knowledge Base Augmentation
In Sect. 4, we have presented techniques for interpreting tables with the help of knowledge bases. The obtained annotations, in turn, can contribute to extending those knowledge bases. Knowledge base augmentation, also known as knowledge base extension, is concerned with generating new instances of relations using tabular data and updating knowledge bases with the extracted information.
Knowledge bases need to be complete, correct, and up-to-date. A precondition of extending knowledge bases using web tables is matching them to those existing knowledge bases. Specific matching problems include table-to-class matching, row-to-instance matching, and attribute-to-property matching. propose an iterative matching method, T2K, to match web tables to DBpedia for augmenting knowledge bases. They also develop and make publicly available the T2D dataset for matching, consisting of 8,700 schema-level and 26,100 entity-level correspondences between web tables and DBpedia, which are extracted and annotated manually. The T2K method utilizes the T2D dataset to execute iterative steps between candidate matching and property matching, to find proper entities/schemas in DBpedia for table rows/columns. However, T2D mainly focuses on large tables and does not work that well for small-sized tables (Lehmberg and Bizer, 2017). To counter this problem, Lehmberg and Bizer (2017) propose to combine tables from each website into larger tables for table matching, building on the intuition that tables from the same website are created in a similar fashion.
Strictly speaking, we classify the work in (Wang et al., 2015a) as row extension. Nevertheless, since they map table entities to a knowledge base with the purpose of collecting more entities from other tables that belong to the same concept in the knowledge base, their work can also be regarded as a knowledge base augmentation approach.
Knowledge Base Construction
Instead of augmenting existing knowledge bases, web tables contain abundant information to be turned into knowledge bases themselves.
Even though there exists a number of large-scale knowledge bases, they are still far from complete (Dong et al., 2014). Therefore, Dong et al. (2014) introduce a web-scale probabilistic knowledge base named Knowledge Vault (KV) that fuses different information sources. For web tables, Dong et al. (2014) firstly identify the relation that is expressed in a table column by checking the column's entities, and reason about which predicate each column could correspond to. This latter task is approached using a standard schema matching method (Venetis et al., 2011), with Freebase as the knowledge base. The extracted relations, together with relational data from other sources, are converted into RDF triples, along with associated confidence scores. The confidence scores are computed based on a graph-based method. Specifically, the triples are fused by machine learning methods from multiple sources, including an existing knowledge base, (i.e., Freebase) and web tables. Consequently, 1.6B triples are generated, of which 324M have a confidence score above 0.7 and 271M have a confidence score above 0.9. Figure 6: Illustration of three table augmentation tasks: row extension, column extension, and data completion. Tables Table search Row for an illustration. One might envisage these functionalities being offered by an intelligent agent that aims to provide assistance for people working with tables (Zhang and Balog, 2017b).
Table Augmentation
Row Extension
Row extension aims to extend a given table with more rows or row elements (see Fig. 7). It mainly focuses on a particular type of tables, namely, relational tables. More specifically, row extension primarily targets horizontal relational tables, where rows represent entities and columns describe the attributes of those entities. In such tables there usually exists a core column (or key column) containing mostly entities (Bhagavatula et al., 2015;Venetis et al., 2011). Instead of directly providing a complete tuple (row), existing work has focused on identifying entities for populating such core columns (i.e., the Upper scenario in Fig. 7). Table 12 provides an overview of methods that will be covered below. As we shall see, table search is inherently involved here.
Populating entities in the core column of a table is similar to the problem of concept expansion, also known as entity set expansion, where a given set of seed entities is to be completed with additional entities Bron et al. (2013) Sarma et al. (2012) Schema complement Search join Zhang and Balog (2017b) Column population Xin (2011); Metzger et al. (2013Metzger et al. ( , 2014. Existing methods for concept expansion suffer from two main issues: input ambiguity and semantic drift (i.e., entities belonging to different concepts are mixed during expansion). Motivated by the intuition that tables tend to group entities that belong to a coherent concept, Wang et al. (2015a) leverage web tables for the concept expansion task, thereby aiming to prevent semantic drift. They provide both the seed entities as well as a concept name as input. First, they retrieve tables related to the seed entities. Then, they use a graph-based ranking method to rank candidate entities that co-occur with the seed entities in those tables. Specifically, they first expand the set by iteratively adding the most relevant tables based on concept likelihood, and collecting entities there. Then, they refine the earlier estimation and remove less relevant tables based on more complete information. Wang et al. (2015a) find that adding an input concept can address the semantic drift problem for tail concepts. While this method is developed for concept expansion, it is directly applicable to the problem of populating entities in a core column.
Das Sarma et al. (2012) search for entity complement tables that are semantically related to entities in the input table (as we have already discussed in Sect. 5.2). Then, the top-k related tables are used for populating the input table. Das Sarma et al. (2012), however, stop at the table search task. A similar approach is taken in InfoGather (Yakout et al., 2012), where this task is referred to as the augmentation by example operation. There, they first search for related tables (cf. Sect. 5.2), and then consider entities from these tables, weighted by the table relatedness scores. Yakout et al. (2012) build a schema matching graph among web tables, based on pairwise table similarity. Despite the use of scalable techniques, this remains to be computationally very expensive, which is a main limitation of the approach. Instead of relying only on related tables from a table corpus, Zhang and Balog (2017b) also consider a knowledge base (DBpedia) for identifying candidate entities. Specifically, they collect entities sharing the same types or categories with the input entities from DBpedia, and entities from similar tables (i.e., tables sharing seed entities, having similar captions, or including the same headings) as candidates. They find that entity type information in DBpedia is too general to help identify relevant candidates, and end up using only category information when extracting candidates from DBpedia. It is also shown that using related tables and using a knowledge base are complementary when identifying candidate entities. They develop a generative probabilistic model for the subsequent ranking of candidate entities based on their similarity to (1) other entities in the table, (2) column headings, and (3) the caption of the input table. Among the three table elements, seed entities are the most important component for entity ranking, followed by table headings and caption. A combination of the three table elements performs the best in the end. In recent work, Deng et al. (2019) utilize Word2vec to train table embeddings for core column entities. Combining the embedding-based similarity scores with the probability-based scores from (Zhang and Balog, 2017b) results in further performance improvements.
Column Extension
The most widely studied subtask in table augmentation is column extension: extending a table with additional columns. This task roughly corresponds to the join operation in databases. In this context, the set of column heading labels is also often referred to as the table schema. Commonly, column extension is approached by first locating similar tables and then considering the column headings/values in those tables. Table 13 provides an overview of the methods discussed in this section.
One particular variant of column extension aims to identify additional column heading labels (see Fig. 8 (Upper)). As table columns often correspond to entity attributes, this task is also referred to as attribute discovery (Yakout et al., 2012) or schema auto-complete Cafarella et al. (2008a). The WebTables system Cafarella et al. (2008a) implements this functionality based on the attribute correlation statistics database (ACSDb). ACSDb contains frequency statistics of attributes and co-occurring attribute pairs in a table corpus. ACSDb comprises 5.4M unique attribute names and 2.6M unique schemas. With these statistics at hand, the next probable attribute can be chosen using a greedy algorithm. The statistics-based method in Cafarella et al. (2008a) was the first approach to column extension, and was found to be able to provide coherent heading suggestions. However, later research has proven that considering additional features can further improve performance. Das Sarma et al. (2012) focus on finding related tables, with the aim of schema complement. For ranking tables, they consider two factors: (1) the coverage of entities, and (2) the potential benefits of adding additional attributes from those tables (we discussed the table search method in Sect. 5.2). Again, they stop at the table search task. The task of identifying potential attributes or column labels is also known as schema matching or column population (Zhang and Balog, 2017b). Zhang and Balog (2017b) try to find the headings that can be placed as the next column in an input table. They first find candidate headings from similar tables (the same strategy that they also use for row population). Zhang and Balog (2017b) observe that input entities and table caption contribute comparably to the identification of relevant candidates, while table headings are the least important component. However, similar to row population, all these sources are complementary, i.e., each source can identify candidate headings that none of the others could. In a subsequent ranking step, the candidates are scored based on table similarity, by aggregating element-wise similarities between (corresponding elements of) the input table and related tables. In (Deng et al., 2019), they utilize Word2vec to train embeddings for table headings. Similar to row population, combining the embedding similarity scores with the probabilities from (Zhang and Balog, 2017b) yields further performance improvements. The above approaches differ in what they use as input, i.e., whether they use only table headings (Cafarella et al., 2008a; or the entire table (Das Sarma et al., 2012;Zhang and Balog, 2017b).
Another variant attempts to augment the input table with entire columns, that is, including both the heading label as well as the corresponding cell values for each row within that column (see Fig. 8 (Lower)). Bhagavatula et al. (2013) present the relevant join task, which returns a ranked list of column triplets for a given input table. Each triplet consists of SourceColumn, MatchedColumn, and CandidateColumn. SourceColumn is from the query table, while MatchedColumn and CandidateColumn are from the candidate tables. They propose a semantic relatedness measure to find candidate tables from related Wikipedia pages, where page relatedness is estimated based on in-link intersections. Their idea is to compute similarity between columns, such that if SourceColumn and MatchedColumn share largely similar values, then the input table may be extended with CandidateColumn. These candidate columns are classified as relevant or non-relevant, using a linear ranking model, before performing the actual join. To reduce the number of candidate columns, some are filtered out in a pre-processing stage using simple heuristics. Columns that are kept are required to be non-numeric, have more than four rows, and an average string length larger than four. Bhagavatula et al. (2013) find that columns containing numeric data make more relevant additions than non-numeric ones. Additionally, more distinct values in the SourceColumn and a higher match percentage lead to better quality joins. The join operation is also supported by the Mannheim Search Join Engine . It first searches for related tables based on column headings (cf. Sec. 5.2), then applies a series of left outer joins between the query table and the returned tables. Afterwards, a consolidation operation is performed to combine attributes. Specifically, they employ a matching operator that relies on data from knowledge bases. Given two columns, similar match (Levenshtein distance) and exact match are used for matching headings. observe that similar match returns on average 3.4 times more tables than exact match. Among different table corpora, web tables provide the largest number of relevant tables, and Wikipedia tables tend to be populous on certain topics, such as countries and films.
Data Completion
Data completion for tables refers to the task of filling in the empty table cells. Table 14 summarizes the methods we discuss here. One variant of this task attempts to find the cell values for an entire column (see Fig. 9 (Upper)). This is known as the augmentation by attribute name operation in the InfoGather system (Yakout et al., 2012). This is typical of a scenario where the core entity column as well as the column headings are given in a relational table, and the values for the corresponding attributes (augmenting attributes) are to be filled in. The system in (Yakout et al., 2012) Figure 9: Illustration of data completion tasks: join (Upper) and data imputation (Lower). (2019a) undirected graphical models and build a semantic graph that labels columns with units, scales, and timestamps, and computes semantic matches between columns. Their experiments are conducted on three types of tables: company (revenue and profit), country (area and tax rate) and city (population). Zhang and Chakrabarti (2013) find that the conversion rules (manually designed unit conversion mapping) achieve higher coverage than string-based schema matching methods. Similar to InfoGather's augmentation by attribute name operation, the extend operation in the OCTOPUS systems Cafarella et al. (2009) enables the user to add more columns to a table by performing a join. It takes a keyword query and a given (existing) table column as input, where the keyword describes the newly added column. Different from a regular join, the added column is not necessarily an existing column. It may be formed row-by-row by combining information from multiple related tables (see Sect. 5.1 for the table search operation). However, Cafarella et al. (2009) rely on simple methods like edit distance for schema matching, which leaves room for improvement.
Another flavor of the data completion task focuses on filling in missing values for individual cells, referred to as data imputation (see Fig. 9 (Lower)). present a hybrid imputation method that combines a lookup-based approach, based on a corpus of web tables, and a model-based approach that uses machine learning (e.g., k-nearest neighbors or linear regression) to predict the value for a missing cell. It is worth noting that all the above methods rely only on tables and ignore the cases where no similar tables can be found. The method in is shown to be able to improve coverage. However, being able to automatically decide when to do simple lookup and when to employ a machine learned model remains an open challenge. CellAutoComplete is a recent framework proposed by Zhang and Balog (2019a) to tackle several novel aspects of this problem, including: (1) enabling a cell to have multiple, possibly conflicting values, (2) supplementing the predicted values with supporting evidence, (3) combining evidence from multiple sources, and (4) handling the case where a cell should be left empty. This framework makes use of a large table corpus and a knowledge base as data sources, and consists of preprocessing, candidate value finding, and value ranking components.
Conclusions and Future Directions
Tables are a powerful and popular tool for organizing and manipulating data. Research on web tables has seen nearly two decades of development. During this long period, the research focus has evolved considerably, from low level tasks (table extraction) to tapping more and more into the actual knowledge contained in tables (for search and for augmenting existing knowledge bases). Below, we review past progress and identify open research directions for each of the six main categories of tasks.
Table Extraction
In the early years, research was mainly focused on detecting, identifying, and extracting tables from webpages, and classifying them according to some type taxonomy. Gradually, spreadsheet documents were also considered for table extraction, and type taxonomies became more fine-grained. With the advancement of table extraction and classification methods, large-scale table corpora were constructed, which became available as resources to be utilized in other tasks. One open issue is that the available table corpora are all a result of a one-off extraction effort. As such, these collections get quickly outdated.
Table Interpretation
The problem of uncovering table semantics, including but not limited to identifying table column types, linking entities in tables, and extracting relational data from tables, represents an active research area. It is also an important one, as the resulting semantic annotations are heavily utilized in other table-related tasks, such as knowledge base augmentation and question answering. While there exist methods for high-precision extraction, there is plenty of room for improvement in terms of recall, as most existing methods can only interpret a small portion of tables. For instance, find that only 2.85% of web tables can be matched to DBpedia. Further, most of the emphasis has been on relational tables; other table types (e.g., entity tables) bring about a different set of challenges. Another line of future research concerns the development of user interfaces and tools for facilitating and visualizing the annotations Mazumdar and Zhang ([n. d.]).
Table Search
The task of retrieving relevant tables from a table corpus for an input query is one of the core tasks that was started in the early days and remains to be an active research topic ever since. One limitation of existing work is that it often makes assumptions about underlying query intent and the preferred answer table types. For example, Zhang and Balog (2018a) assume that queries follow a class-property pattern, which can be successfully answered by relational tables. As a result, relational tables with this pattern are preferred, which might therefore result in lower coverage. TableNet (Fetahu et al., 2019), a recent study on the interlinking of tables with subPartOf and equivalent relations, can provide a better understanding of table patterns. In the future, it would be desirable if an automatic query intent classifier were to identify the type of result table sought, which does not need to be limited to relational tables. Another topic that deserves attention in our opinion, but has not been explored yet, is the presentation of table search results. For example, for large tables, how should appropriate snippets (summaries) be generated for search result pages? Following the two main lines of approaches to automatic summarization, summaries could be extraction-based, by selecting relevant portions of the table to be displayed, or abstraction-based, by generating a natural language summary of its contents Hancock et al. (2019).
Question Answering
Facts extracted from tabular data can be used for answering natural language questions. Previous work has looked at answering questions using a single table or multiple tables. Much of the research emphasis has been directed to parsing the questions and on extracting the facts from tables. While studying these, certain simplifications were made concerning other aspects of the problem. For example, works that address QA on a single table all take a carefully selected table (which is to be treated as a knowledge base) for granted. Locating a proper KB table is a challenging table search task that remains to be solved. There also seems to be a lack of understanding of when tables can actually aid QA. Existing research has found that even though QA on tables suffers from low coverage, it can complement QA on text. Yet, there has not been any systematic study on understanding what are the types of questions where tables can help or what is the scope of facts or relations where web tables have sufficient coverage. Finally, the heterogeneity of web tables limits the applicability of current methods to a small portion of tables. In the future, more generic methods would need to be developed to be able to deal with heterogeneous tables.
Knowledge Base Augmentation
The knowledge mined from tables can (also) be utilized for knowledge base (KB) population/construction. Existing methods address one of two main problems: (1) matching tables to a knowledge base or (2) discovering new facts/relations from tables by utilizing these "table-to-KB" matching results. Yet, all these approaches seem to ignore "out-of-KB" data, that is, entities or properties that are not linked to a knowledge base. Wikipedia tables are one specific example that contain many unlinked entity mentions. Those entities/properties could also be potentially useful for populating KBs with new information. Shortcomings of current approaches include (1) the lack of consideration for temporal information and (2) identifying entities at the right level of granularity (e.g., location may be given as a city or as a state or country) . The former is especially important, as it may promote further utilization of tables to help keep KBs up-to-date.
Table Augmentation
There is a solid body of work on augmenting existing tables with additional data, extracted either from other tables or from knowledge bases. However, there are at least two issues that remain. One is tapping into the large volumes of unstructured sources (e.g., webpages). The other is combining data from multiple sources, which brings about a need for techniques to draw users' attention to conflicting information and help them to deal with those cases. Existing work on augmentation assumes the presence of an input table that the user needs helps with completing. An exciting and ambitious research direction is to automatically generate a table, which can answer the user's information need, from scratch (Zhang and Balog, 2018b).
; Lautert et al. (2013); Nishida et al. (2017); Venetis et al. (2011); Mulwad et al. (2010); Fan et al. (2014); Bhagavatula et al. (2015); Wu et al. (2016); Efthymiou et al. (2017); Zhang et al. (2013); Hassanzadeh et al. (2015); Mulwad et al. (2013); Sekhavat et al. (2014); Ibrahim et al. (2016); Limaye et al. (2010); Muñoz et al. (2014); Ritze et al. (2016); Ritze and Bizer (2017) Table search Query Ranked list of tables Cafarella et al. (2009); Pimplikar and Sarawagi (2012); Cafarella et al. (2008a); Bhagavatula et al. (2013); Ahmadov et al. (2015); Lehmberg et al. (2015); Das Sarma et al. (2012); Yakout et al. (2012); Nguyen et al. (2015); Zhang and Balog (2018a); Limaye et al. (2010); Nargesian et al. (2018); Zhang and Balog (Liang (2015); Sun et al. (2016); Berant et al. (2013); Fader et al. (2014); Neelakantan et al. (2015); Sarawagi and Chakrabarti (2014); Banerjee et al. et al. (2015); Ritze et al. (2015); Ritze and Bizer (2017); Ritze et al. (2016); Lehmberg et al. (2016); Ibrahim et al. (2016); Zhang et al. (2013); Sekhavat et al. (2014); Fan et al. (2014); Dong et al. (2014)
Sarma et al. (2012); Yakout et al. (2012); Cafarella et al. (2008a); Lehmberg et al. (2015); Bhagavatula et al. (2013); Zhang and Balog (2017b); Bhagavatula et al. (2015); Venetis et al. (2011); Cafarella et al. (2009); Zhang and Chakrabarti
Figure 1 :
1Table-related information access tasks and their relationships.
Figure 2 :
2Illustration of
Figure 3 :
3Illustration of table interpretation: (A) Column Type Identification. (B) Entity Linking. (C) Relation extraction.
Bhagavatula et al. (2015) Chen and Cafarella (2013) Efthymiou et al. (2017) Fan et al. (2014) Hassanzadeh et al. (2015) Ibrahim et al. (2016) Lehmberg and Bizer (2016) Limaye et al. (2010) Muñoz et al. (2014) Mulwad et al. (2013) Mulwad et al. (2010) Ritze and Bizer (2017) Ritze et al. (2016) Sekhavat et al. (2014) Venetis et al. (2011) Wang et al. (2012) Wu et al. (2016) Zhang and Chakrabarti(2013)Zhang(2017)
Figure 5 :
5Illustration in(Chirigati et al., 2016), showing an example of knowledge exploration for the query of "kentucky derby" through Knowledge Carousels: (a) a downward showing the winners of Kentucky Derby; (b) a sideway representing the famous Triple Crown horse races in the US, of which Kentucky Derby is a member.
population Das Sarma et al. (2012) Wang et al. (2015a) * Yakout et al. (2012) Zhang and Balog (2017b) * Originally developed for concept expansion, but can be used for row population.
Figure 7 :
7Illustration of row extension by adding only an entity (Upper) or an entire row, i.e., an entity as well as cell values (Lower). Existing work has focused on the former task.
Figure 8 :
8Illustration of column extension by adding only a heading label (Upper) or an entire column, i.e., heading label and cell values (Lower).
Table extraction refers to the process of detecting tables in webpages, extracting them, and storing them in a consistent format, resulting in a table corpus. • Table interpretation aims to uncover the semantics of the data contained in a table, with the aim of making tabular data intelligently processable by machines. This entails classifying tables according to some taxonomy,arXiv:2002.00207v2 [cs.IR] 5 Feb 2020
Table 1 :
1Overview of table-related information access tasks.Task
Input
Output
Key references
Table extraction
extractionWebpages
Tabular data
Table interpretation Table
interpretation(s)
Structured data
Table augmentation is directed at expanding an existing table with additional data. Specific subtasks include populating a table with new rows or columns, or finding missing cell values.
table elements in a web table: table page title (T p ), table caption (T c ), table headings (T H ), table cell (T [i,j] ), table row (T [i,:] ), table column (T
Table page title
pageThe table page title T p is the title of the webpage which embeds the table T . Table caption The caption of a table, T c , is a short textual label summarizing what the table is about.
Table headings Table
headingsheadings, T H , is a list of labels defining what each table row/column is about. Headings are typically in the first row/column in a table. In case of relational tables (see below, in Sect. 2.2), table headings are also referred to as table schema or attribute names. Table cell A table cell T
Table row A
rowtable row T [i,:] is a list of table cells lying horizontally in line i of a table. column A table row T [:,j] is a list of table cells lying vertically in column j of a table. entities Tables often mention specific entities, such as persons, organizations, locations. Table entities T E is a set consisting of all the entities that are mentioned in the table.Table Table
Table 2 :
2Classification of table types in this paper. Our primary focus is on relational tables.Dimension Type
Description
Content
Relational
Describes a set of entities with their attributes
Entity
Describes a specific entity
Matrix
A three dimensional data set, with row and column headers
Other
Special-purpose tables, including lists, calendars, forms, etc.
Layout
Navigational
Table 2 .
2In the remainder of this paper, we shall follow this classification when referring to a certain type of table. Among all table types, relational tables have received the bulk of attention in the literature. Accordingly, we focus primarily on relational tables and the tasks based on them in this survey.
Table 3 :
3Overview of table corpora.
Table corpora
corporaType
#tables
Source
WDC 2012 Web Table Corpus Web tables
147M
Web crawl (Common Crawl)
WDC 2015 Web Table Corpus Web tables
233M
Web crawl (Common Crawl)
Dresden Web Tables Corpus
Web tables
174M
Web crawl (Common Crawl)
WebTables
Web tables
154M
Web crawl (proprietary)
WikiTables
Wikipedia tables
1.6M
Wikipedia
TableArXiv
Scientific tables
0.34M
arxiv.org
Table Corpus
Corpuswere labeled as relational tables. Tables in this corpus are not classified further and neither is table context data provided. The WDC 2015 Web Table Corpus, constructed by, contains 10.24 billion genuine tables. The extraction process consists of two steps: table detection and table classification. The percentages of relational, entity, and matrix tables are 0.9%, 1.4%, and 0.03%, respectively. The remaining 97.75% accounts for layout tables. When storing a table, its orientation is also detected, indicating how the attributes are placed. In horizontal tables, the attributes are placed in columns, while in vertical tables they represent rows. There are 90.26 million relational tables in total. Among those, 84.78 million are horizontal and 5.48 million are vertical. The average number of columns and rows in horizontal tables are 5.2 and 14.45. In vertical tables, these numbers are 8.44 and 3.66, respectively. also extract the column headers and classify each table column as being numeric, string, data, link, boolean, or list. The percentages of the numeric and string columns are 51.4% and 47.3%, respectively. Besides, the text surrounding the table (before and after) is also provided. Furthermore, provide the English-language Relational Subset, comprising of relational tables that are classified as being in English, using a naive Bayesian language detector. The language filter considers atable's page title, table header, as well as the text surrounding the table to classify it as English or non-English. The average number of columns and rows in this subset are 5.22 and 16.06 for horizontal tables, and 8.47 and 4.47 for vertical tables. The percentages of numeric and string columns are 51.8% and 46.9%. A total of 139 million tables in the WDC 2015 Web Table Corpus are classified as entity tables. also extracted tables from the Common Crawl web corpus. The total number of tables is 174 million, which is reduced to 125 million after filtering with regards to content-based duplication. The Dresden WebTable Corpus contains only the core table data, and not the entire HTML page. Even though the corpus is not available for download directly, the table extraction framework (extractor code and companion library for working with the data set) is made publicly available. 3, 1 which were released in 2012 and 2015, respectively. The 2012
version contains 147 million web tables, which were extracted from the 2012 Common Crawl corpus (consisting of
3.5 billion HTML pages). Tables in this corpus are roughly classified as relational or non-relational in terms of layout.
Statistically, 3.3 billion HTML pages were parsed and 11.2 billion tables were identified; tables that are not innermost
(that is, contain other tables in their cells) were discarded. 1.3% of the remaining tables (originating from 101 million
different webpages) Out of these, 76.70
million are horizontal and 62.99 million are vertical tables. The average number of columns and rows are 2.40 and
9.08 for horizontal tables, and 7.53 and 2.06 for vertical tables. The column data types are quite different from that of
relational tables. String columns are the most popular, amounting to 86.7% of all columns, while numeric columns
account for only 9.7%.
The complete corpus as well as the different subcorpora are made publicly available. 2
2.3.2 Dresden Web Table Corpus
2.3.3 WebTables
Cafarella et al. (2008b) extracted 154 million high-quality relational web tables from a (proprietary) general-purpose
web crawl. Unfortunately, this corpus is not made public. However, frequency statistics of attributes, known as the
ACSDb dataset (cf. Sec. 8.2), is available for download. 4
2.3.4 Wikipedia Tables
Bhagavatula et al. (2015) focused on Wikipedia and extracted 1.6 million high-quality relational tables. Each table
is stored as a JSON file, including table body, table caption, page title, column headers, and the number of row and
columns. The existing links in the tables are also extracted and stored in a separate file. The corpus is available for
download. 5
Table Corpus
Corpus(DWTC, cf. Sect. 2.3.2).
To obtain metadata for relational tables, Eberius et al. (2015) consider whether tables have a header row or not. They find
that 71% of the tables in the corpus have a relational header. For the remaining 29%, they attempt to generate synthetic
labels by comparing the column content to similar columns that have proper labels. Cafarella et al. (2009) design
a system called OCTOPUS, which combines search, extraction, data cleaning, and integration. Further challenges
related applying WebTables in practice, including table identification and table semantics recovery, are detailed in
(Balakrishnan et al., 2015). The resulting system, Google Fusion Tables, is made publicly available. 7
Table type
typeclassification is the task of
classifying tables according to a pre-defined type taxonomy (cf. Sect. 2.2 for the discussion of various classification
schemes). Additional metadata extracted for tables includes the embedding page's title, the table's caption, and the text
surrounding the table.
The same features that are intended for relational table classification and header detection can also be used for table
type classification Wang and Hu (2002b,a); Lehmberg et al. (2016); Chen and Cafarella (2013); Cafarella et al. (2008b).
For example, the features listed in
Table 4 :
4volume, like image data, in the first step. The CNN encoders encode the3D table volume to capture table semantics, which is used for table type classification by the Classifier. Even though TabNet is designed to capture table structure, itcan be applied to any matrix for type classification.Selected features for relational table classification (RTC), header detection (HD), and table type classification
(TTC) (Part 1/2).
Features
Explanation
Task
Source
Global layout features
Max rows
Maximal number of cells per row
RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
Max cols
Maximal number of cells per column RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
Max cell length
Maximal number of characters per cell RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
#rows
Number of rows in the table
RTC, HD
Cafarella et al. (2008b)
#cols
Number of columns in the table
RTC, HD
Cafarella et al. (2008b)
%rows
Percentage of rows that are mostly
NULL
RTC
Cafarella et al. (2008b)
#cols non-string
Number of columns with non-string
data
RTC
Cafarella et al. (2008b)
µ
Average length of cell strings
RTC
Cafarella et al. (2008b)
δ
Standard deviation of cell string length RTC
Cafarella et al. (2008b)
µ
δ
Cell string length
RTC
Cafarella et al. (2008b)
%length one
Percentage of columns with
|len(row 1 ) − µ| > 2δ
HD
Cafarella et al. (2008b)
%length two
Percentage of columns with δ ≤
|len(row 1 ) − µ| ≤ 2δ
HD
Cafarella et al. (2008b)
%length three
Percentage of columns with
|len(row 1 ) − µ| < δ
HD
Cafarella et al. (2008b)
Avg rows
Average number of cells across rows
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Avg cols
Average number of cells across
columns
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Avg cell length
Average number of characters per cell RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015); Wang and
Hu (2002b)
Table 5 :
5Selected features for relational table classification (RTC), header detection (HD), and table type classification
(TTC) (Part 2/2).
Features
Explanation
Task
Source
Layout features
Std dev rows
Standard dev. of the number of cells
per row
RTC
Eberius et al. (2015); Wang and
Hu (2002b)
Std dev cols
Standard dev. of the number of cells
per column
RTC
Eberius et al. (2015); Wang and
Hu (2002b)
Std dev cell length
Standard dev. of the number of charac-
ters per cell
RTC
Crestan and Pantel (2011);
Eberius et al. (2015); Wang and
Hu (2002b)
Local length avg
Average size of cells in segment
RTC
Crestan and Pantel (2011);
Eberius et al. (2015)
Local length variance Variance of size of cells in segment
RTC
Crestan and Pantel (2011);
Eberius et al. (2015)
Content features
%body non-string
Percentage of non-string data in table
body
HD
Cafarella et al. (2008b)
%header non-string
Percentage of non-string data in the
first row
HD
Cafarella et al. (2008b)
%header punctuation Percentage of columns with punctua-
tion in the first row
HD
Cafarella et al. (2008b)
Local span ratio
Ratio of cells with a span tag
RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
Local ratio header
Cells containing a th tag
RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
Local ratio anchor
Cells containing an a tag
RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
Local ratio input
Cells containing an input tag
RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015)
Ratio img
Ratio of cells containing images
RTC, TTC Crestan and Pantel (2011);
Eberius et al. (2015); Wang and
Hu (2002b)
Ratio form
Ratio of cells containing forms
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Ratio hyperlink
Ratio of cells containing hyperlinks
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Ratio alphabetic
Ratio of cells containing alphabetic
characters
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Ratio digit
Ratio of cells containing numeric char-
acters
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Ratio empty
Ratio of empty cells
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
Ratio other
Ratio of other cells
RTC, TTC Eberius et al. (2015); Wang and
Hu (2002b)
4 Table Interpretation
Table interpretation encompasses methods that aim to make tabular data processable by machines. Specifically, it
focuses on interpreting tables with the help of existing knowledge bases. Bhagavatula et al. (2015) identify three main
tasks aimed at uncovering table semantics:
Table 6 :
6Overview of table interpretation tasks addressed in various studies.Reference
Column type
Entity
Relation
Identification Linking Extraction
Table 7 :
7Comparison of column type identification tasks.Reference
Knowledge base
Method
Fan et al. (2014)
Freebase
Concept-based method + crowdsourcing
Lehmberg and Bizer (2016)
DBpedia
Feature-based classification
Mulwad et al. (2010)
Wikitology
Entity search
Wang et al. (2012)
Probase
Heading-based search
Venetis et al. (2011)
Automatically built IS-A database Majority vote
Zhang and Chakrabarti (2013) -
Semantic graph method
Zhang
table semantics, a top-k candidates concepts are returned based on the table headings, which is similar to the idea in Limaye et al. (2010) (cf. Sect. 4.2).
Table 8 :
8Comparison of entity linking tasks.Reference
Knowledge base
Method
Bhagavatula et al. (2015)
YAGO
Graphical model
Efthymiou et al. (2017)
DBpedia
Vectorial representation and ontology matching
Hassanzadeh et al. (2015)
DBpedia, Schema.org, YAGO, Wikidata,
and Freebase
Ontology overlap a
Ibrahim et al. (2016)
YAGO
Probabilistic graphical model
Lehmberg and Bizer (2017) DBpedia
Feature-based method
Lehmberg et al. (2016)
Google Knowledge Graph
-
Limaye et al. (2010)
YAGO catalog, DBpedia, and Wikipedia
tables
Inference of five types of features b
Mulwad et al. (2010)
Wikitology
SVM classifier
Ritze and Bizer (2017)
DBpedia
Feature-based method
Ritze et al. (2015, 2016)
DBpedia
Feature-based method
Wu et al. (2016)
Chinese Wikipedia, Baidu Baike, and
Hudong Baike
Probabilistic method c
Zhang et al. (2013)
DBpedia
Instance-based schema mapping
Zhang (2017)
Freebase
Optimization
a KB comparison
b Designed for table search
c Multiple KBs
Table 9 :
9Comparison of relation extraction tasks.Reference
Knowledge
base
Method
Source of extraction
Chen and Cafarella (2013)
-
Classification
Each value in the value region
Muñoz et al. (2014)
DBpedia
Look-up based
Any pair of entities in the same row
Mulwad et al. (2013)
DBpedia
Semantic passing
Any pair of columns
Mulwad et al. (2010)
DBpedia
Utilizing CTI and EL
Any pair of columns
Sekhavat et al. (2014)
YAGO, PATTY Probabilistic
Any pair of entities in the same row
Venetis et al. (2011)
IS-A database
Frequency-based
Core + attribute columns
Zhang (2017)
Freebase
Optimization
Any pair of columns
above methods tend to perform better for large tables, i.e., tables with several rows. It is considered as one of the main
limitations of linking tabular mentions to DBpedia. To overcome this, Lehmberg and Bizer (2017) stitch tables, i.e.,
merge tables from the same website as a single large table, in order to improve entity linking performance.
table .
.Relation extraction can also be used to augment Linked Data repositories(Sekhavat et al., 2014).Sekhavat et al. (2014) propose a probabilistic approach using under-explored tabular data. Assuming that the entities co-occurring in the same table are related, they focus on extracting relations between pairs of entities appearing in the same row of a table. Entities in table cells are mapped to a knowledge base first. Then, sentences containing both entities from the same table row are collected from a text corpus. Next, textual patterns (describing the relationship between these two entities) are extracted. Finally, the probability of the possible relations is estimated using Bayesian inference. A new relation, which is a triple consisting of two entities and a pattern, can be added to the Linked Data repository for augmentation.Muñoz et al. (2014) utilize entity annotations in Wikipedia tables. Taking existing relations between entities in DBpedia, they look these entities up in Wikipedia tables. This then indicates that the same relation stands between entities in other rows of this table.Chen and Cafarella (2013) introduce a system to automatically extract relational data from spreadsheets instead of the
Web. Most of the methods on spreadsheets require users to provide sheet-specific rules (Ahmad et al., 2003; Hung
Figure 4: Excerpt from a table containing hierarchical attributes. The example is taken from the U.S. Department of
Transportation (http://www.api.faa.gov/CivilAir/index.html).
et al., 2011). In contrast, Chen and Cafarella
table .
.Web tables are embedded in HTML pages, where the surrounding text can help to understand what a given table is about. However, these surrounding sentences are not equally beneficial for table understanding. Wang et al. (2015b) present the Table-Related Context Retrieval system (TRCR) to determine the relevance between a table and each surrounding sentence. Using TRCR, the most relevant texts are selected to uncover table semantics. Another related study is performed in
Table search
searchis the task of returning a ranked list of tables in response to a query. It is an important task on its own and is regarded as a fundamental step in many other table mining and extraction tasks as well, like table integration or data completion.Table search functionality is also available in commercial products; e.g., Microsoft Power Query provides smart assistance features based on table search. Depending on the type of the query, table search may be classified as keyword-based search and table-based search.
Table 10 :
10A selection of features for keyword-based table search.Query features
Table features
features#rows
Number of rows in the table
Cafarella et al. (2008a); Bha-
gavatula et al. (2013)
#cols
Number of columns in the table
Cafarella et al. (2008a); Bha-
gavatula et al. (2013)
#of NULLs in table Number of empty table cells
Cafarella et al. (2008a); Bha-
gavatula et al. (2013)
PMI
ACSDb-based schema coherency score
Cafarella et al. (2008a)
inLinks
Number of in-links to the page embedding the table
Bhagavatula et al. (2013)
outLinks
Number of out-links from the page embedding the table
Bhagavatula et al. (2013)
pageViews
Number of page views
Bhagavatula et al. (2013)
tableImportance
Inverse of number of tables on the page
Bhagavatula et al. (2013)
tablePageFraction
Ratio of table size to page size
Bhagavatula et al. (2013)
Query-table features
#hitsLC
Total query term frequency in the leftmost column cells
Cafarella et al. (2008a)
#hitsSLC
Total query term frequency in second-to-leftmost column
cells
Cafarella et al. (2008a)
#hitsB
Total query term frequency in the table body
Cafarella et al. (2008a)
qInPgTitle
Ratio of the number of query tokens found in page title to
total number of tokens
Bhagavatula et al. (2013)
qInTableTitle
Ratio of the number of query tokens found in table title to
total number of tokens
Bhagavatula et al. (2013)
yRank
Rank of the table's Wikipedia page in web search engine
results for the query
Bhagavatula et al. (2013)
MLM similarity
Language modeling score between query and multi-field
document repr. of the table
Hasibi et al.
) .
)Feature-based methods employ supervised machine learning for table ranking. Features may be divided into three main categories: query features, table features and query-table features. Query features include query length and IDF scores of query terms. Table features characterize the table in terms of its dimensions (number of rows, columns) and schema coherency. With a focus on Wikipedia tables, Bhagavatula et al. (2013) introduce features related to the connectivity of the Wikipedia page (pageViews, inLinks, and outLinks) and the table's importance within the page (table importance and table page fraction). Finally, query-table features capture the degree of matching between the user's information need and the table. Typically, these include similarity scores between the query and various table elements. Table 10 lists a selection of features for keyword table search. In terms of learning algorithm, Cafarella et al. (2008a) train a linear regression classifier, while Bhagavatula et al. (2013) train a linear ranking model learned with Coordinate Ascent.
Table 11 :
11Overview of table elements used when querying by table for various table-related applications.
Table augmentation
augmentationrefers to the task of extending a seed table with more data. Specifically, we discuss three tasks
in this section: row extension (Sect. 8.1), column extension (Sect. 8.2), and data completion (Sect. 8.3). See Fig. 6
l1
e1
l2
…
…
ei
lm
ei+1
l1
e1
l2
…
…
ei
lm
lm+1
t21
…
t2i
l1
e1
l2
…
…
ei
lm
Input Table
R o w E x te n s io n
Column Extension
D a ta C o m p le ti o n
l1
e1
l2
…
…
ei
lm
Table 12 :
12Overview of row population methods. Notice that table search is inherently involved.Data
Tasks
Reference
KB
Table 13 :
13Overview of column population methods.Data
Output
takes the incomplete table as input to search for matching tables, then extracts attribute values from those tables. It is worthwhile to point out that InfoGather focuses on finding values that are entities. An extended version of the system, InfoGather+Zhang and Chakrabarti (2013), focuses on numerical and time-varying attributes. They usel1
e1
l2
…
…
ei
lm
…
l1
e1
l2
…
…
ei
lm
Input Table
tjk
t12
…
ti2
l1
e1
l2
…
…
ei
lm
Jo in
Da ta im pu ta tio n
Table 14 :
14Overview of data completion methods.Data
Output
Reference
Tables Web
T [:,j] T [i,j]
Ahmadov et al. (2015)
Cafarella et al. (2009)
Yakout et al. (2012)
Zhang and Chakrabarti (2013)
Zhang and Balog
http://webdatacommons.org/framework/ 2 http://webdatacommons.org/webtables/#results-2015 3 https://wwwdb.inf.tu-dresden.de/misc/dwtc/
https://web.eecs.umich.edu/~michjc/data/acsdb.html 5 http://websail-fe.cs.northwestern.edu/TabEL/ 6 http://boston.lti.cs.cmu.edu/eager/table-arxiv/
https://research.google.com/tables
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Auto-completion for Data Cells in Relational Tables. Shuo Zhang, Krisztian Balog, Proceedings of the 28th ACM International Conference on Information and Knowledge Management (CIKM'19. the 28th ACM International Conference on Information and Knowledge Management (CIKM'19Shuo Zhang and Krisztian Balog. 2019a. Auto-completion for Data Cells in Relational Tables. In Proceedings of the 28th ACM International Conference on Information and Knowledge Management (CIKM'19). 761-770.
Recommending Related Tables. Shuo Zhang, Krisztian Balog, arXiv:1907.03595Shuo Zhang and Krisztian Balog. 2019b. Recommending Related Tables. CoRR abs/1907.03595 (2019). arXiv:1907.03595 http://arxiv.org/abs/1907.03595
Mapping Entity-Attribute Web Tables to Web-Scale Knowledge Bases. X Zhang, Y Chen, X Du, L Zou, Database Systems for Advanced Applications. X. Zhang, Y. Chen, X. Du, and L. Zou. 2013. Mapping Entity-Attribute Web Tables to Web-Scale Knowledge Bases. Database Systems for Advanced Applications (2013), 108-122.
Effective and efficient Semantic Table Interpretation using TableMiner+. Ziqi Zhang, Semantic Web. 8Ziqi Zhang. 2017. Effective and efficient Semantic Table Interpretation using TableMiner+. Semantic Web 8 (2017), 921-957.
| {'fraction_non_alphanumeric': 0.04259468336241404, 'fraction_numerical': 0.02705532176947552, 'mean_word_length': 4.765583083477987, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 3, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 23, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Tables are a powerful and popular tool for organizing and manipulating data. A vast number of tables can be found on the Web, which represent a valuable knowledge resource. The objective of this survey is to synthesize and present two decades of research on web tables. In particular, we organize existing literature into six main categories of information access tasks: table extraction, table interpretation, table search, question answering, knowledge base augmentation, and table augmentation. For each of these tasks, we identify and describe seminal approaches, present relevant resources, and point out interdependencies among the different tasks.', 'arxivid': '2002.00207', 'author': ['Shuo Zhang [email protected] \nUniversity of Stavanger\nUniversity of Stavanger\n\n', 'Krisztian Balog [email protected] \nUniversity of Stavanger\nUniversity of Stavanger\n\n'], 'authoraffiliation': ['University of Stavanger\nUniversity of Stavanger\n', 'University of Stavanger\nUniversity of Stavanger\n'], 'corpusid': 211011078, 'doi': '10.1145/3372117', 'github_urls': [], 'n_tokens_mistral': 40505, 'n_tokens_neox': 34802, 'n_words': 21779, 'pdfsha': '5e49edaa3c1a24bbcc167145bdec65ade36ffe24', 'pdfurls': ['https://arxiv.org/pdf/2002.00207v2.pdf'], 'title': ['WEB TABLE EXTRACTION, RETRIEVAL AND AUGMENTATION: A SURVEY A PREPRINT', 'WEB TABLE EXTRACTION, RETRIEVAL AND AUGMENTATION: A SURVEY A PREPRINT'], 'venue': []} |
arxiv |
VLSI Layouts and DNA Physical Mappings
Michael J Dinneen
Computer Research and Applications Los Alamos National Laboratory Los Alamos
87545N.M
VLSI Layouts and DNA Physical Mappings
is the minimum vs(L; G) over all layouts L of G.The k-coloring of a graph G = (V; E ) is a mapping color : V ! f1; 2; : : : ; kg. For any subset V 0 V , let C olors(V 0 ) = fcolor(v) j v 2 V 0 g.De nition 3: A colored layout L of a k-colored graph G = (V; E ) is layout L such that for all 1 i < n, color(v i+1 ) 6 2 C olors(V i ).
Introduction
In this short note, we show that an important problem in computational biology is equivalent to a colored version of a well-known graph layout problem. In order to map the human genome, biologists use graph theory, particularly interval graphs, to model the overlaps of DNA clones (cut up segments of a genome) Mir94]. For engineers, Very-Large-Scale-Integrated (VLSI) circuits must be laid out in order to minimize physical and cost constraints. The vertex separation (see below) of a graph layout is one such measurement of how good a layout is.
The NP-complete combinatorial problem of Intervalizing Colored Graphs (ICG) rst de ned in FHW93] (and independently given in GKS93] as the Graph Interval Sandwich problem) is intended to be a limited, rst-step model for nding DNA physical mappings. For this model, it is assumed that the biologist knows some of the overlaps | for instance, overlaps speci ed by some probability threshold based on the physical data. The question asked by the ICG problem is whether other edges can be properly added to di erently colored vertices to form a colored interval graph.
Finding the Vertex Separation (VS) of a graph is related to many diverse problems in computer science besides its importance to VLSI layouts. Lengauer showed that progressive black/white pebble game (important to compiler theory) and vertex separation are polynomially reducible to each other Len81]. Node search number, a variant of search number Par76], was shown equivalent to the vertex separation plus one by Kirousis and Papadimitriou KP86]. From EST94], the search number is informally de ned in terms of pebbeling to be the minimum number of searchers needed to capture a fugitive who is allowed to move with arbitrary speed about the edges of the graph. For node search number, a searcher blocks all neighboring nodes without the need to move along an incident edge. Kin92] has shown that the pathwidth of a graph is identical to the vertex separation of a graph. The concept of pathwidth has been popularized by the theories of Robertson and Seymour (see for example, RS85]). Thus, since the gate matrix layout cost, another well-studied VLSI layout problem KL94, M oh90], equals the pathwidth plus one FL89], it also equals the vertex separation plus one.
Kinnersley in
This paper shows that vertex separation is also related to another area besides computer science, namely computational biology.
Main Result
In this section, we formally de ne our xed-parameter problems k-ICG and k-CVS and then show that they are indeed equivalent.
De nition 1: A layout L of a graph G = (V; E ) is a one to one mapping L : V ! f1; 2; : : : ; jV jg.
If the order of a graph G = (V; E ) is n, we conveniently write a layout L as a permutation of the vertices (v 1 ; v 2 ; : : : ; v n ). For any layout L = (v 1 ; v 2 ; : : : ; v n ) of G let V i = fv j j j i and (v j ; v k ) 2 E for some k > ig for each 1 i n.
De nition 2: The vertex separation of a graph G with respect to a layout L is vs(L; G) = max 1 i jGj fjV i jg. The vertex separation of a graph G, denoted by vs(G), Problem 5: Intervalizing Colored Graphs (ICG) Input: A k-colored graph G = (V; E ). Parameter: k Question: Is there a properly colored supergraph G 0 = (V 0 ; E 0 ) of G, E E 0 , such that V = V 0 and G 0 is an interval graph? Figure 2 below shows a 3-colored graph with an interval supergraph represented on the left and a colored vertex separation layout given on the right. Proof. Let L = (v 1 ; v 2 ; : : : ; v n ) be a colored layout of a k-colored graph G = (V; E ).
We show how to construct a properly colored supergraph G 0 that is also an interval graph. For each vertex v i 2 V , de ne the interval:
I v i = a v i ; b v i ] = i; maxfj j (v i ; v j ) 2 E _ j = ig + 0:5] By de nition, if edge (u; v) 2 E then I u \I v 6 = ;. Let G 0 = (V; E 0 ) where (v i ; v j ) 2 E 0 whenever I v i \ I v j 6 = ;. It su ces to show that color(v i ) 6 = color(v j ) for each edge (v i ; v j ) in E 0 n E . Without loss of generality, assume i < j so that b v i > a v j . Again by the de nition of I v i , there exists a vertex v k such that j < k and (v i ; v k ) 2 E . This implies that v i 2 V j 1 . (This also holds for i = j 1.) Now L is a colored layout so color(v j ) 6 2 C olors(V j 1 ). Thus, color(v i ) 6 = color(v j ). Therefore, G 0 is a properly-colored intervalizable supergraph of G.
For any k-colored graph G = (V; E ) that satis es ICG, let fI v j v 2 V g be an interval graph representation of a supergraph G 0 = (V; E 0 ). Let a v < b v be the endpoints of the interval I v = a v ; b v ] for vertex v. Without loss of generality, assume that a u = a v implies u = v. Let L = (v 1 ; v 2 ; : : : ; v n ) be the unique layout such that i < j if and only if a v i < a v j . We claim that L is a colored layout of G 0 . To prove this claim, we show that color(v i+1 ) 6 2 C olors(V i ), 1 i < n. If there exists a vertex u 2 V i such that color(u) = color(v i+1 ) then by de nition of V i vertex u must be adjacent to a vertex v j for some j > i. Further, j > i + 1 since (u; v i+1 ) would not be a properly colored edge. Since a u < a v j and (u; v j ) 2 G 0 , we must have b u > a v j in order to form an overlap. However, b u < a v i+1 < a v j . This is a contradiction to j > i. So u 6 2 V i if color(u) = color(v i+1 ). Thus L is a colored layout. Now suppose that for some r < s there exist two vertices v r and v s in V i with the same color. Since v r 2 V i , there exists a vertex v j with j > i such that (v r ; v j ) 2 E 0 . This implies v r 2 V s 1 . But this implication contradicts the fact color(v s ) 6 2 C olors(V s 1 ). So color(v r ) 6 = color(v s ). Hence any set V i fv i+1 g has at most one vertex of each color. Since there are k colors, each V i must have k 1 or fewer vertices. Thus, vs(L; G) vs(L; G 0 ) < k.
3 Final Comments
Recently, the corresponding general problem of intervalizing a colored graph to an unit interval graph has been shown to be NP-hard (and x-parameter hard for W 1]) by Kaplan and Shamir KS93] (also see GGKS93,KST94]). The good news from Kaplan and Shamir's paper is that for each xed-parameter k (i.e., k colors) this unit interval problem has a polynomial-time algorithm. It is still unknown if a polynomial-time algorithm exists for k-ICG, or equivalently k-CVS. It is our hope that understanding the original polynomial-time algorithm for the non-colored vertex separation problem may be of some use EST87].
A related approach for nding a practical k-ICG algorithm is based on the easily seen fact that all colored graphs in the k-ICG family have pathwidth less than or equal to k 1. The usual polynomial-time algorithms for these types of bounded pathwidth families are constructed as follows: First nd a path-decomposition of width k 1 and then use some type of dynamic programming approach on the graph using its decomposition. The tricky part for k-ICG is that k-ICG is not nite-state (i.e., not representable by linear/tree automaton) for xed k and hence conventional algorithmic techniques can not be used FHW93].
However, just because k-ICG is not nite-state, we should not avoid altogether the pathwidth structure of the graphs in this family. For small k, Bodlaender and Kloks recently developed an algorithm for recognizing and nding path-decompositions of width k in linear time (see Bod93, BK91, BK93] and CDF]).
Figure 1 :
1Illustrating the k-CVS and k-ICG problems.Theorem 6: For any xed positive integer parameter k, both k-CVS and k-ICG are identical problems.
Better algorithms for pathwidth and treewidth of graphs. L Hans, Ton Boadlaender, Kloks, Proceedings of the 18th International Colloquium on Automata, Languages and Programming. the 18th International Colloquium on Automata, Languages and ProgrammingSpringer VerlagHans L. Boadlaender and Ton Kloks. Better algorithms for pathwidth and treewidth of graphs. In Proceedings of the 18th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, pages 544{555. Springer Verlag, 1991.
Boadlaender and Ton Kloks. E cient and constructive algorithms for the pathwidth and treewidth of graphs. L Hans, preprintHans L. Boadlaender and Ton Kloks. E cient and constructive algorithms for the pathwidth and treewidth of graphs, 1993. preprint.
A linear time algorithm for nding tree-decompostions of small treewidth. L Hans, Bodlaender, Proceedings of the 25th Annual ACM Symposium on Theory of Computing. the 25th Annual ACM Symposium on Theory of ComputingACM PressHans L. Bodlaender. A linear time algorithm for nding tree-de- compostions of small treewidth. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing. ACM Press, 1993.
A simple linear-time algorithm for nding path-decompositions of small width. Submitted to Information Processing Letters. Kevin Cattell, Michael J Dinneen, Michael R Fellows, Kevin Cattell, Michael J. Dinneen, and Michael R. Fellows. A simple linear-time algorithm for nding path-decompositions of small width. Sub- mitted to Information Processing Letters.
Graph separation and search number. J Ellis, I H Sudborough, J Turner, DCS-66-IRP.O. Box. 3055Dept. of Computer Science, University of VictoriaReportJ. Ellis, I. H. Sudborough, and J. Turner. Graph separation and search number. Report DCS-66-IR, Dept. of Computer Science, University of Victoria, P.O. Box 3055, Victoria, B.C. Canada V8W 3P6, August 1987.
The vertex separation and search number of a graph. J Ellis, I H Sudborough, J Turner, Information and Computation. 1131J. Ellis, I. H. Sudborough, and J. Turner. The vertex separation and search number of a graph. Information and Computation, 113(1):50{79, 1994.
DNA physical mapping: Three ways di cult. Michael R Fellows, Michael T Hallett, H Todd Wareham, Proceedings of European Symposium on Algorithms (ESA'93). T. LengauerEuropean Symposium on Algorithms (ESA'93)BerlinSpringer-Verlag726Michael R. Fellows, Michael T. Hallett, and H. Todd Wareham. DNA physical mapping: Three ways di cult. In T. Lengauer, editor, Proceed- ings of European Symposium on Algorithms (ESA'93), volume 726 of Lec- ture Notes in Computer Science, pages 157{168. Springer-Verlag, Berlin, 1993.
On search, descision and the e ciency of polynomial-time algorithms. R Michael, Michael A Fellows, Langston, Proceedings of the 21th. the 21thMichael R. Fellows and Michael A. Langston. On search, descision and the e ciency of polynomial-time algorithms. In Proceedings of the 21th
Annual ACM Symposium on Theory of Computing. 501to appearAnnual ACM Symposium on Theory of Computing, pages 501{512, 1989. to appear Journal of Computer and System Sciences.
Four strikes against physical mapping of DNA. Paul W Goldberg, Martin C Golumbic, Haim Kaplan, Ron Shamir, 287/93Frida Eskenasy Insitute of Computer Sciences, Tel Aviv UniversityTechnical reportPaul W. Goldberg, Martin C. Golumbic, Haim Kaplan, and Ron Shamir. Four strikes against physical mapping of DNA. Technical report 287/93, The Moise and Frida Eskenasy Insitute of Computer Sciences, Tel Aviv University, December 1993.
The Moise and Frida Eskenasy Insitute of Computer Sciences. C Martin, Haim Golumbic, Ron Kaplan, Shamir, 271/93Advances in Applied Mathematics. Tel Aviv UniversityTechnical reportOn the complexity of DNA physical mapping. to appearMartin C. Golumbic, Haim Kaplan, and Ron Shamir. On the complexity of DNA physical mapping. Technical report 271/93, The Moise and Frida Eskenasy Insitute of Computer Sciences, Tel Aviv University, January 1993. to appear Advances in Applied Mathematics.
The vertex separation number of a graph equals its path-width. Nancy G Kinnersley, Information Processing Letters. 42Nancy G. Kinnersley. The vertex separation number of a graph equals its path-width. Information Processing Letters, 42:345{350, 1992.
Obstruction set isolation for the Gate Matrix Layout problem. Nancy G Kinnersley, Michael A Langston, Discrete Applied Mathematices. 54Nancy G. Kinnersley and Michael A. Langston. Obstruction set isolation for the Gate Matrix Layout problem. Discrete Applied Mathematices, 54:169{213, 1994.
Searching and pebbling. L M Kirousis, C H Papadimitriou, Theoretical Computer Science. 47L. M. Kirousis and C. H. Papadimitriou. Searching and pebbling. Theo- retical Computer Science, 47:205{216, 1986.
Pathwidth, bandwidth and completion problems to proper interval graphs. Haim Kaplan, Ron Shamir, 285/93Frida Eskenasy Insitute of Computer Sciences, Tel Aviv UniversityTechnical reportHaim Kaplan and Ron Shamir. Pathwidth, bandwidth and completion problems to proper interval graphs. Technical report 285/93, The Moise and Frida Eskenasy Insitute of Computer Sciences, Tel Aviv University, November 1993.
Tractability of parameterized completion problems on chordal and interval graphs: minimum ll-in and physical mapping. Haim Kaplan, Ron Shamir, Robert E Targan, Proceedings of the 35th Annual IEEE Conference on the Foundations of Computer Science. the 35th Annual IEEE Conference on the Foundations of Computer ScienceHaim Kaplan, Ron Shamir, and Robert E. Targan. Tractability of param- eterized completion problems on chordal and interval graphs: minimum ll-in and physical mapping. In Proceedings of the 35th Annual IEEE Conference on the Foundations of Computer Science, 1994.
Black-white pebbles and graph separation. T Lengauer, Acta Informatica. 16T. Lengauer. Black-white pebbles and graph separation. Acta Informatica, 16:465{475, 1981.
Graphs and Genes. B G Mirkin, H.L. BeusSpringer-VerlagB.G. Mirkin. Graphs and Genes. Springer-Verlag, 1994. (translated from Russian by H.L. Beus.
Graph problems releted to gate matrix layout and PLA folding. H Rolf, Ohring, Computational Graph Theory. G. Tinhofer, E. Mayr, H. Noltemeier, and M. SysloSpringer-VerlagM oh90] Rolf H. M ohring. Graph problems releted to gate matrix layout and PLA folding. In G. Tinhofer, E. Mayr, H. Noltemeier, and M. Syslo, editors, Computational Graph Theory, pages 17{51. Springer-Verlag, 1990.
Pursuit-evasion in a graph. T D Parsons, Theory and applications of graphs. Y. Alavi and D. R. LickSpringer-VerlagT. D. Parsons. Pursuit-evasion in a graph. In Y. Alavi and D. R. Lick, ed- itors, Theory and applications of graphs, pages 426{441. Springer-Verlag, 1976.
Graph Minors { A Survey. Neil Robertson, Paul D Seymour, Surveys in Combinatorics. Cambridge University Press103Neil Robertson and Paul D. Seymour. Graph Minors { A Survey. In Sur- veys in Combinatorics, volume 103, pages 153{171. Cambridge University Press, 1985.
| {'fraction_non_alphanumeric': 0.055676928169480305, 'fraction_numerical': 0.02608407811982787, 'mean_word_length': 3.9045454545454548, 'pattern_counts': {'":': 0, '<': 12, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 14, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'is the minimum vs(L; G) over all layouts L of G.The k-coloring of a graph G = (V; E ) is a mapping color : V ! f1; 2; : : : ; kg. For any subset V 0 V , let C olors(V 0 ) = fcolor(v) j v 2 V 0 g.De nition 3: A colored layout L of a k-colored graph G = (V; E ) is layout L such that for all 1 i < n, color(v i+1 ) 6 2 C olors(V i ).', 'arxivid': 'math/9503221', 'author': ['Michael J Dinneen \nComputer Research and Applications Los Alamos National Laboratory Los Alamos\n87545N.M\n'], 'authoraffiliation': ['Computer Research and Applications Los Alamos National Laboratory Los Alamos\n87545N.M'], 'corpusid': 10150243, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 4663, 'n_tokens_neox': 4128, 'n_words': 2533, 'pdfsha': '1cc8851c3cf40ac2b9bc6583a54103782a2d0fc0', 'pdfurls': ['https://arxiv.org/pdf/math/9503221v1.pdf'], 'title': ['VLSI Layouts and DNA Physical Mappings', 'VLSI Layouts and DNA Physical Mappings'], 'venue': []} |
arxiv |
Effective field theory for magnetic compactifications
31 Mar 2017
Wilfried Buchmuller 1e-mail:[email protected]:[email protected]:[email protected]:[email protected]
Markus Dierigl
Emilian Dudas
Centre de Physique Théorique
École Polytechnique
CNRS
Université Paris-Saclay
F-91128PalaiseauFrance
Julian Schweizer
Deutsches Elektronen-Synchrotron DESY
22607HamburgGermany
Effective field theory for magnetic compactifications
31 Mar 2017
Magnetic flux plays an important role in compactifications of field and string theories in two ways, it generates a multiplicity of chiral fermion zero modes and it can break supersymmetry. We derive the complete four-dimensional effective action for N = 1 supersymmetric Abelian and non-Abelian gauge theories in six dimensions compactified on a torus with flux. The effective action contains the tower of charged states and it accounts for the mass spectrum of bosonic and fermionic fields as well as their level-dependent interactions. This allows us to compute quantum corrections to the mass and couplings of Wilson lines. We find that the one-loop corrections vanish, contrary to the case without flux. This can be traced back to the spontaneous breaking of symmetries of the six-dimensional theory by the background gauge field, with the Wilson lines as Goldstone bosons.
Introduction
Magnetic flux plays a crucial role in the compactification of field theories and string theories 1 in several ways. First of all, it leads to a multiplicity of fermion zero modes, which can be used to explain the number of quark-lepton generations [4]. Moreover, it is an important source of supersymmetry breaking [5] and, together with a nonperturbative superpotential, it can stabilize the compact dimensions [6] consistent with four-dimensional Minkowski or de Sitter vacua [7,8].
In this paper we study the effect of flux on quantum corrections. We consider the simplest case, a six-dimensional (6d) gauge theory with N = 1 supersymmetry compactified to four dimensions (4d) on a torus T 2 . Following techniques developed in [9,10], we start from the 6d Lagrangian written in terms of 4d chiral and vector superfields. Before we consider the constant magnetic flux background we derive a supersymmetric effective action for the Kaluza-Klein (KK) states of a 6d Abelian gauge theory and show how the KK excitations of the vector multiplet obtain their masses from a supersymmetric Stückelberg mechanism. In the flux background the covariant derivatives of the charged fields satisfy a harmonic oscillator algebra [5,6,[11][12][13], which allows to encode their dynamics in the compact dimensions via ladder operators. Applying this harmonic oscillator analogy to the full superfields we derive a 4d supersymmetric effective action that incorporates the complete tower of charged states. Even though the explicit form of the field profiles in the flux background is known [5,14] our analysis only uses their orthonormality. The 4d effective action contains the masses of all charged fields, which are reminiscent of Landau levels, as well as their interactions, see also [15]. A similar treatment is carried out for non-Abelian flux where also components of the gauge field are charged and affected by the flux.
Internal magnetic fields were largely discussed in the string literature, see e.g. [16,17], starting with [5], followed by its T-dual interpretation of D-branes at angles [18][19][20]. Global string theory models of this type with 4d supersymmetry or completely broken supersymmetry were constructed in [21]. In the case where 4d supersymmetry is broken by the internal magnetic flux, however, a NS-NS tadpole appears at the disk level, which signals a change of the ground state of the theory (see e.g. [22]). As a result, most quantum corrections at the string theory level cannot be reliably computed. On the other hand, the effective field theory action we construct in this paper is adapted to compute quantum corrections, as we will exemplify in the following.
Our main interest concerns the effect of flux on the quantum corrections to massless scalar particles. Without flux it is well-known that the mass of Wilson lines, i.e. the scalar zero modes associated with the higher-dimensional gauge field, are protected from quadratic divergences by the invariance under a discrete shift symmetry, a consequence of the higher-dimensional gauge invariance on the torus [23,24]. For 6d gauge theories the one-loop effective potential has been explicitly computed in [23], and it has been shown that the squared mass of the Wilson line is proportional to the volume of the compact dimensions.
In the following we compute the one-loop correction to the Wilson line mass for an Abelian gauge theory in the magnetic flux background. As we shall see, due to the effect of the flux on the KK spectrum and couplings various cancellations occur, and the total one-loop mass vanishes. The same is true for the one-loop quartic coupling. This result can be understood by considering the 6d Lagrangian rather than the 4d Lagrangian: the Wilson lines are the Goldstone bosons of symmetries of the 6d Lagrangian, which are spontaneously broken by the background gauge field. The paper is organized as follows. To introduce some formalism we first consider a supersymmetric Abelian 6d gauge theory without flux and derive the 4d Lagrangian of all KK modes in terms of chiral and vector superfields in Sec. 2. The 4d Lagrangian in the case of flux is derived for an Abelian and a non-Abelian gauge theory in Sec. 3 and Sec. 4, respectively. Sec. 5 deals with the one-loop effective potential for the Wilson line of an Abelian gauge theory, and the mass and the quartic coupling of the Wilson line are computed in the case with flux. The role of Wilson lines as Goldstone bosons is discussed in Sec. 6, and we conclude with an outlook in Sec. 7.
Abelian effective action without flux
We consider a globally supersymmetric U (1) gauge theory in six dimensions. Two of the dimensions are compactified on a square torus T 2 of area L 2 . Following [10], we decompose the 6d vector multiplet into an N = 1 vector multiplet V and a chiral multiplet φ. The scalar part of φ contains the internal components of the vector field,
φ| θ=θ=0 = 1 √ 2 (A 6 + iA 5 ) .(1)
The six-dimensional gauge action can then be written in 4d superspace as [10]
S 6 = d 6 x 1 4 d 2 θ W α W α + h.c. + d 4 θ ∂V ∂V + φφ + √ 2V ∂φ + ∂φ ,(2)
with ∂ = ∂ 5 − i∂ 6 . Note that compared to [10] we have performed an integration by parts in the last term. We further include a hypermultiplet of charge q that decomposes into two 4d chiral multiplets of opposite charge, Q andQ. The corresponding matter action can be written as
S 6 = d 6 x d 2 θQ(∂ + √ 2gqφ)Q + h.c. + d 4 θ Qe 2gqV Q +Qe −2gqVQ .(3)
It is straightforward to compute the 4d effective action, keeping the full KK tower in the gauge sector as well as the matter sector. The superfields, which depend on all six coordinates, can be decomposed in terms of modes of fixed internal momenta,
φ(x M , θ, θ) = n,m φ n,m (x µ , θ, θ) ψ n,m (x m ) , V (x M , θ, θ) = n,m V n,m (x µ , θ, θ) ψ n,m (x m ) ;(4)
here the index M runs over all spacetime dimensions, whereas µ and m only run over non-compact and compact dimensions, respectively. The ψ n,m are a complete set of mode functions that we choose as
ψ n,m (x m ) = 1 L exp 2πi L (nx 5 + mx 6 ) ;(5)
thay satisfy the orthonormality condition
T 2 d 2 x ψ n,m ψ k,l = δ n,k δ m,l .(6)
The reality of the vector field, V = V , implies for the mode functions V n,m = V −n,−m . Using the expansion (4) for vector and chiral superfields and integrating over the compact dimensions we obtain from Eq. (2) the equivalent 4d gauge action containing the full KK tower,
S 4 = d 4 x n,m d 2 θ 1 4 W α n,m W α,−n,−m + h.c. + d 4 θ |M n,m | 2 V n,m V n,m + φ n,m φ n,m − √ 2 M n,m V n,m φ n,m + M n,m V n,m φ n,m ,(7)
where
M n,m = 2π L (m + in) .(8)
The vector bosons of the KK tower acquire mass via the Stückelberg mechanism. At each KK level they absorb the imaginary part of of the complex field φ whereas the real part corresponds to the mass degenerate scalar that is needed to complete the massive vector multiplet. This can be made manifest by means of a shift of the vector field 2 2 For a discussion in component form see, for example [26].
(M n,m = 0),
V n,m → V n,m + 1 √ 2 1 M n,m φ n,m − 1 M n,m φ −n,−m .(9)
Performing the shift of V n,m and neglecting a total derivative, one obtains for the 4d gauge action
S 4 = d 4 x n,m d 2 θ 1 4 W α n,m W α,−n,−m + h.c. + d 4 θ |M n,m | 2 V n,m V n,m + ϕϕ .(10)
This is the standard N = 1 supersymmetric action for a massless vector multiplet together with a tower of massive KK vector multiplets. A massless chiral multiplet ϕ ≡ φ 0,0 remains since the vector multiplet can only be shifted if M n,m = 0. In order to include the matter sector we have to evaluate integrals of three and four mode functions. This yields the couplings of the different KK levels and guarantees momentum conservation in the internal space. The relevant integrals are T 2 d 2 x ψ n,m ψ k,l ψ r,s = 1 L δ n,k+r δ m,l+s ,
T 2 d 2 x ψ n,m ψ k,l ψ r,s ψ u,v = 1 L 2 δ n,k+r+u δ m,l+s+v .(11)
The complete effective 4d action, including gauge and matter KK towers, is given by
S 4 = d 4 x n,m d 2 θ 1 4 W α n,m W α,−n,−m + M n,mQn,m Q n,m + h.c. + d 4 θ |M n,m | 2 V n,m V n,m + φ n,m φ n,m + Q n,m Q n,m +Q n,mQ n,m + d 4 x n,m,k,l d 2 θ √ 2qgQ n+k,m+l φ k,l Q n,m + h.c. + d 4 θ 2qg Q n+k,m+l V k,l Q n,m −Q n,m V k,lQn+k,m+l + d 4 x
n,m,k,l,r,s d 4 θ 2q 2 g 2 Q n+k+r,m+l+s V k,l V r,s Q n,m +Q n,m V k,l V r,sQn+k+r,m+l+s .
In addition to the gauge field V n,m and ϕ, it describes massless and massive matter fields that are formed from pairs of chiral multipletsQ and Q with Dirac mass terms M n,m . At all KK levels the vector and matter fields are mass degenerate.
In the following sections we shall restrict the discussion to massless fields in the uncharged sector, which are denoted by V 0 and ϕ. The action then takes the simplified form
S * 4 = d 4 x d 2 θ 1 4 W α 0 W α,0 + n,m M n,m + √ 2qgϕ Q n,m Q n,m + h.c. + d 4 θ ϕϕ + n,m Q n,m e 2qgV 0 Q n,m +Q n,m e −2qgV 0Q n,m .(13)
Note that the zero mode ϕ, the Wilson line of the gauge field, couples to matter like the mass terms.
Abelian effective action with flux
Now we turn the attention to the flux background in the internal dimensions. Since they are compact the flux is quantized. Moreover, the mass spectrum and field profiles of charged fields will be changed drastically and resemble that of Landau levels. Due to the magnetic field the charged fields will be localized in the extra dimensional space. A harmonic oscillator analogy, based on the work of [5] and used in [6,11,13], allows to explicitly construct the shape of the charged field profiles [5,14]. In this way we obtain the four-dimensional effective action in terms of 4d superfields, restricted to the zero modes of the uncharged fields.
Flux and the harmonic oscillator
Before we derive the full supersymmetric effective action we want to elucidate the harmonic oscillator approach in a minimal example. For that reason we only consider the six-dimensional gauge field A M as a background for a charged scalar field Q of charge q. Consequently, the 6d action reads
S 6 = d 6 x −D M QD M Q ,(14)
with the gauge covariant derivative acting as D M Q = (∂ M + iqg A M )Q. The gauge field background accounts for a constant flux density f in the internal dimensions, which in our choice of gauge reads 3
A 5 = − 1 2 f x 6 , A 6 = 1 2 f x 5 , F 56 = ∂ 5 A 6 − ∂ 6 A 5 = f .(15)
As mentioned above, for the square torus of volume L 2 the flux is quantized. In the presence of particles with charge q the flux density can take the values
qg 2π T 2 F = qg 2π T 2 dx 5 dx 6 F 56 = qg 2π L 2 f ∈ Z(16)
Using a product space metric for M 4 × T 2 , and splitting the kinetic terms into 4d and 2d parts, the six-dimensional action (14) decomposes as
S 6 = d 6 x −η µν D µ QD ν Q − QH 2 Q ,(17)
where after integration by parts in the internal coordinates we define the 2d Hamiltonian
H 2 = −D 2 5 − D 2 6 = − ∂ 5 − i 2 qgf x 6 2 − ∂ 6 + i 2 qgf x 5 2 .(18)
In analogy to the quantum harmonic oscillator with Hamiltonian H = 1 2m p 2 + 1 2 mω 2 x 2 and the standard commutator relation [x, p] = i , we identify
p = iD 6 , x = iD 5 , m = 1 2 , ω = 2 ,(19)
with the commutator relation
[iD 5 , iD 6 ] = −iqgf .(20)
This leads to the further identification = −qgf [6], since we choose f to be negative for left-handed zero modes, c.f. [25]. One now defines the ladder operators
a = 1 −2qgf (iD 5 − D 6 ) , a † = 1 −2qgf (iD 5 + D 6 ) ,(21)
3 The calculations in the following sections are equally valid for other gauge choices.
which satisfy the canonical commutator relation [a, a † ] = 1. The internal Hamiltonian can be written in terms of the ladder operators as
H 2 = −qgf a † a + aa † = −2qgf a † a + 1 2 .(22)
Therefore, the energy eigenvalues of H 2 and thus the 4d Landau level masses show the typical spectrum of an harmonic oscillator. All levels are |N |-fold degenerate, with N the number of flux quanta on the torus, in analogy to Landau levels. We denote the internal field profiles as ψ n,j , see [14], where n refers to the Landau level and j accounts for the |N |-fold degeneracy. The field profiles corresponding to the lowest mass can then be constructed from the condition
a ψ 0,j = 0 , a † ψ 0,j = 0 .(23)
Applying the ladder operator we obtain the higher mode functions
ψ n,j = 1 √ n! (a † ) n ψ 0,j , ψ n,j = 1 √ n! (a) n ψ 0,j .(24)
The explicit form of the lowest wave function was obtained in [5,14]. In our consideration the specific form of the field profile is irrelevant and we will only need the orthonormality condition 4
T 2 d 2 x ψñ , ψ n,j = δ n,ñ δ j, .(25)
Instead of the KK decomposition in Sec. 2 we now decompose the charged fields with respect to the Landau levels,
Q(x M ) = n,j Q n,j (x µ )ψ n,j (x m ) = n,j Q n,j (x µ ) 1 √ n! a † n ψ 0,j (x m ) , Q(x M ) = n,j Q n,j (x µ )ψ n,j (x m ) = n,j Q n,j (x µ ) 1 √ n! (a) n ψ 0,j (x m ) .(26)
4 Note that the charged wave functions in the flux background are not orthonormal with respect to the standard KK states discussed in Sec. 2. Therefore, to discuss the interaction of the charged states with higher excitations of the uncharged sector one has to evaluate the overlaps explicitly, see e.g. [15].
The 6d action (14) then becomes
S 6 = d 4 x n,j,m,k −η µν D µ Q n,j D ν Q m,k T 2 d 2 x ψ n,j ψ m,k −Q n,j Q m,k T 2 d 2 x (−2qgf )ψ n,j a † a + 1 2 ψ m,k .(27)
The four-dimensional effective action is derived by using the harmonic oscillator algebra and the orthonormality of the internal field profiles in the gauge field background,
S 4 = d 4 x n,j −D µ Q n,j D µ Q n,j + (2qgf ) n + 1 2 Q n,j Q n,j .(28)
The masses for the 4d fields are given by
m 2 n,j = −2qgf n + 1 2 = 2π|N | L 2 (2n + 1) ,(29)
as discussed in [5]. For fields with an internal helicity the mass formula is supplemented by a term (−2qgf )Σ, where Σ is the internal helicity, see [5]. This leads to the appearance of |N | chiral fermion zero modes as predicted by the index theorem for the flux background (Σ = 1 2 ) and a tachyonic mode in the presence of charged gauge fields with Σ = 1, as discussed in Sec. 4.
Supersymmetric effective action for Abelian flux
The field profiles for charged fermions and bosons are identical because both arise as solutions to the gauge covariant Laplace equation on the torus. Therefore, instead of decomposing only the component fields with respect to the Landau levels we can decompose the superfield as a whole, similar to the procedure for the standard KK tower in Sec. 2. As mentioned above, the six-dimensional hypermultiplet can be written in terms of two chiral multiplets with opposite charge,
Q(x M , θ, θ) = n,j Q n,j (x µ , θ, θ) ψ n,j (x m ) , Q(x M , θ, θ) = n,jQ n,j (x µ , θ, θ) ψ n,j (x m ) .(30)
Furthermore, the index theorem guarantees |N | fermion zero modes. In our convention, c.f. [25], we choose f to be negative which corresponds to zero modes contained in thẽ Q multiplet.
The uncharged 6d vector multiplet has the usual KK expansion on the torus, see Sec. 2. Here, we concentrate on its background value and the zero mode, which are encoded in V 0 and φ 0 . The scalar component of φ 0 contains the internal component of the gauge field and therefore encodes the magnetic flux on the torus. We split this contribution into the background gauge field generating the flux and perturbations ϕ, which are constant with respect to the internal dimensions. Hence, ϕ corresponds to the continuous Wilson lines on the torus, ϕ = 1 √ 2 (a 6 + ia 5 ). In the symmetric gauge (15) the scalar component reads
φ 0 | θ=θ=0 = f 2 √ 2 (x 5 − ix 6 ) + ϕ .(31)
The coupling of the hypermultiplet to the internal components of the 6d gauge field can then be written in N = 1 notation as in Eq. (3). Plugging in the expressions for the ladder operators (21) and the mode expansion (30), we obtain
S 6 ⊃ d 6 x d 2 θQ(∂ + √ 2qgφ 0 )Q + h.c. = d 6 x d 2 θ −i −2qgfQa † Q + √ 2qgQ ϕ Q + h.c. (32) = d 4 x d 2 θ n,ñ,j,Qñ , Q n,j T 2 d 2 x ψñ , (−i −2qgf a † + √ 2qgϕ)ψ n,j + h.c.
After using the orthonormality condition (25) of the states, we find the contribution to the 4d effective action after integration over the torus,
S * 4 ⊃ d 4 x d 2 θ W + h.c. (33) = d 4 x d 2 θ n,j −i −2qgf (n + 1)Q n+1,j Q n,j + √ 2qgQ n,j ϕ Q n,j + h.c.
The superpotential contains a mass term for the charged superfields and an interaction term which couples them to the internal components of the gauge field, i.e. the Wilson lines. The kinetic terms of the charged fields can be treated similarly, which yield
S * 4 ⊃ d 4 x d 4 θ T 2 d 2 x Qe 2qgV 0 Q +Qe −2qgV 0Q = d 4 x d 4 θ n,j Q n,j e 2qgV 0 Q n,j +Q n,j e −2qgV 0Q n,j .(34)
Finally, the 4d zero modes of the gauge field are included, leading to the same effective action as in Sec. 2,
S * 4 ⊃ d 4 x T 2 d 2 x d 2 θ 1 4 W α W α + h.c. = d 4 x d 2 θ 1 4 W α 0 W α,0 + h.c. .(35)
The last contribution we have to add leads to a kinetic term for the complex Wilson line ϕ as well as a Fayet-Iliopoulos (FI) term 5
S * 4 ⊃ d 4 x T 2 d 2 x d 4 θ ∂V 0 ∂V 0 + φ 0 φ 0 + √ 2V 0 ∂φ 0 + √ 2V 0 ∂φ 0 = d 4 x d 4 θ (ϕϕ + 2f V 0 ) .(36)
Note again that compared to [10] our action differs by an integration by parts. This is important since the boundary terms do not vanish in the flux background. In summary, the 4d effective action with the complete tower of charged states and a restriction to the zero modes in the uncharged sector reads
S * 4 = d 4 x d 4 θ ϕϕ + n,j (Q n,j e 2gqV 0 Q n,j +Q n,j e −2qgV 0Q n,j ) + 2f V 0 + d 2 θ 1 4 W α 0 W α,0(37)+ n,j −i −2qgf (n + 1)Q n+1,j Q n,j + √ 2qgQ n,j ϕ Q n,j + h.c. .
In order to obtain the mass spectrum of the charged fields and their interactions with the uncharged field ϕ one has to integrate out the auxiliary fields. The bosonic mass terms receive contributions from F -and D-terms, whereas only the F -terms enter for the fermion masses. The couplings of the auxiliary field D are given by
L D = f D + |Q n,j | 2 qgD − |Q n,j | 2 qgD + 1 2 D 2 ,(38)
yielding
D = −f − qg n,j |Q n,j | 2 − |Q n,j | 2 .(39)
5 Here, we use ∂φ = ∂φ = f / √ 2 in the flux background, since ∂ϕ = 0 = ∂ϕ, and ∂V = 0 = ∂V .
Similarly, the F -terms appear in the component action as
L F =|F ϕ | 2 + n,j |F n,j | 2 + |F n,j | 2 + n,j −i −2qgf (n + 1) F n+1,j Q n,j +Q n+1,j F n,j + √ 2qg F n,j ϕ Q n,j +Q n,j F ϕ Q n,j +Q n,j ϕ F n,j + h.c. ,(40)
leading to
F n,j = −i −2qgf (n + 1)Q n+1,j − √ 2qgQ n,j ϕ , F n+1,j = −i −2qgf (n + 1) Q n,j − √ 2qg Q n+1,j ϕ , F ϕ = − √ 2qg n,jQ n,j Q n,j .(41)
Plugging the F -and D-terms back into the component Lagrangian we find the bosonic mass terms
L b M = − n,j −2qgf (n + 1) |Q n+1,j | 2 + |Q n,j | 2 + qgf |Q n,j | 2 − qgf |Q n,j | 2 . (42)
Therefore, the two scalars of different charge have the same tower of massive states due to a charge dependent shift induced by the D-term, leading to the bosonic masses evaluated in Eq. (29),
m 2Q n,j = −2qgf n − qgf = −qgf (2n + 1) = 2π|N | L 2 (2n + 1) , m 2 Q n,j = −2qgf (n + 1) + qgf = 2π|N | L 2 (2n + 1) .(43)
The fermionic mass terms can be directly read off the superpotential (33),
L f M = n,j −i 2qgf (n + 1)χ n+1,j χ n,j + h.c. .(44)
We find the |N | chiral zero modesχ 0,j predicted by the index theorem. The rest of the chiral fermions pair up to form massive Dirac fields,
Ψ n,j = χ n+1,j χ n,j ,(45)
with masses
m 2 Ψ n,j = −2qgf (n + 1) = 4π|N | L 2 (n + 1) .(46)
Including the interactions among fermions and bosons we arrive at the full component Lagrangian
L ef f = L kin + L M + L int − 1 2 f 2 ,(47)
with the bilinear kinetic and mass terms
L kin = − ∂ µ ϕ∂ µ ϕ − 1 4 F µν F µν − n,j D µ Q n,j D µ Q n,j + D µQn,j D µQ n,j − i λ 1 σ µ ∂ µ λ 1 + λ 2 σ µ ∂ µ λ 2 − i n,j χ n,j σ µ D µ χ n,j +χ n,j σ µ D µχ n,j ,(48)L M = − n,j (−2qgf ) n + 1 2 |Q n,j | 2 + |Q n,j | 2 + i n,j −2qgf (n + 1)χ n+1,j χ n,j + h.c. ,(49)
and the cubic and quartic interaction terms
L int = − g 2 q 2 2 n,j |Q n,j | 2 − |Q n,j | 2 2 − 2q 2 g 2 n,jQ n,j Q n,j m,kQ m,k Q m,k − i √ 2qg n,j −2qgf (n + 1) Q n+1,j ϕQ n,j − Q n,j ϕ Q n+1,j + h.c. − 2q 2 g 2 n,j |ϕ| 2 |Q n,j | 2 + |Q n,j | 2 + √ 2qg n,j iQ n,j λ 1 χ n,j − iQ n,j λ 1χn,j − Q n,j λ 2χn,j −Q n,j λ 2 χ n,j + h.c. − √ 2qg n,j ϕχ n,j χ n,j + h.c.(50)
Note that the fermionsχ 0,j , j ∈ {1, . . . , |N |}, are the only charged massless fields.
Non-Abelian flux background
In the case of non-Abelian flux we proceed very similar to the Abelian case above. In the following we will consider a six-dimensional super Yang-Mills theory with gauge group SU (2). The generalization to higher rank gauge groups and the inclusion of charged matter fields is straightforward. Again, the starting point is the 6d action expressed in 4d superfields as given in [10]. Moreover, we will always work in the Wess-Zumino (WZ) gauge. The fields of the 6d non-Abelian theory are contained in a vector multiplet V and a chiral multiplet φ that both transform in the adjoint representation,
S 6 = d 6 x 1 2 d 2 θ tr (W α W α ) + h.c.(51)+ d 4 θ 2 g 2 tr √ 2 ∂ + gφ e −gV − √ 2 ∂ + gφ e gV + ∂e −gV ∂e gV ,
with the trace convention tr (T a T b ) = 1 2 δ ab . Expanding the exponentials, integrating some of the terms by part, and bearing in mind that V 3 = 0 in the WZ gauge, this action can be written as
S 6 = d 6 x 1 2 d 2 θ tr (W α W α ) + h.c. + d 4 θ 2 tr φφ + √ 2 ∂φ + ∂φ V(52)+ 2 tr g φ, φ V + ∂V − g √ 2 V, φ ∂V + g √ 2 [V, φ] .
In order to evaluate the group structure we define a new basis of generators {T 3 , T + , T − }, where T i are the properly normalized Pauli matrices and T ± = T 1 ± iT 2 . In this basis one has
tr T 2 ± = 0 , tr (T + T − ) = 1 , tr (T ± T 3 ) = 0 , tr T 2 3 = 1 2 , [T + , T − ] = 2T 3 , [T 3 , T ± ] = ±T ± .(53)
The chiral field φ decomposes as
φ = φ 3 T 3 + φ + 1 √ 2 T − + φ − 1 √ 2 T + , φ = φ 3 T 3 + φ + 1 √ 2 T + + φ − 1 √ 2 T − ,(54)
and analogously the vector field, whose reality condition leads to V 3 = V 3 and V ± = V ∓ . The flux will be encoded as non-trivial background for the field φ 3 corresponding to the Cartan generator T 3 , similar to the Abelian case (15),
φ 3 = f 2 √ 2 (x 5 − ix 6 ) + ϕ 3 .(55)
The commutator identities (53) show that the components φ + , V + and φ − , V − have positive and negative charge with respect to V 3 , respectively. Consequently, their internal field profiles will be the same as the charged wave functions in the Abelian case. In our normalization convention the charge of the chiral fields φ ± is q = ± 1 2 . The fields V 3 and φ 3 are uncharged and we will only take their 4d zero modes into account 6 . Since the wave functions are identical to the Abelian framework we can also adopt the harmonic oscillator analogy and define the ladder operators in terms of the background gauge field, c.f. (21)
a † = i √ −gf ∂ + g √ 2 (φ 3 − ϕ 3 ) , a = i √ −gf ∂ − g √ 2 (φ 3 − ϕ 3 ) .(56)
The full six-dimensional action can then be expressed in the basis (54). After integration by parts one obtains
S 6 = d 6 x d 2 θ 1 4 W α 3 W α,3 + 1 2 W α + W α,− + h.c. + d 4 θ ϕ 3 ϕ 3 + φ + e gV 3 φ + + φ − e −gV 3 φ − + 2f V 3 + V − i −gf a † − g √ 2 ϕ 3 −i −gf a − g √ 2 ϕ 3 V + + V − i −gf a + g √ 2 ϕ 3 −i −gf a † + g √ 2 ϕ 3 V + − √ 2V − 1 − g √ 2 V 3 i −gf a † − g √ 2 ϕ 3 φ − − √ 2φ − 1 − g √ 2 V 3 −i −gf a − g √ 2 ϕ 3 V + − √ 2φ + 1 + g √ 2 V 3 −i −gf a † + g √ 2 ϕ 3 V + − √ 2V − 1 + g √ 2 V 3 i −gf a + g √ 2 ϕ 3 φ + + g 2 2 (V + φ − − V − φ + ) V − φ − − V + φ + .(57)
We clearly identify the kinetic term for ϕ 3 as well as the gauge covariant kinetic terms for the charged chiral multiplets φ ± of charge ± 1 2 . Also the FI-term for the vector multiplet aligned with the flux V 3 is present, as in the Abelian case. The remaining contributions will lead to interaction and mass terms connecting different charged states. Except the last term, that contains four charged fields, we can derive the 4d effective action along the lines of Sec. 3, where φ ± now correspond to the charged chiral multiplets Q andQ. The final result is
S * 4 = d 4 x d 2 θ 1 4 W α 3 W α,3 + 1 2 n,j W α +,n,j W α,−,n,j + h.c. + d 4 θ ϕ 3 ϕ 3 + 2f V 3 + n,j φ +,n,j e gV 3 φ +,n,j + φ −,n,j e −gV 3 φ −,n,j + n,j (2n + 1)(−gf )V −,n,j V +,n,j + i 2n(−gf )gϕ 3 V −,n−1,j V +,n,j −i 2(n + 1)(−gf )gϕ 3 V −,n+1,j V +,n,j + g 2 ϕ 3 ϕ 3 V −,n,j V +,n,j + n,j 1 − g √ 2 V 3 −i 2(n + 1)(−gf )V −,n+1,j φ −,n,j(58)
+i 2n(−gf )φ −,n−1,j V +,n,j + gϕ 3 V −,n,j φ −,n,j + gϕ 3 φ −,n,j V +,n,j
+ 1 + g √ 2 V 3 i 2(n + 1)(−gf )φ +,n+1,j V +,
Integrating out the auxiliary fields in (58) we can work out the masses of the charged fields. The charged vector boson masses in 4d can be evaluated using θσ µ θθσ ν θ = − 1 2 θθθθη µν . They are given by
m 2 A ± ,n,j = 1 2 (−gf )(2n + 1) = 2π|N | L 2 (2n + 1) .(60)
Hence, the charged vector fields have to absorb part of the charged scalar fields via the Stückelberg mechanism, c.f. Sec. 2. The necessary couplings of the charged gauge fields to the derivative of the scalars can be extracted from the action (58)
S * 4 ⊃ d 4 x d 4 θ n,j −i −2gf √ n φ −,n−1,j + √ n + 1 φ +,n+1,j V −,n,j + i −2gf √ n φ −,n−1,j + √ n + 1 φ +,n+1,j V +,n,j ⊃ d 4 x n,j −gf 2 − √ n ∂ µ φ −,n−1,j + √ n + 1 ∂ µ φ +,n+1,j A µ −,n,j + − √ n ∂ µ φ −,n−1,j + √ n + 1 ∂ µ φ +,n+1,j A µ +,n,j .(61)
This identifies the eaten complex Goldstone mode 7
Φ n,j = − n + 1 2n + 3 φ −,n,j + n + 2 2n + 3 φ +,n+2,j ,(62)
for the charged vector bosons A µ ±,n+1,j . The modes A µ ±,0,j eat the complex bosons φ +,1,j . To determine the mass spectrum for the remaining two real charged degrees of freedom we need to evaluate the D-terms. The solutions of the D-term equations read
D 3 = −f − g 2 n,j |φ +,n,j | 2 − |φ −,n,j | 2 ,(63)
D +,n,j = i −gf 2 √ 2n + 1 n 2n + 1 φ −,n−1,j + n + 1 2n + 1 φ +,n+1,j ,
D −,n,j = − i −gf 2 √ 2n + 1 n 2n + 1 φ −,n−1,j + n + 1 2n + 1 φ +,n+1,j .(64)
Substituting the D-terms into the component action we can extract the quadratic part of the scalar Lagrangian and identify the mass terms
L M ⊃ − gf 2 |φ +,0,j | 2 − n,j
φ −,n,j , φ +,n+2,j n + 2 (n + 1)(n + 2) (n + 1)(n + 2) n + 1 φ −,n,j φ +,n+2,j
Since we chose f < 0 we see that there are |N | tachyonic modes φ +,0,j that will acquire vacuum expectation values in the true vacuum (see the comments at the end of this section). This corresponds to the helicity dependent mass shift one expects, as pointed out in Sec. 3.1. The states φ +,1,j have vanishing mass as should be the case for a Stückelberg field for the first level of massive gauge bosons. The rest of the tower has masses corresponding to the eigenvalues of the matrix in (65). Clearly, the determinant of the mass matrix vanishes, which indicates the massless Goldstone modes (62). The remaining complex scalar degree of freedom corresponds to the linear combination orthogonal to (62),Φ n,j = n + 2 2n + 3 φ +,n+2,j + n + 1 2n + 3 φ −,n,j ,
with mass eigenvalues
m 2 Φ n,j = 1 2 (−gf )(2n + 3) = 2π|N | L 2 (2n + 3) .(67)
Note that the physical mass spectrum differs from the one given in [5]. The states with mass squared 2π|N |/L 2 are absorbed by charged vector bosons. The fermion mass terms are L M ⊃ − n,j (n + 1)(−gf ) λ +,n+1,jλ−,n,j − λ −,n,jλ+,n+1,j + h.c. ,
where λ andλ denote the gauginos contained in the vector multiplet and chiral multiplets, respectively. We find 2|N | fermion zero modes λ +,0,j andλ +,0,j and a tower of Dirac fermions Ψ ±,n,j with masses m 2 Ψ ±,n,j = (−gf )(n + 1) = 4π|N | L 2 (n + 1) .
Some comments are in order. The Abelian flux background is perturbatively stable, which means that all fields have a non-negative mass in the background field (15). This situation is different for non-Abelian flux. The flux background can be associated with a gauge field in the Cartan subalgebra. The non-Cartan elements will accordingly be charged under the flux and some of the extra dimensional gauge field components become tachyonic. Therefore, the effective action below does not correspond to an expansion around the ground state of the system but rather around an extremal point. Nevertheless, it might be very interesting to study tachyon condensation in this framework and its interplay with the internal flux background. The study of tachyon condensation should reveal the true ground state of the theory, and the properties of the theory in the ground state could then be studied by shifting the vacuum accordingly, with possible applications to string theory [27,28].
Quantum corrections
In the previous sections we have derived four-dimensional effective actions for sixdimensional gauge theories compactified on a torus without or with magnetic flux, keeping the complete tower of charged excitations. This is a good starting point for computing quantum corrections, in particular for scalar masses which generically are not protected by symmetries. In the case without flux the one-loop effective potential of a Wilson line has been computed, and after subtraction of a divergent contribution a finite mass squared is obtained which is proportional to the inverse volume of the compact dimensions [23,[29][30][31]. Our main interest concerns quantum corrections to the Wilson line mass for a torus compactification with magnetic flux, but for comparison we first reconsider the case without flux.
Quantum corrections without flux
The one-loop corrections to the Wilson line mass are determined by the couplings of ϕ to the matter fields Q andQ. Gauge field contributions only enter at two-loop level. Hence our starting point is the action (13) from which one obtains the Lagrangian in component form,
L 4 ⊃ − 1 4 F µν F µν − ∂ µ ϕ∂ µ ϕ
where the complex mass terms M n,m are defined in Eq. (8) and D µ = ∂ µ + igqA µ . Given the effective action (70) it is straightforward to calculate the one-loop quantum corrections to the Wilson line mass. The relevant bosonic and fermionic contributions are depicted in Fig. 1 and Fig. 2, respectively, from which one obtains after a Wick rotation
δm 2 b = 2g 2 q 2 n,m d 4 k (2π) 4 1 k 2 + |M n,m | 2 − M n,m M n,m (k 2 + |M n,m | 2 ) 2 ,(71)
and As expected the contributions cancel for a supersymmetric spectrum, i.e. for two charged scalars and a pair of charged Weyl fermions with the same masses. The bosonic contribution (71) to the Wilson line mass is an infinite sum of quadratically divergent terms. A consistent treatment of this expression requires a regularization prescription as well as renormalization conditions. In the literature several approaches have been pursued which make use of string inspired Poisson resummation [23,29,32], as well as dimensional regularization [33]. In the following, we shall adopt the treatment in [23], which yields a well-known result for the Wilson line effective potential.
δm 2 f = −4g 2 q 2 n,m d 4 k (2π) 4 k 2 (k 2 + |M n,m | 2 ) 2 = −2δm 2 b . (72) ϕ ϕ Q n,m ϕ ϕ Q n,m Q n,m
Using the Schwinger representation it is conveniently expressed as
δm 2 b =2g 2 q 2 n,m ∞ 0 dt t e −|Mn,m| 2 t d 4 k (2π) 4 k 2 e −k 2 t = g 2 q 2 4π 2 ∞ 0 dt t 2 Θ 3 0; 4πit L 2 2 ,(73)
where we have interchanged summation over KK modes and t-integration, so that the integrand is now given by the square of the Jacobi Θ-function
Θ 3 (z; τ ) = r e iπτ r 2 e 2πizr .(74)
Under modular transformations Θ 3 transforms as
Θ 3 (0; τ ) = (−iτ ) −1/2 Θ 3 (0; −1/τ ) .(75)
From this we obtain
δm 2 b = g 2 q 2 L 2 16π 3 ∞ 0 dt t 3 Θ 3 0; iL 2 4πt 2 = g 2 q 2 L 2 16π 3 ∞ 0 du u Θ 3 0; iL 2 u 4π 2 = g 2 q 2 π 3 L 2 r,s 1 (r 2 + s 2 ) 2 ,(76)
where we have used the explicit form (74) in the last step.
The full divergence of δm 2 b , i.e. summation over KK modes and quadratically divergent momentum integrations, is contained in the r = s = 0 contribution to the sum in Eq. (76). To remove this divergence a counterterm is needed. Following [23,29], we define the finite part of δm 2 b by dropping the r = s = 0 contribution to the sum (76). We have compared this finite part with the result in [33], which has been obtained by using dimensional regularization and Poisson resummation. It is reassuring that both procedures give the same answer.
The mass of ϕ can also be obtained from the second derivative of the Wilson line effective potential which was calculated in [23]. Here one starts from the effective mass of the matter fields Q n,m in a constant Wilson line background ϕ (see Eq. (70)),
M n,m (ϕ) = M n,m + √ 2gqϕ .(77)
From the general expression for the one-loop effective potential,
V eff = 1 2 I (−1) F I d 4 k (2π) 4 log k 2 + M 2 I (ϕ) = − 1 32π 2 I (−1) F I dt t 3 e −M 2 I (ϕ)t ,(78)
with I labeling the bosons and fermions in the theory, one obtains for the contribution of the complete KK tower of a single 6d charged scalar field Q,
V eff = − 1 16π 2 n,m dt t 3 e −|Mn,m| 2 (ϕ)t = − 1 16π 2 n,m ∞ 0 dt t 3 exp − 2πn L + qga 5 2 t − 2πm L + qga 6 2 t ,(79)
where a 5 and a 6 are constant background fields. After a Poisson resummation one finds
V eff = − L 2 64π 3 r,s ∞ 0 dt t 4 exp iqgL (ra 5 + sa 6 ) − L 2 4t r 2 + s 2 = − L 2 64π 3 r,s ∞ 0 du u 2 exp iqgL (ra 5 + sa 6 ) − L 2 4 u r 2 + s 2 .(80)
Performing the u-integration and expressing the effective potential in terms of ϕ yields the final result
V eff = − 2 L 4 π 3 r,s 1 (r 2 + s 2 ) 3 exp i qgL √ 2 ((s − ir)ϕ + (s + ir)ϕ) .(81)
The r = s = 0 contribution is again divergent. It has been argued that dropping this term corresponds to subtracting a divergent cosmological constant. However, since the expression for the effective potential is divergent, omitting the r = s = 0 contribution also subtracts field dependent terms. Indeed, the mass term
∂ ϕ ∂ ϕ V eff | ϕ=0 = g 2 q 2 L 2 π 3 r,s 1 (r 2 + s 2 ) 2(82)
is divergent and identical to the expression (76). On the other hand, the prescription to drop the r = s = 0 is consistent with respect to the finite contributions, since the finite mass terms obtained from the diagrammatic calculation and the effective potential calculation then yield the same result.
Quantum corrections with flux
Given the four-dimensional effective action for the torus compactification with flux, see (38) and (50), containing the complete tower of Landau levels we can again study quantum corrections to the Wilson line effective potential. In the following we shall compute the quantum corrections to the mass term and the quartic coupling. ϕ ϕ Q n,j ,Q n,j ϕ ϕ Q n+1,j ,Q n+1,j Q n,j ,Q n,j
L int = − i √ 2qg
n,j α(n + 1) ϕ Q n+1,jQn,j − Q n,j Q n+1,j + h.c.
− 2q 2 g 2 n,j |ϕ| 2 |Q n,j | 2 + |Q n,j | 2 − √ 2qg n,j ϕχ n,j χ n,j + h.c. ,
where we have introduced the positive parameter α = −2qgf of mass dimension two. Note that the cubic bosonic vertex is proportional to the mass of the charged fields involved. Moreover, the bosonic couplings do not mix the fields Q andQ. On the contrary, the fermionic coupling involves the pair χ andχ at the same Landau level n, analogously to the Dirac mass terms in Eq. (49). As in the case without flux there are two classes of bosonic contributions and one class of fermionic contributions to the Wilson line mass which are depicted in Fig. 3 and Fig. 4, respectively. Using the couplings given in the Lagrangian (83) one obtains for the quantum corrections
δm 2 b = 2q 2 g 2 |N | n d 4 k (2π) 4 2 k 2 + α(n + 1 2 ) − 2α(n + 1) k 2 + α(n + 3 2 ) k 2 + α(n + 1 2 )
,
δm 2 f = −2q 2 g 2 |N | n d 4 k (2π) 4 2k 2 (k 2 + αn) (k 2 + α(n + 1)) ,(84)
which can be brought to the form
δm 2 b = −4q 2 g 2 |N | n d 4 k (2π) 4 n k 2 + α(n + 1 2 ) − n + 1 k 2 + α(n + 3 2 )
,
δm 2 f = 4q 2 g 2 |N | n d 4 k (2π) 4 n k 2 + αn − n + 1 k 2 + α(n + 1) .(85)
Using the Schwinger representation of the propagators and performing the momentum integrations one finds
δm 2 b = − q 2 g 2 4π 2 |N | n ∞ 0 dt 1 t 2 ne −α(n+ 1 2 )t − (n + 1)e −α(n+ 3 2 )t , δm 2 f = q 2 g 2 4π 2 |N | n ∞ 0 dt 1 t 2 ne −αnt − (n + 1)e −α(n+1)t .(86)
As in the case without flux the bosonic as well as the fermionic contribution of each Landau level is quadratically divergent. However, interchanging summation and tintegration and using various identities for geometrical series, one arrives at
δm 2 b = − q 2 g 2 4π 2 |N | ∞ 0 dt 1 t 2 e 1 2 αt (e αt − 1) 2 − e 1 2 αt (e αt − 1) 2 = 0 ,(87)δm 2 f = q 2 g 2 4π 2 |N | ∞ 0 dt 1 t 2 e αt (e αt − 1) 2 − e αt (e αt − 1) 2 = 0 .(88)
We conclude that, contrary to the case without flux, the contributions from the different Landau levels add up to zero and the integrand vanishes. It is remarkable that the bosonic and the fermionic contribution to the Wilson line mass vanish individually. To obtain this result it is important to perform the summation before the momentum integration, as in [23,33]. In this way, the symmetries of the gauge theory in the compact dimensions are kept. Comparing the result with the case without flux suggests that magnetic flux may provide a protection of the Wilson line mass compared to the compactification scale, independent of supersymmetry.
With non-vanishing flux the computation of the complete Wilson line effective potential is not straightforward, unlike in the case without flux. As next step we therefore compute the one-loop contribution to the quartic coupling λ. The calculation is very similar to the one for the mass term, although more cumbersome. The diagrams with charged fermions and bosons in the loops are depicted in Fig. 5 and Fig. 6, respectively. Compared to the computation of the mass term now also fermion propagators appear that mix neighboring Landau levels. As for the mass term we calculate the contributions from bosons and fermions separately. After some manipulations of the integrand one obtains the result
δλ b = − 8q 4 g 4 |N | n d 4 k (2π) 4 n 2 A 2 1/2 − (n + 1) 2 A 2 3/2 + 1 α − n(n + 1) A 1/2 + (n + 1)(n + 2) A 3/2 + n(n + 1) A 3/2 − (n + 1)(n + 2) A 5/2 , δλ f = − 8q 4 g 4 |N | n d 4 k (2π) 4 − n 2 (A 0 ) 2 + (n + 1) 2 (A 1 ) 2 + 1 α n(n + 1) A 0 − (n + 1)(n + 2) A 1 − n(n + 1) A 1 + (n + 1)(n + 2) A 2 ,(89)
where we have introduced the shorthand notation A j = k 2 + α(n + j). Introducing again the Schwinger representation of the propagators, performing the momentum in-ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j 1 t −n 2 e −αnt + (n + 1) 2 e −α(n+1)t + 1 αt 2 n(n + 1)e −αnt − (n + 1)(n + 2)e −α(n+1)t − n(n + 1)e −α(n+1)t + (n + 1)(n + 2)e −α(n+2)t = 0 ,
Q n−1,j , Q n−1,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j Q n+1,j , Q n+1,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j Q n+1,j , Q n+1,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j Q n−1,j , Q n−1,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j Q n+1,j , Q n+1,j Q n+1,j Q n+1,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j Q n−1,j , Q n−1,j Q n+1,j Q n+1,j ϕ ϕ ϕ ϕ Q n,j ,Q n,j Q n,j ,Q n,j Q n+1,j , Q n+1,j Q n−1,j Q n−1,jδλ f = − q 4 g 4 2π 2 |N | ∞ 0 dt n 1(90)
t −n 2 e −αnt + (n + 1) 2 e −α(n+1)t + 1 αt 2 n(n + 1)e −αnt − (n + 1)(n + 2)e −α(n+1)t − n(n + 1)e −α(n+1)t + (n + 1)(n + 2)e −α(n+2)t .
= 0 .(91)
The sums in Eqs. (90) and (91) extend from 0 to +∞. Since they are convergent and the n = 0 contribution vanishes one can perform a shift n → n + 1 in the first term of each line. It is then apparent that the bosonic and the fermionic contribution to the quartic coupling again vanish separately. Hence, no |ϕ| 4 -term is generated at oneloop order. This suggests that the entire one-loop effective potential vanishes. Indeed, this has already been conjectured in the original paper by Bachas [5] based on the independence of the Landau level masses on the the Wilson lines. At the level of the 4d effective action this result appears very surprizing but, as we shall see in the following section, it can be understood in terms of symmetries of the six-dimensional theory.
Wilson lines as Goldstone bosons
From the 4d effective field theory perspective the vanishing of the quantum corrections to the Wilson line effective potential is far from obvious. It is a consequence of an intricate interplay between level-dependent masses and couplings. Furthermore, the separate cancellations in the bosonic and fermionic sectors show that also supersymmetry is not responsible for this protection of scalar masses by magnetic flux. Considering the 6d theory it becomes clear which symmetry lies behind the vanishing of the effective potential for ϕ. The massless Wilson lines are the Goldstone bosons of the translation symmetries that are spontaneously broken by the background gauge field. We subsequently analyze the cases with a single U (1) gauge group and with several U (1) factors.
Goldstone bosons for a single U(1)
The six-dimensional action of a charged matter field that we considered in the previous sections,
S 6 = d 6 x −D M QD M Q , with D M Q = (∂ M + iqg A M )Q,δQ = m ∂ m Q , δA n = m ∂ m A n ,(92)
implying δD M Q = m ∂ m D M Q and therefore δS 6 = 0. In Sec. 3 we considered effective actions where the KK tower of the gauge field was neglected, i.e., the gauge field was replaced by its zero-mode, the complex Wilson line ϕ = 1 √ 2 (a 6 + ia 5 ). The corresponding 6d action is invariant under the transformation
δQ = m ∂ m Q , δa n = 0 .(93)
Let us now include magnetic flux by changing the covariant derivative to
D m Q = ∂ m + iqg a m + f 2 mn x n Q .(94)
The background gauge field A m = f 2 mn x n breaks the translational U (1) × U (1) symmetry spontaneously. Now this symmetry is realized nonlinearly,
δQ = m ∂ m Q , δa n = m f 2 nm ,(95)
and the two real massless scalars a 5 and a 6 are the corresponding Goldstone bosons 8 . Note that the Wilson lines a 5 and a 6 remain massless if the KK tower m of massive scalars is included, i.e., A m = a m + m , with the transformation behavior
δQ = m ∂ m Q , δa n = m f 2 nm , δ m = m ∂ mÂm .(96)
However, 6d gravity effects may modify the Wilson line masses, which remains to be investigated. In this connection also the backreaction of the flux on the geometry has to be taken into account. The background gauge field used in Eq. (94) corresponds to a particular choice of gauge. The same magnetic flux F = d A is generated by the background fields
A(x 5 , x 6 ) = (a 5 − cf x 6 )dx 5 + (a 6 + (1 − c)f x 5 )dx 6 ,(97)
with c ∈ R. However, not all values of c are allowed since the background gauge field has to satisfy the periodicity condition on a torus 9 ,
A(x 5 + kL, x 6 + lL) = A(x 5 , x 6 ) + dΛ , k, l ∈ Z ,(98)
where Λ m,n is a large gauge transformation,
Λ = 2π L (mx 5 + nx 6 ) , m, n ∈ Z .(99)
This means that for all integers k, l other integers m, n have to exist such that the condition (98) is satisfied. Inserting the background field (97) into Eq. (98) yields
− cf lLdx 5 + (1 − c)f kLdx 6 = 2π L (mdx 5 + ndx 6 ) ,(100)
and with f L 2 /(2π) = N ∈ Z this leads to the conditions −cN l ∈ Z and (1 − c)N k ∈ Z, and therefore cN ∈ Z .
For c = 0 and c = 1, it is apparent that the background gauge field (97) breaks both translational symmetries. For c = 0 or 1 one might, at first sight, expect that one of the translations still is an unbroken symmetry, specifically, translations in x 6 for c = 0 and translations in x 5 for c = 1. However, in these cases the seemingly unbroken translational symmetry is broken by the periodicity condition. For instance, consider the case c = 1. The background field A(x 5 , x 6 ) = −f x 6 dx 5 is changed by a torus translation to A(x 5 + kL, x 6 + lL) = −f x 6 dx 5 − f lLdx 5 ≡ −f x 6 dx 5 + dΛ, which yields
Λ = − 2π L N lx 5 .(102)
Clearly, the large gauge transformation that relates the two gauge fields connected by a torus translation breaks the translation symmetry in x 5 -direction. We conclude that also for c = 1 both translation symmetries are broken. One easily confirms that the same is true in the case c = 0.
(Pseudo) Goldstone bosons for more U(1)'s
The situation becomes more subtle in the case of more than one U (1) gauge group and an arbitrary number of charged scalars Q i with different charge assignments. The 9 For a recent discussion and references, see [25].
covariant derivatives then read
D m Q i = ∂ m + iq iα a (α) m + f (α) 2 mn x n Q i ,(103)
with i and α labeling the various U (1) gauge groups and charged matter fields, respectively.
In order to identify the Goldstone bosons, i.e. the nonlinearly transforming Wilson lines a (α) m , we start from the individual translation symmetries for the charged fields,
δQ i = m (i) ∂ m Q i .(104)
As in the previous section, the transformation behavior of the Wilson lines a (α) m is determined by the condition that the Lagrangian transforms into a total derivative, i.e.
δ(D m Q i ) = m (i) ∂ m D m Q i . This implies q iα δa (α) n = q iβ m (i) f (β) 2 nm ,(105)
for all i. This relation expresses the fact that, depending on the matrix q iα , fields charged under U (1) α may feel an effective flux, even though the flux of the gauge group U (1) α vanishes. It is evident that for N U (1) gauge groups and N f charged fields there can be at most min(N, N f ) Goldstone bosons. If there is flux in at least one of the gauge groups the two translation symmetries are spontaneously broken and there are at least two Goldstone boson, as discussed in the previous section. Further symmetries are accidental in the sense that they may be explicitly broken by additional interactions that couple the various matter fields, such as |Q i | 2 |Q j | 2 . Therefore, masses for some Wilson lines a (α) m may be generated beyond one-loop. In order to illustrate the subtleties in identifying the (pseudo) Goldstone bosons we discuss a simple example. Consider the gauge group U (1) 1 × U (1) 2 and two matter fields Q i with the charge matrix
q iα = 1 1 1 −1 .(106)
For vanishing fluxes, f (1) = f (2) = 0, it is obvious from the relation (105) that none of the fields a
q iβ f β = f 2 0 ,(107)
and there is a single Goldstone boson corresponding to a (1) m + a (2) m . Finally, for f (1) ≡ f = 0 and f (2) = 0 one has
q iβ f β = f 1 1 ,(108)
and both Wilson lines transform nonlinearly according to (105), which corresponds to two (pseudo) Goldstone bosons.
Conclusion and Outlook
In this work we have derived the four-dimensional supersymmetric effective action for six-dimensional gauge theories compactified on a torus with various background gauge fields. For non-vanishing background flux we have shown how the Kaluza-Klein excitations of the vector multiplet obtain their masses from a supersymmetric Stückelberg mechanism, and we have determined their couplings to charged chiral multiplets. For non-vanishing flux in the internal dimensions we have restricted the uncharged sector to the zero modes. The entire tower of the charged states, however, is incorporated, and their modified mass spectrum is obtained by solving the D-and F -term equations. As is well known, the massive tower corresponds to a harmonic oscillator spectrum with helicity-dependent shifts, where each level is |N |-fold degenerate. The Abelian flux background is perturbatively stable and we have worked out the full effective action in superfields as well as components. For non-Abelian flux we have clarified the physical mass spectrum. The internal components of the gauge field develop a tachyonic direction, and we expect the derived supersymmetric effective action to prove useful for the treatment of tachyon condensation. Using the explicit expressions for the level-dependent couplings of the charged tower to the Abelian Wilson lines, which are massless at tree-level, we have reproduced the known quantum corrections for vanishing flux. Following a regularization prescription used in the literature, one obtains a finite result. For non-vanishing flux the situation changes drastically. The quantum corrections induced by the charged bosons and fermions separately vanish at one-loop order for the |ϕ| 2 and |ϕ| 4 terms of the effective Lagrangian. This was shown using a diagrammatic approach where it follows from an intricate interplay between level-dependent masses and couplings. Considering the six-dimensional theory one understands that not just the |ϕ| 2 and |ϕ| 4 terms, but the entire effective potential should vanish exactly. The background gauge field associated with the magnetic flux breaks the translation symmetry in the x 5 -and x 6 -directions spontaneously. This leads to two massless Goldstone bosons which can be identified as the Wilson lines a 5 and a 6 contained in the complex field ϕ.
The results described above suggest several extensions of our work. First of all, the analysis of globally supersymmetric gauge theories with magnetic flux should be extended to supergravity theories. This would allow to study the backreaction of the flux on the geometry of the compact dimensions as well as possible mixings between moduli of the metric and the Wilson lines. Very important are also flux compactifications on orbifolds, see e.g. [25,[35][36][37]. In models with gauge-Higgs unification [23,24,[38][39][40] one could then investigate the effect of magnetic flux on quantum corrections to Higgs masses. It is an intriguing possibility that magnetic flux in higher dimensions may contribute significantly to stabilize the electroweak scale.
n,j −i 2n(−gf )V −,n−1,j φ +,n,j − gϕ 3 φ +,n,j V +,n,j − gϕ 3 V −,n,j φ +,I (V +,n,j φ −,ñ, − V −,ñ, φ +,,n,j ) V −,m,l φ −,m,l − V +,m,l φ +,m,l , with I = {n, j,ñ,, m, l,m,l} and C I = T 2 d 2 x ψ n,j ψñ , ψ m,l ψm ,l .
−
D µ Q n,m D µ Q n,m + |M n,m + √ 2gqϕ| 2 Q n,m Q n,m − iχ n,m σ µ D µ χ n,m − iχ n,m σ µ D µχ n,m + (M n,m + √ 2gqϕ)χ n,m χ n,m + h.c. ,
Figure 1 :Figure 2 :
12Bosonic contributions to the Wilson line mass without flux. Fermionic contribution to the Wilson line mass without flux.
Figure 3 :Figure 4 :
34Bosonic contributions to the Wilson line mass with flux. Fermionic contribution to the Wilson line mass with flux. From Eq. (50) one reads off the couplings of the Wilson line ϕ to the towers of charged bosonic and fermionic fields,
Figure 5 :
5Fermionic contributions to the Wilson line quartic coupling with flux.
Figure 6 :
6Bosonic contributions to the Wilson line quartic coupling with flux. tegrations and interchanging summation and t-integration yields δλ b = − q
nonlinearly. Hence, both symmetries (104) are preserved and there are no (pseudo) Goldstone bosons. For the flux assignment f (1) = f (2) ≡ f = 0 one finds
is obviously invariant under translations in the two torus directions,
For a review and references see, for example[1][2][3].
In the following the restriction to the zero mode for V 3 is understood and we do not indicate this with a subscript 0. The zero mode of φ 3 is denoted by ϕ 3 similar to the previous sections.
Here, we denote the scalar component of the superfields φ ± with the same letter as the superfield.
There are other examples where the spontaneous breaking of translational invariance leads to the appearance of Goldstone bosons. For instance, the localization of a Dp-brane in 9 − p dimensions implies the existence of 9 − p massless scalars localized on the Dp-brane. See, for example,[34].
AcknowledgmentsWe thank Fabian Rühle and Yannick Linke for valuable discussions. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Center (SFB) 676 "Particles, Strings and the Early Universe". E.D. was supported in part by the "Agence Nationale de la Recherche" (ANR). M.D. acknowledges support from the "Studienstiftung des deutschen Volkes".
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arxiv |
Homoclinic and Heteroclinic Motions in Hybrid Systems with Impacts
14 Jan 2016
Mehmet Onur Fen
Department of Mathematics
Middle East Technical University
06800AnkaraTurkey
Fatma Tokmak Fen
Department of Mathematics
Gazi University
06500AnkaraTeknikokullarTurkey
Homoclinic and Heteroclinic Motions in Hybrid Systems with Impacts
14 Jan 2016Impulsive systemsStable and unstable setsHomoclinic motionHeteroclinic motionDuffing equation with impacts
In this paper, we present a method to generate homoclinic and heteroclinic motions in impulsive systems. We rigorously prove the presence of such motions in the case that the systems are under the influence of a discrete map that possesses homoclinic and heteroclinic orbits. Simulations that support the theoretical results are represented by means of a Duffing equation with impacts.
Introduction
Impulsive differential equations describe the dynamics of real world processes in which abrupt changes occur. Such equations play an increasingly important role in various fields such as mechanics, electronics, biology, neural networks, communication systems, chaos theory and population dynamics [1,2,4,5,15,17,19,24,25,28]. In this paper, we investigate the existence of homoclinic and heteroclinic motions in systems with impulsive effects.
The main object of the present study is the following impulsive system,
x ′ = A(t)x + f (t, x) + g(t, ζ), t = θ k , ∆x| t=θ k = B k x + J k (x) + ζ k , (1.1)
where {θ k } , k ∈ Z, is a strictly increasing sequence of real numbers such that |θ k | → ∞ as |k| → ∞, A(t) is an n×n continuous matrix function, B k are constant n×n real valued matrices, ∆x| t=θ k = x(θ k +)−x(θ k ),
x(θ k +) = lim t→θ + k x(t), the functions f : R × R n → R n and J k : R n → R n are continuous in all their arguments, the function g(t, ζ) is defined by the equation g(t, ζ) = ζ k , t ∈ (θ k−1 , θ k ], and the sequence ζ = {ζ k } , k ∈ Z, is a solution of the map ζ k+1 = F (ζ k ), (1.2) where the function F : Λ → Λ is continuous and Λ is a bounded subset of R n . Here, R and Z denote the sets of real numbers and integers, respectively. The system under investigation is a hybrid one, since it combines the dynamics of an impulsive differential equation with a discrete map. Our main objective is to prove rigorously the existence of homoclinic and heteroclinic solutions in the dynamics of (1.1) provided that (1.2) possesses such solutions.
The idea of the usage of discontinuous perturbations to generate homoclinic and heteroclinic motions in systems of differential equations was first realized in the papers [3,7] on the basis of functional spaces.
It was shown in [3] that the chaotic attractor of the relay system, which was introduced in the paper [6],
consists of homoclinic solutions. Similar results for impulsive differential equations were obtained in the study [7] by taking advantage of the moments of impulses.
The existence of homoclinic and heteroclinic motions in systems with impulses were also investigated in the papers [9,11,12,14,18,26,27]. The existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects were investigated in [26] by using the mountain pass theorem and the symmetric mountain pass theorem in the critical point theory. The mountain pass theorem was also utilized in [11,18] to show the presence of homoclinic motions in second order impulsive systems. On the other hand, Wei and Chen [22,23] considered the existence of heteroclinic cycles in predator-prey systems with Allee effect and state-dependent impulsive harvesting within the scope of their studies. Zhang and Li [27] proved the existence of at least one non-zero homoclinic solution, which is generated by impulses, under appropriate conditions for a class of impulsive second order differential equations. Han and Zhang [14] obtained the existence of homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulses by using variational methods. It was mentioned in [14] that no homoclinic solutions exist for the system under investigation without impulses. However, in the present study, the emergence of homoclinic and heteroclinic motions are completely provided by the influence of a discrete map instead of impulsive effects. Additionally, our results are valid for systems with arbitrary high dimensions.
The rest of the paper is organized as follows. In Section 2, we discuss bounded solutions of (1.1), and present sufficient conditions for the existence of homoclinic and heteroclinic motions in the system. 2
In the sequel, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for matrices [16].
Let us denote by U (t, s) the transition matrix of the linear homogeneous system
u ′ = A(t)u, t = θ k , ∆u| t=θ k = B k u(θ k ).
(2.
3)
The following conditions are required.
(C1) det (I + B k ) = 0 for all k ∈ Z, where I is the n × n identity matrix;
(C2) There exists a positive number θ such that θ k+1 − θ k ≥ θ for all k ∈ Z;
(C3) There exist positive numbers N and ω such that
U (t, s) ≤ N e −ω(t−s) for t ≥ s; (C4) There exist positive numbers M f , M F and M J such that sup (t,x)∈R×R n f (t, x) ≤ M f , sup σ∈Λ F (σ) ≤ M F , sup k∈Z,x∈R n J k (x) ≤ M J ;
(C5) There exist positive numbers L f and L J such that
f (t, x 1 ) − f (t, x 2 ) ≤ L f x 1 − x 2
for all t ∈ R, x 1 , x 2 ∈ R n , and
J k (x 1 ) − J k (x 2 ) ≤ L J x 1 − x 2 for all k ∈ Z, x 1 , x 2 ∈ R n ; (C6) N L f ω + L J 1 − e −ωθ < 1; (C7) −ω + N L f + 1 θ ln(1 + N L J ) < 0.
Let Θ be the set of all sequences ζ = {ζ k } , k ∈ Z, obtained by equation (1.2). By using the results of [1,21] one can show under the conditions (C1) − (C6) that for a fixed sequence ζ ∈ Θ the system (1.1) possesses a unique bounded on R solution φ ζ (t), which satisfies the following relation,
φ ζ (t) = t −∞ U (t, s) [f (s, φ ζ (s)) + g(s, ζ)] ds + −∞<θ k <t U (t, θ k +) [J k (φ ζ (θ k )) + ζ k ] . (2.4)
One can confirm under the conditions (C1) − (C7) that for a fixed sequence ζ ∈ Θ, the bounded solution φ ζ (t) attracts all other solutions of (1.1), i.e.,
x(t) − φ ζ (t) → 0 as t → ∞ for any solution x(t) of (1.1). Moreover, sup t∈R φ ζ (t) ≤ N M f + M F ω + M J + M F 1 − e −ωθ
for each ζ ∈ Θ.
Homoclinic and heteroclinic motions
In this section, first of all, we will describe the stable, unstable and hyperbolic sets as well as the homoclinic and heteroclinic motions for both system (1.1) and the discrete map (1.2). These definitions were introduced in the papers [3,7]. After that the existence of homoclinic and heteroclinic motions in the dynamics of (1.1) will be proved.
Consider the set Θ described in the previous section once again. The stable set of a sequence ζ ∈ Θ is defined as
W s (ζ) = {η ∈ Θ | η k − ζ k → 0 as k → ∞} ,
and the unstable set of ζ is
W u (ζ) = {η ∈ Θ | η k − ζ k → 0 as k → −∞} .
The set Θ is called hyperbolic if for each ζ ∈ Θ the stable and unstable sets of ζ contain at least one
element different from ζ. A sequence η ∈ Θ is homoclinic to another sequence ζ ∈ Θ if η ∈ W s (ζ)∩W u (ζ). Moreover, η ∈ Θ is heteroclinic to the sequences ζ 1 ∈ Θ, ζ 2 ∈ Θ, η = ζ 1 , η = ζ 2 , if η ∈ W s (ζ 1 ) ∩ W u (ζ 2 ).
On the other hand, let us denote by A the set consisting of all bounded on R solutions of sys-
tem (1.1). A bounded solution φ η (t) ∈ A belongs to the stable set W s (φ ζ (t)) of φ ζ (t) ∈ A if φ η (t) − φ ζ (t) → 0 as t → ∞. Besides, φ η (t) is an element of the unstable set W u (φ ζ (t)) of φ ζ (t) provided that φ η (t) − φ ζ (t) → 0 as t → −∞. We say that A is hyperbolic if for each φ ζ (t) ∈ A the sets W s (φ ζ (t)) and W u (φ ζ (t)) contain at least one element different from φ ζ (t). A solution φ η (t) ∈ A is homoclinic to another solution φ ζ (t) ∈ A if φ η (t) ∈ W s (φ ζ (t)) ∩ W u (φ ζ (t)), and φ η (t) ∈ A is heteroclinic to the bounded solutions φ ζ 1 (t), φ ζ 2 (t) ∈ A , φ η (t) = φ ζ 1 (t), φ η (t) = φ ζ 2 (t), if φ η (t) ∈ W s (φ ζ 1 (t)) ∩ W u (φ ζ 2 (t)).
In what follows, we will denote by i ((a, b)) the number of the terms of the sequence {θ k } , k ∈ Z, which belong to the interval (a, b), where a and b are real numbers such that a < b. It is worth noting
that i((a, b)) ≤ 1 + b − a θ .
The connection between the stable sets of the solutions of (1.1) and (1.2) is provided in the next assertion.
Lemma 3.1 Suppose that the conditions (C1) − (C7) are fulfilled, and let ζ and η be elements of Θ. If
η ∈ W s (ζ), then φ η (t) ∈ W s (φ ζ (t)).
Proof. Fix an arbitrary positive number ǫ, and denote α = ω − N L f − 1 θ ln(1 + N L J ). Assume without loss of generality that ǫ ≤ 2M F . Let γ be a real number such that
γ ≥ 1 + N 1 ω + 1 1 − e −ωθ 1 + N L f (1 + N L J ) α + N L J (1 + N L J ) 1 − e −αθ .
Because the sequence η = {η k } , k ∈ Z, belongs to the stable set W s (ζ) of ζ = {ζ k } , there exists
an integer k 0 such that η k − ζ k < ǫ γ for all k ≥ k 0 . One can confirm that g(t, η) − g(t, ζ) < ǫ γ for t > θ k0−1 .
Making use of the relation
φ η (t) − φ ζ (t) = t −∞ U (t, s) [f (s, φ η (s)) − f (s, φ ζ (s)) + g(s, η) − g(s, ζ)] ds + −∞<θ k <t U (t, θ k +) [J k (φ η (θ k )) − J k (φ ζ (θ k )) + η k − ζ k ] ,
we obtain for t > θ k0−1 that
φ η (t) − φ ζ (t) ≤ θ k 0 −1 −∞ 2N (M f + M F )e −ω(t−s) ds + −∞<θ k ≤θ k 0 −1 2N (M J + M F )e −ω(t−θ k ) + t θ k 0 −1 N ǫ γ e −ω(t−s) ds + θ k 0 −1 <θ k <t N ǫ γ e −ω(t−θ k ) + t θ k 0 −1 N L f e −ω(t−s) φ η (s) − φ ζ (s) ds + θ k 0 −1 <θ k <t N L J e −ω(t−θ k ) φ η (θ k ) − φ ζ (θ k ) ≤ 2N M f + M F ω + M J + M F 1 − e −ωθ e −ω(t−θ k 0 −1 ) + N ǫ γω 1 − e −ω(t−θ k 0 −1 ) + N ǫ γ(1 − e −ωθ ) 1 − e −ω(t−θ k 0 −1 +θ) + t θ k 0 −1 N L f e −ω(t−s) φ η (s) − φ ζ (s) ds + θ k 0 −1 <θ k <t N L J e −ω(t−θ k ) φ η (θ k ) − φ ζ (θ k ) .
(3.5)
Define the functions u(t) = e ωt φ η (t) − φ ζ (t) and h(t) = c 1 + c 2 e ωt , where
c 1 = 2N M f + M F ω + M J + M F 1 − e −ωθ e ωθ k 0 −1 − N ǫ γ e ωθ k 0 −1 ω + e ω(θ k 0 −1 −θ) 1 − e −ωθ and c 2 = N ǫ γ 1 ω + 1 1 − e −ωθ .
The inequality (3.5) implies that
u(t) ≤ h(t) + t θ k 0 −1 N L f u(s)ds + θ k 0 −1 <θ k <t N L J u(θ k ).
The application of the analogue of the Gronwall's inequality for piecewise continuous functions yields
u(t) ≤ h(t) + t θ k 0 −1 N L f (1 + N L J ) i((s,t)) e N L f (t−s) h(s)ds + θ k 0 −1 <θ k <t N L J (1 + N L J ) i((θ k ,t)) e N L f (t−θ k ) h(θ k ).
Since the equation
1 + t θ k 0 −1 N L f (1 + N L J ) i((s,t)) e N L f (t−s) ds + θ k 0 −1 <θ k <t N L J (1 + N L J ) i((θ k ,t)) e N L f (t−θ k ) = (1 + N L J ) i((θ k 0 −1 ,t)) e N L f (t−θ k 0 −1 )u(t) ≤ c 1 (1 + N L J )e (ω−α)(t−θ k 0 −1 ) + c 2 e ωt + t θ k 0 −1 c 2 N L f (1 + N L J )e (ω−α)(t−s) e ωs ds + θ k 0 −1 <θ k <t c 2 N L J (1 + N L J )e (ω−α)(t−θ k ) e ωθ k ≤ c 1 (1 + N L J )e (ω−α)(t−θ k 0 −1 ) + c 2 e ωt + c 2 N L f (1 + N L J ) α e ωt 1 − e −α(t−θ k 0 −1 ) + c 2 N L J (1 + N L J ) 1 − e −αθ e ωt 1 − e −α(t−θ k 0 −1 +θ) .
If we multiply both sides of the last inequality by e −ωt , then we obtain that
φ η (t) − φ ζ (t) ≤ c 1 (1 + N L J )e −ωθ k 0 −1 e −α(t−θ k 0 −1 ) + c 2 + c 2 N L f (1 + N L J ) α 1 − e −α(t−θ k 0 −1 ) + c 2 N L J (1 + N L J ) 1 − e −αθ 1 − e −α(t−θ k 0 −1 +θ) < 2N (1 + N L J ) M f + M F ω + M J + M F 1 − e −ωθ e −α(t−θ k 0 −1 ) + N ǫ γ 1 ω + 1 1 − e −ωθ 1 + N L f (1 + N L J ) α + N L J (1 + N L J ) 1 − e −αθ .
Now, let R > θ k0−1 be a sufficiently large real number such that
2N (1 + N L J ) M f + M F ω + M J + M F 1 − e −ωθ e −α(R−θ k 0 −1 ) ≤ ǫ γ . For t ≥ R, we have φ η (t) − φ ζ (t) < ǫ γ 1 + N 1 ω + 1 1 − e −ωθ 1 + N L f (1 + N L J ) α + N L J (1 + N L J ) 1 − e −αθ ≤ ǫ. Therefore, lim t→∞ φ η (t) − φ ζ (t) = 0. Consequently, φ η (t) ∈ W s (φ ζ (t)).
In the next lemma, we reveal the connection between the unstable sets of the solutions of (1.1) and
(1.2).
η ∈ W u (ζ), then φ η (t) ∈ W u (φ ζ (t)).
Proof. Fix an arbitrary positive number ǫ, and let λ be a real number such that
λ > N (ω + 1 − e −ωθ ) ω(1 − e −ωθ ) − N (L f (1 − e −ωθ ) + L J ω)
.
Since η = {η k } , k ∈ Z, is an element of the unstable set W u (ζ) of ζ = {ζ k } , there exists an integer k 0 such that η k − ζ k < ǫ λ for all k ≤ k 0 .
In this case, we have that g(t, η) − g(t, ζ) < ǫ λ for t ≤ θ k0 .
By using the relation
φ η (t) − φ ζ (t) = t −∞ U (t, s) [f (s, φ η (s)) − f (s, φ ζ (s)) + g(s, η) − g(s, ζ)] ds + −∞<θ k <t U (t, θ k +) [J k (φ η (θ k )) − J k (φ ζ (θ k )) + η k − ζ k ] ,
one can verify for t ≤ θ k0 that
φ η (t) − φ ζ (t) < t −∞ N e −ω(t−s) L f φ η (s) − φ ζ (s) + ǫ λ ds + −∞<θ k <t N e −ω(t−θ k ) L J φ η (θ k ) − φ ζ (θ k ) + ǫ λ ≤ N ω L f sup t≤θ k 0 φ η (t) − φ ζ (t) + ǫ λ + N 1 − e −ωθ L J sup t≤θ k 0 φ η (t) − φ ζ (t) + ǫ λ .
Therefore,
1 − N L f ω − N L J 1 − e −ωθ sup t≤θ k 0 φ η (t) − φ ζ (t) ≤ N ǫ λ 1 ω + 1 1 − e −ωθ .
The last inequality implies that sup
t≤θ k 0 φ η (t) − φ ζ (t) < ǫ. Consequently, lim t→−∞ φ η (t) − φ ζ (t) = 0, and φ η (t) belongs to W u (φ ζ (t)).
The main result of the present paper is mentioned in the following theorem, which can be proved by using the results of Lemma 3.1 and Lemma 3.2. (i) If η ∈ Θ is homoclinic to ζ ∈ Θ, then φ η (t) ∈ A is homoclinic to φ ζ (t) ∈ A ;
(ii) If η ∈ Θ is heteroclinic to ζ 1 , ζ 2 ∈ Θ, then φ η (t) ∈ A is heteroclinic to φ ζ 1 (t), φ ζ 2 (t) ∈ A ;
(iii) If Θ is hyperbolic, then the same is true for A .
Examples
Let us take into account the impulsive Duffing equation
x ′′ + 0.2x ′ + 0.81x + 0.001x 3 = 0.7 cos 2π 3 t + g(t, ζ), t = θ k , ∆x| t=θ k = −0.12x + 0.09 + ζ k , ∆x ′ | t=θ k = −0.12x ′ + 0.015 sin(x),(4.6)
where θ k = 3k, k ∈ Z, the function g(t, ζ) is defined through the equation g(t, ζ) = ζ k , t ∈ (θ k−1 , θ k ], and the sequence ζ = {ζ k } is a solution of the logistic map
ζ k+1 = F µ (ζ k ), (4.7)
where F µ (s) = µs(1 − s) and µ is a parameter.
For 0 < µ ≤ 4, the interval [0, 1] is invariant under the iterations of (4.7) [10,13,20] By using the new variables x 1 = x and x 2 = x ′ one can reduce (4.6) to the system
x ′ 1 = x 2 , x ′ 2 = −0.81x 1 − 0.2x 2 − 0.001x 3 1 + 0.7 cos 2π 3 t + g(t, ζ), t = θ k , ∆x 1 | t=θ k = −0.12x 1 + 0.09 + ζ k , ∆x 2 | t=θ k = −0.12x 2 + 0.015 sin(x 1 ). (4.8)
Denote by U (t, s) the transition matrix of the linear homogeneous system
u ′ 1 = u 2 , u ′ 2 = −0.81u 1 − 0.2u 2 , t = θ k , ∆u 1 | t=θ k = −0.12u 1 , ∆u 2 | t=θ k = −0.12u 2 .
(4.9)
One can verify for t > s that (4.8). It is worth noting that for a periodic solution ζ = {ζ k } of (4.7) the corresponding bounded solution φ ζ (t) of (4.8) is also periodic.
U (t, s) = e −(t−s)/10 22 25 i([s,t)) P cos 2 √ 5 (t − s) − sin 2 √ 5 (t − s) sin 2 √ 5 (t − s) cos 2 √ 5 (t − s) P −1 , where i([s,
Consider the map (4.7) with µ = 3.9. It was demonstrated in [8] that the orbit
η = . . . , h 3 2 (η 0 ), h 2 2 (η 0 ), h 2 (η 0 ), η 0 , F µ (η 0 ), F 2 µ (η 0 ), F 3 µ (η 0 ), . . . ,
where η 0 = 1/3.9, is homoclinic to the fixed point η * = 2.9/3.9 of (4.7). Denote by φ η (t) and φ η * (t) the bounded solutions of (4.8) corresponding to η and η * , respectively. One can conclude by using Theorem 3.1 that φ η (t) is homoclinic to the periodic solution φ η * (t). Figure 1 shows the graphs of the x 1 −coordinates of φ η (t) and φ η * (t). In the figure, the solution φ η (t) is represented in blue color, while φ η * (t) is represented in red color. Figure 1 reveals that φ η (t) is homoclinic to φ η * (t), i.e., Now, we set µ = 4 in equation (4.7). According to [8], the orbit
φ η (t) − φ η * (t) → 0 as t → ±∞.η = . . . , h 3 1 ( η 0 ), h 2 1 ( η 0 ), h 1 ( η 0 ), η 0 , F µ ( η 0 ), F 2 µ ( η 0 ), F 3 µ ( η 0 ), . . . ,
where η 0 = 1/4, is heteroclinic to the fixed points η 1 = 3/4 and η 2 = 0 of (4.7). Suppose that φ η (t), φ η 1 (t) and φ η 2 (t) are the bounded solutions of (4.8) corresponding to η, η 1 and η 2 , respectively. Theorem 3.1 implies that φ η (t) is heteroclinic to the periodic solutions φ η 1 and φ η 2 . Figure 2 shows the graphs of the x 1 −coordinates of φ η (t), φ η 1 (t) and φ η 2 (t) in blue, red and green colors, respectively. The figure supports Theorem 3.1 such that φ η (t) converges to φ η 1 (t) as time increases and converges to φ η 2 (t) as time decreases, i.e., φ η (t) is heteroclinic to φ η 1 (t), φ η 2 (t). Figure 2: Heteroclinic solution of (4.8). The x 1 −coordinates of φ η (t), φ η 1 (t) and φ η 2 (t) are represented in blue, red and green colors, respectively. The figure confirms that φ η (t) is heteroclinic to the periodic solutions φ η 1 (t), φ η 2 (t).
In this study, we rigorously prove the presence of homoclinic and heteroclinic motions in hybrid systems with impacts. The dynamics of the system under consideration consist of an impulsive differential equation and a discrete map, which influences the former. According to our results, homoclinic and heteroclinic orbits of the discrete map give rise to the emergence of homoclinic and heteroclinic motions in the impulsive system. The presented technique is appropriate to design mechanical and electrical impulsive systems with homoclinic and heteroclinic motions, without any restriction in the dimension.
One can take advantage of our approach to investigate the presence of such motions in hybrid systems with impacts. An impulsive Duffing equation is utilized to illustrate the results of the paper. The provided examples show the applicability of our results.
Section 3 is devoted for the main results of the paper. In this part, we show the connection between the stable and unstable sets of the impulsive system (1.1) and the discrete map (1.2), and prove the existence of homoclinic and heteroclinic solutions in (1.1). Examples concerning homoclinic and heteroclinic motions in an impulsive Duffing equation are provided in Section 4. Finally, some concluding remarks are given in Section 5.
is valid and (1 + N L J ) i((a,b)) e N L f (b−a) ≤ (1 + N L J )e (ω−α)(b−a)for any real numbers a and b with a < b, one can confirm that
Lemma 3. 2
2Suppose that the conditions (C1) − (C6) are fulfilled, and let ζ and η be elements of Θ. If
Theorem 3. 1
1Under the conditions (C1) − (C7), the following assertions are valid.
The next section is devoted to examples concerning homoclinic and heteroclinic motions in an impulsive Duffing equation.
t)) is the number of the terms of the sequence {θ k } that belong to the interval [s, It can be calculated that U (t, s) ≤ N e −ω(t−s) , t ≥ s, where ω = 1/10 and N = 1.17.
For 0
0< µ ≤ 4 the bounded solutions of (4.8) lie inside the compact region D = (x 1 , x 2 ) ∈ R 2 : |x 1 | ≤ 2.8, |x 2 | ≤ 1.4 , and the conditions (C1) − (C7) are valid for system
Figure 1 :
1Homoclinic solution of (4.8). The x 1 −coordinates of φη(t) and φ η * (t) are shown in blue and red colors, respectively. The figure manifests that φη(t) is homoclinic to φ η * (t).
AcknowledgmentsThe authors wish to express their sincere gratitude to the referees for the helpful criticism and valuable suggestions, which helped to improve the paper significantly. This work is supported by the 2219 scholarship programme of TÜBİTAK, the Scientific and Technological Research Council of Turkey.
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| {'fraction_non_alphanumeric': 0.09763258631582859, 'fraction_numerical': 0.03742520533690118, 'mean_word_length': 3.2704332645611807, 'pattern_counts': {'":': 0, '<': 37, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 62, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper, we present a method to generate homoclinic and heteroclinic motions in impulsive systems. We rigorously prove the presence of such motions in the case that the systems are under the influence of a discrete map that possesses homoclinic and heteroclinic orbits. Simulations that support the theoretical results are represented by means of a Duffing equation with impacts.', 'arxivid': '1601.03592', 'author': ['Mehmet Onur Fen \nDepartment of Mathematics\nMiddle East Technical University\n06800AnkaraTurkey\n', 'Fatma Tokmak Fen \nDepartment of Mathematics\nGazi University\n06500AnkaraTeknikokullarTurkey\n'], 'authoraffiliation': ['Department of Mathematics\nMiddle East Technical University\n06800AnkaraTurkey', 'Department of Mathematics\nGazi University\n06500AnkaraTeknikokullarTurkey'], 'corpusid': 119148526, 'doi': '10.1515/ms-2017-0041', 'github_urls': [], 'n_tokens_mistral': 10617, 'n_tokens_neox': 9203, 'n_words': 5284, 'pdfsha': '800d0514a56f41ff481b888aa3a39544c9737ae3', 'pdfurls': ['https://arxiv.org/pdf/1601.03592v1.pdf'], 'title': ['Homoclinic and Heteroclinic Motions in Hybrid Systems with Impacts', 'Homoclinic and Heteroclinic Motions in Hybrid Systems with Impacts'], 'venue': []} |
arxiv |
Supplementary Information Chirality locking charge density waves in a chiral crystal
Geng Li
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
CAS Center for Excellent in Topological Quantum Computation
University of Chinese Academy of Sciences
100190BeijingChina
Songshan Lake Materials Laboratory
523808DongguanGuangdongPR China
Haitao Yang
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Peijie 2#
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Jiang
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
2#
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Wang Cong [email protected]
Qiuzhen 5#
Department of Physics
Beijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices
Renmin University of China
100872BeijingPR China
Cheng
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Shangjie Tian
Department of Physics
Beijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices
Renmin University of China
100872BeijingPR China
Guangyuan Han
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Chengmin Shen
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Xiao Lin
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
Hechang Lei [email protected]
Department of Physics
Beijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices
Renmin University of China
100872BeijingPR China
Wei Ji
Department of Physics
Beijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices
Renmin University of China
100872BeijingPR China
Ziqiang Wang
Department of Physics
Boston College
Chestnut HillMAUSA
Hong-Jun Gao [email protected]
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100190BeijingChina
CAS Center for Excellent in Topological Quantum Computation
University of Chinese Academy of Sciences
100190BeijingChina
Songshan Lake Materials Laboratory
523808DongguanGuangdongPR China
Supplementary Information Chirality locking charge density waves in a chiral crystal
# These authors contributed equally to this work. * Correspondence to:
Supplementary Fig. 9: Spatial evolution of dI/dV spectra across an even step. a, STM image of an even step. Scanning settings: Vs=50 mV, It=0.3 nA. b,c, Waterfall plot and intensity map of dI/dV spectra along the dashed arrow in a. On both lower and upper terraces, the asymmetric CDW coherence peaks show larger spectral weight at the positive bias. The asymmetry of the CDW coherence peaks keeps the same characteristics across the even step.
corresponding FT images at another region. a-i, Left panels: dI/dV maps taken from -40 mV to 40 mV of a 20 nm × 20 nm area. Right panels: corresponding FT images of the dI/dV maps. Away from the Fermi energy, the CDW pattern can be resolved with no energy dispersion (a-c, g-i). Inside the CDW gap, the stripe pattern is suppressed by the in-gap states which have not been fully gapped out. The quasi-particle interference pattern is highly energy-dependent, as marked by the red dashed rectangles.
Supplementary
Fig. 1: dI/dV maps under different bias voltages at the region presented in Fig. 2a of the main text. a-k, Left panels: dI/dV maps taken from -30 mV to 30 mV. Scanning area: 30 nm × 30 nm. Right panels: corresponding FT images of the left panels. l, Intensity curves along the Г-QCDW direction (highlighted by green dashed line in j) taken from all the FT images. The curves are vertically offset for clarity. The green arrows highlight the position of the 1 st and 2 nd order of the CDW wave vector. Supplementary Fig. 2: dI/dV maps under different bias voltages and
Fig. 3 :Fig. 4 :
34Comparison of CDW stripes in dI/dV maps and STM topographies. a,b, dI/dV map (a) and corresponding STM topography (b) of the region in Fig. 2a. c,d, dI/dV map (c) and corresponding STM topography (d) of the region in Fig. 2g. While the CDW stripes in dI/dV maps look vague and disordered due to the presence of defects, pronounced and well-ordered CDW stripes can be resolved in STM image and the corresponding FT image of CoSi (001) surface under 80 K. a, 20 nm × 20 nm STM topography. Scanning parameters: Vs=-200 mV, It=0.1 nA. b, FT image of a. The CDW stripes are less ordered, showing that the local defects affect the ordering of the stripes under high temperature.
Fig. 6 :Fig. 7 :Fig. 8 :
678Atomic structure change across odd steps. a, STM image (20 nm × 20 nm) showing the square Co lattice on the terraces. The scanning area is the same as Fig. 3e. b, Zoom-in image of the black square in a. The surface Co and Si atoms are overlaid. The atomic rows on terraces with different index are not aligned with each other as outlined by the red and green lines, indicating the glide-mirror symmetry of these terraces. The unit cells of the upper and lower terraces are outlined by red and green squares, respectively. Scanning settings: Vs=-200 mV, It=0.1 nA. STM image and dI/dV spectra inside the unit cell along the [10] direction of the surface. a, Atomically-resolved STM image showing square CoSi lattice and CDW stripes. b,c, Waterfall plot and intensity map of the dI/dV curves along the red line in a, showing strong periodic modulation of the dI/dV spectra of the lattice. STM image and dI/dV spectra inside the unit cell along the [11] direction of the surface. a, Atomic image of the surface. b-d, Stacking plot and intensity map of the spectra along the red arrow in a, showing strong intra-. 10: Schematic of electron hopping paths in the two crystal enantiomers of CoSi. Left panel, atomic structures of the top-(upper panel) and sideviews (lower panel) of the top three sublayers of enantiomer 1. The red line outlines the electron hopping path of the CDW order, following an atomic sequence of Co_1-Co_2-Si_2-Si_1-Co_1-Co_2…. If we only consider the top two sublayers, an equivalent path is labeled by a grey line. However, they become inequivalent if the third sublayer is considered. In particular, Co_3 lies below the interval between Si_2 and Si_1 along the red path, while it sits below the interval between Si_1 and Co_1 along the grey path, as highlighted by the red and grey dashed ellipses. As a result, the electron hopping path is unidirectional. Right panel, atomic structures of the top-(upper panel) and side-views (lower panel) of the top three sublayers of enantiomer 2. The blue line outlines the electron hopping path of the CDW order.
Fig. 11: FT images of the different CDW wavevector in different enantiomers. a, STM image of the two enantiomers, which is reproduced fromFig. 4c.b,c, FT image of enantiomer 1 (b) and 2 (c) as outlined by the blue and green dashed rectangles in a. CDW wave vectors q1 and q2 can be extracted from the FT images for the two enantiomers. The crystal lattice of the enantiomer 1 region shows a 90° rotation clockwise compared with that inFig. 1. The angle between q1 and q2 is ~69°. The two CDW patterns show mirror symmetry with respect to the horizontal axis.
Fig. 12 :
12Electronic bandstructures of the two crystal enantiomers. a,b, Bulk Brillouin zone of enantiomers 1 (a) and 2 (d). c,d, Theoretical electronic bandstructures, with consideration of spin-orbit coupling, of the CoSi bulk crystal along high-symmetry lines of enantiomers 1 (c) and 2 (d).
Supplementary Fig. 5: LEED pattern of the CoSi (001) surface under different temperatures. Temperature dependent LEED pattern showing the diffraction spots of the CDW pattern taken under 28 K, 50 K, 70 K, 90 K, 110 K, 130 K, 150 K, 170 K, and 190 K, respectively. Beam energy: 14 eV. The CDW spots disappear above 150 K.a
b
Bragg
Q cdw
4 nm
28 K
50 K
70 K
90 K
110 K
130 K
150 K
170 K
190 K
[10]
[01]
| {'fraction_non_alphanumeric': 0.034814638783269965, 'fraction_numerical': 0.03695342205323194, 'mean_word_length': 4.4655844155844155, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Supplementary Fig. 9: Spatial evolution of dI/dV spectra across an even step. a, STM image of an even step. Scanning settings: Vs=50 mV, It=0.3 nA. b,c, Waterfall plot and intensity map of dI/dV spectra along the dashed arrow in a. On both lower and upper terraces, the asymmetric CDW coherence peaks show larger spectral weight at the positive bias. The asymmetry of the CDW coherence peaks keeps the same characteristics across the even step.', 'arxivid': '2110.07813', 'author': ['Geng Li \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nCAS Center for Excellent in Topological Quantum Computation\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongPR China\n', 'Haitao Yang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n', 'Peijie 2# \nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n', 'Jiang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n', '2# \nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n', 'Wang Cong [email protected] ', 'Qiuzhen 5# \nDepartment of Physics\nBeijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices\nRenmin University of China\n100872BeijingPR China\n', 'Cheng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n', 'Shangjie Tian \nDepartment of Physics\nBeijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices\nRenmin University of China\n100872BeijingPR China\n', 'Guangyuan Han \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n', 'Chengmin Shen \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n', 'Xiao Lin \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n', 'Hechang Lei [email protected] \nDepartment of Physics\nBeijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices\nRenmin University of China\n100872BeijingPR China\n', 'Wei Ji \nDepartment of Physics\nBeijing Key Laboratory of Optoelectronic Functional Materials & Micro-Nano Devices\nRenmin University of China\n100872BeijingPR China\n', 'Ziqiang Wang \nDepartment of Physics\nBoston College\nChestnut HillMAUSA\n', 'Hong-Jun Gao [email protected] \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nCAS Center for Excellent in Topological Quantum Computation\nUniversity of Chinese Academy of Sciences\n100190BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongPR China\n'], 'authoraffiliation': ['Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina', 'School of Physical 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Physics\nChinese Academy of Sciences\n100190BeijingChina', 'School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100190BeijingChina', 'CAS Center for Excellent in Topological Quantum Computation\nUniversity of Chinese Academy of Sciences\n100190BeijingChina', 'Songshan Lake Materials Laboratory\n523808DongguanGuangdongPR China'], 'corpusid': 239009589, 'doi': '10.1038/s41467-022-30612-0', 'github_urls': [], 'n_tokens_mistral': 2651, 'n_tokens_neox': 2199, 'n_words': 1308, 'pdfsha': '0de5afe0905408d0f4a24bd74116998699422524', 'pdfurls': ['https://arxiv.org/pdf/2110.07813v2.pdf'], 'title': ['Supplementary Information Chirality locking charge density waves in a chiral crystal', 'Supplementary Information Chirality locking charge density waves in a chiral crystal'], 'venue': []} |
arxiv |
Surface magnetic states of Ni nanochains modified by using different organic surfactants
Weimeng Chen
Department of Physics
Peking University
100871BeijingPeople's Republic of China
Wei Zhou
School of Chemistry and Environment Science
Beijing University of Aeronautics and Astronautics
100191BeijingPeople's Republic of China
Lin He
Department of Physics
Peking University
100871BeijingPeople's Republic of China
Chinping Chen [email protected]
Department of Physics
Peking University
100871BeijingPeople's Republic of China
Lin Guo [email protected]
School of Chemistry and Environment Science
Beijing University of Aeronautics and Astronautics
100191BeijingPeople's Republic of China
Surface magnetic states of Ni nanochains modified by using different organic surfactants
17570Rf7550Lk Keywords: Ni nanochaincore-shell structuresurface magnetic propertyNi@Ni 3 C
Three powder samples of Ni nanochains formed of polycrystalline Ni nanoparticles with an estimated diameter of about 30 nm have been synthesized by a wet chemical method using different organic surfactants. These samples, having magnetically/structurally core-shell structures, all with a ferromagnetic Ni core, are Ni@Ni 3 C nanochains, Ni@Ni SG nanochains with a spin glass (SG) surface layer, and Ni@Ni NM nanochains with a nonmagnetic (NM) surface layer. The average thickness of the shell for these three samples is determined as about 2 nm. Magnetic properties tailored by the different surface magnetism are studied. In particular, suppression in 2 saturation magnetization usually observed with magnetic nanoparticles is revealed to arise from the surface magnetic states with the present samples.3
Introduction
Surface magnetic states have long been a subject of intense studies in the past few decades. It becomes increasingly important with the emerging field of nanomagnetics.
For biomedical applications, magnetic nanoparticles play an important role for the bounding of antibiotics, nucleotides, vitamins, peptides, etc. [1,2,3]. These depend much on the magnetic properties of the nanoparticles, especially, the surface magnetism. In information storage, the surface anisotropy plays an important role. It offers an extra degree of freedom to tune the magnetic anisotropy energy, making it attractive for basic investigation [4,5]. The fundamental role of exchange bias, widely counted on in spin valve and tunneling devices, is directly related to the interaction from the interface or surface [6]. Therefore, the investigations of surface magnetic states are interesting and of great importance.
Theoretical and experimental works reveal that surface atoms may have either enhanced or quenched moments, depending on their chemical environment [7,8].
Enhancement of saturation magnetization, M S , has been reported in small-sized, elemental metallic clusters [9]. However, many more thin films and nanoparticles demonstrate otherwise [10]. It is reported that the coating of organic molecules [11], CO chemisorption [12] or carbonyl ligation [13] on the surface of magnetic nanoparticles dramatically affects the magnetic properties. The low temperature magnetization reaches only 75% of the bulk value at low temperature for NiFe 2 O 4 nanoparticles coated by organic molecules [11]. The chemisorption of CO on the metallic Ni surface leads to the quenching of Ni magnetic moments because electrons 4 of the carbonyl ligation drive the Ni 4s electrons to fill up the 3d shell by repulsive interaction. The Ni atoms in the surface layer, therefore, become nonmagnetic (NM), leaving the magnetism of the inner core unaffected [12,13]. In many cases, surface magnetic effects are usually featured with a surface spin glass (SG) state and an appreciable reduction of saturation magnetization. For instance, surface SG behaviour has often been observed with ferromagnetic (FM) or ferrimagnetic nanoparticles, such as 6.5 nm NiFe 2 O 4 [14], 9-10 nm γ-Fe 2 O 3 [15], 12 nm Fe nanoparticles [16] and Ni nanochains [17]. Even, the surface SG properties are reported with antiferromagnetic (AFM) Co 3 O 4 nanowires having a magnetic core-shell structure of [18]. Surface magnetism has also been observed with the partially-oxidized composite nanoparticles, Cu@(Cu 2 O NM +CuO AFM ) [19].
Co 3 O 4 AFM @Co 3 O 4 SG
Nevertheless, there are other origins leading to surface "dead layer" of magnetic nanoparticles and resulting in the suppression of saturation magnetization [20]. These indicate that the surface magnetic state is one of the most important factors dictating the magnetic properties of nanoparticles.
Ni nanochains with dendritic morphology have been fabricated previously using polyvinyl pyrrolidone (PVP) as a surface modifier [21,22]. These samples show a SG behavior in addition to a FM phase. The SG behaviors are analyzed from the results of magnetization measurements and are revealed to come from the surface layer [17]. It
Sample preparations and characterizations
The detailed processes of synthesis are reported elsewhere. The process and characterizations for S-TOPO are reported in reference [23], S-PVP is Sample B in reference [22]. The process and the chemical reagents used to prepare S-CTAB is the same as that reported in reference [24], however, with a different reaction temperature at 197 ºC. The chemical reagents used for the preparation of S-CTAB are analytical grade without being further purified. In a typical experiment, 0.5 mmol NiCl 2 ·6H 2 O and 3 mmol CTAB were dissolved in the solvent of 60 ml glycol. Afterwards, the 6 solution of hydrazine hydrate (50% v/v) by 1 ml was dropped into the mixture. When the solution mixed homogeneously, it was heated to the boiling point (~197 ºC), and kept for 5 hours. The as-obtained sample was washed with ethanol and deionized water.
The crystal structures were characterized by powder X-ray diffraction (XRD) using a Rigaku Dmax 2200 X-ray diffractometer with Cu Kα radiation (λ=0.1542 nm).
Transmission electron microscopy (TEM) and high-resolution TEM (HRTEM) investigations were carried out by a JEOL JEM-2100F microscope, equipped with EDS (energy dispersive X-ray spectroscopy). The thermogravimetric (TG) analysis was performed using a Diamond thermogravimetric analyzer (Perkin-Elmer instruments) under a stream of air. The product was heated from 50 °C to 600 °C at a scan rate of 10 °C/min. factor has been accounted for in the normalization of the measured magnetization. It is noted that the mass of C atoms in the Ni 3 C shell layer is not taken into account in the above estimation. The NM shell of Ni 3 C is treated as if it were a magnetically dead layer of Ni. This introduces about 2% error in mass for Ni 3 C being treated as Ni.
Magnetic measurements and analysis
The magnetization measurements were performed by a Quantum Design SQUID They account for about 70% of the corresponding bulk value, ~54.2 emu/g, at 300 K [25].
The reduction of the saturation magnetization is attributed to the magnetically inert property of the surface shell. It is noted that the mass effect of the NM organic capped layer has already been corrected for by the TG measurements. Otherwise, the saturation magnetization per unit mass would have been further reduced by 10 to 15%.
For S-TOPO, the reduction in the saturation magnetization is obvious because the Ni 3 C shell is NM [26]. The average diameter of the FM cores is then estimated as D core = 26.6 nm with the Ni 3 C shell thickness of about 1.7 nm. It is consistent with the result of shell thickness, 1-4 nm, observed by the HRTEM investigation [23]. For S-PVP and S-CTAB, although there is no obvious shell structure like the Ni 3 C shell of S-TOPO, a FM core of Ni with roughly the same diameter, D core ~ 26 nm, is also concluded. At T < 80 K, M S (T) of S-PVP increases dramatically, reaching the bulk value of Ni, ~ 55 emu/g. It is attributed to the contribution from the surface SG shell [17]. On the other hand, the shell layer of S-CTAB, although of pure Ni, does not
show any magnetism at all, very much like S-TOPO with a NM Ni 3 C shell layer.
Therefore, S-CTAB exhibits properties of a magnetically core-shell structure, Ni@Ni NM , with a magnetically "dead shell" of Ni NM . The presence of a magnetically "dead layer" has been reported in the early days with the surface of Ni films [27]. In addition, the magnetic moments of the surface Ni are reported to be quenched, e.g., by the carbonyl ligation on the surface [13].
Conclusion
We Figure 1
has been suggested that the surface modifier (PVP) actually tailors the surface magnetic state. In order to further investigate magnetic properties such as the finite size effect of the ferromagnetic (FM) transition point with the Ni nanochains, the 5 diamater-dependent magnetization reversal behavior with the quasi-one dimensional chain-like nanostructure, and the possible transport properties with the nanochain structures, it is important to understand the surface magnetic properties and the thickness of the surface layer. We investigate three powder samples of Ni nanochains with magnetically/structurally core-shell structure, synthesized by a wet chemical method using different surface modifying agents. The estimated outer diameter of the three samples is about 30 nm. They include Ni@Ni 3 C synthesized using trioctylphosphine oxide (TOPO), and Ni nanochains of two kinds synthesized using PVP and hexadecyltrimethyl ammonium bromide (CTAB), forming the magnetic core-shell structure of Ni@Ni SG and Ni@Ni NM , respectively. They are thus labeled as S-TOPO, S-PVP and S-CTAB. For the three samples, the cores are all of FM Ni, however, with a surface shell layer of different magnetic properties, i.e., a NM surface shell of Ni 3 C for S-TOPO, a magnetically dead layer of Ni (also NM) for S-CTAB, and a SG surface shell for S-PVP.
Figure 1
1shows the XRD patterns for S-TOPO, S-PVP and S-CTAB. It reveals that S-PVP and S-CTAB are of pure Ni phase with a face-centered cubic (fcc) structure.The diffraction peaks correspond to the planes of (111), (200) and (220) of fcc Ni (JCPDS 04-0850). Besides the nickel phase, S-TOPO contains the Ni 3 C phase also, which has been characterized in detail and reported in[23].The peaks marked with open triangles could be assigned to Ni 3 C (JCPDS 77-0194), corresponding to the planes of (110), (006), (113), (116), (300), and (119) of Ni 3 C. It is noteworthy that the (111) peak of Ni overlaps the (113) peak of Ni 3 C. Figures 2a, 2b, and 2c show the TEM images of S-TOPO, S-PVP, and S-CTAB, respectively. The morphology is similar, showing chain-like shape with dendritic 7 structure. Their average diameter is estimated about 30 nm. The corresponding insets show the magnified images. In the inset of figure 2a, a HRTEM image of Ni@Ni 3 C with a core-shell structure is presented. There is an almost invisible and very thin capped layer outside the Ni 3 C shell. It is known that a residual nonmagnetic, organic capped layer, TOPO in this case, is inevitable even after several times of thorough rinsing. About 10% mass ratio of the organic capped layer is determined by the TG measurement described below. The Ni 3 C shell is estimated from the inset about 2 to 4 nm in thickness, which is slightly larger than the average thickness of about 2 nm determined by the magnetic measurements. In the inset of figure 2b for S-PVP, the HRTEM image reveals that the Ni chains are covered with a vague layer of organic remnants. A layer of almost invisible organic remnants is also observed for S-CTAB, as shown in the inset of figure 2c. To further confirm the composition of the samples, EDS measurements were conducted, showing pure Ni element without any other magnetic elements.The mass ratio of the nonmagnetic organic capped layer is determined by TG measurements, as shown infigure 3. The mass of the as-prepared samples at room temperature is denoted as m 0 . By heating up the samples gradually, the mass first decrease slightly due to the burning of the organic remnants. Then, the mass increases dramatically arising from the oxidation of Ni, forming NiO, denoted as m(NiO). The mass of Ni is then calculated according to the relative ratio of the formula weight, m(Ni) = (59/75) m(NiO). The correction factor, m(Ni)/m 0 , is then determined as 90%, 87%, and 93% for S-TOPO, S-PVP, and S-CTAB, respectively. The mass correction8
magnetometer. The measurements include, a) temperature dependent saturation magnetization, M S (T), recorded in the field of 20 kOe from T = 380 to 5 K, b) zero-field-cooling (ZFC) and field-cooling (FC) measurements, M ZFC (T) and M FC (T) from T = 5 to 380 K, and c) field dependent hysteresis loops, M(H), at a series of fixed temperature from T = 5 to 380 K.
Figure 4
4shows the temperature dependence of saturation magnetization, M S (T), of the three samples recorded in the applied field of H app = 20 kOe. At T > 80 K, the three M S (T) curves nearly collapse. The values of M S (T) at T = 300 K are determined as 37.7 emu/g (S-TOPO), 38.2 emu/g (S-PVP), and 38.3 emu/g (S-CTAB).
Figure 5
5shows the M ZFC (T) and M FC (T) curves. The ZFC curves were recorded in the applied field, H app = 90 Oe, in the warming process after the sample was cooled down to T = 5 K under zero applied field. For the FC measurement, the procedure of data collection was the same except that the sample was cooled down to 5 K in the applied magnetic field of 20 kOe. For all of the three samples, the M FC (T) and M ZFC (T) curves separate from each other with the temperature going up to 380 K. It indicates that the blocking temperature is higher than 380 K. The inset shows the blown-up M ZFC (T) curves in the low temperature region. A freezing peak appears at T F ~ 8 K with S-PVP, similar to those at about 13 K observed for 50, 75 and 150 nm Ni nanochains also synthesized using the surfactant of PVP [17]. It further confirms that S-PVP exhibits a magnetically core-shell structure of Ni@Ni SG . For the other two samples, S-TOPO and S-CTAB, there is no such characteristic feature, as is expected for the samples with a NM shell layer. The M(H) measurements for the three samples are performed at various temperature from 5 to 380 K. Figure 6a shows the M(H) curves measured at T = 5 K. The magnetization determined in the high field region at H = 10 kOe, and the coercivity, H C , are 45.0 emu/g and 600 Oe for S-TOPO, 52.3 emu/g and 568 Oe for S-PVP, and 43.1 emu/g and 634 Oe for S-CTAB, respectively. The magnetization of S-PVP is higher than that of S-TOPO, or S-CTAB by about 20%, attributed to the contribution from the SG shell. The M(H) curves at T = 300 K is shown in Figure 6b. The difference in saturation magnetization is more or less reduced. The coercivity determined from the M(H) curves at various temperatures is shown in figure 6c. The magnetization reversal can be described by the fanning mode based on the chain of sphere model proposed by Jacobs and Bean[28], as discussed in our previous work for S-TOPO[23].
Figure captions
Figure captions
Figure 1 .
1XRD patterns for samples modified with TOPO (S-TOPO), PVP (S-PVP) and CTAB (S-CTAB).
Figure 2 .
2Morphology revealed by TEM images for (a) S-TOPO, (b) S-PVP and (c) S-CTAB. The insets are the corresponding HRTEM images.
Figure 3 .
3Thermogravimetric analyses for the samples of S-TOPO, S-PVP and S-CTAB. The mass of the as-prepared sample is denoted as m 0 , and the final NiO is denoted as m NiO . The calculated Ni contents, m Ni , is 93%, 90% and 87% for TOPO, CTAB and PVP, respectively.
Figure 4 .
4Saturation magnetization, M S (T), recorded in the applied field of 20 kOe from 380 to 5 K. The magnetization of S-PVP increases dramatically at T < 80 K.
Figure 5 .
5ZFC and FC M(T) curves for S-TOPO, S-PVP and S-CTAB measured in the applied field of 90 Oe from 5 K to 380 K. The inset shows the ZFC curves in the low temperature region. The peak around 8 K in the ZFC curve with S-PVP is attributed to the freezing of the surface SG shell.
Figure 6 . 16 Figures
616M(H) curves for S-TOPO, S-PVP and S-CTAB at (a) T = 5 K, (b) T = 300 K, and (c) coercivity determined from the M(H) curves at different temperatures.
have investigated the magnetic properties of three samples of Ni nanochains, with an estimated outer diameter of 30 nm, synthesized by different surface modifying agents, including CTAB, TOPO and PVP. The surface magnetic properties are modified significantly and differently by the surfactants in the synthesis process, forming different magnetic shell structures. The CTAB leads to the nanochains of Ni@Ni NM with a magnetically dead layer of Ni surface shell, while the TOPO makes the final products of Ni@Ni 3 C nanochains with a shell of NM Ni 3 C. With PVP, a SG shell layer of Ni is formed, resulting in the magnetic structure of Ni@Ni SG . The saturation magnetization of these samples accounts for only 70% of the bulk value at 300 K attributed to the presence of the magnetically inert layer. This value would be further reduced by more than 10% if the mass of the outermost capped layer of organic remnants is not corrected for. The thickness of the shell layer is determined as about 2 nm for all of the three samples.11
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arxiv |
FRACTIONAL CALCULUS AND PATH-WISE INTEGRATION FOR VOLTERRA PROCESSES DRIVEN BY LÉVY AND MARTINGALE NOISE
30 Aug 2016
G Di Nunno
Y Mishura
K Ralchenko
FRACTIONAL CALCULUS AND PATH-WISE INTEGRATION FOR VOLTERRA PROCESSES DRIVEN BY LÉVY AND MARTINGALE NOISE
30 Aug 2016arXiv:1608.08466v1 [math.PR]and Phrases: fractional calculuspathwise integrationVolterra processesLévy processesambit fieldstime changesubordinationfractional Brownian motion
We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.
Introduction
In this paper, we consider Volterra processes, namely processes of the form Y t = t 0 g(t, s) dZ s , t ∈ [0, T ], (1.1) where g(t, s) is a given deterministic Volterra-type kernel, and Z is a Lévy process or a (square integrable) martingale process. The integral in (1.1) is understood in the sense of [24] as taking the limit in probability of elementary integrals. Volterra processes of the type above are widely used in physics for the modelling of turbulence, see e.g. [4], [14]. Also they have been suggested in the context of biology/medicine for the modeling of cancer growth in biological tissues, see [3]. Furthermore, these processes have been used successfully in mathematical finance, specifically in energy finance where the spot prices of electricity and other commodities strongly depend on environmental risk factor, such as temperatures, wind speed, sun coverage, precipitations, etc. Such processes also appear in problems of credit risk and are well suited to fit stochastic volatility models. See e.g. [2,8,9,12,15,28], and reference therein. Finally, Volterra processes (1.1) also provide suitable models in signal processing, see e.g. [29], and for the workload of network devices, see e.g. [31]. The Volterra processes (1.1) are part of the general class of ambit fields, which appear within a space-time framework, while here we have only time, thus a process and the the integrand sees not only a deterministic kernel, but also a stochastic component. Such stochastic component can be also replaced by a time change in Date: August 30, 2016.
1 the driving noise, as it actually done in the present paper in terms of subordination. In the setting of ambit processes the so-called ambit set is here reduced to the real semi-line. See e.g. [22] for a survey on ambit fields. The class of processes defined in (1.1) contains the fractional Brownian motion and its generalisation, namely, the fractional Lévy process. In fact, assume that the function g is the Molchan-Golosov kernel, which is given by
g H (t, s) = C H (t − s) H− 1 2 F 1 2 − H, H − 1 2 , H + 1 2 , s−t s , 0 < s < t < ∞,
and g H (t, s) = 0 otherwise, where H ∈ (0, 1), 1 2 , and F is the Gauss' hypergeometric function. If the driving process Z is a Brownian motion, then the process Y defined by (1.1) with the kernel g H is the fractional Brownian motion, see [20]. If Z is a Lévy process without Gaussian component such that EZ 1 = 0 and EZ 2 1 < ∞, then Y is the fractional Lévy process by Molchan-Golosov transformation (fLpMG), introduced in [28] (see also [12] for multivariate generalization). Let us mention that there exist another definitions of fractional Lévy processes in the literature. In particular, fractional Lévy process by Mandelbrot-van Ness representation (fLpMvN) was defined in [5] and studied in [18]. The comparison between fLpMvN and fLpMG can be found in [28].
C H = 1 Γ(H + 1 2 ) 2HΓ(H + 1 2 )Γ( 3 2 − H) Γ(2 − 2H)
The aim of the present paper is to develop a theory of integration with respect to processes of the form (1.1) applying fractional calculus, thus generalising the famous construction for integrals w.r.t. fractional Brownian motion from [32,33]. The use of fractional calculus allows for a bridging between stochastic and deterministic methods, which is very interesting from the use of models. Indeed our aim is to set the basis for a framework of pathwise calculus for Volterra processes. At present we concentrate on the definition and characterisation of the integrators and the integrands. Future research will focus on the actual calculus rules. It is important to have a manageable calculus from the applied perspective in which beyond the prediction on a model, other questions naturally appear linked e.g. to stochastic control. In this paper we concentrate on the case when the driving noise Z is a Lévy process and when it is possible we will consider Z to be a square integrable martingale. Also, we detail our results in the case in which Z is a subordinate Brownian motion. Indeed subordination is one of the easy way to construct a Lévy process having also advantages from the simulation point of view. See e.g. [10]. In particular, processes with compound Poisson, stable and Gamma subordinators are studied in detail.
Several approaches to the integration with respect to Lévy-driven Volterra processes are known. In [7], a Skorokhod type integral was considered. That construction followed S-transform approach, developed in [6] for fractional Brownian motion. Another approach was proposed in [1] and then extended in [11], where the integration operator was based on Malliavin calculus and described an anticipative integral. Wiener integration with respect to fLpMG was considered in [28]. But it turns out that one of the simplest and natural methods to construct the integral w.r.t. Lévydriven Volterra processes is to apply fractional calculus. This has the advantage that combines deterministic and stochastic techniques and it has a clear relationship with the underlying noise information flow. Hence, in the present paper, following [32], we construct pathwise stochastic integral using fractional integrals and derivatives. We present general conditions for the existence of this integral in terms of fractional derivatives. As an example we consider the case of fLpMG.
The paper is organised as follows. In Section 2 we review the construction of the integral of a deterministic kernel with respect to a Lévy noise and a square integrable martingale. In particular we detail to case of the subordinated Wiener process in Section 3. The elements of fractional calculus are presented in Section 4. Finally, in Section 5 is devoted to pathwise integrals with respect to Volterra processes. Various examples are provided at all stages.
Integration with respect to Lévy processes
In this section we study the stochastic integrals with respect to Lévy processes and square integrable martingales. We review the basic construction and we provide some results on the upper-bounds for the moments of the resulting integrals. Indeed these a priori bounds for the moments of order p ≥ 1 are fundamental results for the development in the sequel. Being very simple and also being a partial case of Bichteler-Jacod inequalities, these bounds with the values of corresponding constants containing the integrals w.r.t. the Lévy measures, are rather elegant therefore we provide the corresponding proofs. We start by the definition of the integral using the approach of Rajput and Rosinski [24].
2.1. Integration of non-random functions with respect to Lévy process. Let Z = {Z t , t ≥ 0} be a Lévy process. Define τ (z) := z, |z| ≤ 1, z |z| , |z| > 1. Then the characteristic function of Z t can be represented in the following form (see, e.g., [27])
E exp {iµZ t } = exp {tΨ(µ)} , where Ψ(µ) = ibµ − aµ 2 2 + R e iµx − 1 − iµτ (x) π(dx), b ∈ R, a ≥ 0, π is a Lévy measure on R, that is a σ-finite Borel measure satisfying R x 2 ∧ 1 π(dx) < ∞,
with π({0}) = 0 for any x ∈ R. The triplet (a, b, π) is shortly called the characteristic triplet of Z. Now we review the construction of integral of non-random function w. r. t. the Lévy process Z proposed in [30] and further developed in [24]. Let the interval [0, T ] be fixed. Consider, for any n, the partition of [0, T ] of the form [0, T ] = n i=1 A i , where A i ∈ B([0, T ]) and are pair-wise disjoint and max i λ(A i ) → 0, n → ∞. Here λ denotes the Lebesgue measure on B(R). Also throughout the paper B(S) stands for the Borel σ-field on the measurable space S equipped with the topology generated by the open sets. Hereafter we construct a measure on B([0, T ]) taking values in the space L 0 (Ω, F, P) as follows. We take a Lévy process Z and consider for any A ∈ B([0, T ]) a random variable Z(A) with the characteristic function of the form
E exp {iµZ(A)} = exp {λ(A)Ψ(µ)} .
Evidently, Z is a measure on B([0, T ]) with the values in L 0 (Ω, F, P) and Z([0, t]) = Z t is the value of the Lévy process Z at point t. Introduce the following definition.
A f dZ = n j=1 f j Z(A ∩ A j ). (ii) A measurable function f : ([0, T ], B([0, T ])) → (R, B(R)) is said to be Z- integrable if there exists a sequence {f n , n ≥ 1} of simple functions as in (i) such that 1) f n → f λ-a. e. 2) for any A ∈ B([0, T ]) the sequence A f n dZ converges in probability (P -lim) as n → ∞. If f is Z-integrable, we put A f dZ = P -lim n→∞ A f n dZ.
The following statement summarises the basic facts about the newly introduced integral. They are established in [24] and [30]. From now on we put 0 · ∞ = 0.
r(u) := au 2 + R |xu| 2 ∧ 1 π(dx) + bu + R τ (xu) − τ (x)u π(dx) .
Then a measurable function f : Ψ(λf (s)) ds
([0, T ], B([0, T ])) → (R, B(R)) is Z-integrable if and only if [0,T ] r(f (s)) ds < ∞. (iii) If f is Z-integrable,= exp ib f λ − a f λ 2 2 + R e iλx − 1 − iλτ (x) F f (dx) , (2.1) where b f = [0,T ] bf (s) + R τ (xf (s)) − τ (x)f (s) π(dx) ds, a f = [0,T ] af 2 (s) ds, F f (B) = [0,T ] R ½ f (s)x∈B\{0} π(dx) ds, B ∈ B(R).
The next lemma contains the modifications of well known properties of the introduced integral in the form that is suitable for our further considerations.
Lemma 2.1. (i) Any function f ∈ L 2 ([0, T ]) is Z-integrable.
In this case the characteristic function of the integral has the following form 2). Suppose that Z satisfies the additional assumptions: a = 0 and |x|≤1 |x| p π(dx) < ∞. Then any function f ∈ L p ([0, T ]) is Z-integrable, and
E exp iλ [0,T ] f dZ = exp ibλ [0,T ] f (s) ds − 1 2 aλ 2 [0,T ] f 2 (s) ds + [0,T ] R e iλf (s)x − 1 − iλf (s)τ (x) π(dx) ds . (2.2) (ii) Let p ∈ [1,E exp iλ [0,T ] f dZ = exp ibλ [0,T ] f (s) ds + [0,T ] R e iλf (s)x − 1 − iλf (s)τ (x) π(dx) ds . (2.3)
Proof. According to paragraph (ii) from Proposition 2.1, in order to establish Zintegrability, we need to prove that
[0,T ] R af 2 (s) π(dx) ds < ∞, [0,T ] R |xf (s)| 2 ∧ 1 π(dx) ds < ∞, and [0,T ] R |τ (xf (s)) − τ (x)f (s)| π(dx) ds < ∞.
The first integral is finite, since f ∈ L 2 ([0, T ]) in the case (i) and a = 0 in the case (ii).
Recall that R x 2 ∧ 1 π(dx) < ∞ by the definition of the Lévy measure. Then |x|≤1 |x| p π(dx) < ∞ and |x|>1 π(dx) < ∞ in both cases (i) and (ii). Since the rest of proof can be carried out similarly for both statements, from now we assume that p ∈ [1,2]. Consider the second integral
[0,T ] R |xf (s)| 2 ∧ 1 π(dx) ds ≤ [0,T ] R (|xf (s)| p ∧ 1) π(dx) ds ≤ [0,T ] |x|≤1 |xf (s)| p π(dx) ds + [0,T ] |x|>1 π(dx) ds = [0,T ] |f (s)| p ds |x|≤1 |x| p π(dx) + T |x|>1 π(dx) < ∞.
The third integral can be rewritten as follows:
[0,T ] R |τ (xf (s)) − τ (x)f (s)| π(dx) ds = [0,T ] |x|≤1 |τ (xf (s)) − τ (x)f (s)| ½ |xf (s)|≤1 π(dx) ds + [0,T ] |x|≤1 |τ (xf (s)) − τ (x)f (s)| ½ |xf (s)|>1 π(dx) ds + [0,T ] |x|>1 |τ (xf (s)) − τ (x)f (s)| π(dx) ds =: I 1 + I 2 + I 3 .
Note that
I 1 = 0, because τ (xf (s)) − τ (x)f (s) = 0 for |x| ≤ 1 and |xf (s)| ≤ 1. Let |x| ≤ 1 and |xf (s)| > 1.
Then
|τ (xf (s)) − τ (x)f (s)| = xf (s) |xf (s)| − xf (s) = |xf (s)| 1 |xf (s)| − 1 ≤ 2 |xf (s)| ≤ 2 |xf (s)| p .
Hence,
I 2 ≤ 2 [0,T ] |f (s)| p ds |x|≤1 |x| p π(dx) < ∞.
Finally, using the inequality |τ (z)| ≤ 1, we can write
|τ (xf (s)) − τ (x)f (s)| ≤ |τ (xf (s))| + |τ (x)f (s)| ≤ 1 + |f (s)| .
Then
I 3 ≤ [0,T ] (1 + |f (s)|) ds |x|>1 π(dx) < ∞.
This concludes the proof of Z-integrability. The formulas (2.2)-(2.3) for the characteristic functions follow directly from the statement (iii) of Proposition 2.1.
Remark 2.1. With no doubt, any function f ∈ L p ([0, T ]), p > 2 is Z-integrable.
The rest of this section is devoted to the upper bounds for the moments of the integral, which are fundamental tools for the analysis in the sequel.
First, assume that f ∈ L 2 ([0, T ]) and R x 2 π(dx) < ∞. Then by differentiation of the characteristic function (2.2), one can deduce that the integral admits second moment. In fact we have
E [0,T ] f dZ 2 = [0,T ] f (s) ds 2 b + R (x − τ (x)) π(dx) 2 + [0,T ] f 2 (s) ds a + R x 2 π(dx) < ∞. Here we use that R |x − τ (x)| π(dx) = |x|>1 |x − τ (x)| π(dx) ≤ 2 |x|>1 |x| π(dx) ≤ 2 |x|>1 x 2 π(dx) < ∞.
In the case when b = 0 and the measure π is symmetric, the formula for the second moment is simplified. Indeed, in this case R (x − τ (x)) π(dx) = 0 and
E [0,T ] f dZ 2 = [0,T ] f 2 (s) ds a + R x 2 π(dx) .
Now let us consider the general case p ≥ 1. The following theorem gives an a priori estimate for the pth moment of the integral. By Lemma 2.1, in order to integrate functions from L p ([0, T ]) with p ∈ [1, 2) we need to assume that a = 0 for the process Z.
Theorem 2.1. (i) Let p ∈ [1, 2). Assume that f ∈ L p ([0, T ]) and that the characteristic triplet of Z satisfies a = b = 0, π is symmetric, R |x| p π(dx) < ∞. Then E [0,T ] f dZ p ≤ C f p Lp([0,T ]) R |x| p π(dx). (2.4) (ii) Let p ≥ 2. Assume that f ∈ L p ([0,
T ]) and that b = 0, π is symmetric and
R |x| p π(dx) < ∞. Then E [0,T ] f dZ p ≤ C a p/2 f p L 2 ([0,T ]) + f p Lp([0,T ]) R |x| p π(dx) . (2.5) Proof. (i) Let Φ p (u) = R |ux| p π(dx). Evidently, Φ p : R → R + is a Young function, i. e. it is a convex function such that Φ p (u) = Φ p (−u), Φ p (0) = 0 and lim u→∞ Φ p (u) = ∞. Therefore, we can consider the Orlicz space L Φp ([0, T ]) = f ∈ L 0 ([0, T ]) : [0,T ] Φ p (|f (s)|) ds < ∞ = L p ([0, T ])
with the Luxemburg norm
f Φp = inf c > 0 : [0,T ] Φ p c −1 |f (s)| ds ≤ 1 = f Lp([0,T ]) R |x| p π(dx) 1 p . (2.6) Then L Φp ([0, T ]) obviously is a Banach space. First, let us prove that [0,T ] f dZ ∈ L p (Ω; P) for any f ∈ L Φp ([0, T ]). Assume that f ∈ L Φp ([0, T ]), that is [0,T ] Φ p (|f (s)|) ds < ∞.
Recall that by F f (·) the Lévy measure in the canonical representation of the characteristic function of [0,T ] f dZ (see Proposition 1 (iii)). Then, by Proposition 1,
|u|>1 |u| p F f (du) = [0,T ] {|f (s)x|>1} |f (s)x| p π(dx) ds ≤ [0,T ] Φ p (|f (s)|) ds < ∞.
Taking into account paragraph (iii) of Proposition 2.1 and the well-known property of Lévy processes (see [27,Theorem 25.3]), we can conclude, as it was done in the proof of inequality (3.6), Theorem 3.3 from [24], that the finiteness of this integral
implies the finiteness of E [0,T ] f dZ p .
Further, let us prove that the linear mapping
L Φp ([0, T ]) ∋ f −→ [0,T ] f dZ ∈ L p (Ω; P) is continuous. Let f n → 0 in L Φp ([0, T ]). This implies that [0,T ] Φ p (|f n (s)|) ds → 0, as n → ∞, (2.7)
see [25,Proposition 3.2.4]. Let b n , a n and F n be, respectively, the centring constant, the variance, and the Lévy measure in the canonical representation of the characteristic function of [0,T ] f n dZ (see Proposition 1 (iii)). Under the assumptions taken, being π symmetric, we have a n = b n = 0, and
|u|>1 |u| p F n (du) = [0,T ] |fn(s)x|>1 |f n (s)x| p π(dx) ds ≤ [0,T ] R |f n (s)x| p π(dx) ds → 0,
as n → ∞, by (2.7). Then the convergence
E [0,T ] f n dZ p → 0,E [0,T ] f dZ p 1 p ≤ C f Φp ,
where the constant C does not depend on f . Taking (2.6) into account, we conclude the proof.
(ii) The statement can be proved similarly, using the function
Φ (a) p (u) = au 2 + R |ux| p π(dx)
instead of Φ p . Arguing as above, we get
E [0,T ] f dZ p ≤ C f p Φ (a) p .
Then it is not hard to see that
f p Φ (a) p = inf c > 0 : a f 2 L 2 ([0,T ]) c 2 + f p Lp([0,T ]) R |x| p π(dx) c p ≤ 1 p ≤ C 1 a 1/2 f L 2 ([0,T ]) + f Lp([0,T ]) R |x| p π(dx) 1/p p ≤ C 2 a p/2 f p L 2 ([0,T ]) + f p Lp([0,T ]) R |x| p π(dx) .
Remark 2.2. The case b = 0 can be considered similarly. If the other assumptions of the above theorem hold, then
E [0,T ] f dZ p ≤ C |b| p f p L 1 ([0,T ]) + f p Lp([0,T ]) R |x| p π(dx) .
for p ∈ [1, 2), and
E [0,T ] f dZ p ≤ C |b| p f p L 1 ([0,T ]) + a p/2 f p L 2 ([0,T ]) + f p Lp([0,T ]) R |x| p π(dx) . (2.8)
for p ≥ 2. In this case the functions Φ p and Φ (a) p in the proof are replaced with Φ
(0,b) p (u) = |bu| + Φ p (u) and Φ (a,b) p (u) = |bu| + Φ (a) p (u), respectively. Remark 2.3. The upper bound (2.8) can be simplified to E [0,T ] f dZ p ≤ C f p Lp([0,T ]) . (2.9)
Inequality (2.9) for predictable stochastic integrands is contained in Theorem 66 [23]. For such integrands the inequality (2.9) is a partial case of Bichteler-Jacod inequality, see, e.g., [17]. However, we prefer to give here the detailed structure of the right-hand side.
Remark 2.4. Volterra processes driven by square integrable martingales. Let p ≥ 2 and R |x| p π(dx) < ∞. Then, taking into account inequality R x 2 ∧ 1 π(dx) < ∞, we get that R x 2 π(dx) < ∞, so that Z is a square-integrable process. Assuming additionally that b = 0, and the measure π is symmetric, we can see that Z is a square-integrable martingale with quadratic characteristics Z t = (a+ R x 2 π(dx))t. However, we can consider the general case. Indeed, let M be a square integrable càdlàg martingale with quadratic variation [M ] and zero mean. Then, using M as integrator, the construction of the integral T 0 f dM coincides with the one of T 0 f dZ, constructed above. Furthermore, according to Burkholder-Davis-Gundy inequalities, for any p ≥ 1 there exists a constant C = C p such that
E T 0 f dM p ≤ C p E T 0 f 2 d[M ] p 2 .
In the simplest case, when p = 2, we obtain that
E T 0 f dM 2 ≤ C p E T 0 f 2 d[M ] = C p E T 0 f 2 d M .
If M t = t 0 m s ds with Em s ≤ C, we obtain the same bound as (2.9), but for a wider class of processes. So, in this case we can use the martingale approach instead of the Lévy-processes approach.
Comparing our a priori estimates with estimates for other classes of integrators, we can consider the process having the form of the sum
Y t = g 1 dM + g 2 dµ = t 0 g 1 dM + (0,t]×R g 2 (s, z)µ(ds, dz),
where M is a square-integrable continuous martingale with quadratic characteristics M , µ = µ − ν, µ is a square-integrable random measure with dual predictable projection ν, integrands g i , i = i, 2 are predictable and such that all integrals are well-defined and square-integrable. Then, according to Burkholder-Davis-Gundy inequalities and the generalisation of Bichteler-Jacod inequalities from [17], the following estimate holds: for any T > 0, α ∈ [1, 2] and any p ≥ 1 there exists a constant C = C α,p,T such that
E( sup t∈[0,T ] |Y t | p ) ≤ C α,p,T E [0,T ] g 2 1 d M p 2 + E [0,T ]×R |g 2 | α dν p α ,
for p ∈ [1, α],and
E( sup t∈[0,T ] |Y t | p ) ≤ C α,p,T E [0,T ] g 2 1 d M p 2 + E [0,T ]×R |g 2 | α dν p α +E [0,T ]×R |g 2 | p dν
for p ∈ (α, ∞). However, we shall not consider such processes in the framework of the present paper.
2.2.
Integration of Volterra-type kernels with respect to a Lévy process. Now, let us have a two-parameter measurable non-random kernel of the form g = g(t, s) : R 2 + → R, and our goal is to construct the integral J(t, g) = t 0 g(t, s) dZ s , for any t ∈ [0, T ]. This construction is the same as for constructed in Subsection 2.1 integral of non-random functions f , therefore we can use Lemma 2.1 and Theorem 2.1 and immediately proceed with the following conclusion.
Theorem 2.2. Let one of the following conditions hold:
(A) for some p ∈ [1, 2), g = g(t, ·) ∈ L p ([0, t]) for any t ∈ [0, T ], a = b = 0, and the measure π is symmetric with R |x| p π(dx) < ∞;
(B) for some p ≥ 2, g = g(t, ·) ∈ L p ([0, t]) for any t ∈ [0, T ], b = 0, and the measure π is symmetric with R |x| p π(dx) < ∞. Then, for any t ∈ [0, T ], g(t, ·) is Z-integrable, in the case when condition (A) holds, we have the a priori estimate
E t 0 g(t, s) dZ s p ≤ C g(t, ·) p Lp([0,t]) R |x| p π(dx),(2.
10)
and in the case when condition (B) holds, we have the a priori estimate
E t 0 g(t, s) dZ s p ≤ C a p/2 g(t, ·) p L 2 ([0,t]) + g(t, ·) p Lp([0,t]) R |x| p π(dx) . (2.11)
Remark 2.5.
(i) It is sufficient for our purposes to consider the restriction of g to the set {0 ≤ s < t ≤ T }. We can simply assume that g : {0 ≤ s < t ≤ T } → R. (ii) The extension to square-integrable martingale considered in Remark 2.4 is valid in the case of the kernel g with evident corrections.
3. An example of Lévy process as integrator: the subordinated Wiener process
Time change and here, in particular, subordination is a feasible way to build Lévy processes from known ones. This constitute one of the simplest ways to simulation and thus it gains particular interest. In this section we shall concentrate on this case. Se e.g. [10].
3.1. Description of subordinate Wiener process. Let W = {W t , t ≥ 0} be a one-dimensional Wiener process. Subordination of the Wiener process consists in time-changing the paths of W by an independent subordinator L = {L t , t ≥ 0}, which is a non-negative, non-decreasing Lévy process starting from 0. The Laplace exponent Φ = Φ(λ) of L, defined by the relation
E exp {−λL t } = exp {−tΦ(λ)} , λ > 0, has the form Φ(λ) = aλ + ∞ 0 (1 − e −λx )ν(dx),
where a > 0 is the drift of the subordinator and ν is its Lévy measure with
∞ 0 (1 ∧ x)ν(dx) < ∞.
Consider the function (2πs) − 1 2 exp − x 2 2s which is bounded in s on R + = (0, +∞) for any fixed x ∈ R. Introduce the following density function
̺(x) = ∞ 0 (2πs) − 1 2 exp − x 2 2s ν(ds), x ∈ R,
and let π be a measure on B(R) with density ̺. For later use we introduce the following condition:
(C) ∞ 0 x 1 2 ν(dx) < ∞.
Lemma 3.1. The following statements are true.
(i) The subordinate Wiener process
W L := W (L) = W L t := W (L t ), t ≥ 0 characteristic function Υ(µ) := E exp iµW L t = E exp − µ 2 2 L t = exp −tΦ µ 2 2 = exp −t aµ 2 2 + ∞ 0 1 − e − xµ 2 2 ν(dx) . (3.1) (ii)
The subordinated Wiener process W L is a Lévy process with zero drift coefficient, its diffusion coefficient equals a, and Lévy measure equal π. Its characteristic function can be represented as
E exp iµW L t = exp t − aµ 2 2 + R e iµx − 1 − iµx1 |x|<1 π(dx) . (3.2)
(iii) Let condition (C) hold. Then R |x|π(dx) < ∞, and therefore,
E|W L t | = tE|W L 1 | < ∞ for any t ≥ 0.
Proof. Statements (i) and (ii) immediately follow from Theorem 30.1 [27]. To prove (iii), note that
R |x| π(dx) = R |x| ̺(x) dx = R ∞ 0 |x| (2πs) − 1 2 exp − x 2 2s ν(ds) dx = 2(2π) − 1 2 ∞ 0 s 1 2 ν(ds) ∞ 0 ze − z 2 2 dz = 2(2π) − 1 2 ∞ 0 s 1 2 ν(ds) < ∞.
(3.3)
Remark 3.1. Note that the density ̺ is a symmetric function therefore the following equality holds: R x1 |x|<1 π(dx) = 0, and we can rewrite Here below we can consider three particular cases.
Subordinate Wiener process as a square integrable martingale. Introduce the natural filtration
F W L = F W L s , s ≥ 0 , where F W L s = σ W L u , 0 ≤ u ≤ s . Intro- duce the condition (D) ∞ 0 xν(dx) < ∞.
Condition (D) is equivalent to the existence of the expectation of L t for any t ≥ 0, because
EL t = tEL 1 = tΦ ′ (λ) λ=0 = t(a + ∞ 0 xν(dx)).
Lemma 3.2. Under condition (D) we have that R x 2 π(dx) < ∞ and the process W L is a square-integrable martingale w.r.t. the natural filtration with the quadratic characteristic W L t = ct, where c = EL 1 .
Proof. Inequality R x 2 π(dx) < ∞ is established similarly to statement (iii) of Lemma 3.1. The second statement is also easy to prove. Let EL 1 < ∞. Then it follows immediately from (3.1) that
EW L 1 = 1 i Υ ′ (µ) µ=0 = − 1 i µEL 1 µ=0 = 0,
and E(W L 1 ) 2 = EL 1 Namely, W L is a square integrable Lévy process W L with zero-mean, hence it is a martingale (see Proposition 3.17 in [10]). By this we complete the proof.
As an illustration, consider the Gamma subordinator L, which is a Lévy process L with zero drift and Lévy measure of the form
ν(dx) = cx −1 e −λx ½ x∈R + dx.
The corresponding subordinate Wiener process W L has no diffusion part, conditions (C) and (D) hold, and it has Lévy measure π with the density of the form
̺(x) = c(2π) − 1 2 ∞ 0 s − 3 2 e −λs− x 2 2s ds.
Evidently, in this case W L is a square-integrable martingale.
Subordinate Wiener process with compound Poisson subordinator.
Let the Lévy process L be a compound Poisson process, that is equivalent to the fulfilment of the conditions a = 0 and ν(R + ) < ∞. In this case the subordinate Wiener process W L is a Lévy process without diffusion component, with characteristic function
E exp iµW L t = exp t R e iµx − 1 π(dx) , and π(R) = R ̺(x) dx = R R + (2πs) − 1 2 e − x 2 2s ν(ds) dx = R + ν(ds) = ν R + < ∞.
Therefore, W L is a compound Poisson process.
3.1.3.
Subordinate Wiener process with a stable subordinator. Consider a measure ν α (dx) on R + of the form
ν α (dx) = c x 1+α ½ {x∈R + } dx.
Then ν α is the Lévy measure of some Lévy process if and only if α ∈ (0, 2), yet ν α is the Lévy measure of some subordinator if and only if α ∈ (0, 1). So, a stable subordinator L with index α ∈ (0, 1) is a subordinator with zero drift and Lévy measure ν α . Its moment generating function is given by
Ee −λLt = e −c 1 tλ α , λ, t > 0.
In this case the subordinate Wiener process W L is 2α-stable and has a characteristic function of the form
E exp iµW L t = E exp − µ 2 2 L t = exp − c 1 2 α µ 2α t .
The moments EL β t , β > 0, for a stable subordinator with index α exist only for β < α and
EL β t = (c 1 t) β α Γ 1 − β α Γ(1 − β) .
Therefore, for α ∈ ( 1 2 , 1), EL t 1/2 < ∞ and consequently E W L t < ∞. For any α ∈ (0, 1) E(W L ) 2 t = ∞.
3.2.
Integration of a non-random kernel with respect to a subordinate Wiener process. Now we apply to a subordinate Wiener process W L the construction of integral w. r. t. a Lévy process Z from Section 2. By Lemma 3.1, W L is a Lévy process with zero drift coefficient, the diffusion coefficient a, and the symmetric Lévy measure π. Similarly to (3.3), one can show that R |x| p π(dx) ≤ C ∞ 0 s p/2 ν(ds).
Then from Theorem 2.1 we immediately get the following result. Remark 3.2. Let L t ∈ L 1 (P) for any t ∈ [0, T ] so that W L t ∈ L 2 (P) for any t ∈ [0, T ], and W L is a square-integrable martingale. We can create the sequence of partitions π n = 0 = t n 0 < t n 1 < . . . < t n kn = T with diam π n → 0, n → 0 and choose in Definition 2.1 the sets A n j = [t n j , t n j+1 ). Then for any f ∈ L 2 ([0, T ]) we see that For any t ∈ [0, T ], the integral t 0 g(t, s) dW L s coincides with the integral of g(t, ·) w. r. t. a square-integrable martingale W L . Moreover, as it will be clarified by the calculations in the sequel, when W L is a square-integrable martingale, it is not really important that W L is a Lévy process and W L t = ct with c = EL 1 . It is actually important that W L is a square integrable martingale. Indeed, wWe can consider any square
-integrable martingale M = {M t , F t , t ∈ [0, T ]} with M t = t 0 σ 2 (s) ds,
where σ is a random measurable adapted function with bounded expectation, Eσ 2 (s) ≤ C, and all the results will be preserved.
Elements of fractional calculus and existence of the generalized
Lebesgue-Stieltjes integrals 4.1. Elements of fractional calculus. In this subsection we describe a construction of the path-wise integral following the approach developed by Zähle [32,33,34]. We start by introducing the notions of fractional integrals and derivatives. See [26] for the details on the concept of fractional calculus. 1 (a, b). The Riemann-Liouville left-and right-sided fractional integrals of order α > 0 are defined for almost all x ∈ (a, b) by
I α a+ f (x) := 1 Γ(α) x a (x − y) α−1 f (y) dy, I α b− f (x) := (−1) −α Γ(α) b x (y − x) α−1 f (y) dy,
respectively, where (−1) −α = e −iπα , Γ denotes the Gamma function.
Denote by I α a+ (L p ) (resp. I α b− (L p )) the class of functions f that can be presented
as f = I α a+ ϕ (resp. f = I α b− ϕ) for ϕ ∈ L p (a, b). Definition 4.2.
For a function f : [a, b] → R the Riemann-Liouville left-and rightsided fractional derivatives of order α (0 < α < 1) are defined by
D α a+ f (x) := ½ (a,b) (x) 1 Γ(1 − α) d dx x a f (y) (x − y) α dy, D α b− f (x) := ½ (a,b) (x) (−1) 1+α Γ(1 − α) d dx b x f (y) (y − x) α dy.
The Riemann-Liouville fractional derivatives admit the following Weyl representation
D α a+ f (x) = 1 Γ(1 − α) f (x) (x − a) α + α x a f (x) − f (y) (x − y) α+1 dy ½ (a,b) (x), D α b− f (x) = (−1) α Γ(1 − α) f (x) (b − x) α + α b x f (x) − f (y) (y − x) α+1 dy ½ (a,b) (x),
where the convergence of the integrals holds pointwise for a. a. x ∈ (a, b) for p = 1 and in L p (a, b) for p > 1.
f (u − δ) exist for a ≤ u ≤ b. Denote f a+ (x) = (f (x) − f (a+))½ (a,b) (x), g b− (x) = (g(b−) − g(x))½ (a,b) (x).
Definition 4.3 ([32]
). Assume that f a+ ∈ I α a+ (L p ),
g b− ∈ I 1−α b− (L q ) for some 1/p + 1/q ≤ 1, 0 < α < 1. The generalized (fractional) Lebesgue-Stieltjes integral of f with respect to g is defined by b a f (x) dg(x) :=(−1) α b a D α a+ f a+ (x) D 1−α b− g b− (x) dx+ + f (a+) g(b−) − g(a+)
.
D α a+ f (x) D 1−α b− g b− (x) dx.
In particular, Definition 4.3 allows to integrate Hölder continuous functions.
Definition 4.4. Let 0 < λ ≤ 1. A function f : R → R belongs to C λ [a, b] if there exists a constant C > 0 such that, for all s, t ∈ [a, b], |f (s) − f (t)| ≤ C |s − t| λ , s, t ∈ [a, b].{R − S} b a f (x) dg(x) := lim |π|→0 i f (x * i )(g(x i+1 ) − g(x i )), where π = {a = x 0 ≤ x * 0 ≤ x 1 ≤ . . . ≤ x n−1 ≤ x * n−1 ≤ x n = b}, and |π| = max i |x i+1 − x i |.
4.2.
Generalised Lebesgue-Stieltjes integral for stochastic processes. Consider two real-valued stochastic processes X = {X t , t ∈ [0, T ]} and Y = {Y t , t ∈ [0, T ]}. We say that X and Y are fractionally α-connected for some t ∈ [0, T ], and for some 0 < α < 1 if the generalised Lebesgue-Stieltjes integral
t 0 X s dY s := t 0 D α 0+ X (s) D 1−α t− Y t− (s) ds
exists with probability 1. Since the above integral is defined ω by ω, it is called a pathwise integral. The next simple result allows us to "separate" X and Y in the pathwise integral.
Lemma 4.1. Assume that for some t ∈ [0, T ] and for some 0 < α < 1 one of the following conditions hold:
(i) t 0 | D α 0+ X (s)|ds < ∞ a.s. and sup 0≤s≤t | D 1−α t− Y t− (s)| < ∞ a.s. (4.2) (ii) sup 0≤s≤t | D α 0+ X (s)| < ∞ a.s. and t 0 | D 1−α t− Y t− (s)|ds < ∞ a.s. (4.3) (iii) for some p > 1, q > 1 such that p −1 + q −1 = 1 t 0 D α 0+ X (s) q ds < ∞ and t 0 D 1−α t− Y t− (s) p ds < ∞.
Then X and Y are fractionally α-connected for this value of t ∈ [0, T ].
Taking the above lemma into account, we introduce the classes of stochastic processes
D + q (α, T ) := X = {X t , t ∈ [0, T ]} : T 0 | D α 0+ X (s)| q ds < ∞ a.s. , 1 ≤ q < ∞, and D + ∞ (α, T ) := X = {X t , t ∈ [0, T ]} : sup 0≤s≤T | D α 0+ X (s)| < ∞ ,
and, correspondingly,
D − p (α, T ) := {Y = {Y t , t ∈ [0, T ]} : t 0 | D 1−α t− Y t− (s)| p ds < ∞ a. s., t ∈ [0, T ] , 1 ≤ p < ∞, and D − ∞ (α, T ) := Y = {Y t , t ∈ [0, T ]} : sup 0≤s≤t | D 1−α t− Y t− (s)| < ∞, t ∈ [0, T ] .
Then it follows that, for the couples (
X ∈ D + 1 (α), Y ∈ D − ∞ (α)), (X ∈ D + ∞ (α), Y ∈ D − 1 (α)), and (X ∈ D + q (α), Y ∈ D − p (α)), p > 1, q > 1, p −1 + q −1 = 1, we have that X and Y are fractionally α-connected for any t ∈ [0, T ].
Let the processes Y ∈ D − p (α) be called appropriate (p, α)-integrators, p ∈ [1, +∞] for X ∈ D + q (α), q = p p−1 (with 1 0 = ∞). It follows from the a priori estimates of Section 2 that it is natural to formulate conditions on the process Y t = t 0 g(t, s) dZ s to be appropriate (p, α)-integrator in terms of expectations. In this connection, we introduce the following classes of processes:
ED − p (α, T ) := {Y = {Y t , t ∈ [0, T ]} : t 0 E| D 1−α t− Y t− (s)| p ds < ∞, t ∈ [0, T ]} ⊂ D − p (α, T ), p ≥ 1, and ED − ∞ (α, T ) := {Y = {Y t , t ∈ [0, T ]} : E sup 0≤s≤t | D 1−α t− Y t− (s)| < ∞, t ∈ [0, T ]} ⊂ D − ∞ (α, T ).
5.
General conditions for Y · = · 0 g(·, s)dZ s to be an appropriate (p, α)-integrator Now we formulate three results supplying the appropriate integrator properties of Y t = t 0 g(t, s) dZ s , t ∈ [0, T ]. Consider the fixed interval [0, T ], and let g = g(t, s) : {0 ≤ s ≤ t ≤ T } → R be a non-random measurable kernel. 5.1. The case p ∈ [1, 2). We immediately formulate the following result.
Theorem 5.1. Let p ∈ [1, 2), α ∈ (0, 1), g = g(t, ·) ∈ L p ([0, t]) for any t ∈ [0, T ], a = b = 0, the measure π is symmetric with R |x| p π(dx) < ∞, and let the following set of conditions hold:
Assumptions (D p ) (1) t 0 (t − s) pα−p t s |g(t, v)| p dv ds < ∞, (2) t 0 (t − s) pα−p s 0 |g(t, v) − g(s, v)| p dv ds < ∞,(3)t 0 t s (u − s) pα−2p u s |g(u, v)| p dv du ds < ∞,(4)t 0 t s (u − s) pα−2p s 0 |g(u, v) − g(s, v)| p dv du ds < ∞. Then Y = Y t = t 0 g(t, s) dZ s , t ∈ [0, T ] ∈ ED − p (α, T ), so, Y is an appropriate (p, α)-integrator for any f ∈ D + q (α, T ).
Proof. Note that the increment of Y are given by
Y t − Y s = t 0 g(t, u)dZ u − s 0 g(s, u)dZ u = t s g(t, u)dZ u + s 0 (g(t, u) − g(s, u))dZ u (s ≤ t). (5.1)
Taking the definitions of fractional derivative and of the class ED − p (α, T ) into account, it is sufficient to prove that
t 0 E |Y t − Y s | p (t − s) p−αp ds < ∞ and t 0 t s E |Y u − Y s | p (u − s) 2p−αp du ds < ∞, t ∈ [0, T ].
According to (5.1) and (2.10),
E |Y t − Y s | p ≤ E t s g(t, u) dZ u p + E s 0 (g(t, u) − g(s, u)) dZ u p ≤ C R |x| p π(dx) t s |g(t, u)| p du + s 0 |g(t, u) − g(s, u)| p du ≤ C t s |g(t, u)| p du + s 0 |g(t, u) − g(s, u)| p du .
The proof immediately follows. Now consider separately the case p = 2 because in this case the martingale structure of the process Y plays a crucial role and it is the most simple case for calculations and estimations. 5.2. The case p = 2. Taking Remark 2.4 and Remark 2.5 into account, we consider a square-integrable càdlàg martingale M = {M t , F t , t ≥ 0} with a quadratic characteristics M that is a càdlàg non-decreasing process. Define also a càdlàg non-decreasing measurable function E t = E M t . The integral t 0 g(t, s) dM s for any t > 0 is defined as a stochastic integral with respect to a square-integrable martingale, or, more exactly, since the kernel g is non-random, as a Wiener integral with non-random integrand and a square-integrable martingale as an integrator. A sufficient condition for its existence is
E t 0 g 2 (t, s) d M s < ∞, t ≥ 0, or, equivalently, t 0 g 2 (t, s) dE s < ∞, t ≥ 0. (5.2)
Under assumption (5.2), define the random process
Y t := t 0 g(t, s) dM s , t ≥ 0. (5.3)
Theorem 5.2. Let p = 2, α ∈ (0, 1). Assume that for any t ∈ [0, T ], the following conditions hold.
Assumptions (D 2 ) (1) t 0 (t − s) 2α−2 t s g 2 (t, u) dE u ds < ∞,(2)t 0 (t − s) 2α−2 s 0 (g(t, u) − g(s, u)) 2 dE u ds < ∞,(3)t 0 t s t v g(u, v) (u − s) 2−α du 2 dE v ds < ∞,(4)t 0 s 0 t s g(u, v) − g(s, v) (u − s) 2−α du 2 dE v ds < ∞.
Then Y ∈ ED − 2 (α, T ), so it is an appropriate (2, α)-integrator for any f ∈ D + 2 (α, T ).
Proof. By the definition of the fractional derivative,
E t 0 D 1−α t− Y t− (s) 2 ds ≤ 2 Γ 2 (α) E t 0 (Y t − Y s ) 2 (t − s) 2−2α ds + (1 − α) 2 E t 0 t s Y u − Y s (u − s) 2−α du 2 ds . (5.4)
Now our goal is to bound from above each of the two terms in the right-hand side of (5.4). Similarly to the proof of the previous theorem, the increments of Y are Therefore, for the first term we have the following upper bound
Y t − Y s =E t 0 (Y t − Y s ) 2 (t − s) 2−2α ds ≤ 2E t 0 (t − s) 2α−2 t s g(t, u) dM u 2 + s 0 (g(t, u) − g(s, u)) dM u 2 ds = 2 t 0 (t − s) 2α−2 E t s g 2 (t, u) d M u + E s 0 (g(t, u) − g(s, u)) 2 d M u ds = 2 t 0 (t − s) 2α−2 t s g 2 (t, u) dE u + s 0 (g(t, u) − g(s, u)) 2 dE u ds. (5.5)
Hence, the first expectation in (5.4) is finite. The second summand in the right-hand side of (5.4) can be bounded as follows:
E t 0 t s Y u − Y s (u − s) 2−α du 2 ds = E t 0 t s (u − s) α−2 u s g(u, v) dM v + s 0 (g(u, v) − g(s, v)) dM v du 2 ds ≤ 2E t 0 t s u s g(u, v) (u − s) 2−α dM v du 2 + t s s 0 g(u, v) − g(s, v) (u − s) 2−α dM v du 2 ds = 2 t 0 E t s t v g(u, v) (u − s) 2−α du dM v 2 + E s 0 t s g(u, v) − g(s, v) (u − s) 2−α du dM v 2 ds = 2 t 0 E t s t v g(u, v) (u − s) 2−α du 2 d M v + E s 0 t s g(u, v) − g(s, v) (u − s) 2−α du 2 d M v ds = 2 t 0 t s t v g(u, v) (u − s) 2−α du 2 dE v + s 0 t s g(u, v) − g(s, v) (u − s) 2−α du 2 dE v ds < ∞.
5.3. The case 2 < p < ∞. Taking inequality (2.9) into account, we can formulate corresponding result similarly to Theorem 5.1. Since the proof follows the same steps, it is omitted.
Theorem 5.3. Let p ∈ (2, ∞), α ∈ (0, 1), g = g(t, ·) ∈ L p ([0, t])
for any t ∈ [0, T ], a = 0, the measure π is symmetric with R |x| p π(dx) < ∞, and let additionally condition (D p ) hold. Then
Y = Y t = t 0 g(t, s) dZ s , t ∈ [0, T ] ∈ ED − p (α, T ),
so, it is an appropriate (p, α)-integrator for any f ∈ D + q (α, T ).
5.4.
The case p = ∞. As in the case p = 2, consider a square-integrable martingale M = {M t , F t , t ≥ 0} with a quadratic characteristics M . In order to give the conditions for Y ∈ ED − ∞ (α, T ), a Hölder continuity of rather high order is required. Therefore, we assume that M and consequently M are continuous processes, and that E t = E M t is a continuous function. Remark immediately that this is not the case for subordinate Wiener process.
Theorem 5.4. Let α ∈ (0, 1), M is a square-integrable continuous martingale with quadratic characteristics M t = t 0 m s ds, where Em s ≤ C. Assume that, for some ̺ ≥ 1, β > 1 ̺ + 1 − α, the following condition holds:
Assumptions (D ∞ ) (1) T 0 T 0 |y − x| −β̺−1 y x g 2 (y, u) du ̺/2 dx dy < ∞ (2) T 0 T 0 |y − x| −β̺−1 x 0 (g(y, u) − g(x, u)) 2 du ̺/2 dx dy < ∞.
Then Y ∈ ED − ∞ (α, T ), so it is an appropriate (∞, α)-integrator for any f ∈ D + 0 (α, T ).
Proof. The proof follows the scheme from [21,Lemma 7.5]. According to the Garsia-Rodemich-Rumsey inequality from [13], for any continuous function f : [0, T ] → R and any ̺ ≥ 1, β > 1 ̺ and 0 ≤ s ≤ t ≤ T , we have
|f (t) − f (s)| ̺ ≤ C β,̺ |t − s| β̺−1 T 0 T 0 |f (x) − f (y)| ̺ |x − y| β̺+1 dx dy.
Consider the increment of Y :
|Y t − Y s | ̺ ≤ C |t − s| β̺−1 T 0 T 0 |Y x − Y y | ̺ |x − y| β̺+1 dx dy (s ≤ t).
Taking the continuity of Y into account, we can apply Burkholder's inequality, and get
E |Y x − Y y | ̺ ≤ CED 1−α t− Y t− (s) ≤ 1 Γ(α) |Y t − Y s | (t − s) 1−α + (1 − α) t s |Y s − Y v | (v − s) 2−α dv .
Combining this with (5.7), and using the inequality β > 1 ̺ + 1 − α, we complete the proof.
Corollary 5.1. Let the function g satisfy the following conditions:
(1) |g(t, s)| ≤ C, t, s ∈ [0, T ];
(2) |g(t, s) − g(v, s)| ≤ C |t − v| 1/2 , t, s, v ∈ [0, T ]. Then the assumptions of Theorem 5.4 are satisfied for any α > 1/2, if we choose ̺ ≥ 2 2α−1 and 1 ̺ + 1 − α < β < 1 2 .
5.5.
Conditions for · = · 0 g(·, s)dW L s to be an appropriate (p, α)-integrator. As an example in line with Section 3, we formulate the results supplying the appropriate integrator properties of Y t = t 0 g(t, s) dW L s , t ∈ [0, T ] for Z = W L a subordinated Wiener process. Consider the fixed interval [0, T ], and let g = g(t, s) : {0 ≤ s ≤ t ≤ T } → R be a non-random measurable kernel.
Theorem 5.5. Let p ∈ [1, ∞), α ∈ (0, 1), g = g(t, ·) ∈ L p ([0, t]) for any t ∈ [0, T ], a = 0 in the case when 1 ≤ p ≤ 2, and ∞ 0 s p 2 dν s < ∞. Also, let conditions (D p ) hold. Then
Y = Y t = t 0 g(t, s) dW L s , t ∈ [0, T ] ∈ ED − p (α, T ),
so, Y is an appropriate (p, α)-integrator for any f ∈ D + q (α, T ). 5.6. Examples of appropriate (p, α)-integrators. Then the process Y from (5.3) is an appropriate (2, α)-integrator for any 1 − H < α < 1, and in addition, has a. s. γ-Hölder trajectories for any 0 < γ < H − 1/2.
Proof. From now on, we denote C different constants whose value is not here important. Start with Hölder continuity. According to Kolmogorov theorem, it is sufficient to prove that This concludes the proof. If M is a Wiener process then the process Y t = t 0 g(1, t, s) ds is the fractional Brownian motion, see [20]. Note, that in this case trajectories of Y are a. s. γ-Hölder for any 0 < γ < H. The pathwise generalized Lebesgue-Stiltjes integrals with respect to fractional Brownian motion was studied in [21]. If M is a Lévy process without Gaussian component, then Y is fLpMG, introduced in [28]. Example 5.3. It is very easy to create examples of processes from ED − 1 (α, T ) and D − ∞ (α, T ). Indeed we can take the same kernel g(1, t, s) and consider any W L satisfying condition (A) with a = 0 to get that Y ∈ ED − 0 (α, T ). Moreover, with the same kernel and M = W we get Y ∈ D − ∞ (α, T ), as it immediately follows from [21].
E(Y t − Y s ) 2 ≤ C(t − s) 2H = C(t −
j ½ A j be a real-valued simple function on [0, T ], where A j ∈ B([0, T ]) are pair-wise disjoint and n j=1 A j = [0, T ]. Then, for any A ∈ B([0, T ]), we set
( i )
iThe integral A f dZ is well defined, i.e., for any Z-integrable function f : ([0, T ], B([0, T ])) → (R, B(R)), the integral does not depend on the choice of approximating sequence {f n , n ≥ 1}. (ii) Define
then the characteristic function of the integral can be rewritten as the characteristic function of a Lévy process: E exp iλ [0,T ] f dZ = exp [0,T ]
E
exp iµW L t = exp {tΨ(µ)} := exp t − aµ 2 2 + R e iµx − 1 π(dx).(3.4)
)
Let p ∈ [1, 2), f ∈ L p ([0, T ]), a = 0, and ∞ 0 s p/2 ν(ds) < ∞. Then f is W L
)
Let p ≥ 2, f ∈ L p ([0, T ]), and ∞ 0 s p/2 ν(ds) < ∞. Then f is W L -integrable and E [0,T ] f dW L p ≤ C a p/2 f p L 2 ([0,T ]) + f p Lp([0,T ]) ∞ 0 s p/2 ν(ds) . (3.6)
[ 0 ,
0T ] f dW L coincides with the Wiener integral T 0 f (s) dW L s of the non-random function w. r. t. a square-integrable martingale. Now, let us have a two-parameter measurable non-random kernel of the form g = g(t, s) : R 2 + → R. Using Theorem 2.2, we obtain the following statement.
Let for some p ∈ [1, 2) g = g(t, ·) ∈ L p ([0, t]) for any t ∈ [0, T ], a = 0, and∞ 0 s p/2 ν(ds) < ∞. Then for any t ∈ [0, T ] g(t, ·) is W L -Let for some p ≥ 2 g = g(t, ·) ∈ L p ([0, t]) for any t ∈ [0, T ], and ∞ 0 s p/2 ν(ds) < ∞. Then for any t ∈ [0, T ] g(t, ·) is W L -Remark 3.3. Let g = g(t, ·) ∈ L 2 ([0, t])for any t ∈ [0, T ], and let the subordinate Wiener process satisfy condition (D). Then for any t ∈ [0, T ] g(t, ·) is W L -integrable, and
Definition 4. 1 .
1Let f ∈ L
Let f, g : [a, b] → R. Assume that the limits f (u+) := lim δ↓0 f (u + δ) and g(u−) := lim
this definition is independent of the choice of α ([32, Prop. 2.1]). If αp < 1, then (4.1) can be simplified to b a f (x) dg(x) := (−1) α b a
Proposition 4. 1 ([ 32 ,
132Th 4.2.1]). Let f ∈ C λ [a, b] and g ∈ C µ [a, b] with λ + µ > 1.Then the assumptions of Definition 4.3 are satisfied with any α ∈ (1 − µ, λ) and p = q = ∞. Moreover, the generalised Lebesgue-Stieltjes integral b a f (x) dg(x) defined by (4.1) coincides with the Riemann-Stieltjes integral
t, u) dM u − s 0 g(s, u) dM u
t, u) dM u +
(t, u) − g(s, u)) dM u (s ≤ t).
.
t − Y s | ≤ C |t − s| β−1/̺ ξ,By the upper bound (5.6) and condition (D ∞ ), E |ξ| ̺ < ∞. By the definition of the fractional derivative,
Example 5 . 1 .
51Assume that on interval [0, T ] two following properties hold:(i) E M t = t 0 σ 2 (s) ds, where |σ(s)| ≤ σ, where σ > 0 is some constant; (ii) g(t, s) = g(j(·), t, s) = c H s 1 2 −H t s u H− 1 2 (u − s) H−3/2 j(u) du,where H ∈ 1 2 , 1 , j is a measurable bounded function, |j(u)| ≤ G, where G > 0 is some constant.
2H (t − s) 2H+2α−2 ds < ∞.
and j(u) ≡ 1 n Example 5.1. Then g(1, t, s) is the Molchan-Golosov kernel.
as n → ∞, follows from [24, Lemma 3.2]. Thus, the continuity is proved. Since any continuous linear operator is bounded [16, § 29, Theorem 1], we have
AcknowledgementsThe authors thank the EU project Ukrainian Mathematicians for Life Sciences for providing the framework for this research. Giulia Di Nunno acknowledges financial support from the Norwegian Research Council Project 239019 FINEWSTOCH.To establish (5.8), we note that, similarly to (5.5), for s ≤ t, (g(j(·), t, u) − g(j(·), s, u)) 2 d M u = 2 E t s g 2 (j(·), t, u)σ 2 (u) du +E s 0 (g(j(·), t, u) − g(j(·), s, u)) 2 σ 2 (u) du ≤ 2σ 2 (I 1 + I 2 ), where I 1 = t s g 2 (j(·), t, u) du, I 2 = s 0 (g(j(·), t, u) − g(j(·), s, u)) 2 du.Consider I 1 .Changing the order of integration and using the equality f (u, v, z) = f (u, z, v), we get Similarly I 2 can be rewritten as followsSummarizing, we get thatUsing [20, Lemma 2.2(i)] we can calculate the inner integral:ThenThus, (5.8) is proved. Now it remains to check the conditions of Theorem 5.2. Since dE M s = σ 2 (s) ds and |σ(s)| ≤ C, it is very easy to understand that we need to show the existence of the following four integrals:where we replaced j(·) with the constant function identically equal to 1. Using (5.10), we can bound the first two integrals byConsider J 3 .The convergence of the integral for some α = α 0 ∈ (0, 1) implies its convergence for all α ∈ [α 0 , 1). Therefore, we can assume without loss of generality that α < 3 2 − H. ThenBy changing the order of integration, we getConsider J 4 .
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Box 1053 Blindern, N-0316 Oslo Norway. Email: [email protected] G. Di Nunno, Norwegian School of Economics and Business Administration (NHH), Helleveien. G , Di Nunno, 30Bergen, NorwayDepartment of Mathematics, University of Oslo, P.OG. Di Nunno, Department of Mathematics, University of Oslo, P.O. Box 1053 Blin- dern, N-0316 Oslo Norway. Email: [email protected] G. Di Nunno, Norwegian School of Economics and Business Administration (NHH), Helleveien 30, N-5045 Bergen, Norway.
Y Mishura, Department of Probability Theory, Statistics and Actuarial Mathematics. Kyiv, Ukraine. E-mail641601Taras Shevchenko National University of KyivY.Mishura, Department of Probability Theory, Statistics and Actuarial Mathe- matics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine. E-mail: [email protected]
K Ralchenko, Department of Probability Theory, Statistics and Actuarial Mathematics. Kyiv; Ukraine641601Taras Shevchenko National University of KyivK. Ralchenko, Department of Probability Theory, Statistics and Actuarial Mathe- matics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine. Email: [email protected]
| {'fraction_non_alphanumeric': 0.1145899331221401, 'fraction_numerical': 0.03426610348468849, 'mean_word_length': 3.233738171522241, 'pattern_counts': {'":': 0, '<': 109, '<?xml version=': 0, '>': 38, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 54, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in this there is a bridging of the stochastic and deterministic techniques. The present paper aims at setting the basis for a framework in which further computational rules can be devised. Our results are general in the choice of driving noise. Additionally we propose some further details in the relevant context subordinated Wiener processes.', 'arxivid': '1608.08466', 'author': ['G Di Nunno ', 'Y Mishura ', 'K Ralchenko '], 'authoraffiliation': [], 'corpusid': 119335108, 'doi': '10.1515/fca-2016-0071', 'github_urls': [], 'n_tokens_mistral': 22423, 'n_tokens_neox': 19839, 'n_words': 11182, 'pdfsha': '1cb3d86c44e6baa00acad0059530be1d6dc23ffb', 'pdfurls': ['https://arxiv.org/pdf/1608.08466v1.pdf'], 'title': ['FRACTIONAL CALCULUS AND PATH-WISE INTEGRATION FOR VOLTERRA PROCESSES DRIVEN BY LÉVY AND MARTINGALE NOISE', 'FRACTIONAL CALCULUS AND PATH-WISE INTEGRATION FOR VOLTERRA PROCESSES DRIVEN BY LÉVY AND MARTINGALE NOISE'], 'venue': []} |
arxiv |
Critical network effect induces business oscillations in multi-level marketing systems
25 Sep 2012 (Dated: May 2, 2014)
Dranreb Earl
Department of Mathematics
School of Science and Engineering
Ateneo de Manila University
Loyola Heights1108Quezon CityPhilippines
Juanico
Department of Mathematics
School of Science and Engineering
Ateneo de Manila University
Loyola Heights1108Quezon CityPhilippines
Critical network effect induces business oscillations in multi-level marketing systems
25 Sep 2012 (Dated: May 2, 2014)
Business-cycle phenomenon has long been regarded as an empirical curiosity in macroeconomics. Regarding its causes, recent evidence suggests that economic oscillations are engendered by fluctuations in the level of entrepreneurial activity[1,2]. Opportunities promoting such activity are known to be embedded in social network structures[3,4]. However, predominant understanding of the dynamics of economic oscillations originates from stylised pendulum models on aggregate macroeconomic variables[5,6], which overlook the role of social networks to economic activity-echoing the so-called aggregation problem of reconciling macroeconomics with microeconomics[7,8]. Here I demonstrate how oscillations can arise in a networked economy epitomised by an industry known as multi-level marketing or MLM[9], the lifeblood of which is the profit-driven interactions among entrepreneurs. Quarterly data (over a decade) which I gathered from public MLMs reveal oscillatory time-series of entrepreneurial activity that display nontrivial scaling and persistence[10,11]. I found through a stochastic population-dynamic model, which agrees with the notion of profit maximisation as the organising principle of capitalist enterprise, that oscillations exhibiting those characteristics arise at the brink of a critical balance between entrepreneurial activation and inactivation brought about by a homophily-driven network effect. Oscillations develop because of stochastic tunnelling permitted through the destabilisation by noise of an evolutionarily stable state. The results fit altogether as evidence to the Burns-Mitchell conjecture that economic oscillations must be induced by the workings of an underlying "network of free enterprises searching for profit"[12]. I anticipate that the findings, interpreted under a mesoeconomic framework[13], could open a viable window for scrutinising the nature of business oscillations through the lens of the emerging field of network science. Enquiry along these lines could shed further light into the network origins of the business-cycle phenomenon.
Business-cycle phenomenon has long been regarded as an empirical curiosity in macroeconomics. Regarding its causes, recent evidence suggests that economic oscillations are engendered by fluctuations in the level of entrepreneurial activity [1,2]. Opportunities promoting such activity are known to be embedded in social network structures [3,4]. However, predominant understanding of the dynamics of economic oscillations originates from stylised pendulum models on aggregate macroeconomic variables [5,6], which overlook the role of social networks to economic activity-echoing the so-called aggregation problem of reconciling macroeconomics with microeconomics [7,8]. Here I demonstrate how oscillations can arise in a networked economy epitomised by an industry known as multi-level marketing or MLM [9], the lifeblood of which is the profit-driven interactions among entrepreneurs. Quarterly data (over a decade) which I gathered from public MLMs reveal oscillatory time-series of entrepreneurial activity that display nontrivial scaling and persistence [10,11]. I found through a stochastic population-dynamic model, which agrees with the notion of profit maximisation as the organising principle of capitalist enterprise, that oscillations exhibiting those characteristics arise at the brink of a critical balance between entrepreneurial activation and inactivation brought about by a homophily-driven network effect. Oscillations develop because of stochastic tunnelling permitted through the destabilisation by noise of an evolutionarily stable state. The results fit altogether as evidence to the Burns-Mitchell conjecture that economic oscillations must be induced by the workings of an underlying "network of free enterprises searching for profit" [12]. I anticipate that the findings, interpreted under a mesoeconomic framework [13], could open a viable window for scrutinising the nature of business oscillations through the lens of the emerging field of network science. Enquiry along these lines could shed further light into the network origins of the business-cycle phenomenon.
Known widely in literature as network marketing, MLM executes through embedded social networks its essential business functions such as goods distribution, consumption, marketing, and direct selling [9,14]. That makes MLM a stark microcosm of a networked economy. One salient yet so far overlooked feature of MLM dynamics is the aperiodic oscillations in firm size N (t), quantified by the number of participating entrepreneurs ( Fig. 1). Empirical quarterly firm-size data have been collected from four public MLMs (Supplementary Data; Supplementary Methods, S1): NuSkin Enterprises (NUS), Nature Sunshine (NATR), USANA Health Sciences (USNA), and Mannatech Inc. (MTEX). Publicly-listed firms are chosen because they are required to disclose accurate business data on a regular basis. The average revenue (i.e., total revenue divided by N for any given quarter) does not proportionately rise with N ( Fig. 1c,d), implying that firm-size expansion does not inevitably translate into revenue growth. Firm size is thus a more reliable quantifier for entrepreneurial activity than is total revenue.
The scaling property of the N (t) time-series is examined via Hurst analysis [10,11] (Methods; Supplementary Methods, S1). The Hurst exponents, H(1) and H(2), quantify the scaling of the absolute increments and of the power spectrum, respectively. If time series were generated by a Wiener process, such as in the Black-Scholes model [15], then H(1) = 0.5. But H(1) > 0.5 indicates persistence, i.e., changes in one direction usu-ally occur in consecutive periods; whereas H(1) < 0.5 suggests anti-persistence, i.e., changes in opposite directions usually appear in sequence [11]. Ideally, a single scaling regime means H(1) = H(2), which applies to time-series generated by unifractal processes such as the Wiener process and the fractal Brownian motion. Table 1 presents Hurst exponents for different MLMs. Generally, H(1) > 0.5 except for NUS North Asia and NATR with H(1) ∼ 0.5 (within standard deviation); and H(1) = H(2) within standard deviation. Overall, these features of the time-series suggest that MLM firm dynamics is a non-Wiener unifractal process [11]. Unifractality implies self-similarity such that conclusions drawn at one timescale remains statistically valid at another timescale.
An MLM firm is considered as a population of profitseeking entrepreneurs. This population exhibits disorder through the presence of two entrepreneur types distinguished by socio-economic status (SES). Let A and B denote these types, wherein A has higher SES than B, and N A and N B denote their subpopulation size, respectively. The total population at any given time t is thus N (t) = N = N A + N B . Three major processes run the population dynamics: entrepreneurial activation by recruitment; competitive inactivation; and catalytic inactivation due to a network effect. Recruitment is expressed in the following reaction equations: wherein µ and λ are per-capita rates of recruitment of types A and B, respectively, and µ > λ because A's higher SES implies a faster rate of entrepreneurial activation through the support of bigger capital and vaster social resources. Competitive inactivation occurs due to market overlap, or niche overlap [16], as participants can go head-to-head over the same clientele or market. An encounter rate δ, which can be related to the density of the embedding social network, quantifies the probability of market overlaps. Thus, competitive inactivation is:
A µ −→ 2A, B µ −→ B + A, B λ −→ 2B, A λ −→ A + B,(1)¡ ¢ £ ¤ ¥ ¦ § ¨ § © § § ¨ © ¢ ¤ § ! " # $ ! " " $ " " $ $ " " " " % " $ " " " # # $ " " $ $ $ " " $ " $ ! $ " " # " & $ ! $ " " ' " ' " # $ " " ( " " $ " " ( " " $ " " & " % " $ " " % " # # $ " " % $ $ $ " " ! " $ % $ " " " % " ' $ " " " ' " # $ " " " $ " " " ) 0 1 2 1 3 1 4 5 6 7 8 9 1 2 3 9 @ A B C @ D E F G B A @ E C H I D P Q R S @ D A @ T U V W X Y ` a b c d e f g h i p b f q q d g i f h b q r ) s t u v 1 w 4 5 6 7 8 9 1 2 3 9 x y y y y C £ ¢ ¥ § ¦ ¨ § § ¨ © £ ¤ § r r " # # $ # $ " " " " # " $ " " " & # " $ " " $ " # # $ " " $ $ # $ " " # " # " $ " " ' " & # " $ " " ( " # # $ " " ( $ # $ " " & " # " $ " " % " & # " $ " " ! " # # $ " " ! $ # $ " " " # " $ " " " & # " $ " " # # $ " $ # ) d 1 e d 0 6 f 2 1 4 5 6 7 8 9 1 2 3 9 @ A B C @ D E F G B A @ E C H I D P Q R S @ D A @ T U V W X Y ` a b c d e b g b g Y h i p b h i q h c i a Y h d c q r i j ) k 7 e 6 u 9 f 1 4 5 6 7 8 9 1 2 3 9 l m n y o n m x p q y r s m o p t q y u £ ¤ v w ¥ x y © z © r i ' " # " ( " # " & " # " % " # " ! " # " " # " $ " " " " # " $ " " " # " $ " " $ " # " $ " " # " # " $ " " ' " # " $ " " ( " # " $ " " & " # " $ " " % " # " $ " " ! " # " $ " " " # " $ " " " # " $ " " # " 5 6 7 8 9 1 2 3 9 @ A B C @ D E F G B A @ E C H I D P Q R S @ D A @ T U V { h | W X Y ` a b c d e h i q h c i a Y h d c q r r } { h | W f p b c f ~ b c b p b X Y b b c h i q h c i a Y h d c { | v ¥ x y © z © r r ! " ' # " ! $ # " # " $ " " " " & # " $ " " " # # $ " " $ # $ " " $ " # " $ " " # " & # " $ " " ' " # # $ " " ' $ # $ " " ( " # " $ " " & " & # " $ " " % " # # $ " " % $ # $ " " ! " # " $ " " " & # " $ " " " # # $ " " $ # $ " " # " @ A B C @ D E F G B A @ E C H I D P Q R S @ D A @ T U V { h | W X Y ` a b c d e i X h b b X h b X h f q q d g i f h b q r r } { h | W f p b c f ~ b c b p b X Y b b c f q q d g i f h b { |Z + A δ −→ Z, Z + B δ −→ Z for Z ∈ {A, B} .(2)
Lastly, catalytic inactivation, which denotes the network effect (Methods), is expressed as:
Z νΦ − −−−− → MLM ∅, for Z ∈ {A, B} .(3)
The network structure of the MLM catalyses inactivation of existing participants at the rate ν Φ, where
Φ = Z∈{A,B} N Z (N Z − 1) N (N − 1)
is a measure of the probability that two members drawn randomly from the MLM belong to the same type. It has been widely used in literature as a diversity index [17]. Due to interconnectedness and homophily [22], the inactivation of one could (like a contagion) infect another to follow suit.
Combining equations (1-3) results to a Master equation (Supplementary Equation 1) for the state probability density. Perturbation analysis accounts for the fluctuations arising from demographic stochasticity [26]. In terms of the system size Ω (roughly the size of that part of the overall population considered fit for entrepreneurial activities) the following ansatz is made: N A = Ωα+ √ Ω a and N B = Ωβ + √ Ω b, where α and β are the average concentrations, and a and b are the magnitude of the fluctuations of the stochastic variables N A and N B , respectively (Supplementary Methods, S2). The highest order in the expansion expresses the macroscopic rate equations for α and β:α
= µ(α + β) − ν Φ (α + β) − δ (α + β)α; β = λ(α + β) − ν Φ (α + β) − δ (α + β)β.(4)= µ b (ν), where µ b (ν) = 1 2 + (1 − ν) 3 27ν , 0 < ν < 1.(5)
Equation (5) coincides with an evolutionarily stable state (ESS) of a population game between the types (Supplementary Methods, S3-S4). The fraction of A-entrepreneurs is
x A = x A + Ω −1/2 ξ, where x A = α α + β ,
and ξ is the fluctuation component. Analysis of the second moments from the FPE shows that the variance diverges as ξ 2 ∼ δ |µ − µ b | −1 as µ → µ − b (Supplementary Methods, S5). That is a signature of criticality through which the ESS, where x A = x * and N = N * = 1 − 2ν Φ(x * ) ∆ , is (quite counterintuitively) destabilised as the bifurcation manifold is approached. This mechanism is hereby referred as stochastic tunnelling wherein noise enables the state trajectory to cross a phase barrier that could not have been otherwise traversed without actively tuning the bifurcation parameter (Supplementary Methods, S4).
Stochastic tunnelling drives the business oscillations (Fig. 2). Time series is generated by solving the model using a numerical technique, known as Gillespie's stochastic simulation algorithm [27], which directly integrates the master equation (Supplementary Methods, S6). Diverging variance indeed allows the solution to wander far enough from the ESS and closer to an unstable point (UEP) which pushes that solution toward the boundary state, where x * = 1 and N * = µ − νΦ ∆ (Fig. 2a). That noise also enables the solution to sling back to the ESS consequently forming loops in the phase portrait, hence, oscillations in the time series (Fig. 2b). The time series consist of upswings associated with increasing diversity and downswings with decreasing diversity, i.e., Φ → 1 as x A → 1. The remarkable observation is that recovery from low points of the series coincide with periods when A is dominant-a case of the fitter entrepreneurs surviving through "recessions" [28].
Profit maximisation is an axiom of capitalist enterprise [28]. MLM may enhance profitability by maximising the proportion of A -entrepreneurs (Supplementary Methods, S7). Thus, the time-average value x A t is examined for various pairs of µ and ν which consequently depicts the phase diagram of the model (Fig. 3a). Business oscillations come about as a result of stochastic tunnelling through the critical boundary µ = µ b . Phase II, where A stably dominates ( Supplementary Fig. S2b), can be considered Pareto-optimal as the MLM maximises profitability as a whole. But high levels of targeted recruitment, i.e., µ > 4 7 , are required. Entrepreneurial activation, however, might in reality be less discriminatory and thus µ ≈ 1 2 , which denotes higher entropy (Supplementary Discussion). The critical boundary delineates, for any magnitude ν of the network effect, the minimum µ that promotes long-run dominance of A -entrepreneurs. Nevertheless, a stronger network effect tends to frustrate that dominance as catalytic inactivation increasingly outpaces activation, leading to degradation of entrepreneurial activity (Supplementary Fig. S2d). The ESS at the III-IV boundary (Fig. 3a) is therefore Paretodominated [29].
The Hurst maps (Fig. 3b, c) locate where the real MLMs are on the phase diagram. The Hurst exponents are determined from the same exact method. Clusters appear in the vicinity of the III-IV boundary. On these clusters H(1) and H(2) are approximately between 0.5 and 0.8, about the same range of values found in real MLMs (see Table 1). A correlative plot (Fig. 3d) between H(1) and H(2) further confirms not only agreement between model and empirical data, but also their unifractality. Overall, these findings suggest that realworld MLMs are Pareto-dominated economic systems [8], which are operating in an environment characterised by high entropy, i.e., µ ≈ 1 2 , and by a strong network effect (i.e., ν > µ).
The study paints an illuminating insight about the nature of MLM operations. MLMs have been accused in several instances by discontent participants for ethical violations concerning its business practices [30]. The model justifies such disgruntlement for two reasons. First, that profit is closely associated with recruitment implies less selective entrepreneurial activation. Second, that recruitment proceeds through embedded networks connotes strong network effects. Less-fit entrepreneurs can join the market in droves but are weeded out too soon [28] because of the Pareto-dominated nature of the venture. The feeling of being victimised is thus not at all surprising.
The mesoeconomic framework (i.e., linking microeconomic foundations with macroeconomic phenomena [13]) puts the present study in a broader economic context. A more network-dynamic approach to viewing business cycles is hereby encouraged. Lastly, the mathematical model could be extended or refined, such as by generalising the network effect using the Hölder mean such Supplementary Fig. S1); whereas empirical data of higher temporal resolution may become available in the future, to further test the implications that came forth.
that Φ(x A , x B ) = q−1 x q A + x q B , ∀q > 1 (Supplementary Discussion,
METHODS
Network effect.
The local network effect, which is a relatively new idea in economics [18][19][20][21], means that the decision of one entity can influence those by whom that entity is connected to. Particularly, the network effect manifests through inactivation as the value of exiting the enterprise is enhanced through the catalytic action of the connections between participants. Homophily [21,22] spells that "a contact between similar people occurs at a higher rate than among dissimilar people" [23], and strongly influences contagions that diffuse through social links [24]. MLM participants are thus more likely to connect with others of the same SES, consequently elevating homogeneity (or depressing diversity) in the firm. Diversity here is measured by the Simpson index Φ serving as a dimensionless potential function minimised when N A = N B (at highest diversity). The network effect is constituted as ν Φ, for ν > 0; hence, the network effect is stronger at less diversity.
Hurst analysis.
The generalized Hurst method [10,11] has been coded by one of its authors, T. Aste.
The code was downloaded from Matlab File Exchange website, http://www.mathworks.com/matlabcentral/fileexchange/30076 and was used with default settings in the calculation of H(1) and H(2).
Nondimensionalisation.
Dimensionless time is defined as t = t/t c , wherein t c = (µ + λ) −1 . Population numbers are in units of u:
N A = N A u and N B = N B u.
Consequently, equation (4) becomes:
d N A d t = (µt c ) N A − (νt c ) Φ( N A , N B ) N A + N B − (ut c δ/Ω) N A + N B N A , d N B d t = (λt c ) N B − (νt c ) Φ( N A , N B ) N A + N B − (ut c δ/Ω) N A + N B N B .
Characteristic timescale is chosen at t c = 5 days (i.e., 1 month ≡ 20 days). Assuming that the system-size parameter is of the order, Ω ∼ 10 6 individuals, and the unit u ∼ 10 3 individuals, then setting 10 −4 < ∆ = u t c δ Ω < 10 −3 implies an average per-capita encounter rate δ between 1 and 10 per month, which is a reasonable estimate. the critical manifold µ = µ b (ν) is inaccessible due to the constraint xA ≤ 1. Phase II denotes the Pareto-optimal region µ > µ b (ν) and µ > ν where xA ≈ 1. Phase III, where µ < µ b is similar to phase I except that here the critical manifold is accessible. Phase IV is where µ > µ b but µ < ν resulting to degradation of N due to µ − ν Φ < 0 (Supplemen-
FIG. 1 .
1Empirical data for different publicly-listed MLM firms. N (t) is the firm size, R(t) is average revenue per member, and t is time in quarters. a, USANA Health Sciences (USNA) in Canada, and Southeast Asia and Pacific (including Australia) from March 1998 to March 2012. b, NuSkin (NUS) in Greater China (including Hong Kong) and North Asia (Japan and South Korea) from March 1999 to March 2012. c, Nature Sunshine (NATR) worldwide showing N and R from September 1994 to March 2012. d, Mannatech Inc. (MTEX) worldwide showing N and R from April 1998 to March 2012. Data sets are provided in Supplementary Data.
Meanwhile, the next highest order term gives the Fokker-Planck equation, or FPE (Supplementary Equation 2), governing the dynamics of the probability density for the magnitude of the fluctuations. From the FPE, the expectation values of the stationary fluctuations are a = b = 0, which supports the interpretation that the deterministic solutions to (4) are the correct average values. The model is nondimensionalised by setting the characteristic timescale at t c = 5 days (Methods). Consequently, the rates can be squarely related to empirical data by rescaling to appropriate units. Dimensionless rates take on simplified yet meaningful values: λ = 1 − µ; µ, ν ∈ (0, 1); and ∆ = ut c δ Ω ∈ 10 −4 , 10 −3 . Bifurcation analysis (Supplementary Methods, S4) of the nondimensionalised equation (4) unveils a bifurcation manifold µ
FIG. 2 .
2high-SES type, x A (t) Business oscillations in the MLM. The model is simulated with the following parameters: µ = 0.57, ν = 0.57, ∆ = 5 × 10 −4 for a total time of t = 240 1−day periods ≈ 1 year. a, Phase portrait showing one stochastic realization for N (t) (in units u = 10 3 ) versus xA(t); ⊗ marks the initial condition: N (0) = 400 and xA(0) = 0.54. The dashed curves and lines are the nullclines of the replicator equations from the evolutionary game (Supplementary Methods, S3), which intersect at the evolutionarily stable state (ESS, •) and at an unstable equilibrium point (UEP, •). The trajectory of the solution forms a loop indicating the oscillations. b, Time series for N (t) and xA(t). Hurst analysis gives H(1) = 0.6668 ± 0.0235 and H(2) = 0.6564 ± 0.0204 for the N (t) series.
FIG. 3 .
3Phase diagram and Hurst maps of the MLM. a, xA t for different pairs of µ and ν (Resolution: ∆µ × ∆ν = 0.01 × 0.02). Phase I represents the regime 0 ≤ ν ≤ 1 4 where
tary Fig. 2d). Phase boundaries: I-II,III, ν = 1 4 ; III-II,IV, µ = µ b (ν); and II-IV, µ = ν. b, c, Map of the Hurst exponents H(1) and H(2), respectively. The phase boundaries are superimposed. The data are generated by simulating the model for t = 420 5−day periods ≈ 8.75 years with initial population N (0) = 100 for different pairs of µ and ν (Resolution: ∆µ×∆ν = 0.01×0.02). Each data pixel is an average of four stochastic realisations. d, Correlative plot between H(1) and H(2). Empirical data are those listed in Table 1. The dashed line H(1) = H(2) denotes unifractality of the time series.
TABLE I .
IHurst exponents H(1) and H(2) for different
MLMs estimated using the generalised Hurst method[10, 11].
H(1) > 0.5 indicates persistence, whereas H(1) < 0.5 indi-
cates anti-persistence. H(2) values, which are closely related
to the scaling of the power spectrum, are also shown. Gener-
ally, H(2) = H(1) within standard deviation. The standard
deviation values are determined from a pre-testing procedure
(Supplementary Methods, S1).
M LM
H(1)
H(2)
MTEX
0.8328 ± 0.1140 0.7425 ± 0.1081
USNA United States 0.7602 ± 0.1140 0.6823 ± 0.1081
USNA Canada
0.6619 ± 0.1140 0.6013 ± 0.1081
USNA SEA-Pacific 0.6383 ± 0.1140 0.6139 ± 0.1081
NUS Greater China 0.6159 ± 0.1098 0.5387 ± 0.1038
NATR
0.5128 ± 0.1032 0.5126 ± 0.0876
NUS North Asia 0.4780 ± 0.1098 0.4789 ± 0.1038
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| {'fraction_non_alphanumeric': 0.07982631930527723, 'fraction_numerical': 0.03473613894455578, 'mean_word_length': 3.8020850040096232, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 11, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 157, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Business-cycle phenomenon has long been regarded as an empirical curiosity in macroeconomics. Regarding its causes, recent evidence suggests that economic oscillations are engendered by fluctuations in the level of entrepreneurial activity[1,2]. Opportunities promoting such activity are known to be embedded in social network structures[3,4]. However, predominant understanding of the dynamics of economic oscillations originates from stylised pendulum models on aggregate macroeconomic variables[5,6], which overlook the role of social networks to economic activity-echoing the so-called aggregation problem of reconciling macroeconomics with microeconomics[7,8]. Here I demonstrate how oscillations can arise in a networked economy epitomised by an industry known as multi-level marketing or MLM[9], the lifeblood of which is the profit-driven interactions among entrepreneurs. Quarterly data (over a decade) which I gathered from public MLMs reveal oscillatory time-series of entrepreneurial activity that display nontrivial scaling and persistence[10,11]. I found through a stochastic population-dynamic model, which agrees with the notion of profit maximisation as the organising principle of capitalist enterprise, that oscillations exhibiting those characteristics arise at the brink of a critical balance between entrepreneurial activation and inactivation brought about by a homophily-driven network effect. Oscillations develop because of stochastic tunnelling permitted through the destabilisation by noise of an evolutionarily stable state. The results fit altogether as evidence to the Burns-Mitchell conjecture that economic oscillations must be induced by the workings of an underlying "network of free enterprises searching for profit"[12]. I anticipate that the findings, interpreted under a mesoeconomic framework[13], could open a viable window for scrutinising the nature of business oscillations through the lens of the emerging field of network science. Enquiry along these lines could shed further light into the network origins of the business-cycle phenomenon.', 'arxivid': '1209.5576', 'author': ['Dranreb Earl \nDepartment of Mathematics\nSchool of Science and Engineering\nAteneo de Manila University\nLoyola Heights1108Quezon CityPhilippines\n', 'Juanico \nDepartment of Mathematics\nSchool of Science and Engineering\nAteneo de Manila University\nLoyola Heights1108Quezon CityPhilippines\n'], 'authoraffiliation': ['Department of Mathematics\nSchool of Science and Engineering\nAteneo de Manila University\nLoyola Heights1108Quezon CityPhilippines', 'Department of Mathematics\nSchool of Science and Engineering\nAteneo de Manila University\nLoyola Heights1108Quezon CityPhilippines'], 'corpusid': 58771904, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 9698, 'n_tokens_neox': 8308, 'n_words': 4774, 'pdfsha': 'faab69ef22ffa24621a4be62a8fbe5da60256306', 'pdfurls': ['https://arxiv.org/pdf/1209.5576v1.pdf'], 'title': ['Critical network effect induces business oscillations in multi-level marketing systems', 'Critical network effect induces business oscillations in multi-level marketing systems'], 'venue': []} |
arxiv |
Intersection Bodies of Polytopes: Translations and Convexity
Marie-Charlotte Brandenburg
Chiara Meroni
Intersection Bodies of Polytopes: Translations and Convexity
We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P . We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of I(P + t) can be extended to polynomials in variables t ∈ R d within each region of the arrangement. Establishing the convexity space as the set of translations such that I(P + t) is convex, we fully characterize it for two-dimensional polytopes and partially characterize it for higher dimensions, revealing unexpected finite behavior in the two-dimensional case and for the d-dimensional cube.We will rely on methods and results which were developed in[BBMS22]. In this section we review the most important concepts and results we are going to make use of.
Introduction
In the field of convex geometry, intersection bodies have been widely studied from an analytical viewpoint, and mainly in the context of volume inequalities. Originally introduced by Lutwak [Lut88], they have played a significant role in solving the Busemann-Petty problem, which asks to compare the volume of two convex bodies based on the volumes of their linear sections [Gar94a; Gar94b; Kol98; GKS99;Zha99]. Unlike its more famous counterparts, the projection body, the intersection body IK of a star body K is not invariant under affine translation. Furthermore, an intersection body can be both convex and non-convex. Convexity is certified Busemann's theorem [Bus49], which states that IK is convex if K is a convex body centered at the origin (i.e., K is centrally symmetric, where the center of symmetry is the origin), and this statement has been generalized to L p -intersection bodies [Ber09]. On the other hand, given a convex body K ⊆ R d , there always exists some t ∈ R d such that I(K + t) is not convex [Gar06, Thm. 8.1.8].
The occurrence of non-convex intersection bodies has motivated considerations of various measures for capturing the magnitude of their non-convexity, leading to the study of p-convexity of intersection bodies both over the complex numbers and over the reals [KYZ11;HHW12]. Another direction of research concerns an adaptation of the construction of intersection bodies in order to get convexity, which resolves in convex intersection bodies [MR11;Ste16]. A different relative of intersection bodies is the cross-section body [Mar92;Mar94]; however, this starshaped set turned out to be non-convex as well, in the general case [Bre99]. Summarizing, many of the positive results towards convexity in all these works concern intersection bodies of centrally symmetric star bodies. In contrast, we focus on affine translates, and consider objects which are not necessarily centrally symmetric.
The goal of this article is to investigate the behavior of intersection bodies of polytopes under translations, and to determine under which translations the intersection body is convex. In our previous work [BBMS22] we exhibit rich semialgebraic structures of intersection bodies of polytopes. However, in general, the intersection body IP of a polytope P is not a basic semialgebraic set, and there exists a central hyperplane arrangement which describes the regions in which the topological boundary of IP is defined by a fixed polynomial. Taking advantage of these combinatorial and semialgebraic structures opens up new possibilities to study the question of convexity in the present work. In particular, exploiting this semialgebraicity, we are able to characterize convexity by using elementary geometric arguments.
In this article we introduce an affine hyperplane arrangement associated to a fixed polytope P . We prove that for translation vectors t ∈ R d within a region of this arrangement the polynomials defining the boundary of I(P + t) can be extended to polynomials in t 1 , . . . , t d (Theorem 3.5). Establishing the convexity space CS(P ), namely all those translation vectors t such that I(P +t) is convex, we give a full characterization of the convexity space in dimension 2, and a partial characterization for general dimensions. Surprisingly, it turns out that the convexity space of two-dimensional polytopes is always a finite set, and we exhibit the same behavior in higher dimensions for the d-dimensional cube. In particular, this implies that the convexity space is itself non-convex. For higher dimensions d > 2, the convexity space of a polytope may be infinite an even full-dimensional. We summarize our results as follows.
Results. Let CS(P ) = {t ∈ R d | I(P + t) is convex} be the convexity space of P .
(i ) If d = 2 then CS(P ) is finite and consists of at most 5 points.
(ii ) If P = [−1, 1] d is a cube, then CS(P ) is finite and consists of precisely 2d + 1 points.
(iii ) If IP is strictly convex then CS(P ) contains a full-dimensional open ball.
A full classification of the 2-dimensional case is given in Corollary 4.8, and the remaining statements can be found in Proposition 5.4 and Remark 5.5. An example of a strictly convex intersection body is given in Example 5.6.
Overview. The article is structured as follows. In Section 2 we review the main concepts and notation from [BBMS22]. In Section 3 we introduce an affine hyperplane arrangement and describe how it governs the behavior of IP under translation of P . We then turn to the characterization of the convexity space CS(P ), where Section 4 concerns the 2-dimensional case, and Section 5 the case of general dimensions.
Let P ⊆ R d be a convex polytope. The intersection body IP of P is the starshaped set
IP = x ∈ R d ρ IP (x) ≥ 1 , where the radial function ρ IP : R d → R of IP is ρ IP (x) = 1 x vol d−1 (P ∩ x ⊥ ).
Here, vol d−1 denotes the (d − 1)-dimensional Euclidean volume, and x ⊥ ⊆ R d denotes the linear hyperplane which is orthogonal to x ∈ R d , namely the set
x ⊥ = {y ∈ R d | x, y = 0}.
To obtain meaningful results, we may thus assume that P ⊆ R d is a d-dimensional polytope throughout this article. The topological boundary of the intersection body IP is defined by the equation
∂IP = {x ∈ R d | ρ IP (x) = 1}.
Since the radial function satisfies ρ IP (λx) = 1 λ ρ IP (x) for every λ > 0, it is completely determined by its restriction to the unit sphere.
The intersection body IP of a polytope is governed by the central hyperplane arrangement
H(P ) = v =0 is a vertex of P v ⊥ .
We denote the set of vertices of P by vert(P ), and the origin is denoted by 0 ∈ R d . An open chamber C of H(P ) is a connected component of R d \ H(P ). Given such a chamber C, all hyperplanes x ⊥ for x ∈ C intersect P in the interiors of a fixed set of edges. The radial function restricted to such a chamber is a quotient of polynomials
ρ IP | C = p C x 2 q C ,(1)
where p C is divisible by x 2 . Therefore, the topological boundary ∂IP ∩ C is the zero-set of the (irreducible) polynomial p C x 2 − q C . We repeat a key argument in the proof of (1). Let x ∈ C and Q = P ∩ x ⊥ . The value ρ IP (x) is by definition the volume of Q. This computation is done by considering a triangulation T of the boundary of Q. We extend this to a covering of conv(Q, 0) by considering the set conv(∆, 0) for every simplex ∆ ∈ T such that 0 ∈ ∆. Note that if 0 ∈ P , then this induces a central triangulation of Q. Denoting v 1 , . . . , v d the vertices of a simplex ∆ ∈ T , the volume of conv(∆, 0) = conv(v 1 , . . . , v d , 0) is, up to a constant scaling factor, given by the determinant of the matrix
M ∆ (x) = b i 1 ,x a i 1 − a i 1 ,x b i 1 b i 1 −a i 1 ,x . . . b i d−1 ,x a i d−1 − a i d−1 ,x b i d−1 b i d−1 −a i d−1 ,x x ,
where the vertices v i arise as intersection of x ⊥ with edges of P , i.e., v i = conv(a i , b i ) ∩ x ⊥ for a i , b i ∈ vert(P ). Assigning sgn(∆) ∈ {−1, 1} to each simplex, this gives
ρ IP (x) = vol d−1 (Q) = 1 x 2 (d − 1)! ∆∈T sgn(∆) det(M ∆ (x)).
Translations and Affine Hyperplane Arrangements
Let P ⊆ R d be a polytope. In this section we consider how the intersection body of P + t transforms under variation of t ∈ R d . Recall from Section 2 that the combinatorial structure of the boundary of I(P + t) is described by the central hyperplane arrangement H(P + t). We thus begin by studying the behavior of this hyperplane arrangement under translation of P . For this, we introduce a new affine hyperplane arrangement L (P ), which captures the essence of H(P + t) under variation of t. We show that within a region R of L (P ) the polynomials describing the boundary of I(P + t), for t ∈ R, can be extended to polynomials in the variables t 1 , . . . , t d .
Let P ⊆ R d be a polytope and let vert(P ) be the set of its vertices. Denote by H v = v ⊥ ⊆ R d the hyperplane though the origin that is orthogonal to a vertex v ∈ vert(P ). As described in the previous section, the collection of all such hyperplanes forms a central hyperplane arrangement H(P ) in R d . For each such hyperplane we define its positive and negative side as
H + v = x ∈ R d | x, v > 0 and H − v = x ∈ R d | x, v < 0 .
We now choose a translation vector t ∈ R d and consider the vertices {v + t | v ∈ vert(P )} of the translated polytope P + t. The hyperplane arrangement H(P + t) is given by the
hyperplanes (v + t) ⊥ , where v ranges over the vertices of P . The hyperplane H v+t can be obtained from H v by a rotation r v,t : R d → R d such that r v,t v ||v|| = v+t ||v+t|| , and thus r v,t (H v ) = H v+t , r v,t (H + v ) = H + v+t and r v,t (H − v ) = H − v+t .
We label each maximal chamber C of H(P + t) with a sign vector s(C) ∈ {+, −} vert(P +t) indexed by the vertices w = v + t of P + t, where
s(C) w = + if C ⊆ H + w , s(C) w = − if C ⊆ H − w .
The set {s(C) | C maximal chamber of H(P + t)} describes the chirotope or signed cocircuits of the underlying oriented matroid of the hyperplane arrangement [GOT18, Chapter 6.2.3].
H 1 H 2 H 3 (+, −, +) (+, −, −) (+, +, −) (−, +, −) (−, +, +) (−, −, +) H(P ) : H 1 H 2 H 3 (+, −, +) (+, +, +) (+, +, −) (−, +, −) (−, −, −) (−, −, +) H(P t 1 ) : H 1 H 2 H 3 (−, −, +) (−, −, −) (−, +, −) (+, +, −) (+, +, +) (+, −, +)
H(P t 2 ) : Figure 1: The hyperplane arrangements of P + t from Example 3.1. Figure 1 shows the hyperplane arrangements H(P +t) for t 0 = 0, t 1 = (0, 2), and t 2 = (0, −2). Note that the underlying oriented matroids of H(P + t) for t = t 1 and t = t 2 are the same. We continue with this in Example 3.3.
Example 3.1. Let P = conv (v 1 , v 2 , v 3 ) be the triangle with vertices v 1 = 0 1 , v 2 = −1 −1 , v 3 = 1 −1 .
We begin by showing that the signed cocircuit s(C) of a chamber C fully determines the set of edges of P which are intersected by x ⊥ for any x ∈ C.
Lemma 3.2. Let P ⊆ R d be a polytope and let t ∈ R d . Let C be a maximal open chamber of H(P ), and C t be a maximal open chamber of H(P + t) such that s(C) = s(C t ), i.e., their signed cocircuits agree. Given x ∈ C, x t ∈ C t consider
E = {e ⊆ P | e is an edge of P, x ⊥ ∩ e = ∅}, E t = {e t ⊆ P + t | e t is an edge of P + t, x ⊥ t ∩ e t = ∅}
.
Then E t = {e + t | e ∈ E}. Proof. Let e = conv (v 1 , v 2 ) ∈ E be an edge of P . Since x ⊥ ∩ e = ∅, we have that v 1 , v 2 lie on different sides of x ⊥ . Equivalently, we have s(C) v 1 = −s(C) v 2 , and without loss of generality s(C) v 1 = +. Thus, x ∈ H + v 1 ∩ H − v 2 . Since H(P + t) is obtained from H(P ) by rotating the hyperplanes individually, and s(C) = s(C t ), it follows that x t ∈ H + v 1 +t ∩ H − v 2 +t .
Since e + t is an edge of P + t if and only if e is an edge of P , the claim follows.
We consider the affine hyperplane arrangement
L (P ) = {aff(−v 1 , . . . , −v d ) | v 1 , . . . , v d are affinely independent vertices of P }, where aff(−v 1 , . . . , −v d ) denotes the unique affine hyperplane containing the points −v 1 , . . . , −v d . An open region R of L (P ) is a connected component of R d \ L (P )
. We emphasize that there are two hyperplane arrangements in R d which which we consider simultaneously. We have the central hyperplane arrangement H(P + t), which depends on the choice of t, and subdivides R d into open d-dimensional cones, which we call chambers of H(P + t). On the other hand, we have the affine hyperplane arrangement L (P ), which subdivides R d into open d-dimensional components, which we call regions of L (P ). Note that L (P + t) = L (P ) − t by construction.
Example 3.3. Let P be the triangle from Example 3.1. The affine line arrangement L (P ) is shown in Figure 2. Note that the translation vectors t = t 0 , t 1 , t 2 all lie in different regions of the arrangement, despite the fact that the signed cocircuits of H(P + t 1 ) and H(P + t 2 ) agree, as displayed in Figure 1.
In the following we show that L (P ) captures the characteristics of H(P + t) under variation of t. More precisely, we show that within a region R of L (P ) the polynomials describing the boundary of I(P + t), for t ∈ R, can be extended to polynomials in t 1 , . . . , t d .
Proposition 3.4. Let P ⊆ R d be a polytope and R be an open region of L (P ). Then the set of signed cocircuits of H(P + t) is fixed for all t ∈ R. Proof. Let v 1 , . . . , v d be affinely independent vertices of P . By construction of L (P ), R does
t 0 t 1 t 2 −v 3 −v 2 −v 1not intersect A = aff(−v 1 , . . . , −v d ), i.e.
, R is strictly contained in one side of this hyperplane. Without loss of generality, we assume R ⊆ A + . The points w k = v k + t, for k = 1, . . . , d, are linearly independent vertices of P +t for all t ∈ R d \A. Hence, the subarrangement of H(P +t) consisting of hyperplanes w ⊥ 1 , . . . , w ⊥ d is a simplicial arrangement which dissects R d into 2 d open chambers, where each chamber is the image of an orthant of R d under the linear map f defined by e i → w i for all i = 1, . . . , d. Note that the signed cocircuits are fixed for every t ∈ A + . We now consider H(P + t) as common refinement of all subarrangements formed by d hyperplanes with linearly independent normals. The signed cocircuit of a chamber of H(P + t) is uniquely determined by the signed cocircuits of all subarrangements, and the cocircuits of the subarrangements are fixed for all t ∈ R. Thus, the cocircuits of H(P + t) are fixed for all t ∈ R.
Theorem 3.5. Let R be an open region of L (P ), t ∈ R, and let C t be an open chamber of H(P + t). Then the radial function ρ I(P +t) | Ct of I(P + t) restricted to the chamber C t and for t ∈ R is a polynomial in the variables t 1 , . . . , t d of degree at most d − 1.
Proof. By Proposition 3.4, for a fixed region R the set of signed cocircuits of H(P +t) is fixed. Lemma 3.2 then implies that given a region R, t ∈ R, and a chamber C t of H(P + t), for any vector x ∈ C t the set of edges of P + t which intersect x ⊥ is fixed. Let Q = P ∩ x ⊥ , for a certain x ∈ C t , and let T be a triangulation of ∂Q, as explained in Section 2. Let ∆ ∈ T be a maximal simplex with vertices v 1 , . . . v d−1 such that 0 ∈ ∆ and, for each i = 1, . . . , d − 1,
let a i , b i ∈ vert(P ) such that v i = conv(a i + t, b i + t) ∩ x ⊥ . The volume of the d-dimensional simplex conv(∆, 0) is, up to a multiplicative factor of ±1 x (d−1)! , the determinant of the matrix M ∆ (x, t) = b 1 +t,x (a 1 +t)− a 1 +t,x (b 1 +t) b 1 −a 1 ,x . . . b d−1 +t,x (a d−1 +t)− a d−1 +t,x (b d−1 +t) b d−1 −a d−1 ,x x = M ∆ (x, 0) + t,x (a 1 −b 1 ) b 1 −a 1 ,x + t . . . t,x (a d−1 −b d−1 ) b d−1 −a d−1 ,x + t 0 .
The determinant of this matrix is a polynomial in the variables t 1 , . . . , t d of degree at most d − 1. Since the volume of Q can be computed as
vol(Q) = 1 x (d − 1)! ∆∈T sgn(∆) det(M ∆ (x, t)),
the claim follows.
Example 3.6. Figure 3 shows the continuous deformation of the intersection body I(P + t) of the unit square P = [−1, 1] 2 under translation by t ∈ R 2 within each bounded region of the affine line arrangement L (P ).
Convexity in Dimension 2
For each fixed region R of the affine line arrangement L (P ), Theorem 3.5 implies that, as we move t ∈ R continuously, the intersection body I(P + t) deforms continuously as well. We now characterize under which circumstances the intersection body of a polygon is convex. Note that IP cannot be convex if the origin lies outside of P or is a vertex of P (the argument for general dimensions will be given in Remark 5.1). We thus consider the distinct cases of when the origin lies in the interior of P , and when the origin lies in the interior of an edge. Figure 3 indicates that in the case of the square, the intersection body of P + t is convex for precisely 5 translation vectors: the center of symmetry, as well as the midpoints of the four edges. In Theorem 4.6 we show that the number of such translation vectors is always finite, and that parallelograms maximize this number.
Although in this section we focus on 2-dimensional polytopes, we make the following definitions for polytopes of general dimensions.
Definition 4.1. Let P ⊆ R d be a polytope. The convexity space of P is the set
CS(P ) = {t ∈ R d | I(P + t) is convex}.
The goal of this section is to give a characterization of the convexity space of a polygon. In the following Propositions 4.2 and 4.3 we consider polygons with the origin in the interior, and characterize the geometry of the boundary of IP . More precisely, we will see that the chambers in which IP is convex correspond to pairs of parallel edges of P , and that the polynomials defining the boundary of IP are linear in this case.
Proposition 4.2. Let P ⊆ R 2 be a polygon. Let C be a chamber of H(P ), and let x ∈ C. We denote by v 1 (x), v 2 (x) the points of intersection conv (a 1 , b 1 ) , conv (a 2 , b 2 ) be edges of P such that v 1 (x) ∈ conv (a 1 , b 1 ) and v 2 (x) ∈ conv (a 2 , b 2 ). Then the polynomial defining ∂IP in the chamber C is linear if and only if the segments conv (a 1 , b 1 ) and conv (a 2 , b 2 ) are parallel.
x ⊥ ∩ ∂P = {v 1 (x), v 2 (x)}. Let
Proof. We want to prove that {x ∈ C | ρ IP | C (x) = 1} is a line segment if and only if the two edges conv (a i , b i ) are parallel. Assume that v 1 (x) = λa 1 +(1−λ)b 1 and v 2 (x) = µa 2 +(1−µ)b 2 for some λ, µ ∈ (0, 1).
Since v 1 (x), v 2 (x) ∈ x ⊥ , we have λ = b 1 , x b 1 − a 1 , x , µ = b 2 , x b 2 − a 2 , x .
We compute the length of conv(v 1 (x), v 2 (x)), or equivalently of conv(0, v 1 (x) − v 2 (x)). We do this via the area of the triangle with vertices 0, v 1 (x) − v 2 (x) and x x 2 . Hence, the radial function can be computed by the determinantal expression
ρ IP | C (x) = 1 x 2 det v 1 (x) − v 2 (x) x .
We compute the radial function explicitly. First,
v 1 (x) − v 2 (x) = ( b 2 −a 2 ,x ( b 1 ,x a 1 − a 1 ,x b 1 )− b 1 −a 1 ,x ( b 2 ,x a 2 − a 2 ,x b 2 )) b 1 −a 1 ,x b 2 −a 2 ,x .
The boundary ∂P ∩ C is given by the set of points x ∈ C such that ρ IP | C (x) = 1, i.e., the points which satisfy
1 x 2 det b2 − a2, x ( b1, x a1 − a1, x b1) − b1 − a1, x ( b2, x a2 − a2, x b2) x = b1 − a1, x b2 − a2, x ,(2)
assuming that the determinant in the left hand side is positive in C (otherwise it gets multiplied by −1). This determinant is a cubic polynomial in x, which by [BBMS22, Prop. 5.5] is divisible by x 2 . Hence, the left hand side of (2) is a homogeneous linear polynomial in x. It divides the right hand side if and only if (b 2 − a 2 ) = κ(b 1 − a 1 ) for some κ ∈ R, i.e., if the the two edges conv (a i , b i ) are parallel. In this case (2) is a linear equation, and hence the curve defined by (2) is a line; otherwise it is a conic, passing through the origin.
Proposition 4.3. Let P ⊆ R 2 be polygon with the origin in its interior. If there exists a line through the origin which intersects ∂P in two non-parallel edges, then IP is not convex.
Proof. Let C be a chamber of of H(P ) such that x ⊥ intersects two non-parallel edges 1 , 2 of P . Consider u a , u b ∈ C ∩ S 1 . As shown in Figure 4, we denote for some positive real numbers α, β > 0. Since 1 and 2 are not parallel, we have α = β. We can choose a, b such that u a = 1 a ( a 2 −a 1 ) and
u ⊥ a ∩ 1 = a = ( a 1 a 2 ) , u ⊥ b ∩ 1 = b = b 1 b 2 , u ⊥ a ∩ 2 = −αa, u ⊥ b ∩ 2 = −βb, a b −αa −βb a+b 2 0 1 2 u ⊥ a u ⊥ b u ⊥ a+b p a p bu b = 1 b b 2 −b 1 . The lengths of the line segments u ⊥ a ∩ P = conv (a, −αa) and u ⊥ b ∩ P = conv (b, −βb) are u ⊥ a ∩ P = a − (−αa) = (1 + α) a , u ⊥ b ∩ P = b − (−βb) = (1 + β) b .
Thus, the boundary points of IP in directions u a , u b are p a := ρ IP (u a ) u a = (1 + α) a u a = (1 + α) a 2 −a 1 ,
p b := ρ IP (u b ) u b = (1 + β) b u b = (1 + β) b 2 −b 1
respectively. Consider the midpoint a+b 2 ∈ 1 and let u a+b be the unit vector in C orthogonal to a + b (and thus also to a+b 2 ). Then u a+b = 1 a+b a 2 +b 2 −a 1 −b 1 , u ⊥ a+b ∩ 2 = − αβ α+β (a + b) and the boundary point of IP in direction u a+b is
p a+b = ρ IP (u a+b ) u a+b = 1 2 + αβ α + β a + b u a+b = 1 2 + αβ α + β a 2 + b 2 −a 1 − b 1 .
Let q = conv (p a , p b ) ∩ cone (u a+b ), as in Figure 4. We want to prove that IP ∩ C is not convex, by showing that q > p a+b . Indeed, we can compute that
q = (1 + α)(1 + β) 2 + α + β (a 2 + b 2 , −a 1 − b 1 ),
and therefore
q − p a+b = (1 + α)(1 + β) 2 + α + β a + b − 1 2 + αβ α + β a + b = (α − β) 2 2(2 + α + β)(α + β) a + b .
Since α = β, this expression is strictly positive, and so q ∈ IP . This proves that p a , p b ∈ IP , but the segment conv (p a , p b ) is not contained in IP . Hence, IP is not convex.
We are now ready to move towards a full classification of convexity of intersection bodies of polygons for any translation. Note that if P is centrally symmetric, then the convexity of P and the description of IP follow from the following classical statement.
Theorem 4.4 ([Gar06, Theorem 8.1.4]). Let K ⊆ R 2 be a centrally symmetric convex body centered at the origin. Then IK = r π 2 (2K), where r π 2 is a counter-clockwise rotation by π 2 . Our goal is to classify also the cases in which P is not centrally symmetric and centered at the origin. A key argument in the proof of the following Theorem 4.4 is done via the chordal symmetral of P . The chordal symmetral ∆K of a star body K ⊆ R d is the union of segments conv (−c u u, c u u), where u ∈ S d−1 and c u = 1 2 vol d−1 (K ∩ u ⊥ ) [Gar06, Definition 5.1.3]. The chordal symmetral is a starshaped set with respect to the origin. We will make use of the following statements. (ii ) the origin is the midpoint of an edge of P , and P ∪ −P is convex.
Proof. As noted in Remark 5.1, IP is not convex if the origin lies in R 2 \ P , or if the origin is a vertex of P . We are left to analyze the cases in which the origin lies in the interior of P or in the interior of an edge of P .
We first consider the case in which the origin lies in the interior of P and show that IP is convex if and only if P = −P . If P = −P , then Theorem 4.4 implies that IP is convex. Assume now that IP is convex, and the origin lies in the interior of P . Then C ∩ IP is convex for every chamber C of H(P ). In particular, by Proposition 4.3, every line u ⊥ , u ∈ S 1 , which does not intersect a vertex of P intersects ∂P in the interior of two parallel edges. Hence, the edges of P come in pairs of parallel edges. We rotate u ∈ S 1 continuously. Whenever u ⊥ crosses a vertex of one edge, it must also cross a vertex in the parallel edge, since otherwise this results in a pair of non-parallel edges. This implies that for every vertex v of P , there exists a vertex w of P such that w = −λv for some λ > 0. Since all edges are pairwise parallel, this positive scalar λ is the same for all vertices. Therefore, we also get that v = −λw, which implies that λ = 1. Hence, P = −P .
Consider now the case in which the origin lies in the interior of an edge of P . Since the origin lies on the boundary of P , we have that IP = 1 2 I(P ∪ −P ). Using Proposition 4.5 we deduce the following chain of equalities:
IP = 1 2 I(P ∪ −P ) (ii) = 1 2 · 2 ∆(P ∪ −P ) (i) = P ∪ −P.
Therefore, IP is convex if and only if P ∪ −P is convex. In order for this to happen, the origin must be the midpoint of the edge, and additionally P ∪ −P must be convex.
Example 4.7. By Corollary 4.8 for each polygon P there are only finitely many positions of the origin such that the intersection body of P is convex. Figure 5 shows , an acute triangle (k = 3), a diamond shape (k = 2), a panettone shape, and a centrally symmetric polygon which is not a parallelogram (k = 1). The case k = 4 is not realizable.
Corollary 4.8. The convexity space CS(P ) of a polygon P ⊆ R 2 is finite. More precisely, let k = |CS(P )| = |{t ∈ R 2 | I(P + t) is convex}|.
Then k ≤ 5 and the equality is realized exactly when P is a parallelogram.
Proof. By Theorem 4.6, I(P + t) is convex if and only if −t is the center of symmetry of P (if it exists), or a midpoint of an edge such that (P + t) ∪ −(P + t) is convex. Thus, the number of such t ∈ R 2 is finite.
If −t is the midpoint of an edge, then (P − t) ∪ −(P − t) is convex if and only if the sum of the angles adjacent to this edge is at most π. Let v 1 , . . . , v n be the vertices of P , ordered cyclically, and let α i be the interior angle of P at v i (and α n+1 = α 1 ). Assume that there are m pairs of consecutive interior angles (α i , α i+1 ), i = 1, . . . , m such that α i + α i+1 ≤ π. Recall that for any polygon with n vertices, the sum of all interior angles is (n − 2)π. Furthermore, for any angle in a polygon, α i ≤ π holds. We thus obtain that
2(n − 2)π = 2 n i=1 α i = n i=1 (α i + α i+1 ) = m i=1 (α i + α i+1 ) + n i=m+1 (α i + α i+1 ) ≤ m i=1 π + n i=m+1 2π ≤ mπ + 2(n − m)π.
This implies m ≤ 4, hence k ≤ 5. A similar computation, with the exterior angles of P , implies that if k = 5 then n = m = 4 and all pairs of consecutive angles sum up to π. Hence, the unique maximizers of k are parallelograms.
We close this section by pointing out that many arguments made in this section do not generalize to higher dimensions: In constrast to Propositions 4.2 and 4.3, in higher dimensions there exist convex pieces IP ∩ C which are not linear. Furthermore, the identification with the chordal symmetral body, as in Theorem 4.6, does not hold in general. However, these insigts in the 2-dimensional case will turn out to be essential for arguments on the general case in the following section.
Convexity in Higher Dimensions
We devote this section to discuss the convexity space of polytopes of dimension d > 2. We make use of the results obtained in Section 4 to show that, similar to the 2-dimensional case, the convexity space of the d-dimensional cube is finite. In contrast, we give a sufficient condition under which the convexity space is infinite and contains a full-dimensional ball.
Remark 5.1. To obtain an intersection body IP which is convex, the origin must either lie in the interior of P , or in the interior of a facet of P . Otherwise, there exists a hyperplane x ⊥ intersecting P at most in a lower-dimensional face, and thus the radial function of IP in direction x has value 0. The set of such x is a cone V = C ∪ −C, where C ⊂ R d is a convex pointed cone. Then, given x ∈ C, there exist x 1 , x 2 ∈ R d \ V such that x is a convex combination of x 1 and x 2 . Since ρ IP (x) = 0, the segment with extrema ρ IP (x 1 ) x 1 and ρ IP (x 2 ) x 2 is not entirely contained in the intersection body IP , but its extrema are.
The next result connects the intersection body of a convex body to the intersection body of a prism over the given convex body. Proof. Let u = ( u, 0) ∈ H and consider its orthogonal complement u ⊥ ⊆ R d , which in this case can be interpreted as u ⊥ × R ⊆ R d−1 × R. Then
K ∩ u ⊥ = (L × [a, b]) ∩ ( u ⊥ × R) = (L ∩ u ⊥ ) × [a, b].
We can therefore compute the radial function of IK as
ρ IK (u) = vol d−1 (K ∩ u ⊥ ) = vol d−1 (L ∩ u ⊥ ) × [a, b] = (b − a) · ρ IL ( u) for u ∈ H. Equivalently, IK ∩ H = (b − a) IL.
It follows that if IL is non-convex, then so is IK. This behavior can be observed in the following example.
Example 5.3. Consider the unit cube P = [−1, 1] 3 , which is a prism over a square. With the translation t = (1, 1, 1) we obtain the cube P + t = [0, 2] 3 , and I(P + t) is displayed in Figure 6, from two different points of view. Proposition 5.2 implies that I(P + t) ∩ (0, 0, 1) ⊥ is the second dilation of the intersection body of the square [0, 2] 2 , which is also displayed at the bottom left of Figure 3 in red. We can now use Proposition 5.2 to describe the convexity space of a cube in any dimension.
Proposition 5.4. Let P = [−1, 1] d be the centrally symmetric d-dimensional cube. The convexity space CS(P ) is finite, and |CS(P )| = |{t ∈ R d | I(P + t) is convex}| = 2d + 1.
These positions correspond to placing the origin in the center of symmetry of the cube or in the center of symmetry of one of its 2d facets.
Proof. We prove this statement by induction on d. The base case of d = 2 follows from Theorem 4.6. Let now P = [−1, 1] d and consider P + t for some t such that at least one of its coordinates is not in {−1, 0, 1}. Without loss of generality let that coordinate be t 1 . We first prove that in this case P + t is not convex.
Let Q = (P + t) ∩ H, where H = {x ∈ R d | x d = 0}
. Then Q is a translation of a (d − 1)dimensional cube. By the assumption on t 1 , the origin is not the center of symmetry of Q, and not the center of any of its facets. Therefore, by induction, IQ is not convex. Notice that P + t = Q × [a, b] for some a, b ∈ R with b − a = 2. Proposition 5.2 implies that I(P + t) ∩ H = 2IQ. Therefore, I(P + t) ∩ H is not convex, hence I(P + t) itself is not convex.
Consider now the case in which all the coordinates of t are in {−1, 0, 1}. Then the origin is either the center of symmetry of P + t, or it is the center of one of its faces. Recall from Remark 5.1 that if the origin lies on a face of dimension at most d − 2, then the intersection body is not convex.
We are left with the cases in which the origin is the center of the cube or of one of its facets. For t = (0, . . . , 0), P + t = P is centrally symmetric hence I(P + t) is convex. If instead the origin is the center of a facet, then P ∪ −P is a centrally symmetric parallelepiped and by construction I(P + t) = 1 2 I(P ∪ −P ), which is convex. This concludes the proof.
Remark 5.5. We note that whenever the intersection body is strictly convex, then the convexity space contains an open ball around the origin. Indeed this holds in more generality for the intersection body IK of any star body K ⊆ R d , with 0 in its interior, and follows directly from the continuity of the volume function, and therefore of the radial function, with respect to t. Let x, y ∈ R d and p = ρ I(K+t) ( ) · for ∈ {x, y, x + y}, so that p ∈ ∂I(K + t).
Denote by q x+y the point of the segment conv (p x , p y ) which is a multiple of x + y, namely q x+y = ρ I(P +t) (x) ρ I(K+t) (y) ρ I(K+t) (x)+ρ I(K+t) (y) (x + y). Then, I(K + t) is strictly convex if and only if ρ I(P +t) (x) ρ I(K+t) (y) ρ I(K+t) (x) + ρ I(K+t) (y) = q x+y x + y < p x+y x + y = ρ I(K+t) (x + y).
This gives a quadratic condition in ρ I(K+t) , which is continuous in t. Therefore, if (3) holds for IK, it holds also for I(K + t) with t ∈ B ε (0), for some ε > 0.
The next example shows that strictly convex intersection bodies of polytopes as in Remark 5.5 do indeed exist.
Example 5.6. The intersection body of the 3-dimensional centrally symmetric icosahedron P is strictly convex. Indeed, using HomotopyContinuation.jl [BT18] one can check that the algebraic varieties that define the boundary of IP do not contain lines (this is expected, since the generic quintic and sextic surface in 3-dimensional space do not contain lines). Moreover, because of the central symmetry, the intersection body is convex. Hence, it is strictly convex. This intersection body is displayed in [BBMS22, Figure 1], and our computations can be verified using the code on MathRepo [BBMS21].
To summarize, the convexity space can be finite or infinite. Indeed, for d = 2 we have shown that it is always finite, while in higher dimensions it is sometimes infinite, but other times finite, as for a cube. We note that proving non-convexity is a much easier task then proving convexity, as the first can be achieved by showing the non-convexity of a small curve on the boundary, while convexity is a global condition. A possible approach to tackle this problem in the case of polytopes might be studying the curvature of the algebraic hypersurfaces defining the boundary of the intersection body, as in [BRW22].
Another interesting direction of research concerns the topology of the convexity space. We collect here some open questions.
Questions:
1. If the convexity space of P is finite, what are the possible values of |CS(P )|?
2. If the convexity space of P is infinite, how many connected components does it have? What is the dimension of these connected components?
Figure 2 :
2The arrangement L (P ) of the triangle from Examples 3.1 and 3.3.
Figure 3 :
3The arrangement L (P ) of affine lines for P = [−1, 1] 2 , in black, together with I(P + t) for different choices of t, in red, as in Example 3.6.
Figure 4 :
4The proof of Proposition 4.3 in a picture. Left: the lines orthogonal to u a , u b , u a+b and their intersections with the edges 1 , 2 of P . Right: the points p a , p b , p a+b ∈ ∂IP , and the point q ∈ conv (p a , p b ), but q ∈ IP .
Proposition 4.5 ([Gar06, Chapter 5.1]). Let K ⊆ R d be a star body. Then
(i ) K is centrally symmetric and centered at the origin if and only if K = ∆K,(ii ) if K ⊆ R 2 then IK = 2 ∆KWe now prove the main result of this section.
Theorem 4 . 6 .
46Let P ⊆ R 2 be a polygon. Then IP is a convex body if and only if (i ) P = −P , or
a collection of examples of polygons, together with the possible positions of the origin.
Figure 5 :
5Examples in which IP is convex; the bullets represent admissible positions of the origin. From left to right: a parallelogram (k = 5)
Proposition 5. 2 .
2Let L ⊆ R d−1 be a convex body and K = L × [a, b] ⊆ R d−1 × R ∼ = R d be a prism over L. Then, the intersection of IK with the hyperplane H = {x ∈ R d | x d = 0} is the (b − a)th dilate of IL, i.e., IK ∩ H = (b − a) IL.
Figure 6 :
6The intersection body of the 3-dimensional cube P = [0, 2] 3 (blue) and the intersection body of the square Q = [0, 2] 2 (red).
Acknowledgements. We are thankful to Christoph Hunkenschröder for posing a question during a seminar discussion which inspired this work. We thank Andreas Bernig and Jesús De Loera for inspiring conversations about intersection bodies and convexity. We are thankful to the organizers of the conference "Geometry meets Combinatorics in Bielefeld", where most of our ideas fell into place.
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arxiv |
A BLOB METHOD FOR DIFFUSION
José Antonio Carrillo
ANDKaty Craig
Francesco S Patacchini
A BLOB METHOD FOR DIFFUSION
As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles and apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the Fokker-Planck equation, and the Keller-Segel equation and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exist and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under sufficient regularity assumptions, we prove that gradient flows of the regularized porous medium energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method. We conclude by considering a range of numerical examples to demonstrate our method's rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller-Segel equation.2010 Mathematics Subject Classification. 35Q35 35Q82 65M12 82C22;
Introduction
For a range of partial differential equations, from the heat and porous medium equations to the Fokker-Planck and Keller-Segel equations, solutions can be characterized as gradient flows with respect to the quadratic Wasserstein distance. In particular, solutions of the equation
∂ t ρ = ∇ · (∇V ρ) drift + ∇ · ((∇W * ρ)ρ) interaction + ∆ρ m diffusion V : R d → R, W : R d → R, m ≥ 1, (1)
where ρ is a curve in the space of probability measures, are formally Wasserstein gradient flows of the energy
E(ρ) = V dρ + 1 2 (W * ρ) dρ + F m (ρ), F m (ρ) = ρ log(ρ) dL d for m = 1, ρ L d , ρ m m − 1 dL d for m > 1, ρ L d +∞ otherwise,(2)
where L d is d-dimensional Lebesgue measure. This implies that solutions ρ(t, x) of (1) satisfy
∂ t ρ = −∇ W 2 E(ρ),
for a generalized notion of gradient ∇ W 2 , which is formally given by
∇ W 2 E(ρ) = −∇ · ρ∇ δE δρ ,
where δE/δρ is the first variation density of E at ρ (c.f. [3,27,28,81]). Over the past twenty years, the Wasserstein gradient flow perspective has led to several new theoretical results, including asymptotic behavior of solutions of nonlinear diffusion and aggregationdiffusion equations [27,28,68], stability of steady states of the Keller-Segel equation [9,11], and uniqueness of bounded solutions [26]. The underlying gradient flow theory has been well developed in the case of convex (or, more generally, semiconvex) energies [2-4, 24, 54, 76, 81, 82], and more recently, is being extended to consider energies with more general moduli of convexity [5,26,28,35].
Wasserstein gradient flow theory has also inspired new numerical methods, with a common goal of maintaining the gradient flow structure at the discrete level, albeit in different ways. Recent work has considered finite volume, finite element, and discontinuous Galerkin methods [8,16,21,60,79]. Such methods are energy decreasing, positivity preserving, and mass conserving at the semidiscrete level, leading to high-order approximations. They naturally preserve stationary states, since dissipation of the free energy provides inherent stability, and often also capture the rate of asymptotic decay. Another common strategy for preserving the gradient flow structure at the discrete level is to leverage the discrete-time variational scheme introduced by Jordan, Kinderlehrer, and Otto [54]. A wide variety of strategies have been developed for this approach: working with different discretizations of the space of Lagrangian maps [42,55,[65][66][67], using alternative formulations of the variational structure [43], making use of convex analysis and computational geometry to solve the optimality conditions [7], and many others [10,17,23,29,31,46,47,83].
In this work, we develop a deterministic particle method for Wasserstein gradient flows. The simplest implementation of a particle method for equation (1), in the absence of diffusion, begins by first discretizing the initial datum ρ 0 as a finite sum of N Dirac masses, that is,
ρ 0 ≈ ρ N 0 = N i=1 δ x i m i , x i ∈ R d , m i ≥ 0,(3)
where δ x i is a Dirac mass centered at x i ∈ R d . Without diffusion and provided sufficient regularity of V and W , the solution ρ N of (1) with initial datum ρ N 0 remains a sum of Dirac masses at all times t, so that
ρ N (t) = N i=1 δ x i (t) m i ,(4)
and solving the partial differential equation (1) reduces to solving a system of ordinary differential equations for the locations of the Dirac masses,
x i = −∇V (x i ) − N j=1
∇W (x i − x j )m j , i ∈ {1, . . . , N }.
The particle solution ρ N (t) is the Wasserstein gradient flow of the energy (2) with initial data ρ N 0 , so in particular the energy decreases in time along this spatially discrete solution. The ODE system (5) can be solved using range of fast numerical methods, and the resulting discretized solution ρ N (t) can be interpolated in a variety of ways for graphical visualization.
This simple particle method converges to exact solutions of equation (1) under suitable assumptions on V and W , as has been shown in the rigorous derivation of this equation as the mean-field limit of particle systems [22,24,51]. Recent work, aimed at capturing competing effects in repulsive-attractive systems and developing methods with higher-order accuracy, has considered enhancements of standard particle methods inspired by techniques from classical fluid dynamics, including vortex blob methods and linearly transformed particle methods [19,36,45,48]. Bertozzi and the second author's blob method for the aggregation equation obtained improved rates of convergence to exact solutions for singular interaction potentials W by convolving W with a mollifier ϕ ε . In terms of the Wasserstein gradient flow perspective this translates into regularizing the interaction energy (1/2) (W * ρ) dρ as (1/2) (W * ϕ ε * ρ) dρ.
When diffusion is present in equation (1), the fundamental assumption underlying basic particle methods breaks down: particles do not remain particles, or in other words, the solution of (1) with initial datum (3) is not of the form (4). A natural way to circumvent this difficulty, at least in the case of linear diffusion (m = 1), is to consider a stochastic particle method, in which the particles evolve via Brownian motion. Such approaches were originally developed in the classical fluids case [33], and several recent works have considered analogous methods for equations of Wasserstein gradient flow type, including the Keller-Segel equation [49,51,52,61]. The main practical disadvantage of these stochastic methods is that their results must be averaged over a large number of runs to compensate for the inherent randomness of the approximation. Furthermore, to the authors' knowledge, such methods have not been extended to the case of degenerate diffusion m > 1.
Alternatives to stochastic methods have been explored for similar equations, motivated by particle-in-cell methods in classical fluid, kinetic, and plasma physics equations. These alternatives proceed by introducing a suitable regularization of the flux of the continuity equation [34,74]. Degond and Mustieles considered the case of linear diffusion (m = 1) by interpreting the Laplacian as induced by a velocity field v, ∆ρ = ∇ · (vρ), v = ∇ρ/ρ, and regularizing the numerator and denominator separately by convolution with a mollifier [40,73]. For this regularized equation, particles do remain particles, and a standard particle method can be applied. Well-posedness of the resulting system of ordinary differential equations and a priori estimates relevant to the method were studied by Lacombe and Mas-Gallic [57] and extended to the case of the porous medium equation by Lions and Mas-Gallic [59,62]. In the case m = 2 on bounded domains, Lions and Mas-Gallic succeeded in showing that solutions to the regularized equation converge to solutions of the unregularized equation, as long as the initial data has uniformly bounded entropy. Unfortunately, this assumption fails to hold when the initial datum is given by a particle approximation (3), and consequently Lions and Mas-Gallic's result doesn't guarantee convergence of the particle method. An alternative approach, now known as the particle strength exchange method, incorporates instead the effects of diffusion by allowing the weights of the particles m i to vary in time. Degond and Mas-Gallic developed such a method for linear diffusion (m = 1) and proved second order convergence with respect to the initial particle spacing [38,39]. The main disadvantage of these existing deterministic particle methods is that, with the exception of Lions and MasGallic's work when m = 2, they do not preserve the gradient flow structure [59]. Other approaches that respect the method's variational structure have been recently proposed in one dimension by approximating particles by non-overlapping blobs [25,30]. For further background on deterministic particle methods, we refer the reader to Chertock's comprehensive review [32].
The goal of the present paper is to introduce a new deterministic particle method for equations of the form (1), with linear and nonlinear diffusion (m ≥ 1), that respects the problem's underlying gradient flow structure and naturally extends to all dimensions. In contrast to the above described work, which began by regularizing the flux of the continuity equation, we follow an approach analogous to Bertozzi and the second author's blob method for the aggregation equation and regularize the associated internal energy F. For a mollifier ϕ ε (x) = ϕ(x/ε)/ε d , x ∈ R d , ε > 0, we define
F m ε (ρ) = log(ϕ ε * ρ) dρ for m = 1, (ϕ ε * ρ) m−1 m − 1 dρ for m > 1.(6)
For more general nonlinear diffusion, we define F ε (ρ) = F (ϕ ε * ρ) dρ, F : (0, ∞) → R.
As ε → 0, we prove that the regularized internal energies F m ε Γ-converge to the unregularized energies F m for all m ≥ 1; see Theorem 4.1. In the presence of a confining drift or interaction potential, so that minimizers exist, we also show that minimizers converge to minimizers; see Theorem 4.5. For m ≥ 2 and semiconvex potentials V, W ∈ C 2 (R d ), we show that the gradient flows of the regularized energies E m ε are well-posed and are characterized by solutions to the partial differential equation ∂ t ρ = ∇ · ((∇V + ∇W * ρ)ρ) + ∇ · ρ ∇ϕ ε * (ϕ ε * ρ) m−2 ρ + (ϕ ε * ρ) m−2 (∇ϕ ε * ρ) .
Under sufficient regularity conditions, we prove that solutions of the regularized gradient flows converge to solutions of equation (1); see Theorem 5.8. When m = 2 and the initial datum has bounded entropy, we show that these regularity conditions automatically hold, thus generalizing Lions and Mas-Gallic's result for the porous medium equation on bounded domains to the full equation (1) on all of R d ; see Corollary 5.9 and [59, Theorem 2]. For this regularized equation (8), particles do remain particles; see Corollary 5.5. Consequently, our numerical blob method for diffusion consists of taking a particle approximation for (8). We conclude by showing that, under sufficient regularity conditions, our blob method's particle solutions converge to exact solutions of (1); see Theorem 6.1. We then give several numerical examples illustrating the rate of convergence of our method and its qualitative properties.
A key advantage of our approach is that, by regularizing the energy functional and not the flux, we preserve the problem's gradient flow structure. Still, at first glance, our regularization of the energy (6) may seem less natural than other potential choices. For example, one could instead consider the following more symmetric regularization Although studying the above regularization is not without interest, we focus our attention on the regularization in (6) and (7) for numerical reasons. Indeed, computing the first variation density of U ε gives δU ε δρ = ϕ ε * (U • (ϕ ε * ρ)),
as compared to δF ε δρ = ϕ ε * (F • (ϕ ε * ρ)ρ) + (ϕ ε * ρ)F • (ϕ ε * ρ)
for F ε . In the first case, one can see that replacing ρ by a sum of Dirac masses still requires the computation of an integral convolution with ϕ ε , whereas in the second case, all the convolutions reduce to finite sums, which are numerically less costly. Another advantage of our approach, in the m = 2 case, is that our regularization of the energy can naturally be interpreted as an approximation of the porous medium equation by a very localized nonlocal interaction potential. In this way, our proof of the convergence of the associated particle method provides a theoretical underpinning to approximations of this kind in the computational math and swarming literature [56,58]. Further advantages our blob method include the ease with which it may be combined with particle methods for interaction and drift potentials, its simplicity in any dimension, and the good numerical performance we observe for a wide choice of interaction and drift potentials.
Our paper is organized as follows. In Section 2, we collect preliminary results concerning the regularization of measures via convolution with a mollifier, including a mollifier exchange lemma (Lemma 2.2), and relevant background on Wasserstein gradient flow and weak convergence of measures. In Section 3, we prove several results on the general regularized energies (7), which are of a novel form from the perspective of Wasserstein gradient flow theory, combining aspects of the well-known interaction and internal energies. We show that these regularized energies are semiconvex and differentiable in the Wasserstein metric and characterize their subdifferential with respect to this structure; see Propositions 3.10-3.12. In Section 4, we prove that F ε Γ-converges to F as ε → 0 and that minimizers converge to minimizers, when in the presence of a confining drift or interaction term; see Theorems 4.1 and 4.5. With this Γ-convergence in hand, in Section 5 we then turn to the question of convergence of gradient flows, restricting to the case m ≥ 2. Using the framework introduced by Sandier and Serfaty [75,77], we prove that, under sufficient regularity assumptions, gradient flows of the regularized energies converge as ε → 0 to gradient flows of the unregularized energy, recovering a generalization of Lions and Mas-Gallic's results when m = 2; see Theorem 5.8 and Corollary 5.9. Finally, in Section 6, we prove the convergence of our numerical blob method, under sufficient regularity assumptions, when the initial particle spacing h scales with the regularization like h = o(ε); see Theorem 6.1.
We close with several numerical examples, in one and two dimensions, analyzing the rate of convergence to exact solutions with respect to the 2-Wasserstein metric, L 1 -norm, and L ∞ -norm and illustrating qualitative properties of the method, including asymptotic behavior of the Fokker-Planck equation and critical mass of the two-dimensional Keller-Segel equation; see Section 6.3. In particular, for the heat equation and porous medium equations (V = W = 0, m = 1, 2, 3), we observe that the 2-Wasserstein error depends linearly on the grid spacing h ∼ N −1/d for m = 1, 2, 3, while the L 1 -norm depends quadratically on the grid spacing for m = 1, 2 and superlinearly for m = 3. We apply our method to study long time behavior of the nonlinear Fokker-Planck equation (V = |·| 2 /2, W = 0, m = 2), showing that the blob method accurately captures convergence to the unique steady state. Finally, we conduct a detailed numerical study of equations of Keller-Segel type, including a one-dimensional variant (V = 0, W = 2χ log |·| , χ > 0, m = 1, 2) and the original two-dimensional equation (V = 0, W = ∆ −1 , m = 1). The one-dimensional equation has a critical mass 1, and the two-dimensional equation has critical mass 8π, at which point the concentration effects from the nonlocal interaction term balance with linear diffusion (m = 1) [12,41]. We show that the same notion of criticality is present in our numerical solutions and demonstrate convergence of the critical mass as the grid spacing h and regularization ε are refined.
There are several directions for future work. Our convergence theorem for m ≥ 2 requires additional regularity assumptions, which we are only able to remove in the case m = 2 when the initial data has bounded entropy. In the case of m > 2 or more general initial data, it remains an open question how to control certain nonlocal norms of the regularized energies, which play an important role in our convergence result; see Theorem 5.8. Formally, we expect these to behave as approximations of the BV -norm of ρ m , which should remain bounded by the gradient flow structure; see equations (24) and (25). When 1 ≤ m < 2, it is not clear how to use these nonlocal norms to get the desired convergence result or whether an entirely different approach is needed. Perhaps related to these questions is the fact that our estimate on the semiconvexity of the regularized energies (6) deteriorates as ε → 0, while we expect that the semiconvexity should not deteriorate along smooth geodesics; see Proposition 3.11. Finally, while our results show convergence of the blob method for diffusive Wasserstein gradient flows, they do not quantify the rate of convergence in terms of h and ε. In particular, a theoretical result on the optimal scaling relation between h and ε remains open, though we observe good numerical performance for ε = h 1−p , 0 < p 1. In a less technical direction, we foresee a use of the presented ideas in conjunction with splitting schemes for certain nonlinear kinetic equations [1,20], as well as in the fluids [48], since our numerical results demonstrate comparable rates of convergence to the particle strength exchange method, which has already gained attention in these contexts [40].
Preliminaries
2.1. Basic notation. For any r > 0 and x ∈ R d we denote the open ball of center x and radius r by B r (x). Given a set S ⊂ R d , we write 1 S : R d → {0, 1} for the indicator function of S, i.e., 1 S (x) = 1 for x ∈ S and 1 S (x) = 0 otherwise. We say a function A : R d → R has at most quadratic growth if there exist c 0 , c 1 > 0 so that |A(x)| ≤ c 0 + c 1 |x| 2 for all x ∈ R d .
Let P(R d ) denote the set of Borel probability measures on R d , and for, any p ∈ N, P p (R d ) denotes elements of P(R d ) with finite pth moment, M p (R d ) := R d |x| p dµ(x) < +∞. We write L d for the d-dimensional Lebesgue measure, and for given µ ∈ P(R d ), we write µ L d if µ is absolutely continuous with respect to the Lebesgue measure. Often we use the same symbol for both a probability measure and its Lebesgue density, whenever the latter exists. We let L p (µ; R d ) denote the Lebesgue space of functions with pth power integrable against µ.
Given σ a finite, signed Borel measure on R d , we denote its variation by |σ|. For a Borel set E ⊂ R d we write σ(E) for the σ-measure of set E. For a Borel map T : R d → R d and µ ∈ P(R d ), we write T # µ for the push-forward of µ through T . We let id :
R d → R d denote the identity map on R d and define (id, T ) : R d → R d × R d by (id, T )(x) = (x, T (x)) for all x ∈ R d .
For a sequence (µ n ) n ⊂ P(R d ) and some µ ∈ P(R d ), we write µ n * µ if (µ n ) n converges to µ in the weak- * topology of probability measures, i.e., in the duality with bounded continuous functions.
Convolution of measures.
A key aspect of our approach is the regularization of the energy (2) via convolution with a mollifier. In this section, we collect some elementary results on the convolution of probability measures, including a mollifier exchange lemma, Lemma 2.2. For any µ ∈ P(R d ) and measurable function φ, the convolution of φ with µ is given by
φ * µ(x) = R d φ(x − y) dµ(y) for all x ∈ R d ,
whenever the integral converges. We consider mollifiers satisfying the following assumption.
6 ASSUMPTION 2.1 (mollifier). Suppose ζ ∈ C 2 (R d ; [0, ∞)) is even, ζ L 1 (R d ) = 1, and ζ(x) ≤ C ζ |x| −q , |∇ζ(x)| ≤ C ζ |x| −q for some C ζ , C ζ > 0 and q > d + 1, q > d. Let ϕ = ζ * ζ.
This assumption is satisfied by both Gaussians and smooth functions with compact support. Assumption 2.1 also ensures that ϕ has finite first moment. For any ε > 0, we write
ϕ ε = ε −d ϕ(·/ε) and ζ ε = ε −d ζ(·/ε).
Throughout, we use the fact that the definition of convolution allows us to move mollifiers from the measure to the integrand. In particular, for any φ bounded below and ψ ∈ L 1 (R d ) even,
R d φ d(ψ * µ) = R d φ * ψ dµ.
Likewise, we often use the following lemma which provides sufficient conditions for moving functions in and out convolutions with mollifiers within integrals. (See also [59] for a similar result.) The proof is contained in Appendix A.
LEMMA 2.2 (mollifier exchange lemma). Let f : R d → R be Lipschitz continuous with constant L f > 0, and let σ and ν be finite, signed Borel measures on R d . There is p = p(q, d) > 0 so that
ζ ε * (f ν) dσ − (ζ ε * ν)f dσ ≤ ε p L f (ζ ε * |ν|) d|σ| + C ζ |σ|(R d )|ν|(R d ) for all ε > 0.
We conclude this section with a proposition stating that if a sequence of measures converges in the weak- * topology of P(R d ), then the mollified sequence converges to the same limit. We refer the reader to Appendix A for the proof. PROPOSITION 2.3. Let µ ε be a sequence in P(R d ) such that µ ε * µ as ε → 0 for some µ ∈ P(R d ). Then ϕ ε * µ ε * µ.
2.3.
Wasserstein metric. For µ, ν ∈ P(R d ), we denote the set of transport plans from µ to ν by Γ(µ, ν) := {γ ∈ P(R d × R d ) | π 1 # γ = µ, π 2 # γ = ν}, where π 1 , π 2 : R d × R d → R d are the projections of R d × R d onto the first and second copy of R d , respectively. The Wasserstein distance W 2 (µ, ν) between two probability measures µ, ν ∈ P 2 (R d ) is given by
W 2 (µ, ν) = min γ∈Γ(µ,ν) R d ×R d |x − y| 2 dγ(x, y) 1/2 ,(9)
and a transport plan γ o is optimal if it attains the minimum in (9). We denote the set of optimal transport plans by Γ o (µ, ν). If either µ is absolutely continuous with respect to Lebesgue measure, then there is a unique optimal transport plan γ o , and
γ o = (id, T o ) # µ, for a Borel measurable function T o : R d → R d .
T o is unique up to sets of µ-measure zero is known as the optimal transport map from µ to ν. Convergence with respect to the Wasserstein metric is stronger than weak- * convergence. In particular, if (µ n ) n ⊂ P 2 (R d ) and µ ∈ P 2 (R d ), then W 2 (µ n , µ) → 0 as n → ∞ ⇐⇒ µ n * µ and M 2 (µ n ) → M 2 (µ) as n → ∞ .
In order to define Wasserstein gradient flows, we will require the following notion of regularity in time with respect to the Wasserstein metric.
DEFINITION 2.4 (absolutely continuous). µ ∈ AC 2 loc ((0, ∞); P 2 (R d )) if there is f ∈ L 2 loc ((0, ∞)) so that W 2 (µ(t), µ(s)) ≤ t s f (r) dr for all t, s ∈ (0, ∞) with s ≤ t.
Along such curves, we have a notion of metric derivative. DEFINITION 2.5 (metric derivative). Given µ ∈ AC 2 loc ((0, ∞); P 2 (R d )), its metric derivative is
|µ |(t) := lim s→t W 2 (µ(t), µ(s)) |t − s|
An important class of curves in the Wasserstein metric are the (constant speed) geodesics. Given µ 0 , µ 1 ∈ P 2 (R d ), geodesics connecting µ 0 to µ 1 are of the form
µ α = ((1 − α)π 1 + απ 2 ) # γ o for α ∈ [0, 1], γ o ∈ Γ o (µ, ν).
If γ o is induced by a map T o , then
µ α = ((1 − α)id + αT o ) # µ 0 .
More generally, given µ 1 , µ 2 , µ 3 ∈ P 2 (R d ), a generalized geodesic connecting µ 2 to µ 3 with base µ 1 is given by
µ 2→3 α = (1 − α)π 2 + απ 3 # γ for α ∈ [0, 1] and γ ∈ P(R d × R d × R d )(10)
such that π 1,2 # γ ∈ Γ o (µ 1 , µ 2 ) and π 1,
3 # γ ∈ Γ o (µ 1 , µ 3 ). with π 1,i : R d × R d × R d → R d × R d
the projection of onto the first and ith copies of R d . When the base µ 1 coincides with one of the endpoints µ 2 or µ 3 , generalized geodesics are geodesics.
A key property for the uniqueness and stability of Wasserstein gradient flows is that the energies are convex, or more generally semiconvex, along generalized geodesics. DEFINITION 2.6 (semiconvexity along generalized geodesics). We say a functional G : P 2 (R d ) → (−∞, ∞] is semiconvex along generalized geodesics if there is λ ∈ R such that for all µ 1 , µ 2 , µ 3 ∈ P 2 (R d ) there exists a generalized geodesic connecting µ 2 to µ 3 with base µ 1 such that
G(µ 2→3 α ) ≤ (1 − α)G(µ 2 ) + αG(µ 3 ) − λ(1 − α)α 2 W 2 2,γ (µ 2 , µ 3 ) for all α ∈ [0, 1], where W 2 2,γ (µ 2 , µ 3 ) = R d ×R d ×R d |y − z| 2 dγ(x, y, z).
For any subset X ⊂ P(R d ) and functional G : X → (−∞, ∞], we denote the domain of G by D(G) = {µ ∈ X | G(µ) < +∞}, and we say that G is proper if D(G) = ∅. As soon as a functional is proper and lower semicontinuous with respect to the weak-* topology, we may define its subdifferential ; see [3, Definition 10.3.1 and Equation 10.3.12]. Following the approach in [24], the notion of subdifferential we use in this paper is, in fact, the following reduced one. DEFINITION 2.7 (subdifferential). Given G : P 2 (R d ) → (−∞, ∞] proper and lower semicontinuous, µ ∈ D(G), and ξ : R d → R d with ξ ∈ L 2 (µ; R d ), then ξ belongs to the subdifferential of G at µ, written ξ ∈ ∂G(µ), if as ν
W 2 − − → µ, G(ν) − G(µ) ≥ inf γ∈Γ 0 (µ,ν) R d ×R d ξ(x), y − x dγ(x, y) + o(W 2 (µ, ν)).
The Wasserstein metric is formally Riemannian, and we may define the tangent space as follows. 8 DEFINITION 2.8. Let µ ∈ P 2 (R d ). The tangent space at µ is
Tan µ P 2 (R d ) = {∇φ | φ ∈ C ∞ c (R d )},
where the closure is taken in L 2 (µ; R d ).
We now turn to the definition of a gradient flow in the Wasserstein metric (c.f.
v : (0, ∞) × R d → [−∞, ∞] d with −v(t) ∈ ∂G(µ(t)) ∩ Tan µ(t) P 2 (R d )
for almost every t > 0 such that µ is a weak solution of the continuity equation
∂ t µ(t, x) + ∇ · (v(t, x)µ(t, x)) = 0;
i.e., µ is a solution to the continuity equation in duality with C ∞ c (R d ). We close this section with the following definition of the Wasserstein local slope.
DEFINITION 2.10 (local slope). Given G : P 2 (R d ) → (−∞, ∞], its local slope is |∂G|(x) = lim sup d(x,y)→0 (G(x) − G(y)) + d(x, y) for all x ∈ D(G),
where the subscript + denotes the positive part.
REMARK 2.11. When the functional G in Definition 2.9 is in addition semiconvex along geodesics the local slope |∂G| is a strong upper gradient for G. In this case a gradient flow of G is characterized as being a 2-curve of maximal slope with respect to |∂G|; see [3, Theorem 11.1.3].
Regularized internal energies
The foundation of our blob method is the regularization of the internal energy F via convolution with a mollifier. This allows us to preserve the gradient flow structure and approximate our original partial differential equation (1) by a sequence of equations for which particles do remain particles. In this section, we consider several fundamental properties of the regularized internal energies F ε , including convexity, lower semicontinuity, and differentiability. In what follows, we will suppose that our internal energies satisfy the following assumption. ASSUMPTION 3.1 (internal energies). Suppose F ∈ C 2 (0, +∞) satisfies lim s→+∞ F (s) = +∞ and either F is bounded below or lim inf s→0 F (s)/s β > −∞ for some β > −2/(d + 2). Suppose further that U (s) = sF (s) is convex, bounded below, and lim s→0 U (s) = 0.
Thanks to this assumption we can define the internal energy corresponding to F by
F(ρ) = F (ρ) dρ if ρ L d , +∞
otherwise.
If F is bounded below, this is well-defined on all of P(R d ). If lim inf s→0 F (s)/s β > −∞ for some β > −2/(d + 2), this is well-defined on P 2 (R d ); see [3, Example 9.3.6]. which leads to F (s) ≥ 0 for all s ∈ (0, ∞). 9 Our assumption does not ensure that F is convex along Wasserstein geodesics, unless F is convex. which, by Remark 3.2, holds when for example F is convex.
We regularize the internal energies by convolution with a mollifier.
DEFINITION 3.4 (regularized internal energies). Given F : (0, ∞) → R satisfying Assumption 3.1, we define, for all µ ∈ P(R d ), the regularized internal energies by
F ε (µ) = F (ϕ ε * µ) dµ for all ε > 0.
Note that, for all µ ∈ P(R d ) and
ε > 0, F ε (µ) < F ( ϕ ε L ∞ (R d ) ) < ∞.
An important class of internal energies satisfying Assumption 3.1 are given by the (negative) entropy and Rényi entropies.
F m (ρ) = F m (ρ) dρ, F m ε (µ) = F m (ϕ ε * µ) dµ, for F m (s) = s log s for m = 1, s m−1 /(m − 1) for m > 1.
Note that, as per our observation just below the definition of F, the entropy F 1 is well-defined on P 2 (R d ) and the Rényi entropies (F m , m > 1) are well-defined on all of P(R d ). Also note that the regularized entropies (F m ε , m ≥ 1, ε > 0) are well-defined on all of P(R d ). In order to approximate solutions of equation (1), we will consider combinations of the above regularized internal energies with potential and interaction energies.
E ε (µ) = V dµ + 1 2 (W * µ) dµ + F ε (µ) for all ε > 0.
When F = F m for some m ≥ 1, then we denote E by E m and E ε by E m ε . The regularized internal energy in Definition 3.4 incorporate a blend of interaction and internal phenomena, through the convolution with the mollifier, or potential, ϕ ε and the composition with the function F . To our knowledge, this is a novel form of functional on the space of probability measures. We now describe some of its basic properties: energy bounds and lower semicontinuity, when F is the logarithm or a power, and differentiability, convexity and subdifferential characterization when F is convex. For the existence and uniqueness of gradient flows associated to this regularized energy, see Section 5. REMARK 3.7. Although the regularized energy in Definition 3.4 is of a novel form, it was noticed in [69, Proposition 6.9] that a previous particle method for diffusive gradient flows leads to a similar regularized internal energy after space discretization [25,30]. The essential difference between these two methods stands in the choice of the mollifier, which, instead of satisfying 2.1, is a very singular potential.
We begin with inequalities relating the regularized internal energies to the unregularized energies. See Appendix A for the proof, which is a consequence of Jensen's inequality and a Carleman-type estimate on the lower bound of the entropy [30, Lemma 4.1].
PROPOSITION 3.8. Let ε > 0. If m = 1, suppose µ ∈ P 2 (R d ), and if m > 1, suppose µ ∈ P(R d ). Then, F m (µ) + C ε ≥ F m ε (µ) ≥ F m (ζ ε * µ) for 1 ≤ m ≤ 2,(11)F m ε (µ) ≤ F m (ζ ε * µ) for m ≥ 2. (12) where C ε = C ε (m, µ) → 0 as ε → 0. Furthermore, for all δ > 0, we have F m ε (µ) ≥ −(2π/δ) d/2 − 2δ(M 2 (µ) + ε 2 M 2 (ζ)) if m = 1, 0 if m > 1.(13)
For all ε > 0, the regularized entropies are lower semicontinuous with respect to weak-* convergence (m > 1) and Wasserstein convergence (m = 1). For m > 2, we prove this using a theorem of Ambrosio, Gigli, and Savaré on the converge of maps with respect to varying measures; see Proposition B.2. For 1 < m ≤ 2, this is a consequence of Jensen's inequality. For m = 1, we apply both Jensen's inequality and a version of Fatou's lemma for varying measures; see Lemma B.3. In this case, we also require that the mollifier ϕ is a Gaussian, so that we can get the bound from below required by Fatou's lemma. We refer the reader to Appendix A for the proof.
PROPOSITION 3.9 (lower semicontinuity). Let ε > 0. Then (i) F m ε is lower semicontinuous with respect to weak- * convergence in P(R d ) for all m > 1; (ii) if ϕ is a Gaussian, then F 1 ε is lower semicontinuous with respect to the quadratic Wasserstein convergence in P 2 (R d ).
When F is convex, the regularized internal energies are differentiable along generalized geodesics. The proof relies on the fact that F is differentiable and ϕ ε ∈ C 2 (R d ), with bounded Hessian; see Appendix A. PROPOSITION 3.10 (differentiability). Let F satisfy Assumption 3.1 and be convex. Given
µ 1 , µ 2 , µ 3 ∈ P 2 (R d ) and γ ∈ P 2 (R d × R d × R d ) with π i #γ = µ i , let µ 2→3 α = (1 − α)π 2 + απ 3 # γ for α ∈ [0, 1]. Then d dα F ε (µ 2→3 α ) α=0 = F (ϕ ε * µ 2 (y)) ∇ϕ ε (y − v) · (z − w − (y − v)) dγ(u, v, w) dγ(x, y, z).(14)
A key consequence of the preceding proposition is that the regularized energies are semiconvex along generalized geodesics, as we now show. PROPOSITION 3.11 (convexity). Suppose F satisfies Assumption 3.1 and is convex. Then F ε is λ F -convex along generalized geodesics, where
λ F = −4 D 2 ϕ ε L ∞ (R d ) F ( ϕ ε L ∞ (R d ) ).(15)
Proof. Let (µ 2→3 α ) α∈[0,1] be a generalized geodesic connecting two probability measures µ 2 , µ 3 ∈ P 2 (R d ) with base µ 1 ∈ P 2 (R d ); see (10). We have, using the above-the-tangent inequality for convex functions,
F ε (µ 3 ) − F ε (µ 2 ) = (F (ϕ ε * µ 3 )(y) − F (ϕ ε * µ 2 )(z)) dγ(x, y, z) ≥ F (ϕ ε * µ 2 (y)) (ϕ ε * µ 3 (z) − ϕ ε * µ 2 (y)) dγ(x, y, z) = F (ϕ ε * µ 2 (y)) (ϕ ε (z − w) − ϕ ε (y − v)) dγ(u, v, w) dγ(x, y, z). 11
Therefore, by Proposition 3.10,
F ε (µ 3 ) − F ε (µ 2 ) − d dα F ε (µ 2→3 α ) α=0 ≥ F (ϕ ε * µ 2 (y)) × [ϕ ε (z − w) − ϕ ε (y − v) − ∇ϕ ε (y − v) · (z − w − (y − v))] dγ(u, v, w) dγ(x, y, z) ≥ − D 2 ϕ ε L ∞ (R d ) 2 F (ϕ ε * µ 2 (y))|z − w − (y − v)| 2 dγ(u, v, w) dγ(x, y, z) ≥ − D 2 ϕ ε L ∞ (R d ) F ( ϕ ε L ∞ (R d ) ) 2 |z − w − (y − v)| 2 dγ(u, v, w) dγ(x, y, z) ≥ −4 D 2 ϕ ε L ∞ (R d ) F ( ϕ ε L ∞ (R d ) )W 2 2,γ (µ 2 , µ 3 )
, which gives the result.
We now use the previous results to characterize the subdifferential of the regularized internal energy. The structure of argument is classical (c.f. [3,24,54]), but due to the novel form of our regularized energies, we include the proof in Appendix A.
PROPOSITION 3.12 (subdifferential characterization). Suppose F satisfies Assumption 3.1 and is convex. Let ε > 0 and µ ∈ D(F ε ). Then v ∈ ∂F ε (µ) ∩ Tan µ P 2 (R d ) ⇐⇒ v = ∇ δF ε δµ , where ∇ δF ε δµ = ∇ϕ ε * F • (ϕ ε * µ)µ + F • (ϕ ε * µ) ∇ϕ ε * µ, µ-almost everywhere.(16)
In particular, we have |∂F ε |(µ) = ∇ δFε δµ L 2 (µ;R d )
.
As a consequence of this characterization of the subdifferential, we obtain the analogous result for the full energy E ε , as in Definition 3.6. See Appendix A for the proof. COROLLARY 3.13. Suppose F satisfies Assumption 3.1 and is convex. Let ε > 0 and µ ∈ D(E ε ). Suppose V, W ∈ C 1 (R d ) are semiconvex, with at most quadratic growth, and W is even. Then
v ∈ ∂E ε (µ) ∩ Tan µ P 2 (R d ) ⇐⇒ v = ∇ δE ε δµ , where ∇ δE ε δµ = ∇ϕ ε * F • (ϕ ε * µ)µ + F • (ϕ ε * µ) ∇ϕ ε * µ + ∇V + ∇W * µ, µ-almost everywhere. In particular, we have |∂E ε |(µ) = ∇ δEε δµ L 2 (µ;R d )
.
Γ-convergence of regularized internal energies
We now turn to the convergence of the regularized energies and, when in the presence of confining drift or interaction terms, the corresponding convergence of their minimizers. In this section, and for the remainder of the work, we consider regularized entropies and Rényi entropies of the form F m ε for m ≥ 1. We begin by showing that F m ε Γ-converges to F as ε → 0. THEOREM 4.1 (F ε Γ-converges to F). If m = 1, consider (µ ε ) ε ⊂ P 2 (R d ) and µ ∈ P 2 (R d ), and if m > 1, consider (µ ε ) ε ⊂ P(R d ) and µ ∈ P(R d ).
(i) If µ ε * µ, we have lim inf ε→0 F m ε (µ ε ) ≥ F m (µ). (ii) We have lim sup ε→0 F m ε (µ) ≤ F m (µ). Proof.
We begin by showing the result for 1 ≤ m ≤ 2, in which case the function F is concave. We first show part (i). By Proposition 3.8, for all ε > 0,
F m ε (µ ε ) ≥ F m (ζ ε * µ ε ). By Proposition 2.3, µ ε * µ implies ζ ε * µ ε * µ. Therefore, by the lower semicontinuity of F m with respect to weak- * convergence [3, Remark 9.3.8], lim inf ε→0 F m ε (µ ε ) ≥ lim inf ε→0 F m (ζ ε * µ ε ) ≥ F m (µ),
which gives the result. We now turn to part (ii). Again, by Proposition 3.8, for all ε > 0,
F m (µ) + C ε ≥ F m ε (µ), where C ε → 0 as ε → 0. Therefore, lim sup ε→0 F m ε (µ) ≤ F m (µ)
. We now consider the case when m > 2. Part (ii) follows quickly: by Proposition 3.8, Young's convolution inequality, and the fact that
ζ ε L 1 (R d ) = 1, for all ε > 0 we have F m ε (µ) ≤ F m (ζ ε * µ) = 1 m−1 ζ ε * µ m L m (R d ) ≤ 1 m−1 ζ ε m L 1 (R d ) µ m L m (R d ) = 1 m−1 µ m L m (R d ) = F m (µ)
. Taking the supremum limit as ε → 0 then gives the result. Let us prove part (i). Without loss of generality, we may suppose that lim inf ε→0 F m ε (µ ε ) is finite. Furthermore, there exists a positive sequence (ε n ) n such that ε n → 0 and lim n→+∞ F m εn (µ εn ) = lim inf ε→0 F m ε (µ ε ). In particular, there exists C > 0 for which F m εn (µ εn ) < C for all n ∈ N. By Jensen's inequality for the convex function x → x m−1 and the fact that ζ ε * ζ ε = ϕ ε for all ε > 0,
(m − 1)F m ε (µ ε ) = (ϕ ε * µ ε ) m−1 dµ ε ≥ ϕ ε * µ ε dµ ε m−1 = R d |ζ ε * µ ε (x)| 2 dx m−1 .
Thus, since F m εn (µ εn ) < C for all n ∈ N, we have ζ εn * µ εn L 2 (R d ) < C := (C(m − 1)) 1/2(m−1) . We now use this bound on the L 2 -norm of ζ εn * µ εn to deduce a stronger notion of convergence of ζ εn * µ εn to µ. First, since (µ εn ) n converges weakly- * to µ as n → ∞, Proposition 2.3 ensures that (ζ εn * µ εn − µ εn ) n converges weakly- * to 0. Since the L 2 -norm is lower semicontinuous with respect to weak- * convergence [64, Lemma 3.4], we have
C ≥ lim inf n→∞ ζ εn * µ εn L 2 (R d ) ≥ µ L 2 (R d ) ,
so that µ ∈ L 2 (R d ). Furthermore, up to another subsequence, we may assume that (ζ εn * µ εn ) n converges weakly in
L 2 to some w ∈ L 2 (R d ). Since ζ εn * µ εn * µ, for all f ∈ C ∞ c (R d ), f w = lim n→∞ f dζ εn * µ εn = f dµ,
so (ζ εn * µ εn ) n converges to µ weakly in L 2 . By the Banach-Saks theorem (c.f. [71, Section 38]), up to taking a further subsequence of (ζ εn * µ εn ) n , the Cesàro mean (v k ) k defined by
v k := 1 k k i=1 ζ ε i * µ ε i for all k ∈ N, converges to µ strongly in L 2 . Finally, for any f ∈ C ∞ c (R d ), this ensures f (v k ) 2 dL d − f µ 2 dL d ≤ |f ||v k − µ||v k + µ| dL d ≤ f L ∞ (R d ) v k − µ L 2 (R d ) v k + µ L 2 (R d ) , 13 so that lim k→∞ f (v k ) 2 dL d = f µ 2 dL d .(17)
We now use this stronger notion convergence to conclude our proof of part (i). Since m > 2 and
ϕ εn * µ εn m−1 L m−1 (µε n ;R d ) = (m − 1)F m εn (µ εn ) < C for all n ∈ N, by part (i) of Proposition B.2, up to another subsequence, there exists w ∈ L 1 (µ; R d ) so that for all f ∈ C ∞ c (R d ), lim n→∞ f (ϕ εn * µ εn ) dµ εn = f w dµ.(18)
Furthermore, recalling the definition of the regularized energy and applying [3,
Theorem 5.4.4(ii)], lim inf ε→0 F m ε (µ ε ) = lim n→∞ F m εn (µ εn ) = lim n→∞ 1 m − 1 (ϕ εn * µ εn ) m−1 dµ n ≥ 1 m − 1 w m−1 dµ.
Therefore, to finish the proof, it suffices to show that w(x) ≥ µ(x) for µ-almost every x ∈ R d . By Lemma 2.2 and the fact that ζ εn * ζ εn = ϕ εn for all n ∈ N, there exists p > 0 and C ζ > 0 so that
for all f ∈ C ∞ c (R d ), f (ϕ εn * µ εn ) dµ εn − f (ζ εn * µ εn ) 2 dL d = ζ εn * (f µ εn ) dζ εn * µ εn − (ζ εn * µ εn )f dζ εn * µ εn ≤ ε p n ∇f L ∞ (R d ) (ζ εn * µ εn ) 2 L 2 (R d ) + C ζ
Combining this with equation (18), we obtain
lim n→∞ f (ζ εn * µ εn ) 2 dL d = f w dµ.(19)
Finally, using equation (17) and the definition of (v k ) k as a sequence of convex combinations of the family
{ζ ε i * µ ε i } i∈{1,...,k} , for all f ∈ C ∞ c (R d ) with f ≥ 0 we have f µ 2 dL d = lim k→ ∞ f (v k ) 2 dL d = lim k→∞ R d f 1 k k n=1 ζ εn * µ εn (x) 2 dx ≤ lim k→∞ 1 k k n=1 f (ζ εn * µ εn ) 2 dL d .
Since the limit in (19) exists, it coincides with its Cesàro mean on the right-hand side of the above
equation. Thus, for all f ∈ C ∞ c (R d ) with f ≥ 0, f µ 2 dL d ≤ f w dµ.
This gives w(x) ≥ µ(x) for µ-almost every x ∈ R d , which completes the proof. Now, we add a confining drift or interaction potential to our internal energies, so that energy minimizers exist and we may apply the previous Γ-convergence result to conclude that minimizers converge to minimizers. For the remainder of the section we consider energies of the form E m ε given in Definition 3.6, with the following additional assumptions on V and W to ensure that the energy is confining. 14 ASSUMPTION 4.2 (confining assumptions). The potentials V and W are bounded below and one of the following additional assumptions holds:
V has compact sublevel sets;
(CV)
V (x) ≥ C 0 |x| 2 + C 1 for all x ∈ R d for some C 0 > 0, C 1 ∈ R; (CV ) V = 0 and W is radial satisfying lim |x|→∞ W (x) = +∞; (CW)
Under these assumptions, the regularized energies E m ε are lower semicontinuous with respect to weak- * convergence (m > 1) and Wasserstein convergence (m = 1), where for the latter we assume ϕ is a Gaussian (c.f. Proposition 3.9, and [3, Lemma 5.
P(R d ) | V dµ ≤ C} is tight for all C > 0; c.f. [3, Remark 5.1.5]. Likewise, Assumption (CW) on W ensures that the set {µ ∈ P(R d ) | W * µ dµ ≤ C} is tight up to translations for all C > 0; c.f. [78, Theorem 3.1].
We now prove existence of minimizers of E m ε , for all ε > 0.
PROPOSITION 4.4. Let ε > 0. For m > 1, if either Assumption (CV) or (CW) holds, then minimizers of E m ε over P(R d ) exist. For m = 1, if (CV ) holds and ϕ is a Gaussian, then minimizers of E 1 ε over P 2 (R d ) exist. Proof. First suppose m > 1, so that F ε ≥ 0 and E m
ε is bounded below. By Remark 4.3, if (CV) holds, then any minimizing sequence of E m ε has a subsequence that converges in the weak- * topology. Likewise, if (CW) holds, then any minimizing sequence of E m ε has a subsequence that, up to translation, converges in the weak- * topology. By lower semicontinuity of E m ε , the limits of minimizing sequences are minimizers of E m ε . Now, suppose m = 1. By Proposition 3.8, for all δ > 0 and µ ∈ P 2 (R d ),
F m ε (µ) ≥ −(2π/δ) d/2 − 2δ(M 2 (µ) + ε 2 M 2 (ζ))
, Consequently, by the assumption in (CV ) and the fact that W is bounded below by, say,C ∈ R, we can choose δ = C 0 /2 and obtaiñ
C + C 0 M 2 (µ) + C 1 − (4π/C 0 ) d/2 − C 0 ε 2 M 2 (ζ) ≤ E m ε (µ) for all µ ∈ P 2 (R d ),(20)
Hence any minimizing sequence (µ n ) n ⊂ P 2 (R d ) has bounded second moment. Thus, (µ n ) n has a subsequence that converges in the weak- * topology to a limit with bounded second moment. By the lower semicontinuity of E m ε the limit must be a minimizer of E m ε . Finally, we conclude that minimizers of the regularized energy converge to minimizers of the unregularized energy.
(µ ε ) ε ⊂ P(R d ) such that µ ε is a minimizer of E m ε for all ε > 0, we have, up to a subsequence, µ ε * µ, where µ is minimizes E m . Alternatively, if Assumption (CW)
holds, then we have µ ε * µ, up to a subsequence and translation, where again µ minimizes E m . Now suppose m = 1. If Assumption (CV ) holds and ϕ is a Gaussian, then for any sequence
(µ ε ) ε ⊂ P 2 (R d ) such that µ ε is a minimizer of E 1 ε for all ε > 0, we have, up to a subsequence, µ ε * µ, where µ minimizes E 1 .
Proof. The proof is classical. We include it for completeness. We only prove the result under Assumptions (CV)/(CV ) since the argument for (CW) is analogous. For any ε > 0, since µ ε is a minimizer of E m ε we have that E m ε (µ ε ) ≤ E m ε (ν) for all ν ∈ P(R d ) 15 if m > 1, and for all ν ∈ P 2 (R d ) if m = 1. Taking the infimum limit of the left-hand side and the supremum limit of the right-hand side, Theorem 4.1(ii) ensures that
lim inf ε→0 E m ε (µ ε ) ≤ lim sup ε→0 E m ε (ν) ≤ E m (ν).(21)
Since E m is proper there exists ν ∈ P(R d ) if m > 1 and ν ∈ P 2 (R d ) if m = 1 so that the right-hand side is finite. Thus, up to a subsequence, we may assume that {E m ε (µ ε )} ε is uniformly bounded. When m > 1, F ε (µ) ≥ 0 for all ε ≥ 0, and this implies that { V dµ ε } ε is uniformly bounded, so {µ ε } ε is tight. When m = 1, the inequality in (20) ensures that {M 2 (µ ε )} ε is uniformly bounded, so again {µ ε } ε is tight. Thus, up to a subsequence, (µ ε ) ε converges weakly- * to a limit µ ∈ P(R d ) if m > 1 and µ ∈ P 2 (R d ) if m = 1. By Theorem 4.1(i) and the inequality in (21), we obtain
E m (µ) ≤ lim inf ε→0 E m ε (µ ε ) ≤ E m (ν) for all ν ∈ P(R d ) if m > 1 and for all ν ∈ P 2 (R d ) if m = 1. Therefore, µ is a minimizer of E m .
REMARK 4.6 (convergence of minimizers). One the main difficulties for improving the topology in which the convergence of the minimizers happen is that we do not control L m -norms of the regularized minimizing sequences due to the special form of our regularized energy. This is the main reason we only get weak- * convergence in the previous result and the main obstacle to improve results for the Γ-convergence of gradient flows, as we shall see in the next section.
Γ-convergence of gradient flows
We now consider gradient flows of the regularized energies E m ε , as in Definition 3.6, for m ≥ 2 and prove that, under sufficient regularity assumptions, gradient flows of the regularized energies converge to gradient flows of the unregularized energy as ε → 0. For simplicity of notation, we often write E m ε and F m ε for ε ≥ 0 when we refer jointly to the regularized and unregularized energies. We begin by showing that the gradient flows of the regularized energies are well-posed, provided that V and W satisfy the following convexity and regularity assumptions.
ASSUMPTION 5.1 (convexity and regularity of V and W ). The potentials V, W ∈ C 1 (R d ) are semiconvex, with at most quadratic growth, and W is even. Furthermore, there exist C 0 , C 1 > 0 so
|W (x)|, |∇V (x)|, |∇W (x)| ≤ C 0 + C 1 |x| m−1 for all x ∈ R d .
REMARK 5.2 (ω-convexity). More generally, our results naturally extend to drift and interaction energies that are merely ω-convex; see [35]. However, given that the main interest of the present work is approximation of diffusion, we prefer the simplicity of Assumption (5.1), as it allows us to focus our attention on the regularized internal energy.
PROPOSITION 5.3. Let ε ≥ 0 and m ≥ 2. Suppose E m
ε is as in Definition 3.6 and V and W satisfy Assumption 5.1. Then, for any µ 0 ∈ D(E m ε ), there exists a unique gradient flow of E m ε with initial datum µ 0 .
Proof. It suffices to verify that E m ε is proper, coercive, lower semicontinuous with respect to 2-Wasserstein convergence, and semiconvex along generalized geodesics; c.f. [3,Theorem 11.2.1]. (See also [3, Equation (2.1.2b)] for the definition of coercive.) If ε > 0, then F m ε is finite on all of P 2 (R d ), and if ε = 0, then F m is proper. Thus, our assumptions on V and W ensure that E m ε is proper. Clearly F m ε is bounded below. Hence, since the semiconvexity of V and W ensures that their negative parts have at most quadratic growth, E m ε is coercive. For ε > 0, Proposition 3.9 ensures that F m ε is lower semicontinuous with respect to weak- * convergence, hence also 2-Wasserstein convergence. For ε = 0, the unregularized internal energy F m is also lower semicontinuous with respect to weak- * and 2-Wasserstein convergence [64,Lemma 3.4]. Since V and W are lower semicontinuous and their negative parts have at most quadratic growth, the associated potential and interaction energies are lower semicontinuous with respect to 2-Wasserstein convergence [3, Lemma 5.1.7, Example 9.3.4]. Therefore, E m ε is lower semicontinuous for all ε ≥ 0.
For ε > 0, Proposition 3.11 ensures that F m ε is semiconvex along generalized geodesics in P 2 (R d ). For ε = 0, the unregularized internal energy F m is convex [64,Theorem 2.2]. For V and W semiconvex, the corresponding drift V dµ and interaction (1/2) (W * µ) dµ energies are semiconvex [3, Proposition 9.3.2], [24,Remark 2.9]. Therefore, the resulting regularized energy E m ε is semiconvex.
In the case ε = 0, gradient flows of the energies E m are characterized as solutions of the partial differential equation (1); c.f. [3, Theorems 10.4.13 and 11.2.1], [24,Theorem 2.12]. Now, we show that gradient flows of the regularized energies E m ε can also be characterized as solutions of a partial differential equation.
v = −∇V − ∇W * µ ε − ∇ϕ ε * (ϕ ε * µ ε ) m−2 µ ε − (ϕ ε * µ ε ) m−2 ∇ϕ ε * µ ε .(22)
Moreover,
T 0 v(t) 2 L 2 (µε;R d ) dt < ∞ for all T > 0. Proof. Suppose µ ε ∈ AC 2 loc ((0, +∞); P 2 (R d )) is the gradient flow of E m ε .
Then, by Definition 2.9 and Corollary 3.13, µ ε is a weak solution to the continuity equation with velocity field (22). Conversely, suppose µ ε is a weak solution to the continuity equation with velocity field (22). By
Corollary 3.13, −v(t) ∈ ∂E(µ(t)) ∩ Tan µ(t) P 2 (R d ) for almost every t ∈ (0, ∞). Furthermore, since T 0 v(t) 2 L 2 (µε;R d ) dt < ∞ for all T > 0, µ ε ∈ AC 2 loc ((0, +∞); P 2 (R d )) by [3, Theorem 8.3.1].
A consequence of the previous proposition is that, for the regularized energies E m ε , particles remain particles, i.e. a solution of the gradient flow with initial datum given by a finite sum of Dirac masses remains a sum of Dirac masses, and the evolution of the trajectories of the particles is given by a system of ordinary differential equations.
Ẋ i (t) = −∇V (X i (t)) − j∈I ∇W (X i (t) − X j (t))m j − ∇ δF m ε δµ (Σ j δ X j (t) m j ), t ∈ [0, T ], X i (0) = X 0 i ,(23)
is well-posed for all T > 0. Furthermore, µ ε = i∈I δ X i (·) m i belongs to AC 2 ([0, T ]; P 2 (R d )) and is the gradient flow of E m ε with initial conditions µ ε (0) :
= i∈I δ X 0 i m i .
Proof. To see that (23) is well-posed, first note that the function
(y 1 , . . . , y N ) →∇ δF m ε δµ (Σ j δ y j m j ) = j∈I k∈I ϕ ε (y j − y k )m k m−2 + k∈I ϕ ε (y i − y k )m k m−2 ∇ϕ ε (y i − y j )m j
is Lipschitz. Likewise, Assumption 5.1 ensures y i → ∇V (y i ) and y i → j∈I ∇W (y i − y j ) are continuous and one-sided Lipschitz. Therefore, the ODE system (23) is well-posed forward in time.
Now, suppose (X i ) N i=1 solves (23) with initial data (X 0 i ) N i=1
on an interval [0, T ], for some fixed T . We abbreviate by v i = v i (X 1 , X 2 , . . . , X N ) the velocity field for X i in (23). For any test function ϕ ∈ C ∞ c (R d × (0, T )), the fundamental theorem of calculus ensures that, for all i ∈ I,
T 0 ∇ϕ(X i (t), t)Ẋ i (t) + ∂ t ϕ(X i (t), t) dt = −ϕ(X i (0), 0).
Combining this with (23), we obtain
T 0 ∂ t ϕ(X i (t), t) dt + ϕ(X 0 i , 0) − T 0 ∇ϕ(X i (t), t)v i (t) dt = 0
Multiplying both sides by m i , summing over i, and taking
µ ε = i∈I δ X i (·) m i for t ∈ [0, T ] gives T 0 R d ∂ t ϕ(t, x) dµ ε (t, x)dt + R d ϕ(0, x) dµ ε (0, x) + T 0 R d ∇ϕ(t, x)v(t, x) dµ ε (t, x) dt = 0,
for v as in (22). Therefore, µ ε is a weak solution of the continuity equation with velocity field v.
Furthermore, for all T > 0 T 0 v(t) 2 L 2 (µε;R d ) dt ≤ 2 max (i,j,k)∈I 3 T 0 |∇V (X i (t))| 2 + |∇W (X i (t) − X j (t))| 2 dt + T 0 (ϕ ε (X j (t) − X k (t)) m−2 + (ϕ ε (X i (t) − X k (t)) m−2 2 × |∇ϕ ε (X i (t) − X j (t))| 2 dt < ∞,
by the continuity of ∇V , ∇W , and ϕ ε . Therefore, by Proposition 5.4, we conclude that µ ε ∈ AC 2 ([0, T ]; P 2 (R d )) and µ ε is the gradient flow of E m ε .
We now turn to the Γ-convergence of the gradient flows of the regularized energies, using the scheme introduced by Sandier-Serfaty [75] and then generalized by Serfaty [77], which provides three sufficient conditions for concluding convergence. We will use the following variant of Serfaty's result, which allows for slightly weaker assumptions on the gradient flows of the regularized energies, but follows from the same argument as Serfaty's original result. (See also Remark 2.11 on the correspondence between Wasserstein gradient flows and curves of maximal slope.) THEOREM 5.6 (c.f. [77,Theorem 2]). Let m ≥ 2. Suppose that, for all ε > 0, µ ε belongs to AC 2 ([0, T ]; P 2 (R d )) and is a gradient flow of E m ε with well-prepared initial data, i.e.,
µ ε (0) * µ(0), lim ε→0 E m ε (µ ε (0)) = E m (µ(0)), µ(0) ∈ D(E m ).(S0)
Suppose further that there exists a curve µ in P 2 (R d ) such that, for almost every t ∈ [0, T ], µ ε (t) * µ(t) and
(S1) lim inf
ε→0 t 0 |µ ε |(s) 2 ds ≥ t 0 |µ |(s) 2 ds, (S2) lim inf ε→0 E m ε (µ ε (t)) ≥ E m (µ(t)), (S3) lim inf ε→0 t 0 |∂E m ε | 2 (µ ε (s)) ds ≥ t 0 |∂E m | 2 (µ(s)) ds.
Then µ ∈ AC 2 ([0, T ]; P 2 (R d )), and µ is a gradient flow of E m .
For simplicity of notation, in what follows we shall at times omit dependence on time when referring to curves in the space of probability measures.
In order to apply Serfaty's scheme in the present setting to obtain Γ-convergence of the gradient flows, a key assumption is that the following quantity is bounded uniformly in ε > 0 along the gradient flows µ ε of the regularized energies E m ε :
µ ε BV m ε := R d R d ζ ε (x − y) (∇ζ ε * p ε )(x) + (∇ζ ε * µ ε )(x)(ϕ ε * µ ε )(y) m−2 dµ ε (y) dx ,
where we use the abbreviation p ε := (ϕ ε * µ ε ) m−2 µ ε . This quantity differs from ∇δF m ε /δµ ε L 1 (µε;R d ) merely by the placement of the absolute value sign:
µ ε BV m ε ≥ R d R d ζ ε (x − y)(∇ζ ε * p ε )(x) + (∇ζ ε * µ ε )(x)(ϕ ε * µ ε )(y) m−2 dx dµ ε (y) = (∇ϕ ε * p ε ) + (∇ϕ ε * µ ε )(ϕ ε * µ ε ) m−2 dµ ε = ∇ δF m ε δµ ε L 1 (µε;R d ) .(24)
Serfaty's scheme allows one to assume, without loss of generality, that |F m ε |(µ ε ) is bounded uniformly in ε > 0 for almost everywhere t ∈ [0, T ], and Hölder's inequality ensures that |F m ε |(µ ε ) = ∇δF m ε /δµ ε L 2 (µε;R d ) ≥ ∇δF m ε /δµ ε L 1 (µε;R d ) ; see Proposition 3.12. Consequently, we miss the bound we require on µ ε BV m ε merely by placement of the absolute value sign in inequality (24). Still, µ ε BV m ε has a useful heuristic interpretation. Through the proof of Theorem 5.8, we obtain
lim inf ε→0 T 0 ∇ δF m ε δµ ε L 1 (µε;R d ) dt ≥ m m − 1 T 0 ∇µ(t) m−1 L 1 (µ(t);R d ) dt = T 0 R d |∇µ(t, x) m | dx dt;(25)
see the inequality (33) and Proposition B.2. Consequently, one may think of µ ε BV m ε as a nonlocal approximation of the L 1 -norm of the gradient of µ m .
We begin with a technical lemma we shall use to prove the convergence of the gradient flows.
LEMMA 5.7. Let ε > 0 and m ≥ 2, and let T > 0 and µ ε ∈ AC 2 ([0, T ]; P 2 (R d )). Then for any Lipschitz function f :
[0, T ] × R d → R with constant L f > 0, there exists r > 0 so that [(ζ ε * (f µ ε )) − f (ζ ε * µ ε )] (∇ζ ε * p ε ) + [(ζ ε * (f p ε )) − f (ζ ε * p ε )] (∇ζ ε * µ ε ) L 1 ([0,T ]×R d ) ≤ ε r L f T 0 µ ε (t) BV m ε dt + 2C ζ ∇ζ L 1 (R d ) T 1/(m−1) T 0 F m ε (µ ε (t)) dt m−2 m−1 ,
where C ζ > 0 is as in Assumption 2.1.
Proof. We argue similarly as in Lemma 2.2. Let f :
[0, T ] × R d → R be Lipschitz with constant L f > 0. Then, [(ζ ε * (f µ ε )) − f (ζ ε * µ ε )] (∇ζ ε * p ε ) + [(ζ ε * (f p ε )) − f (ζ ε * p ε )] (∇ζ ε * µ ε ) dL d = R d R d ζ ε (x − y)[f (y) − f (x)] (∇ζ ε * p ε )(x) + (∇ζ ε * µ ε )(x)(ϕ ε * µ ε )(y) m−2 dµ ε (y) dx ≤ L f R d R d ζ ε (x − y)|x − y| (∇ζ ε * p ε )(x) + (∇ζ ε * µ ε )(x)(ϕ ε * µ ε )(y) m−2 dµ ε (y) dx . 19
By Assumption 2.1, C ζ is so that ζ(x) ≤ C ζ |x| −q for q > d + 1 for all x ∈ R d . Chooser so that
0 <r < q − (d + 1) q − 1 .(26)
Now, we break the integral with respect to dµ ε (y) above into integrals over the domain B εr (x) and R d \ B εr (x), bounding the above quantity by
L f R d B εr (x) ζ ε (x − y)|x − y| (∇ζ ε * p ε )(x) + (∇ζ ε * µ ε )(x)(ϕ ε * µ ε )(y) m−2 dµ ε (y) dx + L f R d R d \B εr (x) ζ ε (x − y)|x − y| (∇ζ ε * p ε )(x) + (∇ζ ε * µ ε )(x)(ϕ ε * µ ε )(y) m−2 dµ ε (y) dx, =: I 1 + I 2
First, we consider I 1 . Since, in the integral, |x − y| < εr, we obtain
I 1 < εrL f µ ε BV m ε . Now, we consider I 2 .
We apply the inequality in (52) to obtain ζ ε (x − y)|x − y| ≤ C ζ εr with r :=r(1 − q) + q − d in the integral-the inequality in (26) ensuresr > 1. Consequently,
I 2 ≤ εrL f C ζ |∇ζ ε * p ε | dL d dµ ε + |∇ζ ε * µ ε | dL d p ε dL d ≤ 2εrL f C ζ ∇ζ ε L 1 (R d ) p ε L d ≤ 2εr −1 L f C ζ ∇ζ L 1 (R d ) F m ε (µ) (m−2)/(m−1) ,
where, in the last inequality, we use that ∇ζ ε L 1 (R d ) = ∇ζ L 1 (R d ) /ε and, by Jensen's inequality for the concave function s (m−2)/(m−1) ,
p ε dL d = (ϕ ε * µ ε ) m−2 dµ ε ≤ (ϕ ε * µ ε ) m−1 dµ ε (m−2)/(m−1) = F m ε (µ ε ) (m−2)/(m−1) .(27)
Since 0 ≤ (m − 2)/(m − 1) < 1, Jensen's inequality gives
T 0 F m ε (µ ε (t)) (m−2)/(m−1) dt ≤ T 1 T T 0 F m ε (µ ε (t)) dt (m−2)/(m−1) .(28)
This gives the result by taking r := min(r,r − 1).
With this technical lemma in hand, we now turn to the Γ-convergence of the gradient flows.
that µ ε ∈ AC 2 ([0, T ]; P 2 (R d )) is a gradient flow of E m ε for all ε > 0 satisfying µ ε (0) * µ(0), lim ε→0 E m ε (µ ε (0)) = E m (µ(0)),(A0)
for some µ(0) ∈ D(E m ). Furthermore, suppose that the following hold:
(A1) sup ε>0 T 0 M m−1 (µ ε (t)) dt < ∞; (A2) sup ε>0 T 0 µ ε (t) BV m ε dt < ∞; (A3) there exists µ : [0, T ] → P 2 (R d ) such that ζ ε * µ ε (t) → µ(t) in L 1 ([0, T ]; L m loc (R d )) as ε → 0, and sup ε>0 T 0 ζ ε * µ ε (t) m L m (R d ) dt < ∞. Then µ ε (t) * µ(t) for almost every t ∈ [0, T ], µ ∈ AC 2 ([0, T ]; P 2 (R d ))
, and µ is the gradient flow of E m with initial data µ(0). 20 Proof. First, we prove that µ ε (t) * µ(t) for almost every t ∈ [0, T ]. By assumption (A1),
sup ε>0 T 0 M m−1 (µ ε (t)) dt < ∞. Thus, Fatou's lemma implies T 0 lim inf ε→0 R d |x| m−1 dµ ε (t, x) dt ≤ lim inf ε→0 T 0 R d |x| m−1 dµ ε (t, x)dt < +∞,
and lim inf ε→0 M m−1 (µ ε (t)) < ∞ for almost every t ∈ [0, T ]. Consequently, for almost every t ∈ [0, T ], every sequence (µ ε (t)) ε has a further subsequence along which {M m−1 (µ ε (t))} ε is uniformly bounded in ε; hence, a further subsequence that converges in the weak- * topology to some ν(t) ∈ P(R d ). By Proposition 2.3, this also implies ζ ε * µ ε (t) * ν(t). Since both this and Assumption (A3) imply convergence in weak sense, by uniqueness of limits, ν(t) = µ(t). Thus, µ ε (t) * µ(t) for
almost every t ∈ [0, T ].
It remains to verify conditions (S0), (S1), (S2), and (S3) from Theorem 5.6. Item (S0) holds by hypothesis. Item (S1) follows by the same argument as in [37,Theorem 5.6]. Item (S2) is an immediate consequence of the fact that µ ε (t) * µ(t) for almost every t ∈ [0, T ] and the lower semicontinuity of the potential and interaction energies with respect to weak- * convergence [3, Lemma 5.1.7]. We devote the remainder of the proof to showing Condition (S3). We shall use the following fact throughout: combining Assumption (A3) with Proposition 3.8 implies that
sup ε>0 T 0 F m ε (µ ε (t)) dt ≤ sup ε>0 1 m − 1 T 0 ζ ε * µ ε (t) m L m (R d ) dt < ∞.(29)
To prove (S3) we may assume, without loss of generality, that lim inf ε→0
T 0 |∂E m ε |(µ ε (t)) 2 dt is finite, so by Fatou's lemma ∞ > lim inf ε→0 T 0 |∂E m ε |(µ ε (t)) 2 dt ≥ T 0 lim inf ε→0 |∂E m ε |(µ ε (t)) 2 dt,(30)
so lim inf ε→0 |∂E m ε |(µ ε (t)) < ∞ for almost every t ∈ [0, T ]. In particular, up to taking subsequences, we may assume that, for almost every t ∈ [0, T ], {|∂E m ε |(µ ε (t))} ε is bounded uniformly in ε > 0. By Corollary 3.13,
|∂E m ε |(µ ε ) = ∇V + ∇W * µ ε + ∇ δF m ε δµ ε (µ ε ) L 2 (µε;R d )
.
Furthermore, note that if µ m ∈ W 1,1 (R d ) and ∇µ m + ∇V µ + (∇W * µ)µ = ξµ for some ξ ∈ L 2 (µ; R d ),
ε→0 T 0 ∇V + ∇W * (µ ε (t)) + ∇ δF m ε δµ ε (µ ε (t)) 2 dµ ε (t) dt ≥ T 0 |ξ(t)| 2 dµ(t) dt,(32)lim ε→0 T 0 f (t) ∇V + ∇W * µ ε (s) + ∇ ∂F m ε ∂µ ε (s) dµ ε (s)ds = T 0 f (t)ξ(t) dµ(t) ds,(33)for all f ∈ C ∞ c ([0, T ] × R d ).
Observe that Proposition B.2 is stated for probability measures-we can easily rescale dµ ε ⊗ dL d to be a probability measure by diving the above equations by T > 0.
First, we address the terms with the drift and interaction potentials V and W . Combining Assumption 5.1 on V and W with Assumption (A1) on µ ε ensures that |∇V | is uniformly integrable in dµ ε ⊗ dL d and (x, y) → |∇W (x − y)| is uniformly integrable dµ ε ⊗ dµ ε ⊗ dL d .Therefore, by [3, Lemma 5.1.7], (µ ε ) ε converging weakly- * to µ ensures that
lim ε→0 T 0 f (t) (∇V + ∇W * (µ ε (t))) dµ ε (t) dt = m m − 1 T 0 R d f (t) ∇V + ∇W * (µ(t)) dµ(t) dt.
Now we deal with proving the diffusion part of (31) (that is, for almost every t ∈ [0, T ], we have µ(t) m ∈ W 1,1 (R d ) and ∇µ(t) m = η(t)µ(t) for η ∈ L 2 (µ; R d )), and with proving that
lim ε→0 T 0 f (t)∇ δF m ε δµ ε (µ ε (t)) dµ ε (t) dt = T 0 f (t)η(t) dµ(t) dt,(34)
Recalling the abbreviation p ε := (ϕ ε * µ ε ) m−2 µ ε , we rewrite the inner integral on the left-hand side of (34) as
f ∇ ∂F m ε ∂µ ε dµ ε = f (∇ϕ ε * p ε ) + (ϕ ε * µ ε ) m−2 (∇ϕ ε * µ ε ) dµ ε = (ζ ε * (f µ ε ))(∇ζ ε * p ε ) + (ζ ε * (f p ε ))(∇ζ ε * µ ε ) dL d .
Applying Lemma 5.7 together with (29) and (A3), and integrating by parts, we obtain
lim ε→0 T 0 f (t)∇ δF m ε δµ ε (µ ε (t)) dµ ε (t) dt = lim ε→0 T 0 f (t)(ζ ε * (µ ε (t)))(∇ζ ε * (p ε (t))) dL d dt + T 0 f (t)(ζ ε * (p ε (t)))(∇ζ ε * (µ ε (t))) dL d dt = − lim ε→0 T 0 ∇f (t)(ζ ε * (µ ε (t)))(ζ ε * (p ε (t))) dL d dt = − lim ε→0 T 0 ζ ε * (∇f (t)(ζ ε * (µ ε (t))))p ε (t) dL d dt.
Now we move ∇f out of the convolution. By Lemma 2.2, there exists p > 0 so
ζ ε * (∇f (ζ ε * µ ε ))p ε dL d − ∇f (ζ ε * (ζ ε * µ ε ))p ε dL d ≤ ε p ∇f L ∞ ([0,T ]×R d ) (ϕ ε * µ ε ) m−1 dµ ε + C ζ p ε dL d ≤ ε p ∇f L ∞ ([0,T ]×R d ) F m ε (µ ε ) + C ζ F m ε (µ ε ) (m−2)/(m−1) ,
where we again use (27). Using the inequality in (28) and that
{ T 0 F m ε (µ ε (t)) dt} ε is uniformly bounded in ε, − lim ε→0 T 0 f (t)∇ δF m ε δµ ε (µ ε (t)) dµ ε (t) dt = lim ε→0 T 0 ∇f (t)(ϕ ε * µ ε )p ε dL d dt (35) = lim ε→0 T 0 R d ∇f (t)(ϕ ε * µ ε (t)) m−1 dµ ε (t) dt.(36)
To conclude the proof, we aim to apply Proposition B.2(iii), and we begin by verifying the hypotheses of this proposition. First, note that since ζ ε * µ ε → µ in L 1 ([0, T ]; L m loc (R d )) for m ≥ 2 as ε → 0, we also have ζ ε * µ ε → µ in L 1 ([0, T ]; L 2 loc (R d )). Let w ε = ϕ ε * µ ε . By definition, w ε dµ ε = (ζ ε * µ ε ) 2 dL d . Thus, Assumption (A3) and the fact that ζ ε * µ ε (R d ) = 1 imply
sup ε>0 T 0 |ζ ε * µ ε (t)| 2 dL d dt < ∞,
so that w ε ∈ L 1 ([0, T ], L 1 (µ ε ; R d )). Furthermore, for any h ∈ L ∞ ([0, T ]; W 1,∞ (R d )), the mollifier exchange lemma 2.2 and the convergence of
ζ ε * µ ε to µ in L 1 ([0, T ]; L 2 loc (R d )) give T 0 h(t)w ε (t) dµ ε (t) = T 0 ζ ε * (hµ ε (t)) dζ ε * (µ ε (t)) dt = T 0 h(t)(ζ ε * µ ε (t)) 2 dL d dt(37)+ ε p ∇h L ∞ ([0,T ];W 1,∞ (R d )) T 0 ζ ε * (µ ε (t)) 2 L 2 (R d ) dL d dt + C ζ −→ T 0 h(t)µ(t) 2 dL d dt,
as ε → 0. Thus, w ε ∈ L 1 ([0, T ]; L 1 (µ ε ; R d )) converges weakly to µ ∈ L 1 ([0, T ]; L 1 (dµ)) in the sense of Definition B.1 as ε → 0. As before, while this definition is stated for probability measures, we can easily rescale dµ ε ⊗ dL d to be a probability measure by diving the above equations by T > 0. We now seek to show that, for all
g ∈ C ∞ c ([0, T ] × R d ), lim ε→0 T 0 g(t)|w ε (t)| m−1 dµ ε (t) dt = T 0 g(t)|µ(t)| m−1 dµ(t).
When m = 2, this follows from equation (37). Suppose m > 2. Let κ :
R d → R be a smooth cutoff function with 0 ≤ κ ≤ 1, ∇κ L ∞ (R d ) ≤ 1, D 2 κ L ∞ (R d ) ≤ 4,|κ R w ε (t)| m−1 dµ ε (t) dL d dt ≤ lim sup ε→0 T 0 (ζ ε * (µ ε (t))) m−1 ζ ε * (κ m−1 R µ ε (t)) dL d dt ≤ lim sup ε→0 T 0 κ m−1 R (ζ ε * (µ ε (t))) m dL d dt = T 0 (κ R µ(t)) m−1 dµ(t) dt.
Combining this with (37), where we may choose h = κ R g for any g ∈ C ∞ c (R d ), we have that (κ R w ε ) ε converges strongly in L m−1 (µ ε ; R d ) to κ R µ ∈ L m−1 (µ; R d ) as ε → 0, in the sense of Definition B.1. Finally, since Assumption (A1) ensures that T 0 M m−1 (µ ε (t)) ds is bounded uniformly in ε, we may apply Proposition B.2(iii) to conclude that for all
g ∈ C ∞ c ([0, T ] × R d ), lim ε→0 t 0 R d g|κ R w ε | m−1 dµ ε = t 0 R d g|κ R µ| m−1 dµ.
Taking g = ∇f , choosing R > 1 so that κ R ≡ 1 on the support of ∇f , and combining the above equation with equation (35), we obtain
lim ε→0 T 0 f (t)∇ δF m ε δµ ε (µ ε (t)) dµ ε (t) dt = − T 0 ∇f (t)µ(t) m dL d dt.(38)
We now prove that µ has the necessary regularity. In particular, we show that for almost every t ∈ [0, T ], we have µ m ∈ W 1,1 (R d ) and ∇µ m = ηµ for η ∈ L 2 (µ; R d ). Inequality (30) ensures that, up to subsequences { t 0 |∂F m ε | 2 (µ ε (t)) dt} ε is bounded uniformly in ε > 0. Thus, by Hölder's inequality, there exists C > 0 so that
C > T 0 ∇ δF m ε δµ ε (µ ε (t)) 2 L 2 (µε;R d ) dt ≥ T 0 ∇ δF m ε δµ ε (µ ε (t)) 2 L 1 (µε;R d ) dt ≥ T 1 T T 0 ∇ δF m ε δµ ε (µ ε (t)) L 1 (µε;R d ) dt 2 ,
for all ε > 0. Combining this with (38) gives
CT f L ∞ (R d ) ≥ lim sup ε→0 f L ∞ ([0,T ]×R d ) T 0 ∇ δF m ε δµ ε (µ ε (t)) L 1 (µε;R d ) ≥ T 0 f (t)∇(µ(t) m ) dL d dt.
Hence ∇(µ m ) has finite measure on [0, T ] × R d , so we may rewrite (38) as
lim ε→0 t 0 f ∇ δF m ε δµ ε (µ ε (t)) dµ ε (t) dt = − t 0 f (t) d∇(µ(t) m ) dt.(39)
By another application of Hölder's inequality, this guarantees
√ C t 0 f (t) 2 L 2 (µ;R d ) dt 1/2 ≥ lim sup ε→0 t 0 f (t) L 2 (µε;R d ) ∇ δF m ε δµ ε (µ ε (t)) L 2 (µε;R d ) ≥ t 0 f (t)d∇(µ(t) m ) dt.
Riesz representation theorem then ensures that there exists η ∈ L 2 ([0, t]; L 2 (µ; R d )) so that ηµ = ∇(µ m ). In particular, this implies ∇(µ(t) m ) ∈ L 1 (R d ) for almost every t ∈ [0, T ], so µ m ∈ W 1,1 (R d ) for almost every t ∈ [0, T ] and we may rewrite (39) as
lim ε→0 T 0 R d f (t)∇ δF m ε δµ ε (µ ε (t)) dµ ε (t) dt = − T 0 f (t)η dµ(t) dt,
which completes the proof.
We conclude this section by showing that, in the case when m = 2 and for V, W ∈ C 2 (R d ) with bounded Hessians, whenever the initial data of the gradient flows have bounded second moments and internal energies, we automatically obtain Assumptions (A1)-(A3). Consequently, in this special case, we are able to conclude the convergence of the gradient flows without these additional assumptions.
COROLLARY 5.9. Let ε > 0 and m = 2. In addition to satisfying Assumption 5.1, assume that V, W ∈ C 2 (R d ) have bounded Hessians D 2 V and D 2 W . Fix T > 0, and suppose µ ε ∈ AC 2 ([0, T ]; P 2 (R d )) is a gradient flow of E m ε satisfying
µ ε (0) * µ(0), lim ε→0 E m ε (µ ε (0)) = E m (µ(0)), µ(0) ∈ D(E m ),(40)sup ε>0 M 2 (µ ε (0)) < ∞, sup ε>0 µ ε (0) log(µ ε (0)) dL d < +∞.(41)
Then, there exists µ ∈ AC 2 ([0, T ]; P 2 (R d )) such that µ ε (t) * µ(t) and ζ ε * µ ε (t)
L 2 (R d ) − −−− → µ(t) for all t ∈ [0, T ],
and µ is the gradient flow of E m with initial data µ(0).
24
REMARK 5.10 (Previous work, m = 2). The above theorem generalizes a result by Lions and Mas-Gallic [59] on a numerical scheme for the porous medium equation ∂ t µ = ∆µ 2 on a bounded domain with periodic boundary conditions to equations of the form (1) on Euclidean space.
Proof of Corollary 5.9. First, we show that sup ε>0 ζ ε * (µ ε (0)) L 2 (R d ) < ∞. The fact that D 2 V and D 2 W are bounded ensures |V | and |W | grow at most quadratically. Combining this with equations (40)- (41), which ensure {E m ε (µ(0))} ε and {M 2 (µ ε (0))} ε are bounded uniformly in ε > 0, we obtain
sup ε>0 ζ ε * (µ ε (0)) 2 L 2 (R d ) = sup ε>0 F 2 ε (µ ε (0)) = sup ε>0 E 2 ε (µ ε (0)) − V dµ ε (0) − 1 2 W * (µ ε (0))dµ ε (0) < +∞.
Furthermore, since the energy F 2 ε decreases along solutions to the gradient flow, we have sup
ε>0 ζ ε * (µ ε (t)) 2 L 2 (R d ) ≤ sup ε>0 ζ ε * (µ ε (0)) 2 L 2 (R d ) < ∞ for all t ∈ [0, T ].(42)
Next, we show that our assumption that the initial data has bounded entropy (41) ensures
t 0 ∇ζ ε * (µ ε (s)) 2 L 2 (R d ) ds < C(1 + T ) + M 2 (µ ε (t)) for all t ∈ [0, T ]
, for some C > 0 depending on d, V , W and sup ε>0 log µ ε (0) dµ ε (0). Formally differentiating the entropy F 1 (µ) = log(µ) dµ along the gradient flows µ ε , we expect that, for all t ∈ [0, T ],
d dt F 1 (µ ε (t)) = −2 |∇ζ ε * (µ ε (t))| 2 dL d + ∆V dµ ε (t) + ∆W * (µ ε (t)) dµ ε (t).
Hence, for any t ∈ [0, T ],
F 1 (µ ε (t)) − F 1 (µ ε (0)) = −2 t 0 |∇ζ ε * (µ ε (s))| 2 dL d ds + t 0 ∆V dµ ε (s) ds + t 0 ∆W * (µ ε (s)) dµ ε (s) ds ≤ −2 t 0 |∇ζ ε * (µ ε (s))| 2 dL d ds + t D 2 V L ∞ (R d ) + D 2 W L ∞ (R d )
This computation can be made rigorous by first proving the analogous inequality along discrete time gradient flows using the flow interchange method of Matthes, McCann, and Savaré [63, Theorem 3.2] and then sending the timestep to zero to recover the above inequality in continuous time. Thus, there exists K 0 > 0 depending on V, W and sup ε>0 F 1 (µ ε (0)) so that, for all t ∈ [0, T ],
t 0 ∇ζ ε * (µ ε (s)) 2 L 2 (R d ) ds ≤ −F 1 (µ ε (t)) + K 0 (1 + t).
Finally, by a Carleman-type estimate [30,Lemma 4
.1], we have F 1 (ν) ≥ −(2π) d/2 − M 2 (ν) for any ν ∈ P 2 (R d ). Therefore, t 0 ∇ζ ε * (µ ε (s)) 2 L 2 (R d ) ds ≤ M 2 (µ ε (t)) + C(1 + t).(43)
Now, we use this estimate to show that {M 2 (µ ε (t))} ε is uniformly bounded in ε for all t ∈ [0, T ]. Let κ be a smooth cutoff function with 0 ≤ κ ≤ 1,
∇κ L ∞ (R d ) ≤ 1, D 2 κ L ∞ (R d ) ≤ 4, κ(x) = 1
for all |x| < 1/2 and κ(x) = 0 for all |x| > 2. Given R > 0, define κ R (x) = κ(x/R), so that ∇κ R L ∞ (R d ) ≤ 1/R and D 2 κ R L ∞ (R d ) ≤ 4/R 2 . Then there exists C κ > 0 so that for all R > 1, |∇(κ R (x)x 2 )| ≤ C κ |x| and |D 2 (κ R (x)x 2 )| ≤ C κ for all x ∈ R d . By Proposition 5.4, µ ε is a weak solution of the continuity equation. Therefore choosing κ R (x)|x| 2 as our test function, we obtain, for all t ∈ [0, T ],
R d κ R (x)|x| 2 dµ ε (t, x) − R d κ R (x)|x| 2 dµ ε (0, x) = −2 t 0 R d ∇(κ R (x)x 2 ) (∇ϕ ε * (µ ε (s)) + ∇V (x) + ∇W * (µ ε (s))(x)) dµ ε (s, x).
Since D 2 V and D 2 W are bounded, |∇V | and |∇W | grow at most linearly. Consequently, there exists C > 0, depending on V , W , and C κ so that
−2 t 0 R d ∇(κ R (x)x 2 )(∇V (x) + ∇W * (µ ε (s))(x) dµ ε (s, x) ≤ C 1 + t 0 M 2 (µ ε (s)) ds .
Likewise, by Lemma 5.7, there exists r > 0 so that, for all t ∈ [0, T ],
− 2 t 0 R d ∇(κ R (x)x 2 )∇ϕ ε * (µ ε (s))(x) dµ ε (s, x) = −2 t 0 R d ζ ε * (∇(κ R (x)x 2 )µ ε (s))∇ζ ε * (µ ε (s))(x) dx ds ≤ −2 t 0 R d ∇(κ R (x)x 2 )ζ ε * (µ ε (s))(x)∇ζ ε * (µ ε (s))(x) dx ds + ε r C κ t 0 µ ε (s) BV m ε ds + 2t ∇ζ L 1 (R d ) = t 0 R d ∆(κ R (x)x 2 )(ζ ε * (µ ε (s))) 2 dx ds + ε r C κ t 0 µ ε (s) BV m ε ds + 2t ∇ζ L 1 (R d ) ≤ C κ t 0 F 2 ε (µ ε (s)) ds + 2ε r C κ t 0 ζ ε * (µ ε (s)) L 2 (R d ) ∇ζ ε * (µ ε (s)) L 2 (R d ) ds + 2t ∇ζ L 1 (R d ) ≤ C κ tF 2 ε (µ ε (0)) + 2ε r C κ tF 2 ε (µ ε (0)) M 2 (µ ε (t)) + C(1 + t) + 2t ∇ζ L 1 (R d ) ≤ C (1 + t + ε r M 2 (µ ε (t)))
for C depending on C κ , sup ε>0 F 2 ε (µ ε (0)), and ∇ζ L 1 (R d ) . In the second inequality, we use that
µ ε BV m ε ≤ 2 (∇ζ ε * µ ε )(ζ ε * µ ε ) L 1 (R d ) ≤ ζ ε * µ ε L 2 (R d ) ∇ζ ε * µ ε L 2 (R d )(44)
Therefore, there exists C > 0 so that, for all t ∈ [0, T ],
R d κ R (x)|x| 2 dµ ε (t, x) ≤ M 2 (µ ε (0)) + C t + t 0 M 2 (µ ε (s)) ds + C (1 + t + ε r M 2 (µ ε (t))) .
As the right-hand side is independent of R > 1, by sending R → +∞ by the dominated convergence theorem we obtain that for ε r < 1/(2C ),
M 2 (µ ε (t)) ≤ 2C t + t 0 M 2 (µ ε (s)) ds + 2C (t + 1).
Therefore, by Gronwall's inequality, there existsC depending on C , C and T (and independent of ε) so that 26 We may combine this with the inequality in (43) to obtain, for all t ∈ [0, T ],
M 2 (µ ε (t)) <C for all t ∈ [0, T ].(45)t 0 ∇ζ ε * (µ ε (s)) 2 L 2 (R d ) ds ≤C + C(1 + t) for t ∈ [0, T ].(46)
We now use these results to verify the assumptions of Theorem 5.8 hold, so that we may apply this result to conclude convergence of the gradient flows. Assumption (A1) is a consequence of the inequality in (45). Assumption (A2) is a consequence of the inequalities in (42), (44) and (46).
It remains to show Assumption (A3). First, note that since sup ε>0 ζ ε * µ ε L ∞ ([0,T ]×R d ) < ∞, every subsequence of (ζ ε * µ ε ) ε has a further subsequence, which we also denote by (ζ ε * µ ε ) ε , that converges weakly in L 2 ([0, T ] × R d ) to some ν as ε → 0, and for which ζ ε * µ ε (t) ν(t) weakly in L 2 (R d ) for all t ∈ [0, T ]. By uniqueness of limits and (40), we have ν(0) = µ(0) almost everywhere.
Next, note that (42) and (46) ensure that sup ε>0 ζ ε * µ L 2 ([0,T ];H 1 (R d )) < ∞. In particular we have sup ε>0 κ R ζ ε * µ L 2 ([0,T ];H 1 (R d )) < ∞ for the smooth cutoff function κ R , R > 1. Therefore, by the Rellich-Kondrachov Theorem (c.f. [44,Section 5.7]), for almost every t ∈ [0, T ], up to another subsequence, (κ R ζ ε * µ ε (t)) ε converges strongly in L 2 (R d ) to some ν R (t). In particular, for any
f ∈ C ∞ c (B R/2 (0)), f dν(t) = lim ε→0 f dζ ε * µ ε (t) = f dν R (t) for all t ∈ [0, T ], so ν = ν R almost everywhere in B R/2 (0). Since R > 1 is arbitrary, this shows that for all t ∈ [0, T ], ζ ε * µ ε (t) → ν(t) strongly in L 2 loc (R d ). Finally, using again that { ζ ε * µ ε (t) L 2 (R d ) } t is bounded uniformly in t ∈ [0, T ], the dominated convergence theorem ensures that ζ ε * µ ε (t) → ν(t) in L 1 ([0, T ]; L 2 loc (R d ) as ε → 0.
This completes the proof of assumption (A3). As we have now verified the conditions of Theorem 5.8, we now conclude that µ ε (t) * ν(t) for almost every t ∈ [0, T ], for some ν ∈ AC 2 ([0, T ]; P 2 (R d )) which is the gradient flow of E 2 with initial data µ(0). By Proposition 5.3, the gradient flow of E 2 with initial data µ(0) is unique. Thus, since any subsequence of (µ ε ) ε has a further subsequence which converges to ν, the full sequence must converge to µ, which gives the result.
Numerical results
6.1. Numerical method and convergence. We now apply the theory of regularized gradient flows developed in the previous sections to develop a blob method for diffusion, allowing us to numerically simulate solutions to partial differential equations of Wasserstein gradient flow type (1). We begin by describing the details of our numerical scheme and applying Theorem 5.8 to prove its convergence, under suitable regularity assumptions.
µ ε (0) := i∈Q h R δ ih m i , m i = Q i dµ(0), i ∈ {1, . . . , N },(47)
where Q i is the cube centered at ih of side length h. Next, for ε > 0, define the evolution of these measures by where {X i (t)} i∈Q h R are solutions to the ODE system (23) on a time interval [0, T ] with initial data X i (0) = ih. If h = o(ε) as ε → 0 and Assumptions (A1)-(A3) from Theorem 5.8 hold, then (µ ε (t)) ε converges in the weak- * topology to µ(t) as ε → 0 for almost every t ∈ [0, T ], where µ(t) is the unique solution of (1) with initial datum µ(0).
µ ε (t) = i∈Q h R δ X i (t) m i , t ∈ [0, T ],(48)
Proof. By Corollary 5.5, µ ε ∈ AC 2 ([0, T ]; P 2 (R d )) is the gradient flow of E m ε with initial condition µ ε (0) for all ε > 0. To apply Theorem 5.8 and obtain the result, it remains to show that Assumption (A0) holds. In particular, we must show that, assuming h = o(ε),
lim ε→0 V dµ ε (0) + 1 2 (W * (µ ε (0))) dµ ε (0) + F m ε (µ ε (0)) = V dµ(0) + 1 2 (W * (µ(0))) dµ(0) + F m (µ(0)). Define T : R d → R d by T (y) = ih for y ∈ Q i and i ∈ Q h R .
Then T is a transport map from µ(0) to µ ε (0) and |T (y) − y| ≤ h for all y ∈ R d . By construction,
W 2 (µ ε (0), µ(0)) ≤ {|T (y) − y| | y ∈ supp µ(0)} ≤ h, so µ ε (0) * µ(0) as ε → 0 (and so, as h → 0). Likewise, for all ε, h > 0, supp µ ε (0) ⊆ B R (0).
Consequently, since V and W are continuous,
lim ε→0 V dµ ε (0) + 1 2 (W * (µ ε (0))) dµ ε (0) = V dµ(0) + 1 2 (W * (µ(0))) dµ(0).
Thus, it remains to show that lim ε→0 F m ε (µ ε (0)) = F m (µ(0)).
By Theorem 4.1, we have that lim inf ε→0 F m ε (µ ε (0)) ≥ F m (µ ε (0)). By Proposition 3.8, for all ε > 0 we have
F m ε (µ ε (0)) ≤ F m (µ ε (0)) = 1 m−1 ζ ε * µ ε m L m (R d ) . Consequently, to show that lim sup ε→0 F m ε (µ ε (0)) ≤ F m (µ(0)) = µ(0) m L m (R d ) /(m − 1)
, it suffices to show that ζ ε * µ ε (0) → µ(0) in L m as ε → 0.
For simplicity of notation, we suppress the dependence on time and show ζ ε * µ ε → µ in L m as ε → 0. By the assumptions that µ ∈ D(E m ) with compact support and V and W are continuous, we have µ ∈ L m (R d ). Consequently ζ ε * µ → µ in L m as ε → 0, and it is enough to show that ζ ε * µ ε − ζ ε * µ → 0 in L m . Using that T is a transport map from µ ε to µ,
|ζ ε * µ ε (x) − ζ ε * µ(x)| = R d ζ ε (x − T (y)) − ζ ε (x − y) dµ(y) ≤ 1 0 R d |∇ζ ε (x − (1 − α)T (y) − αy)| |T (y) − y| dµ(y)dα ≤ h 1 0 R d |∇ζ ε (x − (1 − α)T (y) − αy)| dµ(y)dα.
Combining the decay of ∇ζ from Assumption 2.1 with the fact that ∇ζ is continuous, there exists C > 0 so that |∇ζ(x)| ≤ C(1 B (x)+|x| −q 1 R d \B (x)), where B = B 1 (0) is the unit ball centered at the origin. Note that if |x−y| ≥ 2h, then for all α ∈ [0, 1], |x−(1−α)T (y)−αy| ≥ |x−y|−h ≥ |x−y|/2 and |x − (1 − α)T (y) − αy| ≤ 3|x − y|/2. Thus, by the assumptions on our mollifier, we have
|∇ζ ε (x − (1 − α)T (y) − αy)| ≤ C ε d+1 1 B x − (1 − α)T (y) − αy ε 28 + ε q |x − (1 − α)T (y) − αy| −q 1 R d \B x − (1 − α)T (y) − αy ε ≤ C ε d+1 1 B |x − y| 2ε + 2ε 3 q |x − y| −q 1 B\R d 3|x − y| 2ε .
Thus, taking the L m -norm with respect to x, doing a change of variables, and applying Minkowski's inequality, we obtain
ζ ε * µ ε − ζ ε * µ L m (R d ) ≤ h ∇ζ ε ∞ B 2h (x) µ(y) m + Ch ε d+1 B 2h (x) c 1 B |x − y| 2ε + 2ε 3 q |x − y| −q 1 B\R d 3|x − y| 2ε dµ(y) m = h ∇ζ ε ∞ B 2h (0) µ(x − w)dw m + Ch ε d+1 B 2h (0) c 1 B |w| 2ε + 2ε 3 q |w| −q 1 B\R d 3|w| 2ε µ(x − w)dw m ≤ c µ m h d+1 ε d+1 + h ε ,
where c > 0 depends on C, ∇ζ ∞ , and the space dimension. Therefore, provided that h = o(ε) as ε → 0, we obtain that ζ ε * µ ε − ζ ε * µ → 0 in L m . REMARK 6.2 (compact support of initial data). In Theorem 6.1, we assume that the initial datum of the exact solution µ(0) ∈ D(E m ) is compactly supported. More generally, under the same assumptions on V , W , and m, given any ν 0 ∈ D(E m ) ∩ P 2 (R d ) without compact support, there existsν 0 ∈ D(E m ) with compact support such that ν 0 andν 0 are arbitrarily close in the Wasserstein distance. Furthermore, by the contraction inequality for gradient flows of E m , the solution ν with initial data ν 0 and the solutionν with initial dataν 0 satisfy
W 2 (ν(t),ν(t)) ≤ CW 2 (ν 0 ,ν 0 ) for all t ∈ [0, T ],
where C > 0 depends on T and the semiconvexity of V and W [3, Theorem 11.2.1]. In this way, any solution of (1) with initial datum in D(E m ) ∩ P 2 (R d ) can be approximated by a solution with compactly supported initial datum, so that our assumption of compact support in Theorem 6.1 is not restrictive. REMARK 6.3 (Assumptions (A1)-(A3)). In Theorem 6.1, we prove that, as long as Assumptions (A1)-(A3) from Theorem 5.8 hold along the particle solutions {µ ε } ε , then any limit of these particle solutions must be the corresponding gradient flow of the unregularized energy. Verifying these conditions analytically can be challenging; see Theorem 5.9. However, numerical results can provide confidence that these conditions hold along a given particle approximation. A sufficient condition for Assumption (A1) is that the (m − 1)th moment of the particle solution i∈Q h R |X i (t)| m−1 m i is bounded uniformly in t, ε, and h. In particular, this is satisfied if the particles remain compactly supported in a ball. A sufficient condition for Assumption (A2) is that
|∇ζ ε * p ε | dζ ε * µ ε + |∇ζ ε * µ ε | dζ ε * p ε ,(49)
with p ε = (ϕ ε * µ ε ) m−2 µ ε , remains bounded uniformly in t, ε, and h. In fact, for purely diffusive problems, we observe that this quantity not only bounded uniformly in ε and h, but decreases in time along our numerical solutions; see Figure 3 below. For the nonlinear Fokker-Planck equation, we observe that this quantity is bounded uniformly in ε and h and converges to the corresponding norm of the steady state as t → ∞; see Figure 6 below. A sufficient condition for Assumption (A3) is that the blob solution converges to a limit in L 1 and L ∞ , uniformly on bounded time intervals. Again, we observe this numerically, in both one and two dimensions, and both for purely diffusive equations and the nonlinear Fokker-Planck equation; see Figures 4-6 below. In this way, Assumptions (A1)-(A3) may be verified numerically in order to give confidence that the limit of any blob method solution is, in fact, the correct exact solution.
6.2. Numerical implementation. We now describe the details of our numerical implementation. In all of the numerical examples which follow, our mollifiers ζ ε and ϕ ε are given by Gaussians,
ζ ε (x) = 1 (4πε 2 ) d/2 e −|x| 2 /4ε 2 , ϕ ε (x) = ζ ε * ζ ε (x) = 1 (8πε 2 ) d/2 e −|x| 2 /8ε 2 , x ∈ R d , ε > 0.
In addition to Gaussian mollifiers, we also performed numerical experiments with a range of compactly supported and oscillatory mollifiers and observed similar results. In practice, Gaussian mollifiers provided the best balance between speed of computation and speed of convergence. We construct our numerical particle solutions µ ε (t) as described in Theorem 6.1. As a mild simplification, we consider the mass of each particle to be given by m i = µ(0, ih)h d , where µ(0, ih) is the value of the initial datum µ(0) at the grid point ih. For the numerical examples we consider, in which µ(0) is a continuous function, the rate of convergence is indistinguishable from defining m i as in (47).
The system of ordinary differential equations that prescribes the evolution of the particle locations (c.f. (23) and (48)) can be solved numerically in a variety of ways, and we observe nearly identical results independent of our choice of ODE solver. In analogy with previous work on blob methods in the fluids case [6], we find that the numerical error due to the choice of time discretization is of lower order than the error due to the regularization and spatial discretization. We implement the blob method in Python, using the Numpy, SciPy, and Matplotlib libraries [50,53,80]. In particular, we compute the evolution of the particle trajectories via the SciPy implementation of the Fortran VODE solver [15], which uses either a backward differentiation formula (BDF) method or an implicit Adams method, depending on the stiffness of the problem.
Our convergence result, Theorem 6.1, requires that h = o(ε) as ε → 0. Numerically, we observe the fastest rate of convergence with ε = h 1−p , for 0 < p 1, as h → 0. Since computational speed decreases as p approaches 0, we take ε = h 0.99 in the following simulations. In these examples, we discretize the initial data on a line (d = 1) or square of sidelength 5.0 (d = 2), centered at 0.
Finally, to visualize our particle solution (48) and compare it to the exact solutions in L p -norms, we construct a blob solution obtained by convolving the particle solution with a mollifier,
µ ε (t, ·) := ζ ε * µ ε (t, ·) = i∈Q h R ζ ε (· − x i )m i , t ∈ [0, T ](50)
By Proposition 2.3, if µ ε * µ as ε → 0, where µ is the exact solution, then we also haveμ ε * µ.
Consequently our convergence result, Theorem 6.1, also applies to this blob solution.
We measure the accuracy of our numerical method with respect to the L 1 -, L ∞ -, and Wasserstein metrics. To compute the L 1 -and L ∞ -errors, we take the difference between the exact solution and the blob solution (50) and evaluate discrete L 1 -and L ∞ -norms using the following formulas:
f L 1 (Q h R ) = i∈Q h R |f (ih)|h d , f L ∞ (Q h R ) = max i∈Q h R |f (ih)|, for a given function f : R d → R.
We compute the Wasserstein distance between our particle solution µ ε in (48) and the exact solution µ in one dimension using the formula
W 2 (µ ε , µ) = 1 0 |F −1 µε (s) − F −1 µ (s)| 2 ds 1/2 ,(51)
where F −1 µε and F −1 µ are the generalized inverses of the cumulative distribution functions of µ and µ ε , respectively; c.f. [3, Theorem 6.0.2]. We evaluate the integral in (51) numerically using the SciPy implementation of the Fortran library QUADPACK [70]. In two dimensions, we compute the Wasserstein error by discretizing the exact and blob solutions as piecewise constant functions on a fine grid and then using the Python Optimal Transport library to compute the discrete Wasserstein distance between them. In particular, we use the Earth Mover's Distance function in this library, which is based on the network simplex algorithm introduced by Bonneel, van de Panne, Paris, and Heidrich [14].
6.3. Simulations. Using the method described in the previous section, we now give several examples of numerical simulations. We consider initial data given by linear combinations of Gaussian and Barenblatt profiles, which we denote as follows:
ψ m (τ, x) = 1 (4πτ ) d/2 e −|x| 2 /4τ for m = 1, τ −dβ (K − κτ −2β |x| 2 ) 1/(m−1) + for m > 1, x ∈ R d , with β = 1 2 + d(m − 1)
and κ = β 2 m − 1 m , and K = K(m, d) chosen so that ψ m (τ, x)dx = 1.
In Figure 1, we compare exact and numerical solutions to the heat and porous medium equations (V = W = 0, m = 1, 2, 3), with initial data given by a Gaussian (m = 1) or Barenblatt (m = 2, 3) function with scaling τ = 0.0625. The top row shows the evolution of the density on a large spatial scale, at which the exact and numerical solutions are visually indistinguishable for m = 1 and m = 2. However, for m = 3 the fat tails of the numerical simulation peel away from the exact solution at small times. The second row depicts the numerical simulations for m = 3 on a smaller spatial scale, illustrating how the tails of the numerical simulation converge to the exact solution as the spacing of the computational grid is refined.
In Figure 2, we compute solutions of the one-dimensional heat and porous medium equations (V = W = 0, m = 1, 2, 3), illustrating the role of the diffusion exponent m. The initial data is given by a linear combination of Gaussians, ρ 0 (·) = 0.3ψ 1 (· + 1, 0.0225) + 0.7ψ 1 (· − 1, 0.0225), and the grid spacing is h = 0.01. For m = 1, the infinite speed of propagation of support of solutions to the heat equation is reflected both at the level of the density, for which the gap between the two bumps fills quickly, and also in the particle trajectories, which quickly spread to fill in areas of low mass. In contrast, for m = 2 and m = 3, we observe finite speed of propagation of support, as well as the emergence of Barenblatt profiles as time advances. In Figure 3, we compute the evolution of the nonlocal Sobolev norm (49) along the numerical solutions from Figures 1 and 2. In both cases, we observe that the quantity converges as h → 0 and decreases in time. This gives further credence to the heuristic that the nonlocal Sobolev norm is an approximation of the L 1 -norm of the gradient of the mth power of the exact solution, which does decrease in time along the exact solution; see (24) and (25). In particular, this provides numerical evidence that Assumption (A2) from our main convergence theorem, Theorem 5.8, is satisfied.
In Figure 4, we analyze the rate of convergence of our numerical scheme in one dimension. We compute the error between numerical and exact solutions of the heat and porous medium equations (m = 1, 2, 3) in Figure 1 at time t = 0.05, with respect to the 2-Wasserstein distance, L 1 -norm, and L ∞ -norm and examine the scaling of the error with the grid spacing h. (Recall that ε = h 0.99 throughout.) Plotting the errors on a logarithmic scale, we observe that the Wasserstein error depends linearly on the grid spacing for all values of m. The L 1 -norm scales quadratically for m = 1 and 2 and superlinearly for m = 3. Finally, the L ∞ -error scales quadratically for m = 1, superlinearly for m = 2, and sublinearly for m = 3. This deterioration of the rate of L ∞ -convergence for m = 3 is due to the sharp transition at the boundary of the exact solution; see the second row of Figure 1. In Figure 5, we perform the same analysis on the rate of convergence of our method in two dimensions and observe similar rates of convergence as in the one-dimensional case. In Figure 6, we simulate solutions to the nonlinear Fokker-Planck equation (V (·) = |·| 2 /2, W = 0, m = 2) and consider the rate of convergence to the steady state of the equation, ψ 2 (0.25, x). In the top row, we compute the error between the numerical solution at time t = 1.2 and the steady state with respect to the Wasserstein, L 1 -, and L ∞ -norms for various choices of grid spacing h. We consider solutions with Barenblatt initial data (m = 2, τ = 0.15). We plot the error's dependence on h with a logarithmic scale and compute the slope of the line of best fit to determine the scaling relationship between the error and h. We observe similar rates of convergence as in the case of the heat and porous medium equations; see Figure 5. In the middle rows, we give snapshots of the evolution of the blob method solution, as it converges to the steady state. We consider Barenblatt initial data (m = 2, τ = 0.15) and double bump initial data given by a linear combination of Barenblatts, ρ 0 (x) = 0.7ψ 2 (x−(1.25, 0), 0.1)+0.3ψ 2 (x+(1.25, 0), 0.1). The grid spacing is h = 0.02. In the bottom row, we compute the evolution of the nonlocal Sobolev norm (49) along the numerical solutions from the middle rows. In both cases, we observe that this quantity converges for h small. For Barenblatt initial data, it decreases in time along the numerical solution and agrees well with the value of ∇µ 2 L 1 (R d ) along the exact solution µ. For the double bump initial data, it remains bounded in time, converging asymptotically to ∇(ψ 2 (0.25, ·)) 2 L 1 , where ψ 2 (0.25, ·) is the steady state. Again, this supports the interpretation of the nonlocal Sobolev norm as an approximation of the L 1 -norm of the gradient of the mth power of the exact solution and provides numerical evidence for Assumption (A2) from Theorem 5.8.
Heat and Porous Medium Equations: Double Bump Initial Data
34
Convergence Analysis: Two-Dimensional Diffusion Figure 5. Rate of convergence of blob method for two-dimensional heat and porous medium equations.
In the remaining numerical examples, we apply our method to simulate solutions of Keller-Segel type equations, with the interaction potential W given by 2χ log |·| for χ > 0. In one dimension, the derivative of this potential is not integrable, and we remove its singularity it setting it equal to 2χ/ε for all x ∈ R d such that |x| < ε. In two dimensions, the gradient of this potential is integrable, and we regularize it by convolving it with a mollifier ϕ ε , as done in previous work by the second author on a blob method for the aggregation equation [36].
In Figure 7, we consider the one-dimensional variant of the Keller-Segel equation (V = 0, W (·) = 2χ log |·|, m = 1) studied in [18]. Its interest is that it has a defined critical value χ for unit mass leading to the dichotomy of blow-up versus global existence. For χ = 1.5 and initial data of mass one, solutions blow up in finite time. We consider initial data given by a Gaussian ψ 1 (τ, ·), τ = 0.25, discretized on the interval [−4.5, 4.5] with grid spacing h = 0.009. We compare the evolution of the second moment of our blob method solutions with the second moment of the exact solution. We also compare our results with those obtained in previous work via a one-dimensional Discrete Gradient Flow (DGF) particle method [25,30]. By refining our spatial grid with respect to the DGF particle method, we observe modest improvements. (Alternative simulations, with similar spatial and time discretizations as used in the DGF method, yielded similar results as obtained by DGF.) The blow-up of solution is not only evident in the second moment, which converges to zero linearly in time, but also in the evolution of the particle trajectories. In particular, we observe particle trajectories merging on several occasions as time advances. Figure 7. Left: Comparison of the evolution of the second moment along exact solutions (solid line) with blob method solutiosn (dashed line) and previous numerical results by the DGF particle method [25]. Right: Evolution of particle trajectories, with colors indicating the relative mass of each particle. Figure 8. Left: Evolution of the second moment. Right: Evolution of particle trajectories, with colors indicating the relative mass of each particle.
In Figure 8, we consider a nonlinear variant of the Keller-Segel equation (V = 0, W (·) = 2χ log |·|, m = 2) in one dimension, with initial data and discretization as in Figure 7. We observe the convergence to a steady state both at the level of the second moment and the particle trajectories.
In Figures 9-12 we consider the classical Keller-Segel equation (V = 0, W (·) = 1/(2π) log |·|, m = 1) in two dimensions. In Figures 9, 10, and 11, the initial data is given by a Gaussian ψ 1 (τ, ·), τ = 0.16, scaled to have mass that is either supercritical (> 8π), critical (= 8π), or subcritical (< 8π) with respect to blowup behavior. In particular, for supercritical initial data, solutions blow up in finite time [13,41]. In Figure 9, we analyze the blow-up behavior. We compute the evolution of the second moment of solutions for fixed grid spacing h = 0.03 and varying mass 7π, 8π, and 9π, illustrating how initial data with larger mass aggregates more quickly at the origin. In Figure 10, we consider the evolution of the second moment for the solutions from Figure 9. For fixed grid spacing h = 0.03, we observe that the second moment depends linearly on time, and we compute its slope using the line of best fit. We then analyze how the slope of this line converges to the theoretically predicted slope as the grid spacing h → 0.
In Figure 11, we consider the evolution of the second moment for the supercritical mass solution from Figure 9 on a longer time interval. As in the one-dimensional case (see Figure 7), we are able to get approximately halfway to the time when the second moment becomes zero before the second moment of our numerical solution begins to peel away from the second moment of the exact solution. Indeed, one of the benefits of our blob method approach is that the numerical method naturally extends to two and more dimensions, and we observe similar numerical performance independent of the dimension. We also plot the evolution of particle trajectories, observing the tendency of trajectories in regions of larger mass to be driven largely by pairwise attraction, while trajectories in regions of lower mass feel more strongly the effects of diffusion.
Finally, in Figure 12, we consider the evolution of the density and second moment for double bump initial data, with initial mass 7π, 8π, and 9π. The slopes of the second moment agree well with the theoretically predicted slopes given in Figure 10. Figure 6. In particular, we consider constant multiples M = 7π, 8π, and 9π and again observe that larger values of M correspond to faster aggregation at the origin. Bottom: We consider the evolution of the second moment along particle solutions, for each choice of M . We estimate the slope of the line using the line of best fit.
Appendix A. Proofs of preliminary results
We now turn to the proofs of some of the elementary lemmas and propositions from Sections 2 and 3. We begin with the proof of the mollifier exchange lemma. By Jensen's inequality for the convex function s → s log s, the relative entropy is nonnegative, which gives the result. Now, we show the left inequality in (11) for 1 < m ≤ 2. By the above-the-tangent property of the concave function F m and Hölder's inequality, we get
F m (µ) − F m ε (µ) = 1 m − 1 µ m−1 − (ϕ ε * µ) m−1 dµ ≥ (µ − ϕ ε * µ) µ m−2 dµ ≥ − µ − ϕ ε * µ L m (R d ) µ m−1 L m/(m−1) (R d ) = − µ − ϕ ε * µ L m (R d ) µ m−1 L m (R d ) . Since µ ∈ D(F m ) implies µ ∈ L m (R d )
, the first term goes to zero as ε → 0 and the second term remains bounded. This gives the result.
We now turn to the right inequality in (11) in the case 1 ≤ m ≤ 2. By the fact that ϕ ε = ζ ε * ζ ε and Jensen's inequality for the concave function F m , for all x ∈ R d we have
F m (ϕ ε * µ(x)) = F m R d ζ ε (y)ζ ε * µ(x − y) dy ≥ R d ζ ε (y)F m (ζ ε * µ(x − y)) dy = ζ ε * (F m • (ζ ε * µ)) (x). Consequently, we deduce F m ε (µ) = R d F m (ϕ ε * µ(x)) dµ(x) ≥ R d ζ ε * (F m • (ζ ε * µ)) (x) dµ(x) = R d F m (ζ ε * µ(x)) d(ζ ε * µ)(x) = F m (ζ ε * µ) .
Now, we show (12). Since F m is convex for m ≥ 2, this is simply a consequence of reversing the inequalities in the last two inequalities.
Finally, we consider the lower bounds (13). When m = 1, these follow from the right inequality in (11), a Carleman-type estimate [30,Lemma 4.1] ensuring that F m ε (ζ ε * µ) ≥ −(2π/δ) d/2 −δM 2 (ζ ε * µ) for all δ > 0, and the fact that R d ζ ε (y)|x + y| 2 dy ≤ 2|x| 2 + 2M 2 (ζ ε ) =⇒ M 2 (ζ ε * µ) ≤ 2M 2 (µ) + 2M 2 (ζ ε ) = 2M 2 (µ) + 2ε 2 M 2 (ζ).
When m > 1, we simply use that F m ≥ 0.
We now give the proof that, for all ε > 0, the regularized energies are lower semicontinuous with respect to weak-* convergence (m > 1) and Wasserstein convergence (m = 1), where in the latter case, we require ϕ to be a Gaussian.
Proof of Proposition 3.9. We first show (i). Let (µ n ) n ⊂ P(R d ) and µ ∈ P(R d ) be such that µ n * µ; we must show lim inf n→∞ F m ε (µ n ) ≥ F m ε (µ). Without loss of generality, we replace (µ n ) n by a subsequence which attains the limit on the left-hand side, and we may assume that this limit is finite. Consequently, there exists C > 0 so that
F m ε (µ n ) < C for all n ∈ N.(53)
We now consider the case when 1 < m ≤ 2. For any a, b > 0, |a m−1 −b m−1 | ≤ |a−b| m−1 . Combining this with Jensen's inequality for the concave function s → s m−1 ,
F ε (µ n ) = 1 m − 1 (ϕ ε * µ n ) m−1 dµ n − 1 m − 1 (ϕ ε * µ) m−1 dµ n + 1 m − 1 (ϕ ε * µ) m−1 dµ n ≥ − 1 m − 1 |ϕ ε * (µ n − µ)| m−1 dµ n + 1 m − 1 (ϕ ε * µ) m−1 dµ n ≥ − 1 m − 1 |ϕ ε * (µ n − µ)|dµ n m−1 + 1 m − 1 (ϕ ε * µ) m−1 dµ n ≥ − 1 m − 1 |ζ ε * (µ n − µ)|dζ ε * µ n m−1 + 1 m − 1 (ϕ ε * µ) m−1 dµ n .
Since ϕ ε ∈ C b (R d ), µ n * µ ensures that ζ ε * µ n → ζ ε * µ pointwise. The integrand of the first term is bounded above by 2 ζ ε 2 L ∞ (R d ) , so by the dominated convergence theorem, the first integral converges to zero. Since the integrand of the second term is continuous and bounded by ϕ ε m−1 L ∞ (R d ) , the fact that (µ n ) n converges weakly- * to µ ensures this second term converges to F ε (µ). This gives the result for 1 < m ≤ 2. We now deal with the case m > 2. Inequality (53) ensures that F m ε (µ n ) = ϕ ε * µ n L m−1 (µn;R d ) < C for all n ∈ N; so by Proposition B.2 there exists w ∈ L m−1 (µ; R d ) such that, up to another subsequence, for all
f ∈ C b (R d ) we have lim inf n→∞ ϕ ε * µ n L m−1 (µn;R d ) ≥ w L m−1 (µ;R d ) and f (ϕ ε * µ n ) dµ n → f w dµ.(54)
It suffices to show that w ≥ ϕ ε * µ holds µ-almost everywhere. Then, the inequality in (54) gives
lim inf n→∞ F m ε (µ n ) = lim inf n→∞ ϕ ε * µ n L m−1 (µn;R d ) ≥ w L m−1 (µ;R d ) ≥ ϕ ε * µ L m−1 (µ;R d ) = F m ε (µ).
Since µ n * µ and f, ζ ε ∈ C b (R d ), we have ζ ε * (f µ n ) → ζ ε * (f µ) and ζ ε * µ n → ζ ε * µ pointwise.
Consequently, using that ϕ ε = ζ ε * ζ ε and Fatou's lemma, we obtain that for all nonnegative
f ∈ C ∞ (R d ), lim inf n→∞ f (ϕ ε * µ n ) dµ n = lim inf n→∞ ζ ε * (f µ n ) dζ ε * µ n ≥ ζ ε * (f µ) dζ ε * µ = f ϕ ε * µ dµ.
Combining this with the second property in (54) gives
f w dµ ≥ f ϕ ε * µ dµ for all f ∈ C ∞ (R d ) with f ≥ 0.
Therefore, w ≥ ϕ ε * µ holds µ-almost everywhere, which gives the result. We now show (ii). Let (µ n ) n be a sequence in P 2 (R d ) converging in the Wasserstein metric to some µ ∈ P 2 (R d ); we must show lim inf n→∞ F 1 ε (µ n ) ≥ F 1 ε (µ). For all 0 < δ < ε, Jensen's inequality ensures that F 1 ε (µ n ) = R d log(ϕ ε * µ n (x)) dµ n (x) ≥ R d log(ϕ ε−δ * µ n (x))ϕ δ * µ n (x) dx,
and, since ϕ is a Gaussian, there exist x 0 ∈ R d and C 0 , C 1 ∈ R so that log(ϕ ε−δ * µ n (x)) ≥ C 0 |x − x 0 | 2 + C 1 ,
for δ > 0 sufficiently small and n sufficiently large. This ensures that
lim inf n→∞ R d log(ϕ ε−δ * µ n (x))ϕ δ * µ n (x) dx ≥ R d log(ϕ ε−δ * µ(x))ϕ δ * µ(x) dx.(57)
Indeed, let us write f n := log(ϕ ε−δ * µ n ) and q(·) := C 0 | · −x 0 | 2 + C 1 . By continuity of ϕ δ we know that ϕ δ * µ n (E) → ϕ δ * µ(E) as n → ∞ for any Borel set E ⊆ R d . Then, since µ has finite second moment and (µ n ) n converges to µ in the Wasserstein metric, using that f n − q ≥ 0 by (56) and Lemma B.3 implies
lim inf n→∞ R d (f n − q)(x) dϕ δ * µ n (x) ≥ R d f (x) dϕ δ * µ(x) − R d q(x) dϕ δ * µ(x) = R d f (x) dϕ δ * µ(x) − lim sup n→∞ R d q(x) dϕ δ * µ n (x),
Hence,
lim inf n→∞ f n dϕ δ * µ n ≥ f dϕ δ * µ, which is (57). This, together with (55) proves that
lim inf n→∞ F 1 ε (µ n ) ≥ R d log(ϕ ε−δ * µ(x))ϕ δ * µ(x) dx.(58)
We now want to pass to the limit δ → 0. To this end first note that, for every Borel set E ⊆ R d ,
lim δ→0 ϕ δ * µ(E) = lim δ→0 E R d ϕ δ (x − y) dµ(y) dx = R d 1 E (y) dµ(y) = µ(E).(59)
Indeed, for every R > 0, we have
E∩B R R d ϕ δ (x − y) dµ(y) dx = R d R d ϕ δ (x − y)1 E∩B R (x) dµ(y) dx = R d ϕ δ * 1 E∩B R (y) dµ(y), which yields lim δ→0 E∩B R R d ϕ δ (x − y) dµ(y) dx = R d 1 E∩B R (y) dµ(y),
and gives (59) by taking the limit R → ∞. Using Lemma B.3 on the variable δ in (58) gives
lim inf n→∞ F 1 ε (µ n ) ≥ lim δ→0 R d log(ϕ ε−δ * µ(x))ϕ δ * µ(x) dx = R d log(ϕ ε * µ(x)) µ(x) = F 1 ε (µ),
which is the desired result.
Now we turn to the proof that the regularized energies are differentiable along generalized geodesics.
Proof of Proposition 3.10. By definition, for all α ∈ [0, 1], F ε (µ 2→3 α ) = F (ϕ ε * µ α ((1 − α)x + αy)) dγ(x, y).
Therefore, we deduce F ε (µ 2→3 α ) − F ε (µ 2 ) = F ϕ ε * µ 2→3 α ((1 − α)y + αz)) − F (ϕ ε * µ 1 (y)) dγ(x, y, z) = 1 0 F (c s,α (y, z)) ϕ ε * µ 2→3 α ((1 − α)y + αz)) − ϕ ε * µ 1 (y) dγ(x, y, z) ds,
where c s,α (y, z) = (1 − s)ϕ ε * µ 1 (y) + sϕ ε * µ 2→3 α ((1 − α)y + αx). Using Taylor's theorem compute
ϕ ε * µ 2→3 α ((1 − α)y + αz)) − ϕ ε * µ 1 (y) = (ϕ ε ((1 − α)(y − v) + α(z − w)) − ϕ ε * (y − v)) dγ(u, v, w) = (α∇ϕ ε (y − v) · (z − w − (y − v)) + D α (y, z, v, w)) dγ(u, v, w),
where D α (y, z, v, w) is a term depending on the Hessian of ϕ ε satisfying D α (y, z, v, w) dγ(u, v, w) ≤ α 2 2 D 2 ϕ ε L ∞ (R d ) |z − w − (y − v)| 2 dγ(u, v, w)
≤ 2α 2 D 2 ϕ ε L ∞ (R d ) |z| 2 + |y| 2 + |w| 2 dµ 3 (w) + |v| 2 dµ 2 (v) 44 Hence, since F is nondecreasing,
F ε (µ 2→3 α ) − F ε (µ 2 ) = α 1 0
F (c s,α (y, z))∇ϕ ε (y − v) · (z − w − (y − v)) dγ(u, v, w) dγ(x, y, z) ds + C α ,
where |C α | ≤ 4α 2 D 2 ϕ ε L ∞ (R d ) F ( ϕ ε L ∞ (R d ) )( |x| 2 dµ 2 (x) + |x| 2 dµ 3 (x)). Note that c s,α (y, z) converges pointwise to ϕ ε * µ 2 (y) as α → 0 since
ϕ ε * µ 2→3 α ((1 − α)y + αz) −ϕ ε * µ 2 (y)| = (ϕ ε ((1 − α)(y − v) + α(z − w)) − ϕ ε (y − v)) dγ(u, v, w) ≤ α ∇ϕ ε L ∞ (R d ) |z| + |y| + |w| dµ 3 (w) + |v| dµ 2 (v) .
Thus, to complete the result, it suffices to show that there exists g ∈ L 1 (γ ⊗ γ) so that F (c s,α (y, z)) |∇ϕ ε (y − v) · (z − w − (y − v))| ≤ g(y, z, v, w), since the result then follows by the dominated convergence theorem. Since F is nondecreasing we may take g(y, z, v, w)
= F ϕ ε L ∞ (R d ) ∇ϕ ε L ∞ (R d ) |z − w − (y − v)|,
which ends the proof.
Next, we apply the result of the previous proof to characterize the subdifferential of the regularized energies.
Proof of Proposition 3.12. Suppose v is given by equation (16). This part of the proof is closely inspired by that of [24,Proposition 2.2]. For all x, y ∈ R d define G(α) = F (ϕ ε * µ α ((1 − α)x + αy)) for all α ∈ [0, 1], where µ α = ((1 − α)π 1 + απ 2 ) # γ, with some γ ∈ Γ o (µ, µ 1 ), connects µ 0 = µ and µ 1 . Now define f (α) = G(α) − G(0) α − λα 2 |x − y| 2 + W 2 2 (µ 0 , µ 1 ) for all α ∈ [0, 1], where λ = −2F ( ϕ ε L ∞ (R d ) ) D 2 ϕ ε L ∞ (R d ) = λ F /2; see (15). We write [a, b] α := (1 − α)a + αb for any a, b ∈ R d . Let us compute the first two derivatives of G for all α ∈ [0, 1]:
G (α) = F (ϕ ε * µ α ([x, y] α )) R d ×R d (y − x + u − v) · ∇ϕ ε ([x − u, y − v] α ) dγ(u, v),(60)
and
G (α) = F (ϕ ε * µ α ([x, y] α )) R d ×R d (y − x + u − v) · ∇ϕ ε ([x − u, y − v] α ) dγ(u, v) 2 +F (ϕ ε * µ α ([x, y] α )) R d ×R d (y − x + u − v)D 2 ϕ ε ([x − u, y − v] α )(y − x + u − v) dγ(u, v).
Since F ≥ 0, F ≥ 0 and D 2 ϕ ε L ∞ (R d ) is finite, we have
G (α) ≥ −F ( ϕ ε L ∞ (R d ) ) D 2 ϕ ε L ∞ (R d ) R d ×R d |y − x + u − v| 2 dγ(u, v) ≥ −2F ( ϕ ε L ∞ (R d ) ) D 2 ϕ ε L ∞ (R d ) R d ×R d
|y − x| 2 + |u − v| 2 dγ(u, v)
= λ |y − x| 2 + W 2 2 (µ 0 , µ 1 ) . 45 Now, by Taylor's theorem, f (α) = G (0) + α 0 α − s α G (s) ds − λα 2 |x − y| 2 + W 2 2 (µ 0 , µ 1 ) , and therefore, using (61) leads to
f (α) = 1 α 2 α 0 sG (s) ds − λ 2 |x − y| 2 + W 2 2 (µ 0 , µ 1 ) ≥ 0,
which shows that f is nondecreasing, and so f (1) ≥ lim α→0 f (α), which implies (after integrating against dγ(x, y))
F ε (µ 1 ) − F ε (µ 0 ) ≥ R d ×R d lim α→0 G(α) − G(0) α dγ(x, y) + λW 2 2 (µ 0 , µ 1 ) = R d ×R d G (0)
dγ(x, y) + λW 2 2 (µ 0 , µ 1 ).
Then, by (60) and antisymmetry of ∇ϕ ε , compute
R d ×R d G (0) dγ(x, y) = R d ×R d R d ×R d F (ϕ ε * µ 0 (x))(y − x + u − v) · ∇ϕ ε (x − u) dγ(u, v) dγ(x, y) = R d ×R d F (ϕ ε * µ 0 (x))∇ϕ ε * µ 0 (x) · (y − x) dγ(x, y) + R d ×R d ∇ϕ ε * (F • (ϕ ε * µ 0 )µ 0 )(u) · (v − u) dγ(u, v) = R d ×R d ∇ δF ε δµ 0 (x) · (y − x) dγ(x, y). Hence F ε (µ 1 ) − F ε (µ 0 ) ≥ R d ×R d ∇
δF ε δµ 0 (x) · (y − x) dγ(x, y) + λW 2 2 (µ 0 , µ 1 ), which shows that δF ε /δµ 0 ∈ ∂F ε (µ 0 ). We now prove that v ∈ Tan µ P 2 (R d ). Consider a vectorvalued function ξ ∈ C ∞ c (R d ) d , and for any x, y ∈ R d define H(α) = F ( R d ϕ ε (x − y + α(ξ(x) − ξ(y)) dµ(y)) for all α ∈ [0, 1]. Then H (0) = F (ϕ ε * µ(x)) R d (ξ(x) − ξ(y)) · ∇ϕ ε (x − y) dµ(y). Now compute, using the antisymmetry of ∇ϕ ε ,
lim α→0 F ε ((id +αξ) # µ) − F ε (µ) α = lim α→0 R d H(α) − H(0) α dµ(x) = R d H (0) dµ(x) = R d F (ϕ ε * µ(x))∇ϕ ε * µ(x) · ξ(x) dµ(x) + R d ∇ϕ ε * (F • (ϕ ε * µ)µ)(x) · ξ(x) dµ(x) = R d ∇ δF ε δµ (x) · ξ(x) dµ(x),
where passing the limit α → 0 inside the integral in the first line is justified by the fact that H is bounded. Then, by the definition of the local slope of F ε , lim inf α→0 F ε ((id +αξ) # µ) − F ε (µ) W 2 ((id +αξ) # µ, µ) ≥ −|∂F ε |(µ).
Therefore, by the previous computation,
R d ∇ δF ε δµ (x) · ξ(x) dµ(x) ≥ −|∂F ε |(µ) lim inf α→0 W 2 ((id +αξ) # µ, µ) α ≥ −|∂F ε |(µ) ξ L 2 (µ;R d ) ,
since, by definition of the 2-Wasserstein distance, lim sup α→0 W 2 ((id +αξ) # µ, µ) α ≤ ξ L 2 (µ;R d ) .
Then, by replacing ξ with −ξ, by arbitrariness of ξ and by density of C ∞ c in L 2 (µ; R d ), we get v L 2 (µ;R d ) = ∇ δF ε δµ L 2 (µ;R d ) ≤ |∂F ε |(µ), which shows the desired result. Since |∂F ε |(µ) is the unique minimal norm element of ∂F ε , this also shows that we actually have equality in the right-hand side above. Suppose now that v ∈ ∂F ε (µ)∩Tan µ P 2 (R d ). Fix ψ ∈ C ∞ c (R d ) and define µ α = (id +α∇ψ) # µ and µ α = (id −α∇ψ) # µ for all α ∈ [0, 1]. For α sufficiently small, x 2 /2 + αψ(x) is convex and id +α∇ψ is the optimal transport map from µ to µ α , so Γ o (µ, µ α ) = {id ×(id +α∇ψ)}. Similarly, Γ o (μ α , µ) = {id ×(id −α∇ψ)}. Since v ∈ ∂F m ε (µ), taking ν = µ α in Definition 2.7 of the subdifferential, for α sufficiently small, gives
F ε (µ α ) − F ε (µ) ≥ v, α∇ψ dµ + o(α ∇ψ L 2 (µ) ), and F ε (μ α ) − F ε (µ) ≤ v, α∇ψ dµ + o(α ∇ψ L 2 (µ) ),
Combining this with Proposition 3.10, we obtain v, ∇ψ dµ ≤ lim
α→0 F ε (µ α ) − F ε (µ) α = d dα F ε (µ α ) α=0 = d dα F ε (μ α ) α=0 = lim α→0 − F ε (μ α ) − F ε (µ) α ≤ v, ∇ψ dµ.
Rewriting the expression from equation (14) gives v, ∇ψ dµ = d dα F ε (µ α ) α=0 = ∇ϕ ε * F • (ϕ ε * µ)µ + F (ϕ ε * µ)∇ϕ ε * µ, ∇ψ dµ.
Thus, for w = v − ∇ϕ ε * (F • (ϕ ε * µ)µ) + F (ϕ ε * µ)∇ϕ ε * µ, we have w, ∇ψ dµ = 0, i.e. ∇ · (wµ) = 0 in the sense of distribution. By [3,Proposition 8.4.3], since v ∈ Tan µ P 2 (R d ) we get v − w L 2 (µ;R d ) ≥ v L 2 (µ;R d ) . Since we have already shown that the vector in (16) is the element of minimal norm of ∂F ε , we get that v − w L 2 (µ;R d ) ≤ v L 2 (µ;R d ) , and so v − w L 2 (µ;R d ) = v L 2 (µ;R d ) . Again using [3,Proposition 8.4.3], we obtain w = 0, which ends the proof.
Finally, we prove the characterization of the subdifferential of the full regularized energies E m ε . Proof of Corollary 3.13. Write λ V ∈ R and λ W ∈ R the semiconvexity constants of V and W , respectively. The proof follows the same steps as that of Proposision 3.12 with the only difference being the definitions of the functions G, f and H. Given x, y ∈ R d , we define, for all α ∈ [0, 1], G(α) = F (ϕ ε * µ α ((1 − α)x + αy)) + V ((1 − α)x + αy) + 1 2 W * µ α ((1 − α)x + αy),
f (α) = G(α) − G(0) α − (λ + λ W )α 2 |x − y| 2 + W 2 (µ 0 , µ 1 ) − λ V α 2 |x − y| 2 ,
and H(α) = F R d ϕ ε (x − y + α(ξ(x) − ξ(y)) dµ(y) +V (x+αξ(y))+ R d W (x−y +α(ξ(x)−ξ(y))) dµ(y),
where µ 0 , µ 1 , λ and ξ are as in the proof of Proposition 3.12.
DEFINITION 2.9 (gradient flow). Suppose G : P 2 (R d ) → R ∪ {+∞} is proper and lower semicontinuous. A curve µ ∈ AC 2 loc ((0, +∞); P 2 (R d )) is a gradient flow of G if there exists a velocity vector field
REMARK 3. 2
2(nondecreasing). Assumption 3.1 implies that F is nondecreasing. Indeed, by the convexity of U (s) and the fact that lim s→0 sF (s) = 0, sF (s) = s 0 U (r) dr ≤ sU (s) = s 2 F (s) + sF (s) for all s ∈ (0, ∞),
REMARK 3. 3 (
3McCann's convexity condition). McCann's condition[64] on the internal density U for the convexity of the internal energy F can be stated on the function F instead: the function s → F (s −d ) is nonincreasing and convex on (0, ∞), i.e., F (s) ≥ 0 and (d + 1)F (s) + dsF (s) ≥ 0 for all s ∈ (0, ∞),
DEFINITION 3 . 5 .
35The entropy and Rényi entropies, and their regularizations, are given by
DEFINITION 3.6 (regularized energies). Let V, W : R d → (−∞, ∞] be proper and lower semicontinuous. Suppose further that W is locally integrable. For all µ ∈ P(R d ) define
REMARK 4.3 (tightness of sublevels). Assumptions (CV) and (CV ) ensure that the set {µ ∈
THEOREM 4. 5
5(minimizers converge to minimizers). Suppose m > 1. If Assumption (CV) holds, then for any sequence
PROPOSITION 5 . 4 .
54Let ε > 0 and m ≥ 2. Suppose E m ε is as in Definition 3.6 and V and W satisfy Assumption 5.1. Then, µ ε ∈ AC 2 loc ((0, +∞); P 2 (R d )) is the gradient flow of E m ε if and only if µ ε is a weak solution of the continuity equation with velocity field
COROLLARY 5. 5 .
5Let ε > 0 and m ≥ 2, and let V and W satisfy Assumption 5.1. Fix N ∈ N. For i ∈ {1, . . . , N } := I, fix X 0 i ∈ R d and m i ≥ 0 satisfying i∈I m i = 1. Then the ODE system
THEOREM 5. 8 .
8Let m ≥ 2, and let V and W be as in Assumption 5.1. Fix T > 0 and suppose
κ(x) = 1 for all |x| < 1/2 and κ(x) = 0 for all |x| > 2. Given R > 0, define κ R := κ(·/R), so that ∇κ R L ∞ (R d ) ≤ 1/R. Then, by Jensen's inequality for the convex function s → s m−1 , Lemma 2.2, and Assumption (
THEOREM 6. 1 .
1Assume m ≥ 2 and V and W satisfy Assumption 5.1. Suppose µ(0) ∈ D(E m ) is compactly supported in B R (0), the ball of radius R centered at the origin. For fixed grid spacing h > 0, define the grid indices Q h R := {i ∈ Z d : |ih| ≤ R} and approximate µ(0) by the following sequence of measures:
Figure 1 .
1Comparison of exact and numerical solutions to the heat (m = 1) and porous medium (m = 2, 3) equations. Numerical solutions are plotted with thick lines, and exact solutions are plotted with thin lines.
Figure 2 .
2Numerical simulation of the one-dimensional heat and porous medium equations. Top: Evolution of the blob density ρ h ε . Bottom: Evolution of the particle trajectories x i , with colors indicating relative mass of each particle.
Figure 3 .
3Left: Comparison of nonlocal Sobolev norm (49) along numerical solutions (dashed line) with the value of ∇µ m L 1 (R d ) along exact solutions µ (solid line). Right: Evolution of nonlocal Sobolev norm along the numerical solutions.
Figure 4 .
4Rate of convergence of blob method for one-dimensional heat and porous medium equations.
Figure 6 .
6Top row: Error between numerical solutions and steady state. Middle rows: Snapshots of the evolution towards steady state. Bottom left: Comparison of nonlocal Sobolev norm (49) along numerical solution from second row (dashed line) with ∇µ m L 1 (R d ) along exact solution µ (solid line). Bottom right: Comparison of nonlocal Sobolev norm along numerical solution from third row (dashed line) with ∇µ m L 1 (R d ) evaluated at steady state µ (solid line).
Figure 9 .
9Evolution of numerical solutions for the two-dimensional Keller-Segel equation with subcritical, critical, and supercritical initial data.
Figure 10 .
10Left: Evolution of second moment of numerical solutions. Right: Convergence of slope of second moment to theoretically predicted slope (solid line).Two-Dimensional Keller-Segel Equation: Blowup with Supercritical Mass 9πEvolution of Second Moment Evolution of Particle Trajectories
Figure 11 .
11Left: Comparison of second moment of numerical solution (dashed line) to exact solution (solid line). Right: Evolution of particle trajectories, colored according to the relative mass of each trajectory.
Figure 12 .
12Top: We plot the evolution of blob solutions to the two-dimensional Keller-Segel equation, with initial data given by constant multiples of the linear combination of Barenblatts from
1.7], [64, Lemma 3.4] and [78, Lemma 2.2]).
Acknowledgments: The authors would like to thank Andrew Bernoff, Andrea Bertozzi, Eric Carlen, Yanghong Huang, Inwon Kim, Dejan Slepčev, and Fangbo Zhang for many helpful discussions.Proof of Lemma 2.2. By the Lipschitz continuity of f ,Set p := (q−d)/q > 0. Decomposing the domain of the integration of |ν| into B ε p (x) and R d \B ε p (x), we may bound the above quantity byBy the decay assumption on ζ (see Assumption 2.1), for all x, y ∈ R d with |x − y| > ε p we haveThus, we conclude our result by estimating the above quantity byWe now give the proof that if µ ε * µ, then ϕ ε * µ ε * µ.Proof of Lemma 2.3. By [3, Remark 5.1.6], it suffices to show that ϕ ε * µ ε converges to µ in distribution, that is, in the duality with smooth, compactly supported functions.Since µ ε * µ, the second term goes to zero. We bound the first term as follows:which goes to zero as ε → 0.Next, we prove the inequalities relating the regularized internal energies to the unregularized internal energies.Proof of Proposition 3.8. We begin with(11). To prove the left inequality, we may assume without loss of generality that µ ∈ D(F). First, we show the result for the entropy (m = 1). Note thatwhere H is the relative entropy; that is, for all ν ∈ P(R d ),Appendix B. Weak convergence of measuresIn this appendix, we recall several fundamental results on the weak convergence of measures. We begin with a result due to Ambrosio, Gigli, and Savaré on convergence of maps with respect to varying probability measures. This plays a key role in our proofs of both the Γ-convergence of the energies and the Γ convergence of the gradient flows. . Given a sequence (µ n ) n ⊂ P(R d ) converging in the weak- * topology to some µ ∈ P(R d ), we say that a sequence (v n ) n with v n ∈ L 1 (µ n ; R d ) for all n ∈ N converges weakly to some Let (µ n ) n ⊂ P(R d ), µ ∈ P(R d ) and (v n ) n be such that v n ∈ L 1 (µ n ; R d ) for all n ∈ N. Suppose µ n * µ and sup n∈N v n L p (µn;R d ) < ∞ for some p > 1. The following items hold.(i) There exists a subsequence of (v n ) n converging weakly to some w ∈ L 1 (µ; R d ).(ii) If (v n ) n weakly converges to some v ∈ L 1 (µ; R d ), then. (iii) If (v n ) n strongly converges in L p to some v ∈ L p (µ; R d ) and sup n∈N M p (µ n ) < ∞, thenWe close by recalling a generalization of Fatou's lemma, for varying measures. . Given a sequence (µ n ) n ⊂ P(R d ) and µ ∈ P(R d ), suppose that for every Borel set E ⊂ R d we have µ n (E) → µ(E) as n → ∞. Then for any sequence (f n ) n of nonnegative functions on R d with pointwise infimum limit f , we have
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arxiv |
New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary
20 Jun 2014 June 23, 2014
Jimmy Lamboley
Pieralberto Sicbaldi
New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary
20 Jun 2014 June 23, 2014arXiv:1406.5167v2 [math.DG]
We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.
Introduction
New examples of domains with small prescribed volume that are critical points for the first eigenvalue of the Dirichlet Laplace-Beltrami operator are built in [21], under the hypothesis that the Riemannian manifold has at least one nondegenerate critical point of the scalar curvature function. In that case, such domains are given by small perturbations of geodesic balls of small radius centered at a nondegenerate critical point of the scalar curvature. This result has been generalized in [12] to all compact Riemannian manifolds by eliminating the hypothesis of the existence of a nondegenerate critical point of the scalar curvature.
Such examples of critical points for the Laplace-Beltrami operator are parallels to similar shape examples of critical points for the area functional, under the same assumptions, which lead to the construction of constant mean curvature small topological spheres, see [22,28].
The aim of this paper is to give some new examples of domains Ω that are critical points for the first eigenvalue of the Laplace-Beltrami operator (i.e. extremal domains) in some Riemannian manifolds M with boundary. Such examples are new because the boundary of the domain is partially included in the boundary of the manifold. The domains we obtain are close to half-balls centered at a point of ∂M where the mean curvature of ∂M is critical and the criticality is not degenerate. In particular, in the simplest situation, M can be a domain of the Euclidean space, see Fig. 1. Again, we can make a parallel with the case of the area, for which a similar result has been proven in the Euclidian case and dimension 3 in [15], though it is expected to be valid in the general case.
Assume that we are given (M, g) an (n + 1)-dimensional Riemannian manifold, n ≥ 1, with boundary ∂M = ∅. The boundary ∂M is a smooth n-dimensional Riemannian manifold with the metricg induced by g. For a domain Ω contained in the interior of M , Ω ⊂M , the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition is then given by
λ Ω = min u∈H 1 0 (Ω) Ω |∇u| 2 Ω u 2 .
If Ω is a boundary domain (i.e. a domain such that ∂Ω ∩ ∂M = ∅), we consider the first eigenvalue of the Laplace-
where H 1 0 (Ω) denotes the closure of the space {ϕ ∈ C ∞ (Ω), Supp(ϕ) ⊂ Ω ∪ ∂M } for the H 1 -norm. It is very classical that the optimization problem (1) admits a nonnegative solution if Ω has finite volume, and if Ω is connected such a solution is unique among nonnegative functions whose L 2 -norm is 1. This function is then called the first eigenfunction of Ω.
Under smoothness assumption (for example if Ω is a piecewise C 1,α -domain, see Section 2.1 for more detailed definitions, the space H 1 0 (Ω) is equal to the space of functions in H 1 (Ω) with 0 Dirichlet condition on ∂Ω ∩M , and the function u solving (1) satisfies: ∆ g u + λ Ω u = 0 in Ω u = 0 on ∂Ω ∩M , g(∇u, ν) = 0 on ∂Ω ∩ ∂M (2) where ν denotes the outward normal vector to ∂M , which is well-defined as soon as Ω is included in a small enough ball, which will be the case in the whole paper. This will be referred to as a mixed eigenvalue problem over Ω. Moreover, it is also well-known that if there exists (u, λ) a nontrivial solution of (2) for a connected domain Ω such that u is nonnegative, then λ = λ Ω is the first eigenvalue of Ω, and u is the first eigenfunction of Ω, up to a multiplicative constant.
Let us consider a boundary domain Ω 0 ⊂ M . Ω 0 is said to be extremal if Ω −→ λ Ω is critical at Ω 0 with respect to variations of the domain Ω 0 which preserve its volume. In order to make this notion precise, we first introduce the definition of a deformation of Ω 0 . Definition 1.1. We say that (Ω t ) t∈(−t0,t0) is a deformation of Ω 0 , if there exists a vector field V on M , of class C 2 , such that its flow ξ, defined for t ∈ (−t 0 , t 0 ) by dξ dt (t, p) = V (ξ(t, p)) and ξ(0, p) = p , preserves the boundary of the manifold, i.e. ξ(t, p) ∈ ∂M for all (t, p) ∈ (−t 0 , t 0 ) × ∂M , and for which Ω t = ξ(t, Ω 0 ).
The deformation is said to be volume preserving if the volume of Ω t does not depend on t.
If (Ω t ) t∈(−t0,t0) is a deformation of Ω 0 , we denote by λ t the first eigenvalue of the Laplace-Beltrami operator −∆ g on Ω t . We prove in Section 2 that t −→ λ t is smooth in a neighborhood of t = 0. If Ω ⊂M this fact is standard and follows from the implicit function theorem together with the fact that the first eigenvalue of the Laplace-Beltrami operator is simple, see for example [18]. When the boundary ∂M is invariant by the flow of the deformation, as required in Definition 1.1, a similar strategy still works when ∂Ω ∩ ∂M = ∅, but this is less classical since one needs to manage the singularities of the boundary domains under consideration, see Proposition 2.5. The derivative at 0 of t → λ t is then called the shape derivative of Ω → λ Ω at Ω 0 in the direction V .
This remark allows us to give the definition of an extremal domain. Definition 1.2. A domain Ω 0 is an extremal domain for the first eigenvalue of −∆ g if for any volume preserving
deformation {Ω t } t of Ω 0 , we have dλ t dt | t=0 = 0 ,(3)
where λ t = λ Ωt as defined in (1).
All along the paper, we will use a special system of coordinates, that we remind here: let p ∈ ∂M , and let N be the unit normal vector field on ∂M near p that points into M . We fix local geodesic normal coordinates x = (x 1 , ..., x n ) in a neighborhood of 0 ∈ R n to parametrize U p a neighborhood of p in ∂M by Φ. We consider the mapping
Ψ(x 0 , x) = Exp Φ(x) (x 0 N (Φ(x)))(4)
which is a local diffeomorphism from a neighborhood of 0 ∈ R n+1
+ (where R n+1 + = {(x 0 , x) ∈ R n+1 : x 0 > 0}) into V p a neighborhood of p in M . For all ε > 0 small enough, we denote B + ε ⊂ R n+1 +
the half-ball given by the Euclidean ball of radius ε centered at the origin and restricted to x 0 > 0, and we denote Now we can state the main result of the paper: Theorem 1.3. Assume that p 0 ∈ ∂M is a nondegenerate critical point of H, the mean curvature function of (∂M,g).
B + g,ε (p) = Ψ(B + ε ) ⊂M . Φ(p 1 ) p 1 Φ N (Φ(p 2 )) ∂M p 2 x 1 x 2 N (Φ(p 1 )) Φ(p 2 )
Then, for all ε > 0 small enough, say ε ∈ (0, ε 0 ), there exists a boundary domain Ω ε ⊂ M such that :
(i) The volume of Ω ε is equal to the Euclidean volume of B + ε .
(ii) The domain Ω ε is extremal in the sense of Definition 1.2.
(iii) The boundary ∂Ω ε ∩M intersects ∂M orthogonally, (iv) The boundary ∂Ω ε ∩M is analytic if M is analytic.
Moreover, there exists c > 0 and, for all ε ∈ (0, ε 0 ), there exists p ε ∈ ∂M such that ∂Ω ε ∩M is a normal graph over
∂B + g,ε (p ε ) ∩M for some function w ε with w ε C 2,α ∂B + g,ε (pε)∩M ≤ c ε 3 . and dist(p ε , p 0 ) ≤ c ε .
This result will be proven in Section 4.5. The strategy of the proof of this result is inspired by [21]. In order to give the outline of the paper, we recall here the strategy of the proof and insist on the main differences with [21]. The first step is to characterize the extremality of a domain Ω 0 with the Euler-Lagrange equation, that leads to:
g(∇u, ν) = constant on ∂Ω 0 ∩M .(5)
The difficulty here is to prove this characterization for domains that are only piecewise smooth (see Section 2.1 where we introduce the notion of boundary edge domain and analyze the regularity theory of mixed boundary value problem in such domains). In particular, we prove in Section 2.2 that in order to be extremal it is enough for a domain to satisfy (3) for deformations that preserve the contact angle on ∂M ; this important fact will be used in the rest of the paper for the construction of extremal domains. This is an interesting difference with the case of critical points of the area functional, as we explain in Section 2.3: condition (5) contains already the information that the contact angle between ∂Ω 0 ∩M and ∂Ω 0 ∩ ∂M is constant and equal to π/2, see Corollary 2.6; this is due to the non-locality of the Euler-Lagrange equation for this problem. It also implies the analytic regularity of ∂Ω 0 ∩M .
Then, thanks to a dilation of the metric and a control of the volume constraint, we reformulate in Section 3 the problem into solving for any small ε the equation
F (p, ε,v) = 0 (6)
where p ∈ ∂M ,v ∈ C 2,α (S n + ) is a function that parametrize a perturbation of the half-geodesic ball B + g,ε (p), and F (p, ε,v) represents the difference between g(∇u, ν) and its mean value on the boundary of this perturbed half geodesic ball. We then want to solve this equation for ε > 0 by using the implicit function Theorem and therefore study the operator ∂vF (p, 0, 0), which is basically related to the second order shape derivative of λ 1 at the Euclidian half-ball. This is the purpose of Sections 4.1 and 4.2, where we use a symmetrization argument to come down to the study of the same operator in the Euclidian ball, which has been done in [21]. As expected, that operator has a nontrivial kernel (because of the invariance of λ 1 by translation along ∂R n+1 + in the Euclidian setting) and we are only able to solve F (p, ε,v(p, ε)) = k(p, ε)
where k(p, ε) is a linear function induced by an element of ∂R n+1 + , see Proposition 4.4. Here comes the final step of the proof of Theorem 1.3, which takes into account the geometry of ∂M : by studying the expansion of F,v with respect to ε, we prove in the end of Section 4 that close to a point p 0 which is a nondegenerate critical point of the mean curvature of ∂M , one can chose p ε such that k(p ε , ε) = 0 and conclude the proof. We insist on the fact that this step is more involved here than in [21]: indeed, the expansions in ε contain lower order term than in the case without boundary (see Lemma 4.3 and Propositions 4.4,4.5). Nevertheless, thanks to the choice of our coordinates, the strategy still applies because these lower order terms are orthogonal to linear functions induced by elements of ∂R n+1 + .
Characterization of boundary extremal domains
In this section, we focus on an analytic characterization of extremal domains. The main difficulty here is to handle the shape derivative of Ω → λ Ω in a nonsmooth setting. Indeed, because of the presence of a boundary in M , we are naturally led to deal with domains that are only piecewise smooth. First, we will treat the regularity for the mixed problem (2) in some domains called boundary edge domains. We compute then the shape derivative of Ω → λ Ω in this setting. Since we have to deal with possibly nonsmooth eigenfunctions, one needs to carefully prove the differentiability of Ω → λ Ω and compute the shape derivative. We will also insist on some important aspects of the non-locality of the extremality condition for λ 1 , and compare it with the case of critical points for the area functional.
Boundary edge domains and regularity of the eigenfunction
Definition 2.1.
Let Ω be a boundary domain of the manifold M , that is to say ∂Ω ∩ ∂M = ∅. We say that Ω is a boundary edge domain if it satisfies the following condition:
1. ∂Ω ∩M and ∂Ω ∩ ∂M are smooth n-dimensional submanifolds with boundary, 2. Γ := ∂Ω ∩M ∩ ∂M is a (n − 1)-dimensional smooth submanifold without boundary.
In that case, given p ∈ Γ we can define ω(p) the angle between the normal vector to Γ tangent to ∂M and the normal vector to Γ tangent to ∂Ω ∩M . The function ω : Γ → [0, π] will be referred to as the contact angle of the domain Ω, see Fig. 3. Let Ω be a connected boundary edge domain of finite volume such that the contact angle ω is strictly between 0 and π. Then there exists ε > 0 such that for any f ∈ H −1/2+ε (Ω), the solution u of
ω(p 2 ) p 2 ∂M p 1 Γ Ω ω(p 1 ) −∆ g u = f in Ω u = 0 on ∂Ω ∩M , g(∇u, ν) = 0 on ∂Ω ∩ ∂M (7)
is in the space H 3/2+ε (Ω). Remark 2.3. It is important for our purpose to work here with Sobolev regularity: if indeed we work with Hölderregularity, we can only conclude that u ∈ C 0,1/2+ε (Ω), which does not suffice to justify the expression of the shape derivative, which uses the trace of the gradient on ∂Ω, see Section 2.2, while from the fact that u ∈ H 3/2+ε (Ω), we can deduce that ∇u has a trace in L 2 (∂Ω) (we use here a trace theorem, valid since under our assumptions, Ω has a Lipschitz boundary).
Proof. Let f ∈ H s (Ω) where s ∈ (−1, 0). It is well-known from the variational formulation of the problem that there exists a unique u ∈ H 1 (Ω) weak solution of (7). We wonder for which s we can state that u ∈ H s+2 (Ω). To that end, we work locally around a point p ∈ Γ: there exist special cylindrical coordinates (r, θ, y) such that Γ correspond to r = 0, y ∈ Γ parametrizes the edge (p corresponding to y = 0), and Ω corresponds to 0 < θ < ω(y); since Ω is a boundary edge domain, these coordinates are well-defined and C ∞ . From the literature on edge asymptotics, we know that u can be written around p as the sum of a singular function u sing and a remainder term u reg which is more regular that u sing ; more precisely, it is known (see for example [7,8,9,13,14]) that if ω(y) ∈ (0, π/2) then u sing (r, θ, y) = 0 and u reg ∈ H s+2 (Ω)
if ω(y) ∈ (π/2, π) then u sing (r, θ, y) = c(y) r π/2ω(y) ϕ(θ, y),
if ω(y) = π/2 in a neighborhood of y = 0, then u sing (r, θ, y) = r
q≥1 c q (y) ln q (r) ϕ q (y, θ) ,
where c, (c q ) q∈N (containing only a finite number of non-zero terms) and ϕ, (ϕ q ) q∈N are smooth functions (we notice that when n = 1, the set Γ is made of two points, in that case the regularity on Γ is an empty condition). Let us conclude in the last two cases. In the second one, we know that
if α > s ′ − n + 1 2 , then r → r α ∈ H s ′ (R n+1 ),
and therefore the regularity increases with small angles, and the worst regularity is obtained when the angle is close to π, but is always strictly better than H 3/2 which is the limit case when ω = π and n + 1 = 2. In the last case, it is clear that r ln q (r) = o(r 1−δ ) for any small δ, so we obtain that the regularity is also better than H 3/2 , therefore there exists s strictly above −1/2 such that u ∈ H s+2 (Ω). It remains to understand the case where ω(0) = π/2 but ω is not constant in a neighborhood of y = 0. In that case, the asymptotic development is more involved (phenomenon of crossing singularities), but it is explained in [8,9] that up to an arbitrary small loss of regularity, we obtain the same range of regularity as in the case ω = π/2, and therefore again u sing is in H 3/2+ε (Ω).
In the previous proof, we have seen that the regularity is more or less monotone with respect to the contact angle: smaller is the angle, higher is the regularity, and for angles close to π, the regularity decreases up to the space H 3/2 . However, it is also known that there exists some exceptional angles, for which the regularity is higher than expected (see for example [1] for a description of this phenomenon for the angle π/4 in dimension 2). We prove here that the angle π/2 is such an exceptional angle in our situation. More precisely we prove that when the angle is π/2 everywhere on the interface, the regularity is actually C 2,α , whereas it was expected to be C 0,α for every α in the proof of the previous statement. This will be very useful in the proof of Theorem 1.3. This result is related to the fact that one can use a symmetrization argument to conclude that the first expected term in the asymptotic development of u vanishes.
Proposition 2.4.
Let Ω be a boundary edge domain, such that the angle ω defined on Γ is constant and equal to π/2. Then for every α < 1 and any f ∈ C 0,α (Ω), the solution u of (7) is in C 2,α (Ω).
Proof. We use the same setting as in the proof of Proposition 2.2, but now in the class of Hölder spaces, so we consider f ∈ C 0,α (Ω). Around p ∈ Γ, from [8,9,13,14], we know that the exponents in the asymptotic development for the mixed boundary problem are (π/2ω + kπ/ω) k∈N , so for the angle π/2 the first terms are 1 and 3 and since r → r 3 ln q (r) belongs to the space C 2,α (Ω) for every α and any integer q, we conclude that
u(r, θ, y) = r q≥1 c q (y) ln q (r) ϕ q (y, θ) + u reg (r, θ, y),(8)
for y close to 0, r small, θ ∈ (0, π/2) and where functions (c q , ϕ q ) are smooth and u reg is in C 2,α locally around p.
The result will be proven if we prove that c q = 0 for q ≥ 1. To that end, we use a symmetrization procedure through ∂M , using around p ∈ Γ the coordinates (x 0 , x) described in (4). We define
U = Ψ −1 (Ω ∩ B + g,r0 (p)) ⊂ B + r0 , so that ∂U ∩ ({0} × R n ) = Ψ −1 (∂Ω ∩ ∂M ∩ B +
g,r0 (p)). With this choice of coordinates, U is again a boundary edge domain whose contact angle is constant and equal to π/2 on γ = Ψ −1 (Γ).
We now define W = {(x 0 , x) /(|x 0 |, x) ∈ U } and
∀(x 0 , x) ∈ W,ů(x 0 , x) = u(x 0 , x) if x 0 > 0 u(−x 0 , x) if x 0 < 0
and similarly we defineg andf .
Since the contact angle is π/2, the symmetrized domain W is smooth around 0; using that u satisfies a Neumann boundary condition on ∂Ω ∩ ∂M , we deduce thatů satisfies
−∆gů =f in W u = 0 on ∂W ∩ B r0 .
and finally, the symmetrized metricg is no longer C ∞ but has Lipschitz coefficients, andf is again in C 0,α (W ).
Since the Laplace operator can be written in a divergence form
∆gu = 1 |g| ∂ i |g|g ij ∂ jů
we can apply the regularity theory for elliptic PDE in divergence form in a smooth set, with Lipschitz coefficients: precisely, from [19,Theorem 8.34] we know thatů ∈ C 1,α W and therefore (c q ) q≥1 must be zero, and finally u ∈ C 2,α (Ω).
Shape derivative in nonsmooth domains
Proposition 2.5.
Let Ω 0 be a connected boundary domain of finite volume. Assume that (Ω t ) t is a deformation of Ω 0 induced by the vector field V , as defined in Definition 1.2. Then t −→ λ t is C ∞ around t = 0. If moreover Ω 0 is a boundary edge domain such that the contact angle is strictly between 0 and π, then g(∇u 0 , ν 0 ) ∈ L 2 (∂Ω 0 ) and
dλ t dt | t=0 = − ∂Ω0∩M (g(∇u 0 , ν 0 )) 2 g(V, ν 0 ) dvol g ,(9)
where dvol g is the volume element on ∂Ω 0 ∩M for the metric induced by g and ν 0 is the normal vector field on ∂Ω 0 ∩M .
Before proving this result, we give some remarks and consequences. The differentiability of some similar shape functional for mixed boundary value problem is studied in [25,Section 3.9] in the case of a smooth domain, which corresponds to the case of a angle constant and equal to π. In that case formula (9) is not valid since the eigenfunction u is not smooth enough. Also in [3], the case of angles different from π is considered, but for a different shape functional, and restricted to the two-dimensional case. Proposition 2.5 allows us to characterize extremal domains for the first eigenvalue of the Laplace-Beltrami operator under 0 mixed boundary conditions, and state the problem of finding extremal domains into the solvability of an over-determined elliptic problem. As a consequence of the previous result, we obtain indeed:
Corollary 2.6.
Let Ω 0 be a boundary edge domain. Then Ω 0 is extremal if and only if the first eigenfunction u 0 of
Ω 0 satisfies g(∇u 0 , ν 0 ) = constant on ∂Ω 0 ∩M(10)
where ν 0 is the outward normal vector field on ∂Ω 0 ∩M . In that case, ∂Ω 0 ∩M necessarily meets ∂M orthogonally, that is to say the contact angle function ω is equal to π/2 on Γ.
Proof of Corollary 2.6: Let Ω 0 be a boundary extremal domain for the first eigenvalue of the Laplace-Beltrami operator, with 0 Dirichlet boundary condition on ∂Ω 0 ∩M and 0 Neumann boundary condition on ∂Ω 0 ∩ ∂M . Using Proposition 2.5, we obtain
∂Ω0∩M (g(∇u 0 , ν 0 )) 2 g(V, ν 0 ) dvol g = 0
for all field V preserving the volume of the domain, i.e. such that
∂Ω0∩M g(V, ν 0 ) dvol g = 0.(11)
This means that g(∇u 0 , ν 0 ) is constant. On the other hand, if g(∇u 0 , ν 0 ) is constant, by the previous proposition we have that Ω 0 is extremal, because V satisfy (11).
It remains to investigate the angle between ∂Ω 0 ∩M and ∂Ω 0 ∩ ∂M , when (10) is satisfied. Let's assume that y → ω(y) is not constantly equal to π/2; then there exists a neighborhood in Y ⊂ Γ = ∂Ω ∩M ∩ ∂M where ω is different from π/2. We work locally around a point y 0 ∈ Y. We need now a more explicit version of the asymptotic development written in the proof of Proposition 2.2. To that end, we use the results of [10,11,9] which asserts that since the principal part of our operator is the Euclidian Laplacian, we have, up to a smooth change of coordinates, that u 0 (r, θ, y) can be written u reg (r, θ, y) + u sing (r, θ, y) with:
if ω(y) ∈ (0, π/2) in Y, then u sing = 0 and u reg ∈ H s+2 (Ω) is flat at order 2, which means u reg = O(r 2 ) and
∇u reg = O(r), if ω(y) ∈ (π/2, π) in Y, then u sing (r, θ, y) = c(y)r π/2ω(y) cos π 2ω(y) θ ,
and u reg is more flat than u sing , meaning u reg = o(r) and ∇u reg = o(1),
(note that here, with the terminology of [8,9], there is no crossing singularities, since ω(y) = π/2 on Y and we are only interested in the first term of the asymptotic). Therefore in the first case g(∇u 0 , ν 0 ) = O(r) and in the second case g(∇u 0 , ν 0 ) behaves like − π 2ω(y) c(y)r π/2ω(y)−1 sin π 2ω(y) θ , and therefore, in both cases, cannot be a nonzero constant on ∂Ω ∩M = {θ = ω(y)}. This is a contradiction (remind that from maximum principle, the constant g(∇u 0 , ν 0 ) cannot be a zero), and one concludes that ω(y) = π/2 everywhere on Γ.
Proof of Proposition 2.5: Let Ω 0 be a boundary domain, connected and of finite volume. We denote by ξ t = ξ(t, ·) the flow associated to V , ν t the outward unit normal vector field to ∂Ω t . We first remind that, since Ω t is connected, for t small enough λ t the first eigenvalue of Ω t with mixed boundary condition is simple, so one can define t → u t ∈ H 1 0 (Ω t ) the one-parameter family of first eigenfunctions of the Laplace-Beltrami operator, normalized to be positive and have L 2 (Ω t )-norm equal to 1. As usual in the computation of a shape derivative, we
consider u t = u t • ξ(t, ·) ∈ H 1 0 (Ω 0 ). Step 1: ∃ t 0 > 0 such that t ∈ (−t 0 , t 0 ) → ( u t , λ t ) ∈ H 1 0 (Ω 0 ) × R is C ∞ .
The variational formulation of the equation satisfied by u t is:
Ωt g(∇u t , ∇ϕ) = λ t Ωt u t ϕ , ∀ϕ ∈ H 1 0 (Ω t ).
We are going to transport that formulation on the fixed domain Ω 0 , in order to obtain the variational formulation satisfied by u t ∈ H 1 0 (Ω). To that aim, we use the following equality, which relies on the fact that
ξ t (∂Ω 0 ∩ ∂M ) = ∂Ω t ∩ ∂M
and is a consequence of the hypothesis ξ t (∂M ) ⊂ ∂M :
H 1 0 (Ω 0 ) = {ϕ • ξ t , ϕ ∈ H 1 0 (Ω t )}.
With this equality and a change of variable (see for example [18] for details), we obtain:
Ω0 g(A(t) ∇ u t , ∇ϕ) = λ t Ω0 u t ϕ J t , ∀ϕ ∈ H 1 0 (Ω 0 ), where J t = det(Dξ t ), and A(t) := J t Dξ −1 t (Dξ −1 t ) T . We then define G : (−t 0 , t 0 ) × H 1 0 (Ω 0 ) × R −→ H 1 0 (Ω 0 ) ′ × R (t, v, µ) −→ −div g (A(t)∇v) − µvJ t , Ω0 v 2 J t − 1 where H 1 0 (Ω 0 ) ′ is the dual space of H 1 0 (Ω 0 )
, and −div g (A(t)∇v) has to be understood in the weak sense:
−div g (A(t)∇v), ϕ H 1 0 (Ω0) ′ × H 1 0 (Ω0) = Ω0 g(A(t)∇v, ∇ϕ).
It is easy to check that G is C ∞ , see again [18] for more details. In order to apply the implicit function theorem for the equation G(t, u t , λ t ) = 0, we focus on the differential of G at (0, u 0 , λ 0 ) with respect to the couple (v, µ):
∂ (v,µ) G(0, u 0 , λ 0 )(w, ν) = −∆ g w − νu 0 − λ 0 w , 2 Ω0 u 0 w , ∀(w, ν) ∈ H 1 0 (Ω t ) × R.
Because of the Banach isomorphism Theorem, in order to prove to prove that such differential is an isomorphism, it is enough to prove that given (f,
Λ) ∈ H 1 0 (Ω 0 ) ′ × R, the equation −∆ g w − νu 0 − λ 0 w, 2 Ω0 u 0 w = (f, Λ) admits a unique solution (w, ν) ∈ H 1 0 (Ω 0 ) × R. The operator −∆ g − λ 0 ½ has a one-dimensional kernel, spanned by u 0 . Therefore f + νu 0 is in the range of −∆ g − λ 0 ½ if and only if it is orthogonal to u 0 (in the sense of the duality H 1 0 (Ω 0 ) ′ × H 1 0 (Ω 0 )
). This leads to the unique value ν = − f, u 0 .
Moreover, one knows that the solutions w of (
−∆ g − λ 0 ½) w = f + νu 0 form a one-dimensional affine space v 0 + Span(u 0 ), so w = v 0 + αu 0 for some α ∈ R. The equation 2 Ω0 u 0 w = Λ uniquely determines α and so w. We can conclude that ∂ (v,µ) F (0, u 0 , λ 0 ) is an isomorphism, and therefore t → ( u t , λ t ) is C ∞ .
Now and for the rest of the proof, Ω 0 is assumed to be a boundary edge domain whose contact angle is always strictly between 0 and π.
Step 2: Generalized Green formula: we prove in this step that given ε ∈ (0, 1/2) and Ω a Lipschitz domain,
denoting H s (∆ g , Ω) := ϕ ∈ H s (Ω), ∆ g ϕ ∈ L 2 (Ω) for s ∈ (1/2, 3/2) we have: ∀u ∈ H 3/2−ε (∆ g , Ω), ∀v ∈ H 1/2+ε (∆ g , Ω)
,
Ω (v∆ g u − u∆ g v) = g(∇u, ν 0 ), v H −ε (∂Ω)×H ε (∂Ω) − u, g(∇v, ν 0 ) H 1−ε (∂Ω)×H −1+ε (∂Ω) (12)
When u, v are smooth, this equality is just the classical Green formula. The above generalization is easily obtained by a density argument, using the following result from [6, Lemma 2 and 3]:
H 3/2−ε (∆ g , Ω) = {ϕ ∈ H 1 (Ω), ∆ g ϕ ∈ L 2 (Ω) and ϕ |∂Ω ∈ H 1−ε (Ω)},
and H 1/2+ε (∆ g , Ω) = {ϕ ∈ H ε (Ω), ∆ g ϕ ∈ L 2 (Ω) and g(∇ϕ, ν 0 ) |∂Ω ∈ H −1+ε (Ω)} (13) and that C ∞ (Ω) is dense in H 3/2−ε (∆ g , Ω).
Step 3:
Computation of d dt u t : From u t = u t • ξ −1 t , we obtain that u ′ = d dt |t=0 u t is well-defined in Ω 0 and that u ′ = u ′ − g(∇u, V ),(14)
where u ′ = d dt |t=0 u t ∈ H 1 0 (Ω 0 ), well-defined from Step 1. Using that u ∈ H 3/2+ε (Ω 0 ) and that u ′ ∈ H 1 (Ω 0 ), we know from (14) that u ′ ∈ H 1/2+ε (Ω 0 ). We also know that, the domain Ω 0 being piecewise C ∞ , the functions u and u ′ are locally C ∞ on Ω 0 \ Γ. With these regularities, we can compute the equation and the boundary conditions satisfied by u ′ : first, we differentiate with respect to t the identity
∆ g u t + λ t u t = 0.(15)
and evaluate the result at t = 0 to obtain
∆ g u ′ 0 + λ 0 u ′ 0 = −λ ′ 0 u 0 , in Ω 0 .(16)
Moreover, using again (14), we obtain that
u ′ = −g(∇u, V ) on ∂Ω ∩M .
and since u 0 = 0 on ∂Ω 0 ∩M , only the normal component of V plays a rôle in the previous formula. Therefore, we have, again since ξ(t, ∂Ω 0 ∩ ∂M ) = ∂Ω t ∩ ∂M :
u ′ = − g(∇u 0 , ν 0 ) g(V, ν 0 ), on ∂Ω 0 ∩M(17)
About the Neumann part of the boundary, we have:
for all p ∈ ∂Ω 0 ∩ ∂M, g(∇u t (ξ(t, p)), ν t ) = 0.
Since V is tangential on ∂M , using the normal geodesic coordinates we have ν t = −∂ x 0 on ∂Ω t ∩ ∂M , and in particular it does not depend on t and
g(∇u t (ξ(t, p)), ν t ) = −∂ x 0 u t (ξ(t, p)) = 0.(18)
So, differentiating (18) with respect to t and evaluating the result at t = 0 we obtain
0 = −∂ x 0 ∂ t u 0 − g(∇∂ x 0 u 0 , V ) = −∂ x 0 ∂ t u 0 = g(∇∂ t u 0 , ν 0 )(19)
on ∂Ω 0 ∩ ∂M , where we used the facts that ∂ x 0 u 0 = 0 on ∂Ω 0 ∩ ∂M and that g(V, ν 0 ) = 0 in ∂Ω 0 ∩ ∂M .
Step 5: Computation of d dt |t=0 λ t : From (16), multiplying by u and integrating over Ω, we obtain, using the generalized Green formula together with the regularity we have proven on u and u ′ :
λ ′ 0 = Ω (−∆ g u ′ − λ 0 u ′ )u = Ω (−∆ g u − λu)u ′ + u ′ , g(∇u, ν 0 ) H −ε (∂Ω)×H ε (∂Ω) − u, g(∇u ′ , ν 0 ) H 1−ε (∂Ω)×H −1+ε (∂Ω) .
Since u = 0 on ∂Ω ∩M and g(∇u ′ , ν 0 ) = 0 on ∂Ω ∩ ∂M , we have u, g(∇u ′ , ν 0 ) H 1−ε (∂Ω)×H −1+ε (∂Ω) = 0. Finally, since u and u ′ are smooth enough so that g(∇u, ν 0 ) |∂Ω , u ′ |∂Ω ∈ L 2 (∂Ω), we can write
u ′ , g(∇u, ν 0 ) H −ε (∂Ω)×H ε (∂Ω) = ∂Ω u ′ g(∇u, ν 0 ) = − ∂Ω∩M (g(∇u, ν 0 )) 2 g(V, ν),
and we finally obtain
λ ′ = − ∂Ω∩M (g(∇u, ν 0 )) 2 g(V, ν).
Extremal domains versus the isoperimetric problem
As we said, extremal domains are the critical points of the functional Ω → λ Ω under a volume constraint Vol g Ω = κ. The problem of finding extremal domains for the first eigenvalue of the Laplace-Beltrami operator is considered, by the mathematical community, very close to the isoperimetric problem.
Given a compact Riemannian manifold M and a positive number κ < Vol g (M ), where Vol g (M ) denotes the volume of the manifold M , the isoperimetric problem consists in studying, among the compact hypersurfaces Σ ⊂ M enclosing a region Ω of volume κ, those which minimize the area functional Ω → Vol g (∂Ω ∩M ) (note that we do not take in account the area of ∂Ω coming from the boundary of M ). The solutions of the isoperimetric problem are (where they are smooth enough) constant mean curvature hypersurfaces and intersect the boundary of the manifold orthogonally (see for example [24]). In fact, constant mean curvature hypersurfaces intersecting ∂M orthogonally are exactly the critical points of the area functional Ω → Vol g (∂Ω ∩M ) under a volume constraint Vol g Ω = κ.
In the case of a manifod M without boundary, it is well known that the determination of the isoperimetric profile λ Ω (see [4]). For this reason it is natural to expect that the solutions to the isoperimetric problem for small volumes are close in some sense to the solutions of the Faber-Krähn minimization problem. And such closeness can be expected also for the corresponding critical points.
The results known up to now about extremal domains underline such expectations. In the case of a manifold without boundary, the constructions of extremal domains in [21,12] are the parallel of the constructions of constant mean curvature topological spheres in a Riemannian manifold M done in [28,22]. And in the case of a manifold with boundary, our construction is the parallel of the constructions of constant mean curvature topological half-spheres in a Riemannian manifold M done in [15] for dimension 3.
Nevertheless, Proposition 2.5 and Corollary 2.6 show a very interesting difference between extremal domains and critical points of the area functional, based on the following: Remark 2.7. A significant fact contained in the statement of Proposition 2.5 is that the shape derivative for the first eigenvalue of the Laplace-Beltrami operator with mixed boundary condition in the boundary edge domain Ω 0 does not contain a singular term supported by the "corner part" of the boundary ∂Ω 0 , as it is the case for the area functional, see (21).
In order to understand the consequences of this remark, let's compare the Euler-Lagrange equations of the two problems: criticality for λ 1 is written
dλ t dt | t=0 = ∂Ω0∩M (g(∇u 0 , ν 0 )) 2 g(V, ν 0 ) dvol g = 0(20)
whereas for the area functional we have
d dt Vol g (∂Ω t ∩M )| t=0 = ∂Ω0∩M H 0 g(V, ν 0 ) + Γ g(V, τ 0 ) = 0 ,(21)
where (Ω t ) t is a volume preserving deformation of Ω 0 given by the vector field V , H 0 is the mean curvature of ∂Ω 0 ∩M , ν 0 is the normal vector on ∂Ω 0 ∩M , and τ 0 is the normal vector to Γ tangent to ∂Ω 0 ∩M . For the area functional, the consequence of (21) is that in order to be critical Ω 0 must satisfy, denoting ν 1 the normal vector to Γ tangent to ∂M : H 0 ≡ constant, and g(τ 0 , ν 1 ) = 0 or equivalently ω ≡ π/2 on Γ , the first condition being obtained with vector fields V supported inM whereas the second condition is obtained thanks to vector fields V that are supported in a neighborhood of Γ. For λ 1 , only using vector fields V that are supported inM we obtain as a consequence of (20) that in order to be critical Ω 0 must satisfy:
g(∇u 0 , ν 0 ) = constant on ∂Ω 0 ∩M .(22)
The fact that the contact angle is π/2 on Γ is already contained in the above equation (see Corollary 2.6), and therefore domains that are critical domains for λ 1 in the sense of Definition 1.2 (i.e. for any vector field V tangent on ∂M ) are the same as critical domains for λ 1 restricted to vector fields supported inM , which is not the case for the area functional.
In other words, one can easily build surfaces that have a constant mean curvature but intersects the boundary ∂M with an angle different from π/2 (and therefore are not extremal sets for the relative perimeter under volume constraint), whereas every set satisfying (22) intersects the boundary ∂M with angle equal to π/2. These properties lie on the fact that the operator given by the mean curvature is local while the Dirichlet to Neumann operator is nonlocal.
3 Analysis of the problem 3.1 Notations and formulation of the problem Euclidean notations. We define the following notations:
R n+1 + = {x = (x 0 , x ′ ) = (x 0 , x 1 , . . . , x n ) ∈ R n+1 : x 0 > 0}
will be the upper Euclidean half-space,
B + 1 = B 1 ∩ R n+1 +
will be the upper Euclidean unit half-ball and
S n + = {x ∈ S n : x 0 > 0}
will be the upper Euclidean unit hemisphere. Given a continuous function f : S n + −→ (0, ∞), we also denote
B + f := x ∈ R n+1 + : 0 < |x| < f (x/|x|) .
Riemannian notations in (M, g). Let p a point of ∂M . We denote by E 1 , ..., E n the orthonormal base of T p ∂M associated to the geodesic normal coordinates x 1 , ..., x n in ∂M around p. If the point q ∈ ∂M has coordinates x ′ ∈ R n , we set
Θ(x ′ ) := n i=1 x i E i ∈ T p ∂M .(23)
The point q ∈ ∂M whose geodesic coordinates are given by x ′ is
q = Φ(x ′ ) = Exp ∂M p (Θ(x ′ )) .
Given a continuous function f : S n + −→ (0, ∞) whose L ∞ norm is small (say less than the cut locus of p) we define
B + g,f (p) := Exp M Φ(x ′ ) (x 0 N (Φ(x ′ ))) : x ∈ R n+1 + 0 < |x| < f (x/|x|) .
The subscript g is meant to remind the reader that this definition depends on the metric.
Formulation of the problem. Our aim is to show that, for all ε > 0 small enough, we can find a point p ε ∈ ∂M and a (smooth) function v = v(p ε , ε) : S n + −→ R with 0 Neumann condition at the boundary of S n + such that
Vol B + g,ε(1+v) (p) = ε n Vol B + 1(24)
and the over-determined elliptic problem
∆ g φ + λ φ = 0 in B + g,ε(1+v) (p) φ = 0 on ∂B + g,ε(1+v) (p) ∩M g(∇φ, ν) = 0 on ∂B + g,ε(1+v) (p) ∩ ∂M g(∇φ, ν) = constant on ∂B + g,ε(1+v) (p) ∩M(25)
has a nontrivial positive solution, where ν is the normal vector on ∂B + g,ε(1+v) (p). Notice that the 0 Neumann boundary condition on v is justified by Corollary 2.6. Indeed, the half ball B + g,ε (p) intersects ∂M orthogonally, and then, since an extremal domain also intersects ∂M orthogonally, the deformation v should satisfy a fortiori a 0 Neumann boundary condition.
Dilation of the metric
We follow the strategy of [21], paying attention to the fact that we are working in a more general situation because our domains are boundary edge domains. Our first aim is to give a sense to the problem when ε = 0. Observe that, considering the dilated metricḡ := ε −2 g, Problem (24)-(25) is equivalent to finding a point p ∈ ∂M and a function v : S n + −→ R with 0 Neumann condition at the boundary of S n + such that
Vol B + g,1+v (p) = Vol B + 1(26)
and for which the over-determined elliptic problem
∆ḡφ +λφ = 0 in B + g,1+v (p) φ = 0 on ∂B + g,1+v (p) ∩M g(∇φ,ν) = 0 on ∂B + g,1+v (p) ∩ ∂M g(∇φ,ν) = constant on ∂B + g,1+v (p) ∩M(27)
has a nontrivial positive solution, whereν is the normal vector on ∂B + g,1+v (p). The relation between the solutions of the two problems is simply given by φ = ε −n/2φ and λ = ε −2λ .
Let us define the coordinates y = (y 0 , y ′ ) = (y 0 , y 1 , ..., y n ) ∈ B + 1 bȳ Ψ(y) := Exp M Φ(y ′ ) ε y 0N (Φ(y ′ )) whereΦ (y ′ ) := Exp ∂M p ε n i=1 y i E i for p ∈ ∂M , andN is the unit normal vector about ∂M for the metricḡ pointing into M . Using Proposition 5.1 of the Appendix, in the new coordinates y the metricḡ can be written as
g 00 = 1 g 0j = 0 g ij = δ ij + 2 ε g(∇ Ei N, E j ) y 0 + ε 2 R 0i0j (y 0 ) 2 + ε 2 g(∇ Ei N, ∇ Ej N ) (y 0 ) 2 + 2 ε 2 k R k0ij y k y 0 + 1 3 ε 2 k,ℓR ikjl y k y ℓ + O(ε 3 )(28)
for i, j, k, l = 1, ...n, where R andR are respectively the curvature tensors of M and ∂M , and
R 0i0j = g R(N, E i ) N, E j R k0ij = g R(E k , N ) E i , E j R ijkl =g R (E i , E k ) E j , E ℓ .
In the coordinates y and the metricḡ, the problem can be continuously extended for ε = 0 and in this case it becomes
∆φ +λφ = 0 in B + 1+v φ = 0 on ∂B + 1+v ∩ R n+1 + ∇φ,ν = 0 on ∂B + 1+v ∩ ∂R n+1 +(29)
where ∆ denotes the usual Laplacian in R n+1 and ·, · the usual scalar product in R n+1 , with the normalization
B + 1+vφ 2 = 1(30)
and the volume constraint Vol(B + 1+v ) = Vol(B + 1 ). In particular, when v = 0 we have
∆φ 1 + λ 1 φ 1 = 0 in B + 1 φ 1 = 0 on ∂B + 1 ∩ R n+1 + ∇φ 1 , ν = 0 on ∂B + 1 ∩ ∂R n+1 +(31)
where λ 1 is the first eigenvalue of the unit Euclidean ball and φ 1 is the restriction to B + 1 of the solution to
∆φ 1 + λ 1φ1 = 0 in B 1 φ 1 = 0 on ∂B 1 .
chosen in order to be positive and have L 2 (B 1 ) norm equal to 2.
Volume constraint and differentiability with respect to (ε,v)
In this section, we deal with the volume condition (which leads to replace the variable v byv subject to the condition of having a zero mean), and prove the differentiability of (λ,φ) with respect to (ε,v). The result is similar to Proposition 3.2 in [21], and we use the same strategy, though we have to pay attention to the singularities at the boundary of our domain. Let us define the space
C 2,α m,N C (S n + ) := v ∈ C 2,α (S n + ), S n +v = 0 , ∂ N v = 0 on ∂S n + ,
where ∂ N v = 0 denotes the 0 Neumann condition at the boundary of S n + .
Proposition 3.1. Given a point p ∈ ∂M , there exists ε 0 > 0, locally uniform in p, such that for all ε ∈ (0, ε 0 ) and all functionv ∈ C 2,α m,N C (S n + ) such that v C 2,α (S n + ) ≤ ε 0 , there exists a unique positive functionφ =φ(p, ε,v) ∈ C 2,α (B + g,1+v (p)), a constantλ =λ(p, ε,v) ∈ R and a constant v 0 = v 0 (p, ε,v) ∈ R such that
Volḡ(B + g,1+v (p)) = Vol(B + 1 )(32)
where v := v 0 +v andφ is a solution to the problem
∆ḡφ +λφ = 0 in B + g,1+v (p) φ = 0 on ∂B + g,1+v (p) ∩M g(∇φ,ν) = 0 on ∂B + g,1+v (p) ∩ ∂M(33)
which is normalized by
B + g,1+v (p)φ 2 dvolḡ = 1 .(34)
In additionφ,λ and v 0 depend smoothly on the functionv and the parameter ε, can be extended smoothly to ε = 0 by (29), and in particular (φ,λ, v 0 ) = (φ 1 , λ 1 , 0) when (ε,v) = (0, 0).
Proof. The proof of this result is similar to the proof of Proposition 3.2 in [21], basically based on the implicit function Theorem. Therefore we only describe the differences from [21], which are the choice of coordinates and the regularity theory for the Laplace-Beltrami operator in domains with singularities.
For the choice of coordinates we use the following coordinates: given (v 0 ,v) ∈ R × C 2,α m,N C (S n + ) and v = v 0 +v, we consider the parameterization of B + g,1+v (p) = B + g,ε(1+v) (p) given bŷ
Ψ(y) := Exp M Φ(y ′ ) 1 + v 0 + χ(y)v y |y| y 0 N (Φ(y ′ )) whereΦ (y ′ ) = Exp ∂M p 1 + v 0 + χ(y)v y |y| n i=1 y i E i .
Here y = (y 0 , y ′ ) ∈ B + 1 , χ is a cutoff function identically equal to 0 when |y| ≤ 1/2 and identically equal to 1 when |y| ≥ 3/4, introduced to avoid the singularity at the origin of the polar coordinates. In these coordinates the metriĉ
g :=Ψ * ḡ(35)
can be written asĝ
= (1 + v 0 ) 2 i,j (δ ij + C ij ) dy i dy j ,
where the coefficients C ij = C ij ε,v ∈ C 1,α (B + 1 ) are functions of y depending on ε, v = v 0 +v and the first partial derivatives of v. It is important here to notice that
(ε, v 0 ,v) −→ C ij ε,v ∈ C 1,α (B + 1 )
are smooth maps, as in [21]. Now for all ψ ∈ C 2,α (B + 1 ) such that
B + 1 ψ φ 1 = 0 we define N (ε,v, ψ, v 0 ) := ∆ψ + λ 1 ψ + (∆ĝ − ∆ + µ) (φ 1 + ψ) , Volĝ(B + 1 ) − Vol (B + 1 ) where µ is given by µ = − B + 1 φ 1 (∆ĝ − ∆) (φ 1 + ψ) ,
so that the first entry of N is L 2 (B + 1 )-orthogonal to φ 1 (for the Euclidean metric). Thanks to the choice of coordinates, the mapping N is a smooth map from a neighborhood of (0, 0, 0, 0) in [0, ∞)×C 2,α m,N C (S n + )×C 2,α ⊥ , 0 (B + 1 )×R into a neighborhood of (0, 0) in C 0,α ⊥ (B + 1 )×R. Here the subscript ⊥ indicates that the functions in the corresponding space are L 2 (B + 1 )-orthogonal to φ 1 and the subscript 0 indicates that the functions satisfy the mixed condition at the boundary of B + 1 . The differential of N with respect to (ψ, v 0 ), computed at (0, 0, 0, 0), given by
∂ (ψ,v0) N (0, 0, 0, 0) = ∆ + λ 1 , n Vol(B + 1 ) is invertible from C 2,α ⊥,0 (B + 1 ) × R into C 0,α ⊥ (B + 1 ) × R, by Proposition 2.4.
Then the implicit function theorem applies as in [21] and completes the proof of the result.
Strategy for the proof of Theorem 1.3
We define the operator
F (p, ε,v) =ḡ(∇φ,ν) | ∂B + g,1+v (p)∩M − 1 Volḡ ∂B + g,1+v (p) ∩M ∂B + g,1+v (p)∩Mḡ (∇φ,ν) dvolḡ ,
whereν denotes the unit normal vector field to ∂B + g,1+v (p) ∩M , (φ, v 0 ) is the solution of (32)-(33)-(34). Recall that v = v 0 +v. The operator F is locally well defined in a neighborhood of (p, 0, 0) in ∂M × [0, ∞) × C 2,α m,N C (S n + ), and after canonical identification of ∂B + g,1+v (p) ∩M with S n + we can consider that it takes its values in C 1,α (S n + ). Moreover, it is easy to see that the zero mean condition is preserved, and then we will write that F takes its values in C 1,α m (S n + ). Our aim is to find (p, ε,v) such that F (p, ε,v) = 0. Observe that, with this condition,φ =φ(ε,v) will be the solution to the problem (27).
Following the proof of the previous result, we have the alternative expression for F :
F (p, ε,v) =ĝ(∇φ,ν) | ∂B + 1 ∩R n+1 + − 1 Volĝ(∂B + 1 ∩ R n+1 + ) ∂B + 1 ∩R n+1 +ĝ (∇φ,ν) dvolĝ ,
where this timeν is the the unit normal vector field to ∂B + 1 using the metricĝ defined by (35). Our aim is to solve the equation F (p, ǫ,v) = 0 for some (p, ǫ,v). The first question we should consider is the following: if we fix a point p ∈ ∂M , can we find for all ε small enough a functionv =v(ε) in order that
F (p, ǫ,v(ε)) = 0 ?
The answer will be negative, because we will see that the kernel K of ∂vF (p, 0, 0) :
C 2,α m,N C (S n + ) → C 1,α m (S n + )
is nontrivial. Nevertheless, we will obtain a characterization of K proving that it is given by the space of linear functions (restraint to the half-sphere) depending only on the coordinates y 1 , ..., y n , i.e. functions S n + → R y → a, y for some a = (a 0 , a) ∈ R n+1 with a 0 = 0. Moreover we will prove that ∂vF (p, 0, 0) is an isomorphism from K ⊥ to the image of ∂vF (p, 0, 0), and then the implicit function theorem will give the following result: for all ε small enough there exist an element k(ε) ∈ K and a functionv(ε) such that
F (p, ǫ,v(ε)) = k(ε) .
Clearly, since we fixed the point p, the functionv and the element k depend also on p, and in fact we have to write F (p, ǫ,v(p, ε)) = k(p, ε) .
In the last section we will show that it is possible to apply the implicit function theorem to the equation k(p, ε) = 0 obtaining that: for all ε small enough, there exists a point p ε such that k(p ε , ε) = 0 . and this will complete the proof of the result.
4 Solving the problem 4.1 Computation of the linearization of F with respect tov at (p, ε,v) = (p, 0, 0) In Section 3.3 we established the existence of a unique positive functionφ ∈ C 2,α B + 1+v (close to φ 1 ), a constant λ ∈ R (close to λ 1 ) and a constant v 0 ∈ R (close to 0), solutions to (32)-(33)-(34). Recall that λ 1 is the first eigenvalue of −∆ in the half ball B + 1 with 0 mixed boundary condition and φ 1 is the associated eigenfunction which is normalized to be positive and have L 2 (B + 1 ) norm equal to 1. For allv ∈ C 2,α m,N C (S n + ) let ψ be the (unique) solution of Proof. When ε = 0 we have already seen thatḡ in the coordinates y is the Euclidean metric. If v ∈ C 2,α m (S n ) we can define the operator F :F
∆ψ + λ 1 ψ = 0 in B + 1 ψ = −∂ r φ 1v on ∂B + 1 ∩ R n+1 + ∇ψ, ν = 0 on ∂B + 1 ∩ ∂R n+1 + (36) which is L 2 (B + 1 )-orthogonal to φ 1 . We define L 0 (v) := ∂ r ψ + ∂ 2 r φ 1v | ∂B + 1 ∩R n+1 + (37) Clearly we have L 0 : C 2,α m,N C (S n + ) → C 1,α m (S n + ) .(v) = ∇φ,ν | ∂B1+v − 1 Vol ∂B 1+v ∂B1+v ∇φ,ν ,
whereν denotes the unit normal vector field to ∂B 1+v andφ is the solution, with L 2 -norm equal to 2, of
∆φ +λφ = 0 in B 1+ṽ φ = 0 on ∂B 1+v .(38)
After identification of ∂B 1+v with S n we can considered the operatorF well defined from C 2,α m (S n ) into C 1,α m (S n ). In the proof of Proposition 4.3 in [21] it is proved that the linearization ofF with respect to v at v = 0 is given by the operatorL
0 : C 2,α m (S n ) −→ C 1,α m (S n ) v → ∂ rψ + ∂ 2 rφ 1 v | ∂B1(39)
whereφ 1 is the first eigenfunction of −∆ in B 1 with 0 Dirichlet boundary condition and normalized to have L 2 -norm equal to 2, andψ is the (unique) solution of
∆ψ + λ 1ψ = 0 in B 1 ψ = −∂ rφ1 v on ∂B 1 (40)
which is L 2 (B 1 )-orthogonal toφ 1 . Notice that φ 1 and ψ are then the restrictions ofφ 1 andψ to the half-ball B + 1 . Let w be a function in C 2,α m,N C (S n + ). We extend the function w to a functionw over all S n in this way: for (y 0 , y 1 , ..., y n ) ∈ S n + we setw (−y 0 , y 1 , ..., y n ) = w(y 0 , y 1 , ..., y n ) .
Observe thatw ∈ C 2,α (S n ) because the function w satisfies the Neumann condition at the boundary of S n + , and his mean is 0 because are 0 the means over S n + and over the complement of S n + . We conclude thatw ∈ C 2,α m,Sym (S n ), where the subscript Sym means that the function is symmetric with respect to the hyperplane {x 0 = 0}, and m means as usual that the function has mean 0. We have defined the mapping
α : C 2,α m,N C (S n + ) −→ C 2,α m,Sym (S n ) w →w ,(41)
and it is easy to see that this mapping in an isomorphism.
If we consider the operatorF defined only in C 2,α m,Sym (S n ), it is natural that its linearization with respect to v at v = 0 is given by the operatorL 0 restricted to C 2,α m,Sym (S n ) with image in C 1,α m,Sym (S n ). We observe that if v ∈ C 2,α m,Sym (S n ), then the solution of (38) is symmetric with respect to the hyperplain {x 0 = 0} and the normal derivative with respect to x 0 computed at {x 0 = 0} is 0. Then from the definitions of F andF we conclude that
F (p, 0,v) =F (α(v))| ∂B + 1 ∩R n+1 +
where α is the isomorphism defined in (41). We define also the mapping
β : C 1,α m,Sym (S n ) −→ C 1,α m,N C (S n + ) v → v| S n +
and we observe that it is an isomorphism. We claim that
L 0 = β •L 0 • α.
We remark that the operator β •L 0 • α is defined on C 2,α m,N C (S n + ) and his image is contained in C 1,α m,N C (S n + ). We have to prove that L 0 (w) =L 0 (w)| ∂B + 1 ∩R n+1 + By the symmetry of the funcionw with respect to the hyperplane {x 0 = 0}, we conclude that the solution of (40) with v =w is symmetric with respect to the hyperplane {x 0 = 0}, then ∂ x0ψ | {x0=0} = 0 andL 0 (w) is symmetric with respect to the hyperplane {x 0 = 0}. So the restriction ofψ to the half-ball B + 1 is the solution of (36), wherē v = w, and L 0 (w) is exactly the restriction ofL 0 (w) to ∂B + 1 ∩ R n+1 + . This completes the proof of the claim. Using this relation we conclude that that L 0 (w) =L 0 (α(w))| ∂B + 1 ∩R n+1
+ .
This completes the proof of the proposition.
Study of the operator L 0
Proposition 4.2. The operator
L 0 : C 2,α m,N C (S n + ) −→ C 1,α m,N C (S n + )
, is a self adjoint, first order elliptic operator. Its kernel K is given by the space of linear functions depending only on the coordinates y 1 , ..., y n , i.e. functions S n + → R y → a, y for some a = (a 0 , a ′ ) ∈ R n+1 with a 0 = 0. Moreover, L 0 has closed range and is an isomorphism from K ⊥ to Im(L 0 ), where K ⊥ is the space L 2 -orthogonal to K in C 2,α m,N C (S n + ) and Im(L 0 ) denotes the range of L 0 in C 1,α m,N C (S n + ).
Proof. LetL 0 the operator defined in (39) and α the isomorphism defined in (41). In Proposition 4.2 of [21] it is proved that:
•L 0 is a self adjoint, first order elliptic operator,
• its kernel is given by the space of linear functions restraint to S n , and
• there exists a constant c > 0 such that
v C 2,α (S n ) ≤ c L 0 (v) C 1,α (S n ) ,(42)
provided that v is L 2 (S n )-orthogonal to the kernel ofL 0 .
The last elliptic estimate implies that the operatorL 0 has closed range, and using the other two properties we have thatL 0 is an isomorphism from the space L 2 -orthogonal to its kernel and its range.
We are interested in considering the operatorL 0 defined only in the domain C 2,α m,Sym (S n ) and from now onL 0 will be defined only in C 2,α m,Sym (S n ). The image ofL 0 is naturally given by functions that are symmetric with respect to the hyperplane {x 0 = 0}, then we havẽ
L 0 : C 2,α m,Sym (S n ) −→ C 1,α m,Sym (S n )
We can conclude that the new operatorL 0 is a self-adjoint, first order elliptic operator, with kernelK given by the space of linear functions which are symmetric with respect to the hyperplane {x 0 = 0}, i.e. functions S n + → R y → a, y for some a = (a 0 , a ′ ) ∈ R n+1 with a 0 = 0. Inequality (42) holds naturally also for the new operatorL 0 , provided v is L 2 (S n )-orthogonal toK.
From the proof of Proposition 4.1 we have
L 0 = β •L 0 • α .
With this caracterization of the operator L 0 and the properties ofL 0 , we deduce that the kernel of L 0 is given by the space K of functions S n + → R y → a, y for some a = (a 0 , a ′ ) ∈ R n+1 with a 0 = 0, and that L 0 has closed range and is an isomorphism from K ⊥ to Im(L 0 ), where K ⊥ is the space L 2 -orthogonal to K in C 2,α m,N C (S n + ) and Im(L 0 ) denotes the range of L 0 in C 1,α m,N C (S n + ).
4.3
Solving the problem on the space orthogonal to the kernel of L 0 Lemma 4.3. Let p ∈ ∂M . There exists a function f p ∈ C 1,α ([0, 1]) such that
F (p, ε, 0)(y 0 , y ′ ) = ε f p (y 0 ) + O(ε 2 )
for all ε small enough.
Proof. We keep the notations of the proof of the Proposition 3.1 withv ≡ 0. Sincev ≡ 0, we have
N (ε, 0, 0, 0) = (∆ĝ − ∆ + µ) φ 1 , Volĝ(B + 1 ) − Vol (B + 1 ) , and µ = − B + 1 φ 1 (∆ĝ − ∆) φ 1 .
If in addition v 0 = 0, we can estimateĝ ij = δ ij +Ĝ ij ε y 0 + O(ε 2 ) , whereĜ ij are real constants. Hence, by the symmetry of the problem, N (ε, 0, 0, 0)(y 0 , y ′ ) = ε (ϕ(y 0 , |y ′ |),
V ) + O(ε 2 ) ,
where the first component of ϕ ∈ C 0,α ([0, 1] 2 ) and V is a real number. The implicit function theorem immediately implies that the solution of N (ε, v 0 , 0, ψ) = 0 satisfies ψ(ε, p, 0) C 2,α + |v 0 (ε, p, 0)| ≤ c ε but in addition there exist a functionψ p ∈ C 2,α ([0, 1] 2 ) such that ψ(ε, p, 0)(y 0 , y ′ ) = εψ p (y 0 , |y ′ |) + O(ε 2 ) .
To complete the proof, observe thatν = (1
+ v 0 ) −1 ∂ r on ∂B + 1 ∩ R n+1 + whenv ≡ 0. Therefore there exist a function f p ∈ C 2,α ([0, 1] 2 ) such thatĝ (∇φ,ν)(y 0 , y ′ ) = ∂ r φ 1 + εf p (y 0 , |y ′ |) + O(ε 2 ) .
(be careful thatĝ is defined with v 0 = v 0 (ε, p, 0) andv ≡ 0). Since ∂ r φ 1 is constant along ∂B + 1 ∩ R n+1 + , we conclude that there exist a function f p ∈ C 2,α ([0, 1]) such that
F (p, ε, 0)(y 0 , y ′ ) = εf p (y 0 ) + O(ε 2 ) .
This completes the proof of the Lemma.
Proposition 4.4. There exists ε 0 > 0 such that, for all ε ∈ [0, ε 0 ] and for all p in a compact subset of ∂M , there exists a unique functionv =v(p, ε) ∈ K ⊥ such that
F (p, ε,v(p, ε)) ∈ K .
The functionv(p, ε) depends smoothly on p and ε and v(p, ε)(y 0 , y ′ ) = εṽ p (y 0 ) + O(ε 2 ) for a suitable functionṽ p ∈ C 2,α ([0, 1]).
Proof. We fix p in a compact subset of ∂M and definē F (p, ε,v, a) := F (p, ε,v) + a, · By Proposition 3.1,F is a C 1 map from a neighborhood of (p, 0, 0, 0) in M ×[0, ∞)×K ⊥ ×∂R n+1 + into a neighborhood of 0 in C 1,α (S n + ). Moreover we have
•F (p, 0, 0, 0) = 0,
• the differential ofF with respect tov computed at (p, 0, 0, 0) is given by L 0 restricted to K ⊥ , and
• the image of the linear map a −→ a, · , a = (a 0 , a ′ ) with a 0 = 0 coincides with K.
Thanks to the result of Proposition 4.2, the implicit function theorem can be applied to the equation F (p, ε,v, a) = 0 at (p, 0, 0, 0) with respect to the variable ε. We obtain the existence ofv(p, ε) ∈ C 2,α m,N C (S n + ) and a(p, ε) ∈ ∂R n+1 + , smoothly depending on ε such thatF (p, ε,v(p, ε), a(p, ε)) = 0 , that means, by the definition ofF , F (p, ε,v(p, ε)) ∈ K .
The fact thatv depends smoothly on p and ε is standard. The ε-expansion ofv follow at once from Lemma 4.3.
4.4
Projecting over the kernel of L 0 : appearance of the mean curvature of ∂M Thanks to Proposition 4.4 we are able to build, for all p in a compact subset of ∂M and ε small enough, a function v(p, ε) in K ⊥ such that F (p, ε,v(p, ε)) ∈ K .
Now, as natural, we project the operator F over its K and we then we have to find, for each ε, the good point p ε in order that such the projection of F over K is equal to 0. In other words, for all ε small enough we want to find a point p ε ∈ ∂M such that
S n + F (p ε , ε,v(p ε , ε)) b, · = 0 for all b ∈ ∂R n+1 + .
The main result of this section is the following: with |b| = 1, we have the following ε-expansion:
S n + F (p, ε,v(p, ε)) b, · = C ε 2g (∇gH(p), Θ(b ′ )) + O(ε 3 ) .
where C is a real constant, H is the mean curvature of ∂M ,g is the metric of ∂M induced by g and Θ has been defined in (23).
Proof.
Take p ∈ ∂M , ε small enough,v ∈ C 2,α m,N C with small norm, and b ∈ ∂R n+1 + . We denote by L ε the linearization of F with respect tov, and by L 2 ε the second derivative of F with respect tov, both computed at the point (p, ε, 0):
L ε = ∂vF (p, ε, 0) and L 2 ε = ∂ 2 v F (p, ε, 0) . We have S n + F (p, ε,v) b, · = S n + (F (p, ε, 0) + L 0v ) b, · + S n + (F (p, ε,v) − F (p, ε, 0) − L εv ) b, · + S n + (L ε − L 0 )v b, ·
Now we apply this formula for our functionv =v(p, ε) given by Proposition 4.4. We havev ∈ K ⊥ , so L 0v ∈ K ⊥ , and then S n
+ L 0v b, · = 0 .
We obtain that
S n + F (p, ε,v) b, · = S n + F (p, ε, 0) b, · + S n + (F (p, ε,v) − F (p, ε, 0) − L εv ) b, · + S n + (L ε − L 0 )v b, ·(43)
wherev =v(p, ε) is the function given by Proposition 4.4. We need now two intermediate lemmas.
Lemma 4.6. For all p ∈ ∂M , for all b = (0, b ′ ) ∈ ∂R n+1 +
we have the following ε-expansion:
S n + F (p, ε, 0) b, · = C ε 2g (∇gH(p), Θ(b ′ )) + |b| O(ε 3 ) ,
where Θ is defined in (23) and
C = −2 S n + y 0 (y 1 ) 2 1 ∂ r φ 1 (1) B + 1 r |∂ r φ 1 | 2 where r = |y|.
Proof. We recall that
F (p, ε,v) =ĝ(∇φ,ν) | ∂B + 1 ∩R n+1 + − 1 Volĝ(∂B + 1 ∩ R n+1 + ) ∂B + 1 ∩R n+1 +ĝ (∇φ,ν) dvolĝ ,
where the metricĝ has been defined in (35) for the coordinates y. Then
S n + F (p, ε,v) b, · = S n +ĝ (∇φ,ν) b, · . Whenv = 0 we haveν = (1 + v 0 ) ∂ r on ∂B + 1 ∩ R n+1 + , where r = |y|. Then S n + F (p, ε,v) b, · = (1 + v 0 ) S n + ∂φ ∂r b, · = 1 + v 0 ∂ r φ 1 (1) S n + ∂φ ∂r ∇φ 1 , b(44)
where we used the fact that φ 1 is a radial function. Using this last property and the Green's identities we have:
S n + ∂φ ∂r ∇φ 1 , b = B + 1 (∆ + λ 1 )φ ∇φ 1 , b − B + 1φ (∆ + λ 1 ) ∇φ 1 , b = B + 1 (∆ + λ 1 )φ ∇φ 1 , b = B + 1 (∆ − ∆ĝ)φ ∇φ 1 , b + (λ 1 −λ) B + 1φ ∇φ 1 , b = B + 1 (∆ − ∆ĝ) φ 1 ∇φ 1 , b + B + 1 (∆ − ∆ĝ) (φ − φ 1 ) ∇φ 1 , b + (λ 1 −λ) B + 1 (φ − φ 1 ) ∇φ 1 , a
Let compute the first term. Recall that ∆ĝ := n i,j=0ĝ
ij ∂ yi ∂ yj + n i,j=0 ∂ yiĝ ij ∂ yj + 1 2 n i,j=0ĝ ij ∂ yi log |ĝ| ∂ yj .
From (28) we have that the coefficients of the metricĝ can be expanded, for i, k, j, ℓ = 1, ..., n, aŝ
g 00 (y) = (1 + v 0 ) 2 g 0j (y) = 0 g ij (y) = (1 + v 0 ) 2 δ ij + 2(1 + v 0 ) ε g(∇ Ei N, E j ) y 0 + R 0i0j (1 + v 0 ) 2 ε 2 (y 0 ) 2 +(1 + v 0 ) 2 ε 2 g(∇ Ei N, ∇ Ej N ) (y 0 ) 2 + 2(1 + v 0 ) 2 ε 2 k R k0ij y k y 0 + 1 3 (1 + v 0 ) 2 ε 2 k,ℓR ikjℓ y k y ℓ + O(ε 3 )
Keeping in mind that v 0 = v 0 (p, ε) = O(ε), the third equality simplifies slightly obtaininĝ
g 00 (y) = (1 + v 0 ) −2 g 0j (y) = 0 g ij (y) = (1 + v 0 ) −2 δ ij − 2(1 + v 0 ) ε g(∇ Ei N, E j ) y 0 − R 0i0j ε 2 (y 0 ) 2 −ε 2 g(∇ Ei N, ∇ Ej N ) (y 0 ) 2 − 2ε 2 k R k0ij y k y 0 − 1 3 ε 2 k,ℓR ikjℓ y k y ℓ + O(ε 3 ) .
Using the fact that R k0ii = 0, we have
log |ĝ| = 2n log(1 + v 0 ) − 2 ε(1 + v 0 ) H(p) y 0 + ε 2 2 − Ric(N ) + 4 i =j g(∇ Ei N, E i ) g(∇ Ej N, E j ) + i g(∇ Ei N, ∇ Ei N ) − 4 i =j g(∇ Ei N, E j ) g(∇ Ej N, E i ) (y 0 ) 2 + 1 3R kℓ y k y ℓ + O(ε 3 )
where Ric denotes the Ricci curvature of ∂M andR
kℓ = n i=1R ikiℓ .
A straightforward computation (still keeping in mind that v 0 = O(ε)) shows that
∆ − ∆ĝ φ 1 = − λ 1 (1 − (1 + v 0 ) −2 ) φ 1 + 2 (1 + v 0 ) −1 ε i,j g(∇ Ei N, E j ) y 0 y i y j r 2 ∂ 2 r φ 1 + δ i j r ∂ r φ 1 − y i y j r 3 ∂ r φ 1 + ε (1 + v 0 ) −1 H(p) y 0 r ∂ r φ 1 + ε 2 k,i,j,ℓ R 0i0j + g(∇ Ei N, ∇ Ej N ) (y 0 ) 2 + 2 R k0ij y k y 0 + 1 3R ikjℓ y k y ℓ · · y i y j r 2 ∂ 2 r φ 1 + δ i j r ∂ r φ 1 − y i y j r 3 ∂ r φ 1 + ε 2 k,i,j 2R i0ij y 0 + 1 3R ikji y k + 1 6R ik y k y j r ∂ r φ 1 + ε 2 − Ric(N ) + 4 i =j g(∇ Ei N, E i ) g(∇ Ej N, E j ) + i g(∇ Ei N, ∇ Ei N ) −4 i =j g(∇ Ei N, E j ) g(∇ Ej N, E i ) · (y 0 ) 2 r ∂ r φ 1
where i, j, k = 1, ..., n. Observe that we have used the fact that R(X, X) ≡ 0 and the symmetries of the curvature tensor for which R ijkl = R klij . Now, in the computation of
B + 1 (∆ − ∆ĝ) φ 1 ∇φ 1 , b ,
observe that the terms in the expansion of (∆ − ∆ĝ) φ 1 which contain an even number of coordinates different to y 0 , such as y 0 or y i y j y k y ℓ or (y 0 ) 2 y i y j etc. do not contribute to the result since, once multiplied by ∇φ 1 , b (keep in mind that b = (0, b ′ )), their average over S n + is 0. Therefore, we can write
B + 1 (∆ − ∆ĝ) φ 1 ∇φ 1 , b = ε 2 σ =0 B + 1 ∂ r φ 1 a σ y σ r · · 2 k,i,j R k0ij y i y j y k y 0 r 2 ∂ 2 r φ 1 − y i y j y k y 0 r 3 ∂ r φ 1 + 2 k,i,j R i0ij y 0 y j r ∂ r φ 1 + O(ε 3 )
We make use of the technical Lemmas 5.2 and 5.3 of the Appendix to conclude that
B + 1 (∆ − ∆ĝ) φ 1 ∇φ 1 , b =C ε 2g ∇gH(p), Θ(b ′ ) + O(ε 3 ).(45)whereC = −2 S n + y 0 (y 1 ) 2 B + 1 r |∂ r φ 1 | 2 .
Now we have to compute the terms
B + 1 (∆ − ∆ĝ) (φ − φ 1 ) ∇φ 1 , b and (λ 1 −λ) B + 1 (φ − φ 1 ) ∇φ 1 , a .
We observe that the coefficients of the metric, for i, j = 1, ..., n, are given bŷ
g ij (y) = δ ij +Ĝ ij ε y 0 + O(ε 2 )
for some constants G ij . Then the ε-first order term ofφ − φ 1 is radial in the coordinates y 1 , ..., y n , i.e. there exists a function h ∈ C 2,α ([0, 1] 2 ) such that
(φ − φ 1 )(y 0 , y ′ ) = ε h(y 0 , |y ′ |) + O(ε 2 ) .
Let ρ := |y ′ |. Using the same computation given above, we find
∆ − ∆ĝ (φ − φ 1 ) = (1 − (1 + v 0 ) −2 ) ∆(φ − φ 1 ) + O(ε 2 ) y 0 y i y j ρ 2 ∂ 2 ρ h + y 0 ρ δ i j ∂ ρ h − y 0 y i y j ρ 3 ∂ ρ h + ∂ y 0 h + O(ε 3 ) = O(ε 2 ) h (y 0 , ρ) + y 0 y i y j ρ 2 ∂ 2 ρ h + y 0 ρ δ i j ∂ ρ h − y 0 y i y j ρ 3 ∂ ρ h + ∂ y 0 h + O(ε 3 )
for some functionh ∈ C 0,α ([0, 1] 2 ), and the terms O(ε 2 ) do not depend on the coordinates. As in the previous computation, terms which contain an even number of coordinates different to y 0 do not contribute to the result since, once multiplied by ∇φ 1 , b , their average over S n + is 0. Therefore
B + 1 (∆ − ∆ĝ)(φ − φ 1 ) ∇φ 1 , b = O(ε 3 ).
For the last term we have to estimate, the previous computation immediately implies that
B + 1 (φ − φ 1 ) ∇φ 1 , b = O(ε 2 )
and then (λ 1 −λ)
B + 1 (φ − φ 1 ) ∇φ 1 , b = O(ε 3 ) .
We conclude that
S n + ∂φ ∂r ∇φ 1 , b = B + 1 (∆ − ∆ĝ)φ 1 ∇φ 1 , b + |b| O(ε 3 ) =C ε 2g ∇gH(p), Θ(b ′ ) + |b| O(ε 3 ) .
The Lemma follows at once from (44), keeping in mind that v 0 = O(ε).
Lemma 4.7. Letv =v(p, ε) ∈ C 2,α m,N C (S n + ) such that in the coordinates y = (y 0 , y ′ ) we havē v(y 0 , y ′ ) = εṽ p (y 0 ) + O(ε 2 )
for some functionṽ p ∈ C 2,α ([0, 1]). Then there exist two functions δ p , σ p ∈ C 2,α ([0, 1]) such that
((L ε − L 0 )v)(y 0 , y ′ ) = ε 2 δ p (y 0 ) + O(ε 3 )
and
F (p, ε,v) − F (p, ε, 0) − L εv = ε 2 σ p (y 0 ) + O(ε 3 ) .
Proof. Clearly both L ε and L 0 are first order differential operators, and the dependence on ε is smooth. Now, the difference between the coefficients ofḡ written in the coordinates y defined in (28) and the coefficient of the Euclidean metric can be estimated byḡ
ij (y 0 , y ′ ) =Ḡ ij ε y 0 + O(ε 2 ) If the functionv is such thatv (y 0 , y ′ ) = εṽ p (y 0 ) + O(ε 2 )
for some functionṽ p ∈ C 2,α ([0, 1]), it is then clear that
((L ε − L 0 )v) = ε ((L ε − L 0 )ṽ p ) + O(ε 3 )
where now the functionṽ p is considered as a function on the coordinates (y 0 , y ′ ) by the simple relationṽ p (y 0 , y ′ ) = v p (y 0 ). Moreover if we consider the operator F restricted to functionsv that depend only on the first variable y 0 , it is clear that the linearization of F at (p, ε, 0) maps from the subset of functions in C 2,α m,N C that depend only on the first variable y 0 into the subset of functions in C 1,α m,N C that depend only on the first variable y 0 . Then there exists a function δ p ∈ C 1,α ([0, 1]) such that
((L ε − L 0 )ṽ p )(y 0 , y ′ ) = ε δ p (y 0 ) + O(ε 2 ) and then ((L ε − L 0 )v)(y 0 , y ′ ) = ε 2 δ p (y 0 ) + O(ε 3 ) .
Now let us estimate the second term. Taking in account thatv = O(ε) we have
F (p, ε,v) = F (p, ε, 0) + L εv + L 2 ε (v,v) + O(ε 3 )
and then
F (p, ε,v) − F (p, ε, 0) − L εv = L 2 ε (v,v) + O(ε 3 ) . If the functionv is such thatv (y 0 , y ′ ) = εṽ p (y 0 ) + O(ε 2 ) then F (p, ε,v) − F (p, ε, 0) − L εv = ε 2 L 2 ε (ṽ p ,ṽ p ) + O(ε 3 )
where again the functionṽ p is considered as a function on the coordinates (y 0 , y ′ ) byṽ(y 0 , y ′ ) =ṽ(y 0 ), and as for L ε it is easy to see that L 2 ε maps from the subset of functions in C 2,α m,N C that depend only on the first variable y 0 into the subset of functions in C 1,α m,N C that depend only on the first variable y 0 . Then there exists a function
σ p ∈ C 1,α ([0, 1]) such that F (p, ε,v) − F (p, ε, 0) − L εv = ε 2 σ p (y 0 ) + O(ε 3 ) .
This completes the proof of the Lemma.
We are now able to conclude the proof of Proposition 4.5. Using Lemma 4.7 we get
S n + (F (p, ε,v) − F (p, ε, 0) − L εv ) b, · + S n + (L ε − L 0 )v b, · = O(ε 3 ) .
Then, from (43) and using Lemma 4.6, we have that for all p ∈ ∂M and all b ∈ ∂R n+1 + with |b| = 1 the following ε-expansion holds:
S n + F (p, ε,v(p, ε)) b, · = C ε 2g (∇gH(p), Θ(b ′ )) + O(ε 3 ) .
This completes the proof of the Proposition.
Proof of Theorem 1.3
Let b = (0, b ′ ) ∈ ∂R n+1 + with |b| = 1 and define
G b (p, ε) := ε −2 S n + F (p, ε,v(p, ε)) b, · = Cg(∇gH(p), Θ(b ′ )) + O(ε) .
Clearly if ε = 0, we have that
S n + F (p, ε,v(p, ε)) b, · = 0 ⇐⇒ G b (p, ε) = 0 . G b is a function defined on ∂M × [0, +∞) into R.
By the assumption of our main Theorem 1.3, ∂M has a nondegenerate critical point p 0 of the mean curvature. Then the differential of G b with respect to p computed at (p 0 , 0) is invertible and G b (p 0 , 0) = 0. By the implicit function theorem, for all ε small enough there exists p ε ∈ ∂M close to p 0 such that G b (p ε , ε) = 0 for all b ∈ ∂R n+1 + with |b| = 1. In addition we have dist(p 0 , p ε ) ≤ c ε
We conclude then that F (p ε , ε,v(p ε , ε)) ∈ K ⊥ where K is the kernel of the operator L 0 . But by the construction ofv, we have also that F (p ε , ε,v(p ε , ε)) ∈ K and then F (p ε , ε,v(p ε , ε)) = 0 .
This means that the normal derivative of the first eigenfunction of the Laplace-Beltrami operator on Ω ε = B + g,ε (p ε ) with mixed boundary condition is constant on ∂Ω ε ∩M and then Ω ε is extremal.
The only remaining point in the proof of Theorem 1.3, is the analyticity of ∂Ω ε ∩M when M itself is analytic. This is a classical consequence of the extremality condition, see [20].
Appendix
Expansion of the metric
Take the local coordinates x 0 , x 1 , ..., x n in a neighborhood of a point p ∈ ∂M that we introduced in (4). We denote the corresponding coordinate vector fields by X j := Ψ * (∂ x j ) for j = 0, 1, ..., n. We want to write the expansion of the coefficients g ij of the metric Ψ * g in these coordinates. According with our notation, E j are the coordinate vector field X j evaluated at p.
Proposition 5.1. At the point of coordinate x = (x 0 , x 1 , ..., x n ), the following expansion holds :
g 00 = 1
g 0j = 0 g ij = δ ij + 2 g(∇ Ei N, E j ) x 0 + R 0i0j (x 0 ) 2 + g(∇ Ei N, ∇ Ej N ) (x 0 ) 2 +2 k R k0ij x k x 0 + 1 3 k,ℓR ijkl x k x ℓ + O(|x| 3 ) for i, j, k, l = 1, ...n, where
R 0i0j = g R(N, E i ) N, E j R k0ij = g R(E k , N ) E i , E j R ijkl =g R (E i , E k ) E j , E ℓ .
Here R andR are respectively the curvature tensors of M and ∂M .
This result of this proposition is very well known. For example, the same kind of coordinates that we use in this paper are also used in [23], and Proposition 5.1 of [23] combined with the classical expansion of a metric in its geodesic normal coordinate (see for example [27]) immediately implies our Proposition 5.1. Nevertheless, in order to make the reading easier, we write the proof of the proposition.
Proof. We consider the mapping F . The curve x 0 −→ F (x 0 , x) being a geodesic we have g(X 0 , X 0 ) ≡ 1. This also implies that ∇ X0 X 0 ≡ 0 and hence we get ∂ x 0 g(X 0 , X j ) = g(∇ X0 X 0 , X j ) + g(∇ X0 X j , X 0 ) = g(∇ X0 X j , X 0 ) .
The vector fields X 0 and X j being coordinate vector fields we have ∇ X0 X j = ∇ Xj X 0 and we conclude that 2 ∂ x 0 g(X 0 , X j ) = 2 g(∇ Xj X 0 , X 0 ) = ∂ x j g(X 0 , X 0 ) = 0 .
Therefore, g(X 0 , X j ) does not depend on x 0 and since on ∂M this quantity is 0 for j = 1, . . . , n, we conclude that the metric g can be written as g = d(x 0 ) 2 +ḡ x 0 , whereḡ x 0 is a family of metrics on ∂M smoothly depending on x 0 (this is nothing but Gauss' Lemma). Ifg is the metric of ∂M induced by g, we certainly haveḡ
x 0 =g + O(x 0 ) .
We now derive the next term the expansion ofḡ x 0 in powers of x 0 . To this aim, we compute ∂ x 0 g(X i , X j ) = g(∇ Xi X 0 , X j ) + g(∇ Xj X 0 , X i ) , for all i, j = 1, . . . , n. Since X 0 = N on ∂M , we get ∂ x 0ḡ x 0 | x 0 =0 = 2 g(∇ · N, ·) , by definition of the second fundamental form. This already implies that
g x 0 =g + 2g(∇ · N, ·) x 0 + O((x 0 ) 2 ) .
Using the fact that the X 0 and X j are coordinate vector fields, we can compute ∂ 2 x 0 g(X i , X j ) = g(∇ X0 ∇ Xi X 0 , X j ) + g(∇ X0 ∇ Xj X 0 , X i ) + 2 g(∇ Xi X 0 , ∇ Xj X 0 ).
By definition of the curvature tensor, we can write
∇ X0 ∇ Xj = R(X 0 , X j ) + ∇ Xj ∇ X0 + ∇ [X0,Xj ] ,
which, using the fact that X 0 and X j are coordinate vector fields, simplifies into ∇ X0 ∇ Xj = R(X 0 , X j ) + ∇ Xj ∇ X0 .
Since ∇ X0 X 0 ≡ 0, we get ∇ X0 ∇ Xj X 0 = R(X 0 , X j ) X 0 .
Inserting this into (46) yields ∂ 2 x 0 g(X i , X j ) = 2 g(R(X 0 , X i ) X 0 , X j ) + 2 g(∇ Xi X 0 , ∇ Xj X 0 ) .
Evaluation at x 0 = 0 gives ∂ 2
x 0ḡ x 0 | x 0 =0 = 2 g(R(N, ·) N, ·) + 2 g(∇ · N, ∇ · N ). This implies thatḡ
x 0 =g + 2g(∇ · N, ·) x 0 + [g(∇ · N, ∇ · N ) + g(R(N, ·) N, ·)] (x 0 ) 2 + O((x 0 ) 3 )(47)
Now that we have the first terms of the expansion ofḡ x 0 in powers of x 0 we find the expansion of these term with respect to the geodesic coordinates (x 1 , ..., x n ) of ∂M in a neighborhood of p. Recall that for i, j, k, l = 1, ..., ñ
g ij = δ ij + 1 3 k,ℓR ikjℓ x k x ℓ + O(|x| 3 ),(48)
whereR ikjℓ =g R (E i , E k ) E j , E ℓ The proof of this fact can be found for example in [27]. Moreover for k = 1, ..., n we have
∂ x k g(∇ Xi N, X j ) = g(∇ X k ∇ Xi N, X j ) + g(∇ Xi N, ∇ X k X j ) = g(∇ X k ∇ N X i , X j ) + g(∇ Xi N, ∇ X k X j ) = g(R(X k , N )X i , X j ) + g(∇ N ∇ X k X i , X j ) + g(∇ Xi N, ∇ X k X j )
and evaluated at p ∂ x k g(∇ Xi N, X j )| p = g(R(E k , N )E i , E j )
From (47), using (48) and (49), we find the expansion of the metric in the coordinates x 0 , x 1 , ..., x n up to the term of order |x| 2 .
Technical Lemmas
Lemma 5.2. For all σ = 1, . . . , n, we have i,j,k S n + R k0ij x 0 x i x j x k x σ = 0.
Proof. To see that we consider all terms of the above sum, obtained fixing the 4-tuple (i, k, j, σ). We observe that if in such a 4-tuple there is an element that appears an odd number of time then
S n + x 0 x i x j x k x σ = 0. Then i,j,k S n + R k0ij x 0 x i x j x k x σ = i S n + R σ0ii + R i0iσ + R i0σi x 0 (x i ) 2 (x σ ) 2 = 0
by the symmetries of the curvature tensor. Proof. Again, we find that S n +
x 0 x j x σ dvolg = 0 unless the indices j, σ are equal. Hence
i,j S n + R i0ij x 0 x j x σ = S n + x 0 (x σ ) 2 i R i0iσ = − S n + x 0 (x 1 ) 2 H ,σ
This completes the proof of the result.
Figure 1 :
1M can be a Euclidean domain (bounded or not). If p is a nondegenerate critical point for the mean curvature of ∂M , then it is possible to construct an extremal domain as a perturbation of a half-ball centered at p.
Figure 2 :
2Our coordinates are defined as (x 0 , x), x being the normal geodesic coordinates on ∂M and x 0 the coordinate associated to the normal direction.
Figure 3 :
3A boundary edge domain in M Proposition 2.2.
is related to the Faber-Krähn profile, where one looks for the least value of the first eigenvalue of the Laplace-Beltrami operator amongst domains with prescribed volume F K κ := inf Ω⊂M : Volg Ω=κ
Proposition 4. 1 .
1The linearization the operator F with respect tov computed at (p, 0, 0), i.e. ∂vF (p, 0, 0) , is equal to L 0 .
Proposition 4. 5 .
5For all p ∈ ∂M and all b = (0, b ′ ) ∈ ∂R n+1 +
Lemma 5. 3 .
3For all σ = 1, . . . , n, we havei,j S n + R i0ij x 0 x j x σ = − S n + x 0 (x 1 ) 2 H ,σ
Acknowledgements. This work was partially supported by the project Projet ANR-12-BS01-0007 OPTIFORM financed by the French Agence Nationale de la Recherche (ANR).
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arxiv |
Investigating the use of ChatGPT for the scheduling of construction projects
Samuel A Prieto [email protected]
Division of Engineering
Experimental Research Building
S.M.A.R.T. Construction Research Group
New York University Abu Dhabi (NYUAD)
Saadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates
Eyob T Mengiste [email protected]
Division of Engineering
Experimental Research Building
S.M.A.R.T. Construction Research Group
New York University Abu Dhabi (NYUAD)
Saadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates
Borja García De Soto [email protected]
Division of Engineering
Experimental Research Building
S.M.A.R.T. Construction Research Group
New York University Abu Dhabi (NYUAD)
Saadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates
Investigating the use of ChatGPT for the scheduling of construction projects
1Natural Language ProcessingChatGPTSchedulingGenerative Pre-training TransformerProject ManagementConstruction 50GPT 35
Large language models such as ChatGPT have the potential to revolutionize the construction industry by automating repetitive and time-consuming tasks. This paper presents a study in which ChatGPT was used to generate a construction schedule for a simple construction project. The output from ChatGPT was evaluated by a pool of participants that provided feedback regarding their overall interaction experience and the quality of the output. The results show that ChatGPT can generate a coherent schedule that follows a logical approach to fulfill the requirements of the scope indicated. The participants had an overall positive interaction experience and indicated the great potential of such a tool to automate many preliminary and time-consuming tasks. However, the technology still has limitations, and further development is needed before it can be widely adopted in the industry. Overall, this study highlights the potential of using large language models in the construction industry and the need for further research.IntroductionNatural Language Processing (NLP) combines areas such as linguistics, computer science, and Artificial Intelligence (AI) and focuses on the interaction between computers and humans using programs that are developed from large natural language data [1]. Selected applications of NLP in the construction industry include (1) Extracting information from construction documents: NLP techniques can extract relevant information, such as specifications, plans, and contracts, and convert it into a structured format that can be quickly processed by computers[2]. This can help streamline the process of reviewing and comparing construction documents and reduce the risk of errors caused by manual data entry. (2) Analyzing construction site data: NLP can be used to analyze construction site data, such as progress reports, safety inspections, and quality control reports, and extract insights that can help improve project efficiency and mitigate risks[3]. Improving communication on construction sites: NLP-powered chatbots and virtual assistants can be used to improve communication on construction sites by providing a quick and convenient way for project stakeholders to access information and ask questions about the project [4]. (3) Generating or enhancing construction schedules: With enough information about the project being provided to the system, NLP techniques can be used to generate construction schedules based on project details provided by a user. This is not new in the construction field, and there are plenty of tools well developed that generate optimized construction schedules. However, the way this information is presented to the user is always schematized and structured in charts, tables, and diagrams. A natural language discussion could offer a new point of view in this regard[5].In summary, NLP has the potential to significantly improve efficiency, accuracy, and overall communication in the construction industry. Within the field of NLP, research has been focused on developing Large Language representation Models (LLM) that expand the applicability of NLP to more detailed human language understanding tasks such as translation, text classification, and holding conversations[6,7].
Aims and contribution
In this paper, we evaluate the applicability of a Generative Pre-Training Transformer language representation model (GPT) to assist in developing an automated construction schedule based on natural language prompts. The aims of this study are summarized as follows:
• Explore the possible applications and limitations of a rapidly growing and powerful tool for construction scheduling and resource loading.
• Conduct a preliminary case study involving multiple users applying GPT to generate a resource-loaded project schedule for a simple project based on a given detailed natural language description input (i.e., prompt).
• Evaluate the results obtained from the participants in the case study based on parameters such as accuracy, efficiency, clarity, coherence, reliability, relevance, consistency, scalability, and adaptability.
The rest of the paper is organized as follows: Section 2 provides a brief state-of-the-art study on various language representation models and the current trend in project management tools. Section 3 presents the methodology followed to evaluate the results from the case study. Section 4 presents the case study. Section 5 discusses the results and findings from the case study, going into detail about the strengths and limitations of the tool. Finally, Section 6 contains some conclusions and future work by the authors on the application.
Literature review
Language representation models
Large language models have gained significant attention in recent years due to their ability to generate human-like text and perform a wide range of language-based tasks. In the construction industry, large language models have the potential to improve efficiency, accuracy, and communication in several different ways. The fields where NLP technology has been tested and applied are limited. The application of NLP technologies in the construction sector is limited. Some examples include the work by Xue et al. [2] that used NLP techniques to summarize construction contracts. Locatelli et al. [8] studied the potential of combining NLP and BIM, focusing on automated compliance checking and semantic BIM enrichment goals.
Regarding language representation models, different approaches have been developed in the past decade [7]. The major ones can be grouped into three families: the autoregressive language model Generative Pre-trained Transformer (GPT), the Bidirectional Encoder Representations from Transformers (BERT) and the Multi-Task Learning.
Generative Pre-trained Transformer (GPT)
The GPT family (Generative Pre-trained Transformer) of language models was developed by OpenAI. The models are trained on a large dataset of text and, as expected, are able to generate human-like text. The most recent version of the GPT family is GPT-3 [9], which is the base for ChatGPT (considered as GPT-3.5), the tool used for this paper. GPT-3, released in June 2020, has 175 billion parameters, making it one of the largest language models to date. Since its release, it has been proved that GPT-3 is capable of performing some tasks that traditionally require human-level understanding, such as writing essays and programming. Despite the technology not being particularly new [10], GPT-3.5 has been fine-tuned for information retention during the conversation, making it suitable for the scope being tested in this paper. This feature has motivated researchers to study the possibility of incorporating such language models into activities that were solely reserved for human-human interaction, such as healthcare delivery [11]. Floridi and Chiriatti [12] studied the scope, limits and consequences of the GPT-3 model and how society will have to get used to not being able to tell if a text was written by a human or an AI.
Bidirectional Encoder Representations from Transformers (BERT)
BERT (Bidirectional Encoder Representations from Transformers) is a pre-trained transformer-based neural network model developed for NLP tasks, such as text classification and question answering (chatbots). BERT was first introduced by Google in 2018 [13] and has been the state-of-the-art for many of the NLP models developed afterward. Hassan et al. [14] used a BERT-Based model to identify risky and hazardous situations related to construction. Moon et al. [15] used a BERT model for the automated detection of contractual risk clauses from construction specifications. Their approach classified contractual risk categories to provide reviewers with crucial clauses that commonly cause disputes, such as payment, temporal, procedure, safety, role and responsibility, definition and reference. Yao and Garcia de Soto [16] used a BERT model classification for semantic screening trained on construction-specific documentation to investigate main topics related to construction cybersecurity.
Text-to-Text Transfer Transformer (T5) Multi-tasking learning
T5 (Text-to-Text Transfer Transformer) is a multi-task pre-training model for NLP tasks. T5 was introduced by Google in 2020 [17]. T5 models are trained to perform multiple tasks at once, which allows them to learn a general-purpose understanding of language that can be fine-tuned for specific tasks.
Automation of construction scheduling
The area of schedule automation has been heavily researched in the past few decades. Modern construction project scheduling can generally be categorized as BIM Driven and Machine Learning based schedule generations.
BIM-driven schedule generation
BIM models comprise geometric information (special relationships of components), materials and resources. Under the current industrial practice, project management teams utilize the information from the BIM to optimize and generate schedules. Researchers improved the process of manual schedule development by automating task dependency and sequencing using pre-set rules, patterns or pre-set knowledge learned from historic cases [18,19].
These methods extract required data inputs from the pre-built information model to produce a schedule with a logical order [20]. However, BIM models often do not include environmental factors, temporary structures, equipment and material availability, specific construction specifications and methodology of the building site. Therefore, BIM-based automatic generation of schedules requires manual intervention to be practical [20,21].
Machine Learning based schedule generation
Several approaches have been proposed to perform predictive modeling to utilize historical data to predict project outcomes [22]. However, these approaches generally rely on large amounts of data to train the models. The source data could be visual [23], where the algorithm understands the characteristics of the sites and work performance.
Language-based methods were also the focus of research for automated construction schedule development. Natural language models such as GPT [5] or Language clustering methods such as Latent Dirichlet Allocation (LDA) and Latent Semantic Analysis (LSA) [24] were used to understand human language of explaining tasks develop dependency and clustering.
Recently, Hong et al. [20] proposed a graph-based construction scheduling approach. Their proposed approach stores past best practices and recycles the information to optimize resource usage, task sequence and duration of the new project.
In general, Machine Leaning based approaches produced promising results for construction schedules; however, the performance of most methods depends on the availability of data and the data processing technical and infrastructural capacity of the user.
Methodology
A small experiment has been designed to evaluate the potential of ChatGPT as an aid in project management in construction, focusing on project scheduling and task assignment. The experiment consists of a simple construction project (scope) in an existing space. The goal is to retrieve a logical and accurate task breakdown from ChatGPT. The same experiment was performed by different participants, allowing them to challenge the tool and modify the original plan to evaluate the ChatGPT response.
Various parameters were used to evaluate the results of these experiments, including accuracy, efficiency, clarity, coherence, reliability, relevance, consistency, scalability, and adaptability. Accuracy was measured by comparing the ChatGPT-generated project schedules and task assignments to a baseline schedule and assignments created by a human project manager. Efficiency was evaluated by collecting data about the time required to create a schedule and assign tasks using ChatGPT, as well as the time needed to fix the number of errors or mistakes made throughout the process. Clarity and relevance were assessed by evaluating the proposed schedules and responses subject to modifications and complexity changes to ensure they were clear and easy to understand. The responses must be relevant to the input prompt and not deviate from the original task. Coherence was measured by examining the results, such as task dependencies or crew assignments, to ensure that they made sense from a logical point of view in the eye of an expert. Reliability was evaluated by checking conformity with standards and assessing its adequateness if those results were to be used in a real project. Consistency was measured by evaluating the invariance of the ChatGPT responses based on slight modifications to the initial prompt, or multiple instances of the same prompt. The scalability and adaptability of the ChatGPT solution were also evaluated, including its ability to handle larger projects or deal with additional tasks or responsibilities as needed and its ability to handle changing project requirements or unforeseen challenges.
The initial input for the study consisted of a paragraph with enough details of the scope to be completed (see Section 4). Details included the description of the work to be done, dimensions and the type of material to be used, the expected level of completeness of the task, and the due date for the task to be completed. The results expected from the process were a list of tasks and overall scheduling (sequence) of the project, with the ability to interactively react to changes made by the user. An overview of the methodology is shown in Figure 1. The original scope underwent small modifications concerning the original prompt to challenge ChatGPT. Some modifications included adding a new scope (e.g., electrical or plumbing work).
A survey was conducted containing instructions to set the bases of the experiment and a series of questions to evaluate the quality of the output generated by ChatGPT and the participants' experience interacting with it. The results are summarized in Section 5.
Case study
A simple project and scope were used to evaluate the performance of ChatGPT when providing a construction schedule. In general, the work consisted of the addition of a partition wall in an existing space. A simplified floorplan is shown in Figure 2. To ensure that all the participants provided the same information regarding the scope, an initial prompt was given to ChatGPT in all cases. The initial prompt was:
"A set of instructions on a construction project will be provided. You will store the provided information, and you won't provide any answers to the initial prompt until asked otherwise." Table 1.
The initial input describing the scope was:
"A new partition needs to be done in an already existent space, where the new partition is grouted with the existing walls. The details of the room to be partitioned are the following: the room is rectangular shaped, 4 meters by 4 meters in total. The walls are made of concrete masonry units. The height of the walls is 3 meters, and the width is 20 centimeters. The new partition needs to be made out of concrete masonry units as well. The partition is meant to split the original space in half, resulting in two individual spaces of approximately 4 meters by 2 meters. The partition needs to account for the installation of a single solid, two-panel wooden door of 0.8 meters in width by 2.1 meters in height and 35 mm thickness that will communicate the two new spaces. After the partition is made, it needs to be plastered with two layers of stucco and painted with two layers of white latex paint on both sides of the wall. No electrical or plumbing installation is needed. No ceiling work is needed. The floor is cement screed. The work needs to be completed in less than three weeks."
A typical schedule for the proposed scope is shown in Table 1. This was used as the baseline for comparison with the output generated by ChatGPT. Based on the initial prompt, ChatGPT was asked to use that information to generate a construction schedule to complete the scope. To do that, the following prompt was asked to ChatGPT:
"Can you come out with a suitable project schedule?"
To structure the obtained information into usable data, the following structure was asked from ChatGPT: "Based on the details of the work to be completed, extract the information in the following structure 'task name / task priority / task dependencies / number of people needed / expected duration of task'"
The prompt "Add this information into columns" could be used to show the output in a table format. Based on this simple scope, a series of questions with a set of initial instructions were put in a survey form. This form was distributed to several individuals (participants) with different skill sets and qualifications working in the construction and AI fields. The feedback from the six participants is summarized in Section 5.
Results and discussion
The entire output regarding the schedule obtained by the 6 different participants is summarized in Table A.1.
The table contains all the information regarding the different tasks proposed by ChatGPT, their dependencies, the assigned priority, the expected number of people and the time needed to complete them. In general, ChatGPT provided a logical (although very linear) sequence of tasks in all cases. The tool could extrapolate a breakdown of the steps needed without that information being explicitly provided and establish logical and coherent dependences amongst the proposed tasks. At first glance, the output seemed coherent and reasonable, and the participants were awed by the speed at which the responses were provided (in most cases, within a few seconds of entering the prompt). However, with further inspection, it was clear that not all proposed tasks agreed with the scope of work. Table 2 shows a direct comparison of the tasks proposed by ChatGPT in all six cases with respect to the baseline. It is worth noticing that the tasks related to the wooden door (i.e., the frame installation and its protection before painting) were not considered in any of the schedules proposed by ChatGPT. This is due to the fact that ChatGPT has not been trained for specific construction purposes, and it is not aware that for the door installation, the frame needs to be placed first. In some cases, the two layers of plastering are proposed as one single task, and most of the planning and preparation are joined into a single task. In addition, three of the six responses by ChatGPT included the demolition of the existing wall, which is not relevant to the scope provided. This was probably wrongfully inferred by ChatGPT based on the information about a "new partition needs to be done in an already existent space," and demolition might have been assumed because of the existing space condition. Also, incorrect information was provided regarding the tasks that would be expected. For example, in three of the six cases, ChatGPT indicated the installation of a foundation, which might not have been required for a partition wall. In one case, it also suggested the installation of steel rebar for the new partition and the placement ("pouring") of concrete for the new partition. However, minor schedule errors can be fixed by conversing with ChatGPT, instructing the tool to rearrange some of the tasks, asking for a more detailed breakdown, or providing more information that was not properly understood, such as the frame installation for the door. The proposed dependencies and sequence of tasks are logical for the most part, despite being very linear. The proposed sequence for installing the wooden door is arguably not ideal. ChatGPT includes the door installation right after the wall erection. In general, the sequence to install the door would be after the plastering, as indicated in the baseline schedule, to avoid damage from intermediate tasks. If only the door frame is considered, the suggested sequence would be adequate; however, a clear breakdown for that was not provided or inferred by ChatGPT.
In order to evaluate both the duration and crew needed for each task, only those tasks present in the baseline were considered. The durations for each task reported by ChatGPT are summarized in Table A.1. The variation between the baseline durations (Table 1) and the ones from ChatGPT are displayed in Figure 3. The bars represent the difference between the proposed time by ChatGPT in each of the different survey participants' experiments and the one present in the baseline. The bars without data are for tasks that were not considered by ChatGPT. For a given task, a positive increment indicates that the duration from ChatGPT is greater than the one from the baseline and vice versa. Values equal to 0 indicate that the duration from ChatGPT is the same as the one in the baseline. For example, for Task 4, participant 1 (P1) did not have this task, so the value for that is empty. For P2 and P6, the duration obtained from ChatGPT was the same as the baseline (i.e., the deviation is 0). For P3, the duration from ChatGPT was one day more than the baseline (3 days in the baseline vs. 4 days in ChatGPT). For P4 and P5, the duration from ChatGPT was one day less than the baseline (3 days in the baseline vs. 2 days in ChatGPT). Some of the tasks with the biggest difference are those that involve drying time (i.e., Tasks 6, 7, and 10). In the baseline, drying time has been considered both for the plastering and the painting, which is why those tasks take more than three days each. However, ChatGPT does not seem to be considering drying times. The estimation of the number of workers reported by the participants using ChatGPT was compared with the baseline estimation. Most of the number of workers per task from all the ChatGPT responses and the baseline are given as a range of maximum and minimum numbers. Therefore, the comparison was done considering the maximum and the minimum possible difference in workers per task per participant. The highest deviation from the baseline estimation is two workers. The results are shown in Figure 4. Positive increments indicate an increase in the number of workers compared to the baseline and vice versa. For instance, the estimated number of workers for Task 6 in the baseline ranged from 1 to 2. P1 and P4 reported that the ChatGPT estimated the same amount. However, for P3, ChatGPT estimated that Task 6 would require 2 workers. Therefore, based on the difference between the baseline (considering the range) and this response, ChatGPT estimated that Task 6 would require a maximum of 1 more worker (in the case of the lowest baseline estimate) and a minimum of zero additional workers (in the case of a maximum estimate) as represented by the bar for P3 in Task 6. Similarly, P5 and P6 estimated that a minimum of 2 and a maximum of 3 workers would be required to complete task 6. This estimate is one worker higher than the baseline estimate, both in the case of upper and lower baseline estimates.
In order to test the adaptability and scalability of ChatGPT, the original prompt was modified to add additional components to the scope, such as electrical or plumbing work. For example:
"The new wall that needs to be built for the partition needs to have a couple of electrical sockets. Can you include that task in your schedule and regenerate it?"
ChatGPT was able to modify the original list of tasks and update the sequence accordingly. An example of the new task for the example above is shown in Table 3.
To further challenge ChatGPT, information regarding the cost was also asked. The following prompt was used:
"Based on the information above, can you provide a cost estimate for labor and materials needed?"
As expected, the responses from ChatGPT were vague and very generic, constantly advising to consult with a licensed contractor to get a rough cost estimation. A sample response from ChatGPT was, "I'm sorry, but I am not able to provide you with a cost estimate for labor and materials needed based on the information provided. The cost of construction projects can vary greatly depending on a variety of factors such as location, materials, labor costs, and the specific requirements of the project. Additionally, factors such as the cost of permits, inspections, and any necessary equipment rentals should also be considered. It would be best to consult with a licensed contractor or estimator to provide you with an accurate cost estimate for your project." When insisting on providing a rough estimate, the response was still generic (e.g., "As a language model AI, I don't have access to current market pricing and cost may vary greatly depending on location, materials, and labor. However, for a rough estimate for a partition wall and door installation in a standard room, you could expect to spend anywhere from $1000 to $5000 or more for a standard partition wall and door installation. This is a rough estimate and the real cost could vary greatly depending on the location, the specific needs of the project, and the contractor you choose. It is important to consult with a local licensed contractor or estimator to provide you with an accurate cost estimate for your project.") This shows that one of the big limitations of the current model is that the data is not updated since it is not connected to real-time internet data. Since the model is taught to be aware of its own limitations, the default response from ChatGPT is to rely on a licensed contractor for such a query. Regarding the quality of the output generated by ChatGPT, the participants' general impression was positive. They were impressed by the fact that ChatGPT could produce a (for the most part) logical and coherent task breakdown with little initial information, despite being a model not trained specifically for construction purposes. The results are summarized in Figure 5. The overall evaluation is very positive, with ratings of "good" and "very good" predominating in all the different features. Accuracy and reliability got the lowest scores due to the issues related to using tasks that were not related to the scope of work (e.g., excavation, foundation work, rebar, etc.) when compared to the baseline, therefore making the results not as reliable as they would need to be trusted in a professional environment. In addition to the evaluation of the performance of ChatGPT regarding the proposed tasks and sequence (i.e., scheduling component), a set of qualitative questions were asked to evaluate the overall impressions of the participants regarding their experience interacting with ChatGPT. This included their impression of the type of communication over the classical methods (i.e., charts and tables) and their feedback about potential uses for ChatGPT in the construction field. The results regarding the interaction with ChatGPT are presented in Figure 6. Overall, the interaction was very positive, with most participants rating different aspects of the interaction "very good", such as how intuitive it was, how comfortable it was, how efficient it was, and the overall interaction experience.
Regarding the preference for this communication (i.e., in a dialog format) instead of the classical methods (lists, tables, charts), all the surveyed participants shared the common opinion on tables and charts being more intuitive and easier to read than the results provided by ChatGPT. Nonetheless, they all agreed that despite this not being the preferred final form for displaying the results, it might become an important and useful tool to extract the information to be displayed. The possible uses proposed by the surveyed participants ranged from initial consulting, safety and risk identification, basic design, cost estimation, processing and evaluation of contracts or identifying vulnerabilities.
Overall, they all agreed that any task involving text processing could benefit from the inclusion of Natural Language Processing techniques. The full data regarding the performed survey, with all the responses from the participants and their full conversation with ChatGPT, can be made available to interested readers upon reasonable request to the authors.
Limitations
The presented case study has been useful for preliminary testing and evaluating the possible uses and capabilities of ChatGPT when applied to the generation of a schedule for simple projects. The results are promising; however, the complexity of the case study is very limited. Further studies considering increased complexity to resemble actual construction projects should be conducted to further consider this technology in the construction field.
The results provided are based on a limited number of participants and should not be used for generalization purposes. A bigger statistical pool of surveyed people is needed for a wider generalization of the results obtained.
Conclusions and future work
Since ChatGPT has been made available, it has attracted researchers from different fields. The application of such tools in the construction industry should not be overlooked. This study consisted of a simple example to assess the applicability of such a tool in the context of project scheduling. The performance and results were very promising for the simple use case, especially considering that ChatGPT has not been specifically trained for such an application. However, several significant flaws would limit the application of such a tool in a real project. Having said that, the overall performance was reasonable, and the interaction experience was positive. This is important to show that such technology could be relatively easy to integrate if consistent and reliable performance is achieved. Such tools that are specialized in a given field (e.g., project scheduling) could become extremely beneficial and play an important role in the automation of repetitive and time-consuming parts.
Future research needs to be conducted to further explore the applicability and capabilities of Natural Language Processing tools in the construction industry. To do that, a GPT model will be specifically trained for construction purposes and challenged with a more complex scenario. A larger pool of participants will be surveyed in a future study to more accurately generalize the results. In addition, the way the initial information is given to ChatGPT could become a nuisance for larger and more complex projects. This is why it is worth exploring new methods to input information into these models, such as in the form of floorplans and images.
Figure 1 .
1Overview of the main steps to use and assess ChatGPT in this study.
Figure 2 .
2(a) General view of the floorplan and (b) Gantt chart of main activities used as the baseline for the required scope summarized in
Figure 3 .
3Deviation between the durations proposed by ChatGPT and the baseline for each task from each participant.
Figure 4 .
4Deviation of the needed workers proposed by ChatGPT with respect to the baseline.
Figure 5 .
5Results from the survey regarding the evaluation of the ChatGPT output.
Figure 6 .
6Results from the survey regarding the evaluation of the interaction with ChatGPT.
Table 1 .
1Data considered as the baseline for the performed case study. RS Means was considered when estimating the people needed **Total duration, including weekends (equivalent to 11.5 work days)Task
No.
Task name
Task
dependencies
People
needed*
Expected
duration
1
Inspect the existing space and check proposed work is in line with
existing conditions
-
1-2
1 day
2
Prepare the work area and protect surrounding areas as needed
1
1-2
1
3
Measure and mark the location of the new partition, including the
location for opening (door)
1
1
1
4
Install CMU for new partition
1
2
3
5
Install framing for the new door
4
1-2
1
6
Apply the first stucco layer to the CMU wall -includes curing time
4 SS+2
1-2
3
7
Apply the second stucco layer to the CMU wall -includes curing time
6
1-2
4
8
Install and adjust the wooden door
7
1
0.5
9
Protect the door in preparation for the painting of the new CMU
partition wall
8
1
0.5
10
Finish wall (prime, paint, apply 2 layers -allow drying time per
manufacturer's recommendations)
7
2
4
11
Clean up and final inspection
10
1
1
TOTAL
15.5
days**
*
Table 2 .
2Comparison of the tasks proposed in the schedules generated by ChatGPT with the ones from the baseline.Task no. Task name (Baseline)
Table 3 .
3Information regarding the installation of a couple of electrical sockets.Task name
Dependencies
People needed
Expected duration
Install electrical
sockets
Building the new
partition
2
4 hours
Table A
A.1. (continued)Participant
Task name
Dependencies Priority*
People
needed
Expected
duration
* H=High, M=Medium, L=Low
AcknowledgmentThe authors thank the participants who provided their feedback and experience using ChatGPT for the case study.Appendix A
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| {'fraction_non_alphanumeric': 0.04205412297478511, 'fraction_numerical': 0.02974345446151817, 'mean_word_length': 4.775041259362702, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 23, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Large language models such as ChatGPT have the potential to revolutionize the construction industry by automating repetitive and time-consuming tasks. This paper presents a study in which ChatGPT was used to generate a construction schedule for a simple construction project. The output from ChatGPT was evaluated by a pool of participants that provided feedback regarding their overall interaction experience and the quality of the output. The results show that ChatGPT can generate a coherent schedule that follows a logical approach to fulfill the requirements of the scope indicated. The participants had an overall positive interaction experience and indicated the great potential of such a tool to automate many preliminary and time-consuming tasks. However, the technology still has limitations, and further development is needed before it can be widely adopted in the industry. Overall, this study highlights the potential of using large language models in the construction industry and the need for further research.IntroductionNatural Language Processing (NLP) combines areas such as linguistics, computer science, and Artificial Intelligence (AI) and focuses on the interaction between computers and humans using programs that are developed from large natural language data [1]. Selected applications of NLP in the construction industry include (1) Extracting information from construction documents: NLP techniques can extract relevant information, such as specifications, plans, and contracts, and convert it into a structured format that can be quickly processed by computers[2]. This can help streamline the process of reviewing and comparing construction documents and reduce the risk of errors caused by manual data entry. (2) Analyzing construction site data: NLP can be used to analyze construction site data, such as progress reports, safety inspections, and quality control reports, and extract insights that can help improve project efficiency and mitigate risks[3]. Improving communication on construction sites: NLP-powered chatbots and virtual assistants can be used to improve communication on construction sites by providing a quick and convenient way for project stakeholders to access information and ask questions about the project [4]. (3) Generating or enhancing construction schedules: With enough information about the project being provided to the system, NLP techniques can be used to generate construction schedules based on project details provided by a user. This is not new in the construction field, and there are plenty of tools well developed that generate optimized construction schedules. However, the way this information is presented to the user is always schematized and structured in charts, tables, and diagrams. A natural language discussion could offer a new point of view in this regard[5].In summary, NLP has the potential to significantly improve efficiency, accuracy, and overall communication in the construction industry. Within the field of NLP, research has been focused on developing Large Language representation Models (LLM) that expand the applicability of NLP to more detailed human language understanding tasks such as translation, text classification, and holding conversations[6,7].', 'arxivid': '2302.02805', 'author': ['Samuel A Prieto [email protected] \nDivision of Engineering\nExperimental Research Building\nS.M.A.R.T. Construction Research Group\nNew York University Abu Dhabi (NYUAD)\nSaadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates\n', 'Eyob T Mengiste [email protected] \nDivision of Engineering\nExperimental Research Building\nS.M.A.R.T. Construction Research Group\nNew York University Abu Dhabi (NYUAD)\nSaadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates\n', 'Borja García De Soto [email protected] \nDivision of Engineering\nExperimental Research Building\nS.M.A.R.T. Construction Research Group\nNew York University Abu Dhabi (NYUAD)\nSaadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates\n'], 'authoraffiliation': ['Division of Engineering\nExperimental Research Building\nS.M.A.R.T. Construction Research Group\nNew York University Abu Dhabi (NYUAD)\nSaadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates', 'Division of Engineering\nExperimental Research Building\nS.M.A.R.T. Construction Research Group\nNew York University Abu Dhabi (NYUAD)\nSaadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates', 'Division of Engineering\nExperimental Research Building\nS.M.A.R.T. Construction Research Group\nNew York University Abu Dhabi (NYUAD)\nSaadiyat IslandP.O. Box 129188Abu DhabiUnited Arab Emirates'], 'corpusid': 256615316, 'doi': '10.3390/buildings13040857', 'github_urls': [], 'n_tokens_mistral': 11949, 'n_tokens_neox': 10561, 'n_words': 6634, 'pdfsha': '58b98ccd256f1ebe8551faa6798cfbfc80c490a5', 'pdfurls': ['https://export.arxiv.org/pdf/2302.02805v1.pdf'], 'title': ['Investigating the use of ChatGPT for the scheduling of construction projects', 'Investigating the use of ChatGPT for the scheduling of construction projects'], 'venue': []} |
arxiv |
RECOVERY OF THE NONLINEARITY FROM THE MODIFIED SCATTERING MAP
4 Apr 2023
Jason Murphy
RECOVERY OF THE NONLINEARITY FROM THE MODIFIED SCATTERING MAP
4 Apr 2023
We consider a class of one-dimensional nonlinear Schrödinger equations of the form (i∂t + ∆)u = [1 + a]|u| 2 u. For suitable localized functions a, such equations admit a small-data modified scattering theory, which incorporates the standard logarithmic phase correction. In this work, we prove that the small-data modified scattering behavior uniquely determines the inhomogeneity a.
Introduction
We consider one-dimensional nonlinear Schrödinger equations of the form
(i∂ t + ∆)u = [1 + a]|u| 2 u, u| t=0 = u 0 ,(1.1)
where the inhomogeneity a : R → R is a localized function of x ∈ R. For suitable functions a, equation (1.1) admits a small-data modified scattering theory for initial data chosen from a weighted Sobolev space. In this paper, we prove that the modified scattering map uniquely determines the inhomogeneity a.
We first describe the class of inhomogeneities considered in this work: Definition 1.1 (Admissible). We say a : R → R is admissible if a ∈ L 1 ∩ L ∞ , xa ∈ L 2 , and ∂ x a ∈ L 1 .
For admissible inhomogeneities a, we may obtain the following modified scattering result for small initial data in a weighted Sobolev space, which incorporates the typical logarithmic-type phase correction. In the notation below, F denotes the Fourier transform and e it∆ = F −1 e −itξ 2 F is the Schrödinger group. For ε sufficiently small, we may use Theorem 1.2 to define the modified scattering map S a : B ε → L ∞ by S a (u 0 ) = w + , where w + is as in (1.2).
Our main result shows that the modified scattering map uniquely determines the inhomogeneity a. Theorem 1.4 fits in the context of a wide body of work on the recovery of nonlinearities (and external potentials) for nonlinear dispersive equations, particularly the question of recovery from scattering data; we refer the reader to [1-3, 7, 9, 12, 16, 18, 21-24, 26-33] for a broad selection of works in this direction. The chief novelty in our work stems from the fact that we consider a class of equations for which the usual (unmodified) scattering fails. That is, the long-time behavior of solutions is not simply given by the underlying linear dynamics; instead, due to insufficient time decay in the nonlinear term, one must incorporate a logarithmic phase correction in order to describe the long-time asymptotic behavior. Consequently, the structure of the modified scattering map is more complicated to describe. Nonetheless, as we will explain below, this modified map suffices to uniquely determine the inhomogeneity present in the nonlinearity.
Before discussing the proof of Theorem 1.4, let us briefly describe the proof of modified scattering for (1.1) (Theorem 1.2). Modified scattering for cubic nonlinear Schrödinger equations in one dimension is an important topic that has been addressed in many different settings (see e.g. [4][5][6]8,10,11,14,17,19,20]). In the setting of (1.1), the inhomogeneous cubic term may be viewed as a short-range perturbation to the long-range nonlinearity |u| 2 u; indeed, the inhomogeneity a(x) does not appear in the phase correction itself (cf. (1.2)). Our proof of modified scattering follows the basic scheme set out in [11] (based on taking the Fourier transform of the Duhamel formula and using an integrating factor to remove the non-integrable cubic part), using local smoothing estimates (similar to those appearing in [5]) to handle the inhomogeneous cubic term. For the details, see Section 3.
In Section 4, we prove the main result, Theorem 1.4. Before discussing specific details of the proof, let us first recall the standard approach to recovering the nonlinearity from the usual scattering map (going back at least as far as [16,25]). To fix ideas, let us consider the problem of recovering an unknown, localized coefficient in a 1d nonlinear Schrödinger equation of the form
(i∂ t + ∆)u = a|u| 2 u, u| t=0 = u 0 . (1.3)
For a ∈ L 1 ∩ L ∞ , one can prove that the usual (unmodified) scattering behavior holds for small initial data in L 2 (see e.g. [18]); that is, there exists a map S a such that
lim t→∞ u(t) − e it∆ S a (u 0 ) L 2 = 0,
where u is the solution to (1.3). In fact, using the Duhamel formula, one obtains the following implicit formula for S a :
S a (u 0 ) = u 0 − i ∞ 0 e −it∆ [a|u(t)| 2 u(t)] dt.
Specializing to u 0 = εϕ (with ϕ ∈ S and 0 < ε ≪ 1), pairing this identity with ϕ, and approximating u(t) by e it∆ u 0 (the Born approximation), one can show that
S a (εϕ), ϕ = ε ϕ, ϕ − iε 3 ∞ 0 R a(x)|e it∆ ϕ(x)| 4 dx dt + O(ε 4 ).
It follows that knowledge of S a suffices to determine the functionals
∞ 0 R a(x)|e it∆ ϕ(x)| 4 dx dt for ϕ ∈ S. (1.4)
The problem then reduces to showing that knowledge of the functionals (1.4) uniquely determines the coefficient a.
In the setting of Theorem 1.4, the overall structure of the argument is similar; however, the analysis becomes more complicated due to the fact that the form of the modified scattering map is different than that of the standard scattering map. In particular, the modified scattering map is no longer easily viewed as a perturbation of the identity. Instead, in Proposition 4.1, we show that for ϕ ∈ S and 0 < ε ≪ 1, we have the expansion
S a (εϕ),φ = ε φ,φ + 1 2i log(1 + 1 2ε ) |S a (εϕ)| 2 S a (εϕ),φ + ε 3 Q ε [ϕ] − iε 3 ∞ 0 R a(x)|e it∆ ϕ(x)| 4 dx dt + O(ε 4 ),
whereφ is the Fourier transform of ϕ and Q ε is a multilinear expression in ϕ (which, importantly, is independent of a). Thus, despite the more complicated structure of S a , we find that knowledge of S a still essentially determines the functionals appearing in (1.4), and the problem once again reduces to showing that the functionals (1.4) determine the coefficient a.
In earlier works (e.g. [16,25]), this final step is completed by evaluating the functional along a sequence of test functions concentrating at a point and utilizing the dominated convergence theorem in order to determine a pointwise. In the present setting, the low-power nonlinearity poses an additional challenge; indeed, we cannot use dominated convergence directly, as we cannot guarantee that e it∆ ϕ ∈ L 4 t,x (R × R) even for ϕ ∈ S. Instead, inspired in part by [12], we proceed by specializing to the case of Gaussian data, for which the free evolution may be computed explicitly. In this way, we find that knowledge of (1.4) suffices to determine the convolution a * K for an explicit kernel K, and the problem reduces to verifying directly thatK = 0 almost everywhere. This final step is completed by evaluating a Gaussian integral.
The rest of this paper is organized as follows: In section 2, we set up notation and collect some preliminary lemmas. In Section 3, we establish modified scattering for (1.1) (Theorem 1.2). Finally, in Section 4, we prove the main result, Theorem 1.4.
Acknowledgements. J.M. was supported by NSF grant DMS-2137217 and a Simons Collaboration Grant. G.C. would like to thank the Department of Mathematics and Statistics at Missouri S&T, where part of this work was completed, for its hospitality.
Notation and preliminary results
We write A B to denote A ≤ CB for some C > 0. We indicate dependence on parameters via subscripts, e.g. A a B means A ≤ CB for some C = C(a) > 0.
We write H k,ℓ to denote the weighted Sobolev space with norm
u H k,ℓ = ∂ x k x ℓ u L 2 ,
where · is the Japanese bracket notation, i.e. x = √ 1 + x 2 . We write S for Schwartz space.
We denote the Fourier transform of a function f :
R d → C by F d f (ξ) = (2π) − d 2 R d e −ix·ξ f (x) dx,
with the inverse Fourier transform given by
F −1 d f (x) = (2π) − d 2 R d e ix·ξ f (ξ) dξ.
If d = 1, we will omit the subscript. We also write F f =f and F −1 f =f . We caution the reader that factors of 2π will be uniformly omitted throughout the computations below.
The Schrödinger group is given by the Fourier multiplier operator
e it∆ = F −1 e −itξ 2 F .
This operator admits the factorization identity
e it∆ = M (t)D(t)F M (t),
where
M (t) = e i x 2 4t and [D(t)f ](x) = (2it) − 1 2 f ( x 2t ). The Galilean operator J(t) is defined via J(t) = x + 2it∂ x = e it∆ xe −it∆ . (2.1)
Given a solution u to (1.1), we will perform much of the analysis on the associated profile f (t) = e −it∆ u(t). Suitable bounds on the profile imply estimates for the solution itself, as is seen in the following lemma.
Lemma 2.1. Let f (t) = e −it∆ u(t). Then for any 0 < c < 1 4 , u(t) L ∞ x c |t| − 1 2 { f (t) L ∞ + |t| −c f (t) H 1 }.
Proof. We write
u(t) = M (t)D(t)F M (t)f (t) = M (t)D(t)f (t) + M (t)D(t)F [M (t) − 1]f (t).
We now observe that
M (t)D(t)f (t) L ∞ |t| − 1 2 f (t) L ∞ ,
which is acceptable. For the remaining term, we use Hausdorff-Young, the pointwise estimate |M (t) − 1| |x| 2c |t| −c , and Cauchy-Schwarz to obtain
M (t)D(t)F [M (t) − 1]f (t) L ∞ |t| − 1 2 −c |x| 2c f L 1 |t| − 1 2 −c x f L 2 , which is acceptable.
Next we introduce a smoothing estimate, which is the dual of the classical Kato smoothing estimate. This estimate will be used to analyze the inhomogeneous cubic term. Such estimates appear in more general settings in [5].
Lemma 2.2. Let φ : R → C satisfy |φ(k)| |k| 1 2 . (2.2)
Then for all t ≥ 0, we have
t 0 e −iξ 2 s φ(ξ)F (s, ξ) ds L 2 ξ F L 1 x L 2 s (R×[0,t]) . (2.3)
Proof. We argue by duality. We will first prove that
R e ixξφ (ξ)e iξ 2 s h(ξ) dξ L ∞ x L 2 s (R×[0,t]) h L 2 (2.4)
for any h ∈ L 2 . Without loss of generality, we restrict the integral to ξ > 0. Changing variables via ξ 2 = λ and using Plancherel (in time) and (2.2), we obtain
∞ 0 e ixξφ (ξ)e iξ 2 s h(x) dξ 2 L 2 s ([0,t]) ∞ 0 e ix √ λφ ( √ λ)e isλ h( √ λ) 1 √ λ dλ 2 L 2 s (R) ∞ 0 φ ( √ λ) √ λ h( √ λ) 2 dλ h 2 L 2 ,
uniformly in x, which yields (2.4). Now, given h ∈ L 2 and F ∈ L 1 x L 2 s , we use (2.4) and Hölder to estimate
t 0 h(ξ)e iξ 2 sφ (ξ)F (s, ξ) ds dξ = t 0 R R e ixξ e iξ 2 sφ (ξ)h(ξ) dξ F (s, x) dx ds R e ixξφ (ξ)e iξ 2 s h(ξ) dξ L ∞ x L 2 s (R×[0,t]) F L 1 x L 2 s (R×[0,t]) h L 2 F L 1 x L 2 s (R×[0,t])
, which implies the desired estimate.
The Direct Problem
In this section we prove Theorem 1.2. The proof follows largely along standard lines (see e.g. [11]), with some modifications to handle the inhomogeneous cubic term.
We let u 0 ∈ H 1,1 with u 0 H 1,1 = ε > 0, and let u : [0, ∞) × R → C be the corresponding solution to (1.1). We define the profile f (t) = e −it∆ u(t). By standard well-posedness arguments and Sobolev embedding, one can derive that that sup
t∈[0,1] [ u(t) H 1 + J(t)u(t) L 2 ε. (3.1) Using (1.1), we have that i∂ tf (t, ξ) = F e −it∆ (|u| 2 u)(ξ) + F e −it∆ (a|u| 2 u)(ξ).
In particular, we have the following straightforward estimates, which will be useful for t ∈ [0, 1]: by Hausdorff-Young and Plancherel,
∂ tf L ∞ ξ [1 + a]|u| 2 u L 1 x u 3 L 3 x u 3 H 1 x ε 3 and ∂ ξf L 2 ξ J(t)([1 + a]|u| 2 u) L 2 u 2 L ∞ Ju L 2 + u 3 L ∞ t∇a L 2 ε 3 .
Next, we isolate the component of i∂ tf that fails to be integrable as t → ∞. Evaluating the Fourier transform and changing variables via ξ − σ → σ, we obtain
F e −it∆ |u| 2 u (ξ) = e it[ξ 2 −(ξ−η) 2 +(η−σ) 2 −σ 2 )]f (t, ξ − η)f (t, η − σ)f (t, σ) dσ dη = e 2itησ G ξ [f (t), f (t), f (t)](η, σ) dσ dη, where G ξ [f, g, h](η, σ) :=f (ξ − η)ĝ(η − ξ + σ)ĥ(ξ − σ). (3.2)
We continue from above, using Plancherel and the identity
F 2 [e 2itησ ] = 1 2t e −i ησ 2t to obtain F e −it∆ |u| 2 u (ξ) = 1 2t e −i ησ 2s F −1 2 G ξ [f (t), f (t), f (t)] (η, σ) dσ dη.
Noting thatf (−ξ) =f (ξ), so that
G ξ [f (t), f (t), f (t)](0, 0) = |f (t, ξ)| 2f (t, ξ),
we therefore find that
F e −it∆ |u| 2 u (ξ) = 1 2t |f (t, ξ)| 2f (t, ξ) + 1 2t e −i ησ 2t − 1 F −1 2 {G ξ [f (t), f (t), f (t)]}(η, σ) dσ dη.
Combining the computations above, we derive that
i∂ t f (t, ξ) = 1 2t |f (t, ξ)| 2f (t, ξ) + F e −it∆ a|u| 2 u (ξ) + 1 2t e −i ησ 2t − 1 F −1 2 {G ξ [f (t), f (t), f (t)]}(η, σ) dσ dη.
We now define
w(t) = e iB(t)f (t), where B(t) := exp i t 0 |f (s)| 2 ds 2s+1 .(3.
3)
It follows that
i∂ t w(t, ξ) = e iB(t,ξ) i∂ t f (t, ξ) − 1 2t+1 |f (t, ξ)| 2f (t, ξ) (3.4) = e iB(t,ξ) 1 2t(2t+1) |f (t, ξ)| 2f (t, ξ) (3.5) + F e −it∆ (a|u| 2 u)(ξ) (3.6) + 1 2t e −i ησ 2t − 1]F −1 2 {G ξ [f (t), f (t), f (t)]}(η, σ) dσ dη . (3.7)
Using (3.4) and (3.1), we find that
∂ t w H 1 ε 3 uniformly for t ∈ [0, 1]. (3.8)
We obtain estimates for t ∈ [1, ∞) using a bootstrap argument. In particular, assuming that the solution satisfies estimates of the form
f (t) L ∞ ξ ≤ 2Cε and f (t) H 1 ≤ 2C t δ ε (3.9)
uniformly in t ≥ 1, the estimates obtained below will demonstrate that the solution satisfies the improved bounds
f (t) L ∞ ξ ≤ Cε and f (t) H 1 ≤ C t δ ε.
Here δ = O(ε 2 ) is a small parameter. Observe that by Lemma 2.1, the assumptions (3.9) also guarantee that
u(t) L ∞ t − 1 2 ε. Noting that f (t) L ∞ ξ ≡ w(t) L ∞ ξ ,
we begin by using the expansion (3.5)-(3.7) to estimate ∂ t w in in L ∞ ξ . In particular, we will prove that if (3.9) holds, then
∂ t w L ∞ ξ t −1− 1 10 ε 3 uniformly for t ≥ 1. (3.10)
First, by (3.9) we immediately see that
1 2t(2t+1) |f | 2f L ∞ ξ t −2 ε 3 ,
which is acceptable. Next, using (3.9), Hausdorff-Young, and Lemma 2.1, we estimate
F e −it∆ (a|u| 2 u) L ∞ ξ a|u| 2 u L 1 a L 1 u 3 L ∞ a t − 3 2 ε 3 ,
which is acceptable. Finally, we turn to (3.7). We begin by using the pointwise estimate
|e ix − 1| ≤ |x| 1 5 to obtain (3.7) L ∞ ξ |t| −1− 1 5 |η| 1 5 |σ| 1 5 |F −1 2 {G ξ [f, f, f ]}(η, σ)| dσ dη L ∞ ξ . (3.11)
To estimate the right-hand side of (3.11), we rely on the following general trilinear estimate. We state the result in more generality than is needed here, as this formulation will be useful in the next section.
Lemma 3.1 (Trilinear Estimate).
Define G ξ (·, ·, ·) as in (3.2). Then
|η| 1 5 |σ| 1 5 |F −1 2 {G ξ [f, g, h]}(η, σ)| dσ dη f H 0,1 g H 0,1 h H 0,1 uniformly in ξ. Proof. Recall that G ξ [f, g, h](x, y) =f (ξ − x)ĝ(x − ξ + y)ĥ(ξ − y).
Thus, writing e iab db = δ a=0 , we have
F −1 2 {G ξ [f, g, h]}(η, σ) = · · · e i[xη+yσ−v(ξ−x)−z(x−ξ+y)−r(ξ−y)] f (v)ḡ(z)h(r) dx dy dr dv dz = ḡ(z)e izξ f (v)e −ivξ e i[x(v+η−z)] h(r)e −irξ e i[y(r+σ−z)] dy dr dx dv dz = ḡ(z)h(z − σ)e iξσ f (v)e −ivξ e i[x(v+η−z)] dx dv dz = f (z − η)ḡ(z)h(z − σ)e iξ[η+σ−z] dz. (3.12)
It follows that
|F −1 2 {G ξ [f, g, h]}(η, σ)| ≤ |f (z − η)h(z − σ)g(z)| dz
uniformly in ξ, and hence
|x| c f L 1 x f L 2 ,
which is a consequence of Cauchy-Schwarz.
Continuing from (3.11) and applying Lemma 3.1 and (3.9), we obtain
(3.7) L ∞ ξ |t| −1− 1 5 f (t) 3 H 0,1 |t| −1− 1 5 +3δ ε 3 ,
which is acceptable (provided δ is sufficiently small). This completes the proof of (3.10), which suffices to close the bootstrap estimate forf in L ∞ .
To complete the proof of (3.9), it suffices to close the bootstrap estimate for H 1 -norm off . Without loss of generality, we estimate theḢ 1 -norm only.
Using the Duhamel formula, we first write
∂ ξf (t, ξ) = ∂ ξû0 (ξ) (3.13) − i t 0 ∂ ξ F e −is∆ |u| 2 u (ξ) ds (3.14) − i t 0 ∂ ξ F e −is∆ a|u| 2 u (ξ) ds. (3.15)
The term in (3.13) is O(ε) in L 2 ξ , which is acceptable. Using the same computations as above, we may write
(3.14) = −i t 0 e 2isησ ∂ ξ G ξ [f (s), f (s), f (s)](η, σ) dσ dη ds. (3.16)
Recalling the definition of G ξ (see (3.2)), it follows from the product rule that 1)) and undo the computations that led to (3.16) to see that (3.14) may be written as a sum of terms of the form
∂ ξ G ξ [f, f, f ]t 0 F [e −is∆ O(u 2 )Ju](ξ) ds.
In particular, by (3.1) and (3.9), we may estimate
(3.14) L 2 ξ t 0 u(s) 2 L ∞ ξ J(s)u(s) L 2 ds t 0 s −1+δ ε 3 ds t δ ε 3 ,
which is acceptable. It remains to estimate (3.15). We begin by writing
∂ ξ t 0 F e −is∆ a|u| 2 u (ξ) ds = ∂ ξ t 0 e isξ 2 F a|u| 2 u (ξ) ds = t 0 e isξ 2 ∂ ξ F a|u| 2 u (ξ) ds (3.17) + 2i t 0 ξse isξ 2 F a|u| 2 u (ξ) ds. (3.18)
Using (3.1) and (3.9), we first estimate
(3.17) L 2 ξ t 0 x a|u| 2 u L 2 x ds t 0 xa L 2 u 3 L ∞ ds t 0 ε 3 s − 3 2 ds ε 3 ,
which is acceptable. Next, we let ϕ be a smooth cutoff to |ξ| ≤ 1 and decompose
(3.18) = 2i t 0 ξϕ(ξ)se isξ 2 F a|u| 2 u (ξ) ds (3.19) + 2i t 0 [1 − ϕ(ξ)]se isξ 2 ξF a|u| 2 u (ξ) ds. (3.20)
Applying Lemma 2.2 (with φ(ξ) = ξϕ(ξ) andF (s, ξ) = sF (a|u| 2 u)), (3.1), (3.9), and Minkowski's integral inequality, we deduce that
(3.19) L 2 ξ sa|u| 2 u L 1 x L 2 s (R×[0,t]) a L 1 s|u| 2 u L ∞ x L 2 s (R×[0,t]) s|u| 2 u L 2 s L ∞ x ([0,t]×R) ε 3 s s − 3 2 L 2 s ([0,t]) ε 3 log t ,
which is acceptable. Similarly, applying Lemma 2.2 (with φ(ξ) = 1−ϕ(ξ) andF (s, ξ) = sξF (a|u| 2 u)), we find that
(3.20) L 2 ξ s|u| 2 u ∂ x a L 1 x L 2 s (R×[0,t]) + sa|u| 2 ∂ x u L 1 x L 2 s (R×[0,t]) .
(3.21)
For the first term, we proceed as we did for (3.19). This yields
s|u| 2 u∂ x a L 1 x L 2 s (R×[0,t]) ε 3 ∂ x a L 1 log t ,
which is acceptable. For the second term, we write
s∂ x u = 1 2i [J(s)u(s) − xu(s)]
Then, using (3.9) (noting that Ju L 2 = f Ḣ1 by (2.1)), we estimate
sa|u| 2 ∂ x u L 1 x L 2 s (R×[0,t]) x a L 2 |u| 2 x −1 [Ju − xu] L 2 s,x ([0,t]×R) u 2 L 4 s L ∞ x ([0,t]×R) Ju L ∞ s L 2 x ([0,t]×R) + x x u L ∞ s L 2 x ([0,t]×R) ε 3 t δ ,
which is acceptable. Combining the estimates above, we can close the bootstrap for the H 1 -component off . Thus the desired bounds forf hold for all t ≥ 0, and in particular we obtain the bound (3.10).
With (3.10) in hand, we obtain the establish the existence of w + in L ∞ ξ such that
w(t) − w + L ∞ ξ t − 1 10 ε 3 (3.22)
uniformly for t ≥ 0, which suffices to complete the proof of Theorem 1.2.
The Inverse Problem
The goal of this section is to prove Theorem 1.4. Our first step is a careful analysis of the scattering map u 0 → S a (u 0 ) for a fixed admissible inhomogeneity a.
S a (εϕ),φ = ε φ,φ + 1 2i log(1 + 1 2ε ) |S a (εϕ)| 2 S a (εϕ),φ + ε 3 Q ε [ϕ] − iε 3 ∞ 0 R a(x)|e it∆ ϕ(x)| 4 dx dt + O(ε 4 ), (4.1) where Q ε [ϕ] := ∞ ε 1 2it [e −i ησ 2t −1]ϕ(z −η)ϕ(z −σ)φ(z)φ(z −η−σ) dz dη dσ dt. (4.2)
Proof. We write u 0 = εϕ and let u be the solution to (1.1) with u| t=0 = u 0 . We define the profile f (t) = e −it∆ u(t) and the modified profile w(t) = e iB(t)f (t) as in (3.3). In particular, there exists w + ∈ L ∞ ξ such that w(t) → w + = S a (u 0 ) in L ∞ ξ as t → ∞. By construction, we have
w + L ∞ ξ ε.
We begin by using (3.5)-(3.7) from the preceding section to write
iw + (ξ) = iû 0 (ξ) + ε 0 i∂ t w(t, ξ) dt + ∞ ε 1 2t(2t+1) |w(t, ξ)| 2 w(t, ξ) dt (4.3) + ∞ ε e iB(t,ξ) G t [f (t), f (t), f (t)](ξ) dt (4.4) + ∞ ε e iB(t,ξ) F [e −it∆ {a|u(t)| 2 u(t)}](ξ) dt,(4.5)
where
G t [f, g, h](ξ) := 1 2t [e −i ησ 2t − 1]F −1 2 {G ξ [f, g, h]}(η, σ) dη dσ,(4.6)
with G ξ (·, ·, ·) as in (3.2). The termû 0 (ξ) is O(ε). The analysis now proceeds by separating the remaining components in (4.3)-(4.4) that are O(ε 3 ) in L ∞ ξ from those that are o(ε 3 ) as ε → 0. We first observe that by (3.8), we have that
ε 0 ∂ t w dt L ∞ ξ ε 4 .
For the remaining term in (4.3), we claim that
∞ ε 1 2t(2t+1) |w(t, ξ)| 2 w(t, ξ) dt = 1 2 log(1 + 1 2ε )|w + (ξ)| 2 w + (ξ) + O(ε 4 ) (4.7)
in L ∞ ξ . To see this, we use (3.22) to estimate
|w(t)| 2 w(t) − |w + | 2 w + L ∞ ξ { w(t) 2 L ∞ ξ + w + 2 L ∞ ξ } w(t) − w + L ∞ ξ ε 5 t − 1 10 , which yields ∞ ε 1 2t(2t+1) |w(t)| 2 w(t) − |w + | 2 w + dt L ∞ ξ ε 5 ∞ ε 1 2t(2t+1) t − 1 10 dt ε 5 | log ε| = O(ε 4 ).
As ∞ ε 1 2t(2t+1) dt = 1 2 log(1 + 1 2ε ), we conclude that (4.7) holds. Collecting the estimates so far, we have found We turn to the terms in (4.4)-(4.5). We first show that the phase exp{iB(t)} can be removed up to errors that are higher order in ε (at the price of logarithmic time growth). In particular, we have
e iB(t) − 1 L ∞ ξ B(t) L ∞ ξ t 0 f (s) 2 L ∞ ξ ds 2s+1 ε 2 log t .
(4.9)
We now use (4.9) to show that
(4.4) + (4.5) = ∞ ε G t [f (t), f (t), f (t)](ξ) dt + ∞ ε F [e −it∆ {a|u(t)| 2 u(t)}](ξ) dt + O(ε 4 ) (4.10)
uniformly in ξ. To this end, we will verify the following two estimates:
∞ ε log t G t [f (t), f (t), f (t)] L ∞ ξ dt ε 14 5 , (4.11) ∞ ε log t F [e −it∆ {a|u(t)| 2 u(t)}] L ∞ ξ dt ε 3 . (4.12)
Using Lemma 3.1, we first have
|(4.11)| ∞ ε |t| −1− 1 5 log t |η| 1 5 |σ| 1 5 |F −1 2 {G ξ [f (t), f (t), f (t)]}(η, σ)| dη dσ ∞ ε |t| −1− 1 5 log t f (t) 3 H 0,1 dt ε 3 ∞ ε |t| −1− 1 5 t 3δ log t dt ε 14 5 .
Next, by Hausdorff-Young and Lemma 2.1,
|(4.12)| ∞ ε log t a|u(t)| 2 u(t) L 1 dt ∞ ε log t a L 1 u(t) 3 L ∞ dt ε 3 ∞ ε log t t − 3 2 dt ε 3 .
Combining the preceding estimates with (4.9), we derive (4.10). We now analyze each term in (4.10) more closely. We show that up to acceptable errors, we may replace the full solution with its initial data:
Lemma 4.2. The following approximations hold. First, ∞ ε G t [f (t), f (t), f (t)] dt = ∞ ε G t [u 0 , u 0 , u 0 ] dt + O(ε 4 ) (4.13)
in L ∞ ξ . Next, for any test function ψ, Proof. We begin with (4.13). Writing
f (t) = u 0 + t 0 ∂ s f (s) ds,
we find that it suffices to prove that
∞ ε G t g, h, t 0 ∂ s f (s) ds dt = O(ε 4 ) in L ∞ ξ , where g, h ∈ u 0 , t 0 ∂ s f ds .
For each such term, we use Lemma 3.1 to estimate
∞ ε G t g, h, t 0 ∂ s f (s) ds (ξ) dt ∞ ε |t| −1− 1 5 |η| 1 5 |σ| 1 5 F −1 2 G ξ g, h, t 0 ∂ s f (s) ds (η, σ) dη dσ dt ∞ ε |t| −1− 1 5 g H 0,1 h H 0,1 t 0 ∂ s f (s) ds H 0,1 dt uniformly in ξ.
Noting that the estimates in the preceding section imply
x t 0 ∂ s f (s) ds L 2 t 3δ ε 3 ,
we see that
g H 0,1 + h H 0,1 ε + ε 3 t 3δ .
It follows that
∞ ε G t g, h, t 0 ∂ s f (s) ds L ∞ ξ dt ∞ ε |t| −1− 1 5 {ε 5 t 3δ + ε 9 t 9δ } dt ε 24 5 ,
which is acceptable. We turn to (4.14). Fixing a test function ψ, we see that it suffices to prove Thus by the dispersive estimate, Sobolev embedding, unitarity of e it∆ , and Lemma 2.1,
we have ∞ ε a[|u(t)| 2 u(t) − |e it∆ u 0 | 2 e it∆ u 0 ], e it∆ψ dt ∞ 0 a |u(t)| 2 + |e it∆ u 0 | 2 |u(t) − e it∆ u 0 | · e it∆ψ L 1 x dt ∞ 0 a L 2 x N (t) L 2 x u(t) 2 L ∞ x + e it∆ u 0 2 L ∞ x e it∆ψ L ∞ x dt a ε 2 ∞ 0 t − 3 2 ψ H 1,1 t 0 1 + a L ∞ x |u(s)| 2 u(s) L 2 x ds dt a,ψ ε 2 ∞ 0 t − 3 2 t 0 u(s) 2 L ∞ u(s) L 2 ds dt a,ψ ε 5 ∞ 0 t − 3 2 t 0 s −1 ds dt a,ψ ε 5 ∞ 0 t − 3 2 log t dt a,ϕ ε 5 ,
which is acceptable.
We return to the expansion for w + given in (4.3)-(4.5) and pair the expression withφ. We insert (4.8) for (4.3) and combine (4.10) with Lemma 4.2 to replace the terms (4.4)-(4.5). Recalling u 0 = εϕ, this yields
S a (εϕ),φ = ε φ,φ + 1 2i log(1 + 1 2ε ) |w + | 2 w + ,φ − iε 3 ∞ ε G t [ϕ, ϕ, ϕ],φ dt − iε 3 ∞ ε a(x)|e it∆ ϕ(x)| 4 dx dt + O(ε 4 ).
Comparing the identity above with (4.1), we see that to complete the proof of Proposition 4.1 it suffices to verify the following:
ε 0 R a(x)|e it∆ ϕ(x)| 4 dx dt = O(ε), (4.15) ∞ ε 1 i G t [ϕ, ϕ, ϕ],φ dt = Q ε [ϕ],(4.16)
where Q ε is as in (4.2). The estimate (4.15) follows from the straightforward bound ε 0 a(x)|e it∆ ϕ(x)| 4 dx dt ε a L ∞ e it∆ ϕ 4
L ∞ t L 4 x a ε ϕ 4 H 1 ,
where we have applied Sobolev embedding and unitary of e it∆ .
The identity (4.16) follows from a straightforward calculation: recalling the definition in (4.6) and the identity in (3.12), we have [e −i ησ 2t − 1]ϕ(z − η)φ(z)ϕ(z − σ)φ(ξ)e iξ[η+σ−z] dz dξ dη dσ dt
= ∞ ε 1 2it [e −i ησ 2t − 1]ϕ(z − η)φ(z)ϕ(z − σ)φ(z − η − σ) dz dη dσ dt = Q ε [ϕ],
as desired.
We now turn to the proof of our main result, Theorem 1.4.
Proof of Theorem 1.4. We let a and b be admissible in the sense of Definition 1.1 and suppose that the modified scattering maps S a and S b agree on their common domain. We now fix ϕ ∈ S and sufficiently small ε > 0 and apply the main identity (4.1) in Proposition 4.1 to both S a (εϕ) and S b (εϕ). As S a (εϕ) = S b (εϕ), this implies
∞ 0 R a(x)|e it∆ ϕ(x)| 4 dx dt = ∞ 0 R b(x)|e it∆ ϕ(x)| 4 dx dt + O(ε)
for any ε > 0. It follows that exp{− x 2 4(1+it) } (see [34]). In particular,
K ϕ (x) = ∞ 0 1 1+t 2 exp − x 2 1+t 2 dt.
Now suppose that (4.17) holds. Then, by translation invariance for the linear Schrödinger equation, we have that R a(x)K ϕ (x − x 0 ) dx = 0 for all x 0 ∈ R.
Thus, to deduce that a ≡ 0, it suffices to verify thatK ϕ = 0 almost everywhere. In fact, for ξ = 0, we can computeK ϕ (ξ) explicitly as a Gaussian integral:
K ϕ (ξ) = ∞ 0 (1 + t 2 ) −1 R exp −ixξ − x 2 1+t 2 dx dt = √ π ∞ 0 (1 + t 2 ) − 1 2 exp − ξ 2 (1+t 2 ) 4
dt.
AsK ϕ (ξ) is the integral of a positive function, the result follows.
Theorem 1. 2 (
2Modified scattering). Let a : R → R be admissible in the sense of Definition 1.1. If u 0 H 1,1 is sufficiently small, then there exists a unique forwardglobal solution u to (1.1) and w + ∈ L ∞ ξ such that lim t→∞ exp i t 0 |F e −is∆ u(s)| 2 ds 2s+1 F e −it∆ u(t) 1.2, we may define the modified scattering map.
Definition 1. 3 (
3Modified scattering map). Let a be admissible in the sense of Definition 1.1. Given ε > 0, define B ε = {u 0 ∈ H 1,1 : u 0 H 1,1 < ε}.
Theorem 1. 4 (
4The modified scattering map determines the nonlinearity). Suppose a and b admissible in the sense of Definition 1.1. Let S a : B → L ∞ and S b : B ′ → L ∞ denote the corresponding modified scattering maps.If S a = S b on B ∩ B ′ , then a ≡ b.
1 5
1]|f (z − η)h(z − σ)g(z)| dz dσ dη uniformly in ξ. The result now follows from the fact that for any 0 < c < 1 2 ,
is a linear combination of terms of the form G ξ [xf, f, f ]. After distributing the derivative, we can use the identity xf (s) = xe −is∆ u(s) = e −is∆ J(s)u(s) (cf. (2.
Proposition 4 . 1 (
41Structure of S a ). Let a be admissible in the sense of Definition 1.1. Let ϕ ∈ S(R) and ε > 0 be sufficiently small. Let u : [0, ∞) × R → C be the solution to (1.1) with u| t=0 = εϕ. Then
(4.3) = iû 0 (ξ) + 1 2 log(1 + 1 2ε )|w + (ξ)| 2 w + (ξ) + O(ε 4 ).(4.8)
F
[e −it∆ {a|u| 2 u}], ψ dt = ∞ ε a|e it∆ u 0 | 2 e it∆ u 0 , e it∆ψ dt + O(ε 4 ). (4.14)
2 u − |e it∆ u 0 | 2 e it∆ u 0 ], e it∆ψ dt = O(ε 4 ).To prove this we first note that by the Duhamel formula for (1.1), we have u(t) − e it∆ u 0 = N (t) := −i t 0 e i(t−s)∆ [(1 + a)|u| 2 u](s) ds.
x)|e it∆ ϕ(x)| 4 dx dt for all ϕ ∈ S. Thus the proof of Theorem 1.4 reduces to showing that if a is admissible in the sense of Definition 1.1 and ∞ 0 R a(x)|e it∆ ϕ(x)| 4 dx dt = 0 for all ϕ ∈ S, (4.17)then a ≡ 0. Given ϕ ∈ S, we define the functionK ϕ (x) = ∞ 0 |e it∆ ϕ(x)| 4 dtand first prove that K ϕ ∈ L 2 . To see this, we use Minkowski's integral inequality followed by the dispersive estimate and Sobolev embedding to estimate
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Georgia Institute of Technology Email address: gc@math. gatech.eduGeorgia Institute of Technology Email address: [email protected]
Missouri University of Science & Technology Email address: jason. [email protected] University of Science & Technology Email address: [email protected]
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